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} | Modified Function Projective Synchronization between Different Dimension Fractional-Order Chaotic Systems
Ping Zhou¹,² and Rui Ding¹
¹ Research Center for System Theory and Applications, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
² Key Laboratory of Industrial Internet of Things and Networked Control of the Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
Correspondence should be addressed to Ping Zhou, [email protected]
Received 2 August 2012; Accepted 13 August 2012
Academic Editor: Jinhu Lü
A modified function projective synchronization (MFPS) scheme for different dimension fractional-order chaotic systems is presented via fractional order derivative. The synchronization scheme, based on stability theory of nonlinear fractional-order systems, is theoretically rigorous. The numerical simulations demonstrate the validity and feasibility of the proposed method.
1. Introduction
Fractional-order calculus, which can be dated back to the 17th century [1, 2]. However, only in the last few decades, its application to physics and engineering has been addressed. So, the fractional-order calculus has attracted increasing attention only recently. On the other hand, complex bifurcation and chaotic phenomena have been found in many fractional-order dynamical systems. For example, the fractional-order Lorenz chaotic system [3], the fractional-order unified chaotic system [4], the fractional-order Chua chaotic circuit [5], the fractional-order modified Duffing chaotic system [6], and the fractional-order Rössler chaotic system [7, 8], and so on.
Nowadays, synchronization of chaotic systems and fractional-order chaotic systems has attracted much attention because of its applications in secure communication and control processing [9–21]. Many approaches have been reported for the synchronization of chaotic systems and fractional-order chaotic systems [12–19]. In 1999, Mainieri and Rehacek proposed projective synchronization (PS) [12] for chaotic systems, which has
been extensively investigated in recent years because of its proportional feature in secure communications. Recently, a modified projective synchronization, which is called function projective synchronization (FPS) [13–15] has been reported. In FPS, the master and slave systems could be synchronized up to a scaling function, but not a constant. So, the unpredictability of the scaling function in FPS can additionally enhance the security of communication.
To the best of our knowledge, most of the existing FPS scheme for the fractional-order chaotic systems only discuss the same dimension. However, in many real physics systems, the synchronization is carried out through the oscillators with different dimension, especially the systems in biological science and social science [16–21]. Moreover, in some previous works [16, 17], all the nonlinear terms of response system or error system was absorbed. Referring to chaotic synchronization via fractional-order controller, there are a few results reported until now. Inspired by the above discussion, in this paper, we present a modified function projective synchronization (MFPS) scheme between different dimension fractional-order chaotic systems via fractional-order controller. The fractional-order controller is easily designed. The synchronization technique, based on tracking control and stability theory of nonlinear fractional-order systems, is theoretically rigorous. Our modified function projective synchronization (MFPS) scheme need not absorb all the nonlinear terms of response system. This is different from some previous works. Two examples are presented to demonstrate the effectiveness of the proposed MFPS scheme.
This paper is organized as follows. In Section 2, a modified function projective synchronization (MFPS) scheme is presented. In Section 3, two groups of examples are used to verify the effectiveness of the proposed scheme. The conclusion is finally drawn in Section 4.
2. The MFPS Scheme for Different Dimension Fractional-Order Chaotic Systems
The fractional-order chaotic drive and response systems with different dimension are defined as follows, respectively:
\[
\frac{d^{q_d}x}{dt^{q_d}} = F_d(x),
\]
\[
\frac{d^{q_r}y}{dt^{q_r}} = F_r(y) + C(x, y),
\]
where \(q_d (0 < q_d < 1)\) and \(q_r (0 < q_r < 1)\) are fractional order, and \(q_d\) may be different with \(q_r\). \(x \in \mathbb{R}^n, y \in \mathbb{R}^m \ (n \neq m)\) are state vectors of the drive system (2.1) and response system (2.2), respectively. \(F_d : \mathbb{R}^n \rightarrow \mathbb{R}^n, F_r : \mathbb{R}^m \rightarrow \mathbb{R}^m\) are two continuous nonlinear vector functions, and \(C(x, y) \in \mathbb{R}^m\) is a controller which will be designed later.
Definition 2.1. For the drive system (2.1) and response system (2.2), it is said to be modified function projective synchronization (MFPS) if there exist a controller \(C(x, y)\) such that:
\[
\lim_{t \to +\infty} \|e\| = \lim_{t \to +\infty} \|y - M(x)x\| = 0,
\]
\(\)
where \( \| \cdot \| \) is the Euclidean norm, \( M(x) \) is a \( m \times n \) real matrix, and matrix element \( M_{ij}(x) \) \((i = 1, 2, \ldots, m, j = 1, 2, \ldots, n) \) are continuous bounded functions. \( e_i = y_i - \sum_{j=1}^{n} M_{ij}x_j \) \((i = 1, 2, \ldots, m) \) are called MFPS error.
**Remark 2.2.** According to the view of tracking control, \( M(x)x \) can be chosen as a reference signal. The MFPS in our paper is transformed into the problem of tracking control, that is the output signal \( y \) in system (2.2) follows the reference signal \( M(x)x \).
In order to achieve the output signal \( y \) follows the reference signal \( M(x)x \). Now, we define a compensation controller \( C_1(x) \in R^m \) for response system (2.2) via fractional-order derivative \( d^{\nu_i}(M(x)x)/dt^{\nu_i} \). The compensation controller is shown as follows:
\[
C_1(x) = \frac{d^{\nu_i}(M(x)x)}{dt^{\nu_i}} - F_r(M(x)x),
\]
(2.4)
and let controller \( C(x, y) \) as follows:
\[
C(x, y) = C_1(x) + C_2(x, y),
\]
(2.5)
where \( C_2(x, y) \in R^m \) is a vector function which will be designed later.
By controller (2.5) and compensation controller (2.4), the response system (2.2) can be changed as follows:
\[
\frac{d^{\nu_i}e}{dt^{\nu_i}} = D_1(x, y)e + C_2(x, y),
\]
(2.6)
where \( D_1(x, y)e = F_r(y) - F_r(M(x)x) \), and \( D_1(x, y) \in R^{m \times m} \). So, the MFPS between drive system (2.1) and response system (2.2) is transformed into the following problem: choose a suitable vector function \( C_2(x, y) \) such that system (2.6) is asymptotically converged to zero.
In what follows we present the stability theorem for nonlinear fractional-order systems of commensurate order [22–25]. Consider the following nonlinear commensurate fractional-order autonomous system
\[
D^q x = f(x),
\]
(2.7)
the fixed points of system (2.7) is asymptotically stable if all eigenvalues (\( \lambda \)) of the Jacobian matrix \( A = \partial f / \partial x \) evaluated at the fixed points satisfy \( |\arg \lambda| > 0.5 \pi q \). Where \( 0 < q < 1 \), \( x \in R^n \), \( f : R^n \rightarrow R^n \) are continuous nonlinear functions, and the fixed points of this nonlinear commensurate fractional-order system are calculated by solving equation \( f(x) = 0 \).
Now, the following theorem is given based on the above discussion in order to achieve the MFPS between the drive system (2.1) and the response system (2.2).
**Theorem 2.3.** Choose the control vector \( C_2(x, y) = D_2(x, y)e \), and if \( D_1(x, y) + D_2(x, y) \) satisfy the following conditions:
(1) \( d_{ij} = -d_{ji} \) \((i \neq j)\),
(2) \( d_{ii} \leq 0 \) \( \) (all \( d_{ii} \) are not equal to zero),
then the modified function projective synchronization (MFPS) between (2.1) and (2.2) can be achieved. Where $D_1(x, y) \in \mathbb{R}^{m \times m}$, and $d_{ij}$ ($i, j = 1, 2, \ldots, m$, for all $d_{ij} \in \mathbb{R}$) are the matrix element of matrix $D_1(x, y) + D_2(x, y)$.
**Proof.** Using $C_2(x, y) = D_2(x, y)e$, so fractional-order system (2.6) can be rewritten as follows:
$$
\frac{d^\nu e}{dt^\nu} = [D_1(x, y) + D_2(x, y)]e. \tag{2.8}
$$
Suppose $\lambda$ is one of the eigenvalues of matrix $D_1(x, y) + D_2(x, y)$ and the corresponding non-zero eigenvector is $q_r$, that is,
$$
[D_1(x, y) + D_2(x, y)]q_r = \lambda q_r. \tag{2.9}
$$
Take conjugate transpose ($H$) on both sides of (2.9), we yield
$$
\{[D_1(x, y) + D_2(x, y)]q_r\}^T = \overline{\lambda} q_r^H. \tag{2.10}
$$
Equation (2.9) multiplied left by $q_r^H$ plus (2.10) multiplied right by $q_r$, we derive that
$$
q_r^H \left\{[D_1(x, y) + D_2(x, y)] + [D_1(x, y) + D_2(x, y)]^H \right\}q_r = q_r^H q_r \left(\lambda + \overline{\lambda}\right). \tag{2.11}
$$
So,
$$
\lambda + \overline{\lambda} = \frac{q_r^H \left\{[D_1(x, y) + D_2(x, y)] + [D_1(x, y) + D_2(x, y)]^H \right\}q_r}{q_r^H q_r}. \tag{2.12}
$$
Because $d_{ij} = -d_{ji}$ ($i \neq j$, for all $d_{ij} \in \mathbb{R}$) in matrix $D_1(x, y) + D_2(x, y)$, so
$$
\lambda + \overline{\lambda} = \frac{q_r^H \begin{pmatrix} 2d_{11} & 0 & \cdots & 0 \\ 0 & 2d_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 2d_{mm} \end{pmatrix} q_r}{q_r^H q_r}. \tag{2.13}
$$
Because $d_{ii} \leq 0$ (for all $d_{ii} \in \mathbb{R}$), and all $d_{ii}$ are not equal to zero. So,
$$
\lambda + \overline{\lambda} \leq 0. \tag{2.14}
$$
From (2.14), we have
$$
|\arg \lambda [D_1(x, y) + D_2(x, y)]| \geq 0.5\pi > 0.5q_r\pi. \tag{2.15}
$$
Abstract and Applied Analysis
According to the stability theorem for nonlinear fractional-order systems of commensurate order [22–25], system (2.8) is asymptotically stable. That is
$$
\lim_{t \to +\infty} \|e\| = 0.
$$
(2.16)
Therefore,
$$
\lim_{t \to +\infty} \|e\| = \lim_{t \to +\infty} \|y - M(x)x\| = 0.
$$
(2.17)
This indicates that the modified function projective synchronization between drive system (2.1) and response system (2.2) will be obtained. The proof is completed. □
**Remark 2.4.** Theorem 2.3 indicates that the condition of the MFPS between drive system (2.1) and response system (2.2) are \(|\arg \lambda[D_1(x, y) + D_2(x, y)]| > 0.5q_r\pi\). So, in practical applications, we can easily choose the matrix \(D_2(x, y)\) according to the matrix \(D_1(x, y)\). Moreover, in order to reserve all the nonlinear terms in response system or error system, the controller in our work may be complex than the controller reported by [16, 17]. But, all the nonlinear terms in response system or error system are absorbed in [16, 17].
**Remark 2.5.** Perhaps our result can be extended to the modified function projective synchronization of complex networks of fractional order chaotic systems [26–28] and the complex fractional-order multi-scroll chaotic systems [29–31]. But, the modified function projective synchronization for complex networks and complex fractional-order multi-scroll chaotic systems would be much more complex. Further work on this issue is an ongoing research topic in our group.
**3. Applications**
In this section, to illustrate the effectiveness of the proposed MFPS scheme for different dimension fractional-order chaotic systems. Two groups of examples are considered and their numerical simulations are performed.
**3.1. The MFPS between 3-Dimensional Fractional-Order Lorenz System and 4-Dimensional Fractional-Order Hyperchaotic System**
The fractional-order Lorenz [3] system is described as follows:
$$
\begin{align*}
D^{\alpha} y_1 &= 10(y_2 - y_1) \\
D^{\alpha} y_2 &= 28y_1 - y_2 - y_1 y_3 \\
D^{\alpha} y_3 &= y_1 y_2 - \frac{8y_3}{3}.
\end{align*}
$$
(3.1)
The fractional-order Lorenz system exhibits chaotic behavior [3] for \(q_r \geq 0.993\). The chaotic attractor for \(q_r = 0.995\) is shown in Figure 1.
Recently, Pan et al. constructed a hyperchaotic system [17]. Its corresponded fractional-order system is described as follows:
\[
\begin{align*}
D^x_1x_1 &= 10(x_2 - x_1) + x_4 \\
D^x_2x_2 &= 28x_1 - x_1x_3 \\
D^x_3x_3 &= x_1x_2 - \frac{8x_3}{3} \\
D^x_4x_4 &= -x_1x_3 + 1.3x_4.
\end{align*}
\]
The hyperchaotic attractor of system (3.2) for \( q_d = 0.95 \) is shown in Figure 2.
Consider the fractional-order hyperchaotic system (3.2) with fractional-order \( q_d = 0.95 \) as drive system, and the fractional-order Loren system with fractional-order \( q_r = 0.995 \) as response system. According to the above mentioned, we can obtain
\[
F_r(y) - F_r(M(x)x) = D_1(x, y)e = \begin{pmatrix}
-10 & 10 & 0 \\
28 - y_3 & -1 - \sum_{j=1}^{4} M_1jx_j \\
y_2 & \sum_{j=1}^{4} M_1jx_j & -\frac{8}{3}
\end{pmatrix} e. \quad (3.3)
\]
Now, we can choose
\[
D_2(x, y) = \begin{pmatrix}
0 & 0 & -y_2 \\
-38 + y_3 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}.
\] (3.4)
So,
\[
D_1(x, y) + D_2(x, y) = \begin{pmatrix}
-10 & 10 & -y_2 \\
-10 & -1 & -\sum_{j=1}^{4} M_{1j} x_j \\
y_2 & \sum_{j=1}^{4} M_{1j} x_j & \frac{8}{3}
\end{pmatrix}.
\] (3.5)
According to the above theorem, the MFPS between the 3-dimensional fractional-order Lorenz system (3.1) and the 4-dimensional fractional-order hyperchaotic system (3.2) can be achieved. For example, choose \( M(x) = \begin{pmatrix}
1 & x_2 & 2 & 1 \\
0 & x_1 & 2 & 1 \\
0 & 0 & 1 & 1 \\
0 & 0 & 0 & 3
\end{pmatrix} \). The corresponding numerical result is shown in Figure 3, in which the initial conditions are \( x(0) = (2, 1, 2, 1)^T \) and \( y(0) = (18, 13, 13.5)^T \), respectively.
### 3.2. The MFPS between 4-Dimensional Fractional-Order Hyperchaotic Lü System and 3-Dimensional Fractional-Order Arneodo Chaotic System
In 2002, Lü and Chen reported a new chaotic system [32], which be called Lü chaotic system. The Lü chaotic system is different from the Lorenz and Chen system. Based on Lü chaotic system, the hyperchaotic Lü chaotic system and the fractional-order hyperchaotic Lü system have been constructed recently. The fractional-order hyperchaotic Lü system [16] is described by the following
\[
\begin{align*}
D^q y_1 &= 36(y_2 - y_1) + y_4 \\
D^q y_2 &= 20y_2 - y_1y_5 \\
D^q y_3 &= y_1y_2 - 3y_5 \\
D^q y_4 &= y_1y_3 - y_4.
\end{align*}
\] (3.6)
The hyperchaotic attractor of system (3.6) for \( q_r = 0.96 \) is shown in Figure 4.
The fractional order Arneodo chaotic system [16] is defined as follows:
\[
\begin{align*}
D^q x_1 &= x_2 \\
D^q x_2 &= x_3 \\
D^q x_3 &= 5.5x_1 - 3.5x_2 - x_3 - x_1^2.
\end{align*}
\] (3.7)
The chaotic attractor of system (3.7) for \( q_d = 0.998 \) is shown in Figure 5.
Consider the fractional-order Arneodo chaotic system (3.7) with fractional-order \( q_d = 0.998 \) as drive system, and the fractional-order hyperchaotic \( \tilde{\text{L}} \) system (3.6) with fractional-order \( q_r = 0.96 \) as response system. According to the above mentioned, we can yield
\[
F_r(y) - F_r(M(x)x) = D_1(x,y)e = \begin{pmatrix}
-36 & 36 & 0 & 1 \\
-y_3 & 20 & -\sum_{j=1}^{3} M_1 x_j & 0 \\
y_2 & \sum_{j=1}^{3} M_1 x_j & -3 & 0 \\
y_3 & 0 & \sum_{j=1}^{3} M_1 x_j & -1
\end{pmatrix} e. \tag{3.8}
\]
Now, we can choose
\[
D_2(x, y) = \begin{pmatrix}
0 & 0 & -y_2 & 0 \\
-36 + y_3 & -21 & 0 & 0 \\
0 & 0 & 0 & -\sum_{j=1}^{3} M_{1j} x_j \\
-1 - y_3 & 0 & 0 & 0
\end{pmatrix}.
\] (3.9)
So,
\[
D_1(x, y) + D_2(x, y) = \begin{pmatrix}
-36 & 36 & -y_2 & 1 \\
-36 & -1 & -\sum_{j=1}^{3} M_{1j} x_j & 0 \\
y_2 \sum_{j=1}^{3} M_{1j} x_j & -3 & -\sum_{j=1}^{3} M_{1j} x_j & 0 \\
-1 & 0 & \sum_{j=1}^{3} M_{1j} x_j & -1
\end{pmatrix}.
\] (3.10)
According to above theorem, the MFPS between the 4-dimensional fractional-order hyperchaotic Lü system (3.6) and the 3-dimensional fractional-order Arneodo chaotic system (3.7) can be achieved. For example, choose \( M(x) = \begin{pmatrix}
1+x_2 & 0 & 0 \\
0 & 1+x_3 & 0 \\
0 & 0 & 0.5+x_1
\end{pmatrix} \). The corresponding numerical result is shown in Figure 6, in which the initial conditions are \( x(0) = (2, 2, 2)^T \), and \( y(0) = (11, 10, 11, 2)^T \), respectively.
4. Conclusions
In this paper, based on the stability theory of the fractional-order system and the tracking control, a modified function projective synchronization scheme for different dimension fractional-order chaotic systems is addressed. The derived method in the present paper shows that the modified function projective synchronization between drive system and response system with different dimensions can be achieved. The modified function projective synchronization between 3-dimensional fractional-order Lorenz system and 4-dimensional
Fractional-order hyperchaotic system, and the modified function projective synchronization between the 4-dimensional fractional-order hyperchaotic \(\dot{L}\) system, and the 3-dimensional fractional-order Arneodo chaotic system, are chosen to illustrate the proposed method. Numerical experiments shows that the present method works very well, which can be used for other chaotic systems.
Acknowledgments
The authors are very grateful to the anonymous reviewers for their valuable comments and suggestions, which have greatly improved the presentation of this paper. This work is supported by the Foundation of Science and Technology project of Chongqing Education Commission under Grant KJ110525, and by the National Natural Science Foundation of China (61104150).
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} | Nucleon Structure Functions from the NJL-Model Chiral Soliton
I. Takyi, H. Weigel
Institute for Theoretical Physics, Physics Department, Stellenbosch University, Matieland 7602, South Africa
We present numerical simulations for unpolarized and polarized structure functions in a chiral soliton model. The soliton is constructed self-consistently from quark fields from which the structure functions are extracted. Central to the project is the implementation of regularizing the Dirac sea (or vacuum) contribution to structure functions from first principles. We discuss in detail how sum rules are realized at the level of the quark wave-functions in momentum space. The comparison with experimental data is convincing for the polarized structure functions but exhibits some discrepancies in the unpolarized case. The vacuum contribution to the polarized structure functions is particularly small.
I. INTRODUCTION
Perhaps the most convincing evidence for the quark substructure of baryons emerges from Deep Inelastic Scattering (DIS). The conjunction of perturbative Quantum Chromo Dynamics (QCD) and the parton model successfully explains the wealth of DIS data collected over the past decades [1–3]. However, these are not fully first principle calculations as the hadron wave-functions cannot (yet) be directly computed in QCD. Rather, in the spirit of the parton model, quark distributions are parameterized and subjected to perturbative QCD analysis [4]. On the other hand there are many phenomenological models based on various aspects of QCD that (attempt to) describe (static) properties of hadrons with particular focus on the nucleon. Examples are the non-relativistic quark model [5], the MIT bag model [6] relativistic quark-diquark models [7], chiral soliton models [8]; just to name a few. In principle any of these approaches should also be capable to predict nucleon structure functions. This is particular challenging for chiral soliton models that are formulated as bosonized action functionals hiding the quark substructure of hadrons. In this respect the Nambu-Jona-Lasino (NJL) or chiral quark soliton model [9, 10] is special: based on a quark self-interaction the bosonization process can be traced step by step.
In the past nucleon structure functions have indeed been computed from the chiral quark soliton model and mainly two approaches were followed. Within the valence quark only approximation [11, 12] the observation that though the Dirac sea (or vacuum) is essential to form the soliton, the vacuum contribution to nucleon properties is only moderate thereby justifying neglecting its contribution to the structure functions. Though this approximation has empirical support, it is formally incomplete and does not follow from a systematic expansion scheme. In parallel, studies on the quark distributions were performed [13–19] that identified the model quark degrees of freedom with those using the operator product expansion in the analysis of DIS. These studies included vacuum contributions. These are plagued by ultra-violet divergences requiring an a posteriori implementation of regularization by a single Pauli-Villars subtraction imposed onto the distributions. Unfortunately, this is not stringent since there are terms in the action that do not undergo regularization, e.g. to maintain the axial anomaly as measured by the decay of the neutral pion into two photons. Generally, the model has quadratic divergences (most notably the gap equation) and a single subtraction may or may not be sufficient to remove all divergences. In the present project we therefore include the vacuum contributions to the nucleon structure functions (i) without identifying the NJL model quarks field with those of QCD, and (ii) by implementing the mandatory regularization already at the level of the defining action functional. The first issue is addressed by noting that the model emulates the chiral symmetry of QCD and thus produces the same symmetry currents, in particular the electromagnetic one. To address the second issue we recall that DIS is described by the hadronic tensor $W_{\mu\nu}$ which is the Fourier transform of the nucleon matrix element of the commutator of two electromagnetic current operators. Though there is no direct implementation of this commutator when bosonizing the NJL model, we take advantage of its relation to the Compton tensor $T_{\mu\nu}$ which itself is computed from the time-ordered product of these currents. Time-ordered products are straightforwardly included in a path integral formalism within which bosonization is conducted. Regularizing this path integral by multiple Pauli-Villars subtractions proves most appropriate because it allows to trace the quark that carries the large momentum in the Bjorken limit. This formalism was developed already some time ago [20] but its numerical simulation has been long outstanding. It will be central to the study presented here.
This paper is organized as follows: In Section [11] we introduce the NJL model with emphasis on describing the regularization procedure in Minkowski space. The formulation in Minkowski space is advantageous to identify the absorptive part of $T_{\mu\nu}$ which leads to $W_{\mu\nu}$. In Section [11] we review the formalism from Ref. [20] of how to obtain the hadronic tensor in this model and in particular the role of the Bjorken limit. The NJL soliton description is explained...
We discuss the formalism to obtain the structure functions via the hadronic tensor from quark spinors that self-consistently interact with the soliton in Section IV. Subsequently (Section VI) we describe the way in which this formalism builds up the sum rules. Numerical results are presented in Section VII. This analysis also includes the perturbative evolution to the scale at which experimental data are taken. As stressed above, the model structure functions are identified from symmetry currents, not by equating QCD degrees of freedom. However, the evolution makes this unavoidable for the lack of any sensible alternative. We briefly conclude and summarize in Section VIII.
Finally we leave technical details to four Appendixes.
Some preliminary results extracted from this paper have been put forward in Ref. [21]
II. THE MODEL
We formulate the regularized action of the bosonized Nambu-Jona-Lasinio (NJL) model in Minkowski space as the sum of three pieces
\[ A_{\text{NJL}} = A_R + A_I + \frac{1}{4G} \int d^4x \, \text{tr} \left[ S^2 + P^2 + 2m_0 S \right] \]
\[ A_R = -\frac{N_C}{2} \sum_{i=0}^{2} c_i \text{Tr} \log \left[ -DD_5 + \Lambda_i^2 - i\epsilon \right] , \]
\[ A_I = -\frac{N_C}{2} \text{Tr} \log \left[ -D(D_5)^{-1} - i\epsilon \right] . \]
The subscripts \( R \) and \( I \) for real and imaginary refer to the respective properties after Wick-rotation to Euclidean space. In Minkowski space this corresponds to the use of two distinct Dirac operators
\[ D = i\partial \sigma - (S + i\gamma_5 P) \gamma_0 + \partial_0 \gamma_5 =: D^{(\sigma)} + \partial_0 + \partial_5 \gamma_5 \]
\[ D_5 = -i\partial \sigma - (S + i\gamma_5 P) \gamma_0 - \partial_0 - \partial_5 \gamma_5 =: D^{(\sigma)}_5 + \partial_0 - \partial_5 \gamma_5 . \]
Here, as in the local part of Eq. (1), \( S \) and \( P \) are scalar and pseudoscalar fields that represent physical particles. Furthermore \( v_\mu \) and \( a_\mu \) are source fields. When expanding the action with respect to these sources the linear terms couple to the vector and axial vector currents. Since \( A_I \) is (conditionally) finite in the ultra-violet, only \( A_R \) undergoes regularization. Its dynamical content is quadratically divergent\(^1\) and we thus require two subtractions. In the Pauli-Villars scheme they are implemented as
\[ c_0 = 1, \quad \Lambda_0 = 0, \quad \sum_{i=0}^{2} c_i = 0 \quad \text{and} \quad \sum_{i=0}^{2} c_i \Lambda_i^2 = 0 . \]
In practice we will reduce the number of regularization parameters by assuming \( \Lambda_1 \to \Lambda_2 = \Lambda \) which translates into the general prescription \( \sum_i c_i f(\Lambda_i^2) = f(0) - f(\Lambda^2) + \Lambda^2 f'(\Lambda^2) \). Nevertheless we always write the regularization as in Eq. (1).
The variation of the action with respect to the scalar field \( S \) yields the gap equation
\[ \frac{1}{2G} (m - m_0) = -4iN_Cm \sum_{i=0}^{2} c_i \int \frac{d^4k}{(2\pi)^4} \left[ -k^2 + m^2 + \Lambda_i^2 - i\epsilon \right]^{-1} \]
that determines the vacuum expectation value of the scalar field \( \langle S \rangle = m \). Replacing \( S \) by \( \langle S \rangle \) in \( D \) shows that \( m \) is the fermion mass and is thus called the \textit{constituent quark mass}. The chiral field \( U \) defines the non-linear representation for the isovector pion field \( \pi \)
\[ S + iP = mU = m \exp \left[ \frac{g}{m} \pi \cdot \tau \right] . \]
\(^1\) The cosmological constant, computed from setting all fields to their vacuum expectation values, is quartically divergent and must be subtracted when contributing.
We obtain the pion propagator by expanding $A$ to quadratic order and extract the pion mass from the pole
$$m_\pi^2 = \frac{1}{2G} \frac{m_0}{m} \frac{1}{2N_C \Pi(m_\pi^2)}$$ \hspace{1cm} (6)
and requiring a unit residuum at the pole fixes the quark-pion coupling
$$\frac{1}{g^2} = 4N_C \frac{d}{dq^2} \left[ q^2 \Pi(q^2) \right] \bigg|_{q^2=m_\pi^2}.$$ \hspace{1cm} (7)
The above polarization function is
$$\Pi(q^2) = \int_0^1 dx \Pi(q^2, x) \quad \text{with} \quad \Pi(q^2, x) = -i \sum_{i=0}^{2} c_i \frac{d^4 k}{(2\pi)^4} \left[ -k^2 - x(1-x)q^2 + m^2 + \Lambda_i^2 - i\epsilon \right]^2.$$ \hspace{1cm} (8)
Finally we get the pion decay constant $f_\pi$ from the coupling to the axial current. To this end we expand $A$ to linear order in both $\pi$ and $a_\mu$. The result is $f_\pi = 4N_C mg \Pi(m_\pi^2)$. Using the empirical data $m_\pi = 138\text{MeV}$ and $f_\pi = 93\text{MeV}$ fixes two of the three ($m_0$, $G$ and $\Lambda$) model constants. It is customary to use the constituent quark mass $m$ as the single tunable parameter.
III. STRUCTURE FUNCTIONS IN THE NJL MODEL
The hadronic tensor of DIS is the matrix element of the commutator of electromagnetic current operators
$$W_{\mu\nu}(q; H) = \frac{1}{4\pi} \int d^4x e^{iq\cdot x} \left< H \left[ J_\mu(\xi), J_\nu^\dagger(0) \right] \right| H \right>,$$ \hspace{1cm} (9)
where $q$ is the momentum of the virtual photon and $H$ refers to either the pion or the nucleon target. This tensor is decomposed into Lorentz structures whose coefficients are form factors that turn into the structure functions in the so-called Bjorken limit. Labeling the target momentum by $p$ this limit is defined as
$$Q^2 = -q^2 \to \infty \quad \text{with} \quad x = \frac{Q^2}{2p \cdot q} \text{ fixed}.$$ \hspace{1cm} (10)
Often $x$ is referred to as the Bjorken variable.
By the optical theorem, $W_{\mu\nu}$ is proportional to the absorptive part of the Compton amplitude
$$W_{\mu\nu}(q; H) = \frac{1}{2\pi} \text{Im} \, T_{\mu\nu}(q; H).$$ \hspace{1cm} (11)
The latter is the matrix element of a time-ordered product of the currents
$$T_{\mu\nu}(q; H) = i \int d^4\xi e^{iq\cdot \xi} \left< H \left[ T \left\{ J_\mu(\xi), J_\nu^\dagger(0) \right\} \right] H \right>.$$ \hspace{1cm} (12)
With this relation we implement regularization from first principles because the time-ordered product is immediately obtained from the action by functional differentiation
$$T \left\{ J_\mu(\xi), J_\nu^\dagger(0) \right\} = \frac{\delta^2}{\delta \bar{v}_\mu(\xi) \delta \bar{v}_\nu^\dagger(0)} A_{NJL}(v) \bigg|_{v_\mu=0},$$ \hspace{1cm} (13)
where $v_\mu$ is the photon field introduced by minimal coupling. Within the NJL model the photon couples to the quarks inside the hadron. As discussed comprehensively in Ref. [20] the evaluation of $T_{\mu\nu}$ becomes feasible in the Bjorken
\footnote{The hadronic tensor can be written as the matrix element of the commutator for the lowest energy hadron in a given baryon number sector.}
limit. Then the quark propagator that carries the large photon momentum can be identified and thus be taken to be that of a free massless fermion. Thus the functional derivative from Eq. (13) when applied to the real part simplifies to differentiating
\[ A^{(2,\nu)}_{\Lambda_R} = -i \frac{N_C}{4} \sum_{i=0}^{2} c_i \text{Tr} \left\{ \left( -D^{(\nu)} D^{(\nu)}_5 + \Lambda^2_5 \right)^{-1} \left[ Q^2 \phi (\bar{q})^{-1} \gamma_5 \right] \right\}. \]
At this point it is important to explain the crucial role of the subscript `5' attached to the second term in square brackets of Eq. (14). For this second term we have to recall that the (inverse) derivative operator in \( \phi (\bar{q})^{-1} \) is actually associated with the expansion of \( D_5 \). When comparing this \( \gamma_5 \)-odd operator to the ordinary Dirac operator in Eq. (2) one observes immediately that \( D_5 \) has a relative sign between the derivative operator \( i\partial \) and the axial source \( \gamma_5 \). Therefore the axial–vector component of \( (\phi (\bar{q})^{-1} \gamma_5) \) requires a relative sign. With \( S_{\mu\nu\sigma} = g_{\mu\rho}g_{\nu\sigma} + g_{\rho\nu}g_{\mu\sigma} - g_{\mu\sigma}g_{\rho\nu} \), that is
\[ \gamma_\mu \gamma_\rho \gamma_\nu \psi = S_{\mu\nu\sigma} \gamma^\sigma - i\epsilon_{\mu\nu\sigma} \gamma^\sigma \gamma^5 \quad \text{while} \quad (\gamma_\mu \gamma_\rho \gamma_\nu)_5 = S_{\mu\nu\sigma} \gamma^\sigma + i\epsilon_{\mu\nu\sigma} \gamma^\sigma \gamma^5. \]
In Ref. [20] this modification was formally shown to be consistent with the affected sum rules. In Section VI we see on the level of the momentum space quark wave-functions that the structure functions computed on the basis of Eq. (15) indeed fulfill the sum rules.
Similarly, in the Bjorken limit, the imaginary part becomes
\[ A^{(2,\nu)}_{\Lambda_1} = -i \frac{N_C}{4} \text{Tr} \left\{ \left( -D^{(\pi)} D^{(\pi)}_5 + \Lambda^2_5 \right)^{-1} \left[ Q^2 \phi (\bar{q})^{-1} \gamma_5 \right] \right\}. \]
These expressions are still quite formal and we will use them to obtain nucleon structure functions in Section V. We emphasize that these expressions are directly deduced from the regularized action in Eq. (1) and that no further assumption about the regularization has been made. In Ref. [20] it has been shown that applying this formalism to the pion relates its structure function to the spectral function from Eq. (5) as \( F(x) = \frac{5}{2} (4N_C g^2) \frac{d^2}{dp^2} [p^2 \Pi(p^2, x)] \bigg|_{p^2 = m^2_\pi} \), a result that was previously obtained from the analysis of the handbag diagram in Refs. [22, 23].
IV. NJL MODEL SOLITON
We construct the soliton from static meson configurations by introducing a Dirac Hamiltonian \( h \) via
\[ iD^{(\pi)} = \beta (i\partial_t - h) \quad \text{and} \quad iD^{(\pi)}_5 = (i\partial_t - h) \beta. \]
Its diagonalization
\[ h \Psi_\alpha = \epsilon_\alpha \Psi_\alpha, \]
yields eigen-spinors \( \Psi_\alpha = \sum_{\beta} V_{\alpha\beta} \Psi^{(0)}_\beta \) (\( \Psi^{(0)}_\beta \) are free Dirac spinors in a spherical basis, see Appendix A) and energy eigenvalues \( \epsilon_\alpha \). The hedgehog configuration minimizes the action in the unit baryon number sector and introduces the chiral angle \( \Theta(r) \) via
\[ h = \alpha \cdot p + \beta m U_5(r) \quad \text{where} \quad U_5(r) = \exp [i\hat{r} \cdot \tau_5 \gamma_5 \Theta(r)]. \]
With the boundary conditions \( \Theta(0) = -\pi \) and \( \lim_{r \to \infty} \Theta(r) = 0 \) the diagonalization, Eq. (18) yields a distinct, strongly bound level, \( \Psi_\nu \), referred to as the valence quark level in. Its (explicit) occupation ensures unit baryon number. The functional trace in \( A_R \) is computed as an integral over the time interval \( T \) and a discrete sum over the basis levels defined by Eq. (18). In the limit \( T \to \infty \) the vacuum contribution to the static energy is then extracted from \( A_R \to \) \( -TE_{\text{vac}} \). Collecting pieces, we obtain the total energy functional as [3, 10]
\[ E_{\text{tot}}[\Theta] = \frac{N_C}{2} \left[ 1 + \text{sign}(\epsilon_\nu) \right] \epsilon_\nu - \frac{N_C}{2} \sum_{i=0}^{2} c_i \sum_{\alpha} \left\{ \sqrt{\epsilon_\alpha^2 + \Lambda^2_5} - \sqrt{\epsilon^{(0)}_\alpha^2 + \Lambda^2_5} \right\} + m^2 \int d^3 r \left[ 1 - \cos(\Theta) \right]. \]
Here we have also subtracted the vacuum energy associated with the non-dynamical meson field configuration \( \Theta = 0 \) (denoted by the superscript) that is often called the cosmological constant contribution. This subtraction will also
play an important role for the unpolarized isosinglet structure function as it enters via the momentum sum rule. The soliton profile is then obtained as the profile function $\Theta(r)$ that minimizes the total energy $E_{\text{tot}}$ self-consistently subject to the above mentioned boundary conditions.
This soliton represents an object which has unit baryon number but neither good quantum numbers for spin and flavor (isospin). Such quantum numbers are generated by canonically quantizing the time-dependent collective coordinates $A(t)$ which parameterize the spin-flavor orientation of the soliton. For a rigidly rotating soliton the Dirac operator becomes, after transforming to the flavor rotating frame \[ W \]
\[ W(q) = \frac{1}{2} \Omega \cdot \tau, \]
which, according to the quantization rules, are replaced by the spin operator \[ \Omega \rightarrow \frac{1}{\alpha^2} J. \]
The constant of proportionality is the moment of inertia
\[ \alpha^2 = \frac{N_C}{4} \left[ 1 + \text{sign}(\epsilon_v) \right] \sum_{\beta \neq \alpha} |\langle \tau_3 | \beta \rangle|^2 \right] + \frac{N_C}{8} \sum_{\alpha \neq \beta} \sum_{i=1}^2 c_i \left( \frac{|\langle \alpha | \tau_3 | \beta \rangle|^2}{\epsilon^2_{\alpha} - \epsilon^2_{\beta}} \right) \left( \frac{\epsilon^2_{\alpha} + \epsilon_\alpha \epsilon_\beta + 2\Lambda^2}{\sqrt{\epsilon^2_{\alpha} + \Lambda^2}} - \frac{\epsilon^2_{\beta} + \epsilon_\alpha \epsilon_\beta + 2\Lambda^2}{\sqrt{\epsilon^2_{\beta} + \Lambda^2}} \right) \right), \]
which is of the order $N_C$. With Eq. (23) the expansion in $\Omega$ is thus equivalent to the one in $1/N_C$. The nucleon wave-function becomes a (Wigner D) function of the collective coordinates. A useful relation in computing matrix elements of nucleon states is \[ \langle N | \frac{1}{2} \text{tr} (A^\dagger \tau_i A \tau_j) | N \rangle = -\frac{4}{3} \langle N | I_i J_j | N \rangle. \]
V. STRUCTURE FUNCTIONS FROM SOLITON
We first repeat the relation between the structure functions and the hadronic tensor of the nucleon. Its symmetric combination $W_{\mu\nu}^S(q) = \frac{1}{2} (W_{\mu\nu}(q) + W_{\nu\mu}(q))$, is parameterized by two form factors\[ W_{\mu\nu}^S(q) = MW_1(\nu, Q^2) \left( -g_{\mu\nu} + \frac{q_\mu q_\nu}{q^2} \right) + \frac{W_2(\nu, Q^2)}{M} \left( p_\mu - \frac{p \cdot q}{q^2} q_\mu \right) \left( p_\nu - \frac{p \cdot q}{q^2} q_\nu \right). \]
In the Bjorken limit, Eq. (10), these form factors turn into the unpolarized structure functions that we extract by appropriate projections:
\[ F_1(x) = -\frac{1}{2} g^{\mu\nu} W_{\mu\nu}^S(q) \quad \text{and} \quad F_2(x) = -x g^{\mu\nu} W_{\mu\nu}^S(q). \]
It must be noted that the Callan Gross relation, i.e. \[ F_1(x) = 2xF_2(x), \]
is satisfied in this case by construction. Similarly, the anti-symmetric part is also parameterized by two form factors
\[ W_{\mu\nu}^A(q) = i\epsilon_{\mu\nu\lambda\sigma} q^\lambda \left\{ MG_1(\nu, Q^2)s^\nu + \frac{G_2(\nu, Q^2)}{M} ((p \cdot q)s^\nu - (q \cdot s)p^\nu) \right\}. \]
In the Bjorken limit these form factors yield the structure functions
\[ g_1(x) = M^2 \nu G_1(\nu, Q^2) \quad \text{and} \quad g_2(x) = \nu G_2(\nu, Q^2). \]
---
3 Factors of the nucleon mass, $M$, occur for dimensional reasons.
The longitudinal, $g_1(x)$, and transverse, $g_T(x) = g_1(x) + g_2(x)$, structure functions are extracted from the hadronic tensor using the projection operators
$$g_1(x) = \frac{1}{2M} \int \frac{d^4 \rho \rho_{\mu} \rho_{\nu}}{q \cdot s} W_{\mu\nu}^A(q) \quad \text{and} \quad q \parallel s$$
(30)
$$g_T(x) = -\frac{1}{2M} \int \frac{d^4 \rho \rho_{\mu} \rho_{\nu}}{q \cdot s} W_{\mu\nu}^A(q) \quad \text{and} \quad q \perp s$$
(31)
To obtain the hadronic tensor for the nucleon in the soliton model, the functional traces in Eqs. (14) and (16) are computed using the basis defined by the self-consistent soliton, Eq. (13). This calculation has been detailed in Ref. [20] that we adopt directly. We start with the leading order in $M/\Lambda$ to the vacuum (or sea) contribution to $W_{\mu\nu}$
$$W_{\mu\nu}^{(s)}(q) = -\frac{1}{8} \frac{MNc}{2\pi} \int \frac{dw}{2\pi} \sum_\alpha \int d^3 \xi \int \frac{d\lambda}{2\pi} e^{i M x \lambda}$$
(32)
$$\times \langle N, s | \left\{ \left[ \bar{\Psi}_\alpha(\xi) Q_A^2 \gamma_\mu \gamma_\nu \Psi_\alpha(\xi + \lambda \hat{e}_3) e^{-i \lambda \omega} - \bar{\Psi}_\alpha(\xi) Q_A^2 \gamma_\mu \gamma_\nu \Psi_\alpha(\xi - \lambda \hat{e}_3) e^{i \lambda \omega} \right] f^+_\alpha(\omega) \right\} p$$
$$+ \left[ \bar{\Psi}_\alpha(\xi) Q_A^2 (\gamma_\mu \gamma_\nu)_5 \Psi_\alpha(\xi - \lambda \hat{e}_3) e^{-i \lambda \omega} - \bar{\Psi}_\alpha(\xi) Q_A^2 (\gamma_\mu \gamma_\nu)_5 \Psi_\alpha(\xi + \lambda \hat{e}_3) e^{i \lambda \omega} \right] f^-_\alpha(\omega) \right\} \langle N, s \rangle.$$
Here $n^\mu = (1, 0, 0, 1)^\mu$ is the light-cone vector of the photon momentum while $Q_A = A^\dagger Q A$ denotes the flavor rotated quark charge matrix from which we compute nucleon matrix elements as in Eq. (23). Furthermore
$$f^\pm_\alpha = \sum_{i=0}^2 c_i \omega \pm \epsilon_\alpha - \epsilon_\alpha^2 - \Lambda_i^2 + i\epsilon \pm \frac{\omega}{\epsilon_\alpha^2 + \Lambda_i^2 + i\epsilon},$$
are Pauli-Villars regularized spectral functions. The subscript ‘$p$’ indicates their pole contributions that we will explain below.
For the vacuum contribution to the isosinglet unpolarized structure function we then obtain
$$[F_1^{l=0}(x)]_s = \frac{1}{72} \frac{M N c}{2\pi} \int \frac{dw}{2\pi} \sum_\alpha \int d^3 \xi \int \frac{d\lambda}{2\pi} e^{i M x \lambda} \left( \sum_{i=0}^2 c_i \frac{\omega + \epsilon_\alpha}{\omega^2 - \epsilon_\alpha^2 - \Lambda_i^2 + i\epsilon} \right)_p$$
$$\times \int d^3 \xi \left\{ \bar{\Psi}_\alpha(\xi)(1 - \alpha_3) \Psi_\alpha(\xi + \lambda \hat{e}_3) e^{-i \lambda \omega} - \bar{\Psi}_\alpha(\xi)(1 - \alpha_3) \Psi_\alpha(\xi - \lambda \hat{e}_3) e^{i \lambda \omega} \right\}.$$
(34)
Here the pole contributions is
$$\left( \sum_{i=0}^2 c_i \frac{1}{\omega^2 - \epsilon_\alpha^2 - \Lambda_i^2 + i\epsilon} \right)_p = \sum_{i=0}^2 c_i \frac{-i\pi}{\omega_\alpha} [\delta(\omega + \omega_\alpha) + \delta(\omega - \omega_\alpha)],$$
(35)
where (eventually we take the single cut-off limit as described after Eq. (3) and thus omit the label $i$ on $\omega_\alpha$)
$$\omega_\alpha = \sqrt{\epsilon_\alpha^2 + \Lambda_i^2}. \quad (36)$$
We recall that the single cut-off approach requires a derivative with respect to that cut-off. Of course, this also affects the implicit dependence of $\omega_\alpha$ on that cut-off.
We introduce the Fourier transform of the quark wave-function as
$$\tilde{\Psi}_\alpha(p) = \int \frac{d^3 r}{4\pi} \Psi_\alpha(r) e^{ip \cdot r}.$$
(37)
Implementing a full Fourier transform differs from the approaches of Refs. [15, 16] who used the expansion from diagonalizing the Dirac Hamiltonian, Eq. (13). This resulted in discontinuities of the numerically computed quark distributions and required a smoothing procedure.
Performing the frequency $(\omega)$ and lambda$^4$ ‘$\lambda$’ integrals gives the vacuum contribution of the flavor-singlet unpolarized structure function in the nucleon rest frame (RF)
$$[F_1^{l=0}(x)]_s = \frac{5 MN}{144} \sum_\alpha \sum_{i=0}^2 c_i \int_{|M x^\pm|}^\infty p dp \int d\Omega_p \left\{ \pm \bar{\Psi}_\alpha(p) \tilde{\Psi}_\alpha(p) - \frac{\epsilon_\alpha M x^\pm}{\omega_\alpha p} \bar{\Psi}_\alpha(p) \tilde{\Psi}_\alpha(p) \right\}.$$
(38)
$^4$ Technically it is advisable to average the photon direction rather than fixing it along the $z$-axis [15].
where
\[ Mx_\pm^\alpha = Mx \pm \omega_\alpha. \]
(39)
In the above \([F_1^{I=0}(x)]_s^\pm \) refers to the positive (negative) frequency components that are typically referred to as quark and antiquark distributions. In our calculation they arise from the two poles of the \(\delta\)-function in Eq. (35). Then the vacuum part of the isoscalar, unpolarized structure function becomes
\[ [F_1^{I=0}(x)]_s = [F_1^{I=0}(x)]_s^- + [F_1^{I=0}(x)]_s^+. \]
(40)
As a matter of fact, this is still not the full result. Substituting free spinors (not interaction with the soliton) produces a non-zero result. This non-zero result must also be subtracted. In the discussion of the sum rules below we will see that this is nothing but the \(c_\alpha^{(0)}\) type subtraction performed in Eq. (20) and may be considered a cosmological constant type contribution.
The valence quark contribution is obtained by replacing the quark levels in (38) by the cranked valence level \(\psi_v(\tau, t) = e^{-i\epsilon_v t} A(t) \left\{ \Psi_v(r) + \frac{1}{2} \sum_{\alpha \neq v} \Psi_\alpha(r) \langle \frac{\alpha}{\epsilon_v - \epsilon_\alpha} \right\} = e^{-i\epsilon_v t} A(t) \psi_v(r), \)
(41)
In the above \(\psi_v(r)\) is the spatial part of the valence quark wave-function with the rotational correction included and \(\epsilon_v\) is the energy eigenvalue of the valence quark level. Noting that the valence quark wave-function has positive parity and the pole contribution \(f_\pm|_{\text{pole}} = -4i\pi\delta(\omega \mp \epsilon_v)\) gives the valence quark contribution
\[ [F_1^{I=0}(x)]_v^\pm = \frac{-5MN_e}{144} \left[ 1 + \text{sign}(\epsilon_v) \right] \int_0^\infty p dp \int d\Omega_p \left\{ \pm \bar{\Psi}_v(p)\tilde{\Psi}_v(p) - \frac{Mx_\pm^\alpha}{p} \tilde{\Phi}_v(p)\tilde{\Phi}_v(p) \right\}, \]
(42)
where \(Mx_\pm^\alpha = Mx \pm \epsilon_v.\) Again, we have separated positive and negative frequency components.
The quark spinors \(\Psi_v(r)\) separate into radial and angular pieces \([20].\) At the end, the structure functions, as in Eq. (38) are computed as integrals over the (Bessel-)Fourier transforms of the radial functions in the quark spinors. In Appendix [A] we list examples explicitly.
In quite an analogous manner, the isovector components of the polarized structure functions are extracted from the anti-symmetric combination \(W_{\mu\nu}(q)\). Explicitly we find the vacuum contribution to the longitudinal polarized structure function to be
\[ \left[ g_1^{I=1}(x) \right]_s^\pm = \frac{2MN_e}{72} I_3 \sum_{\alpha} \sum_{i=0}^{2} C_i \left\{ \mp \int_{|Mx_\pm^\alpha|}^\infty dp Mx_\alpha \int d\Omega_p \bar{\tilde{\Phi}}_\alpha(p)\hat{\Phi}_\alpha(p) \right\}, \]
(43)
and the isovector transverse structure function as
\[ \left[ g_1^{I=1}(x) \right]_s^\pm = \frac{2MN_e}{72} I_3 \sum_{\alpha} \sum_{i=0}^{2} C_i \left\{ \frac{\epsilon_\alpha}{\omega_\alpha} \int_{|Mx_\pm^\alpha|}^\infty dp p^2 \left[ A_\pm \int d\Omega_p \bar{\tilde{\Phi}}_\alpha(p)\hat{\Phi}_\alpha(p) + B_\pm \int d\Omega_p \bar{\tilde{\Phi}}_\alpha(p)\hat{\Phi}_\alpha(p) \right] \right\}. \]
(44)
In these formulas we have introduced the abbreviations, see also Eq. (39)
\[ A_\pm = \frac{1}{2p} \left( 1 - \frac{(Mx_\pm^\alpha)^2}{p^2} \right), \quad B_\pm = \frac{1}{2p} \left( 3\frac{(Mx_\pm^\alpha)^2}{p^2} - 1 \right). \]
(45)
The total (vacuum contribution to the) polarized structure functions is the sum of the positive (+) and negative (−) frequency components. Again, some details in terms of the Fourier transformed radial functions are given in Appendix [C]. For completeness we also list the formulas for the valence quark contribution [12]. The contribution to
the longitudinal polarized structure function is obtained as
\[
[g_{l1}^T(x)]^\pm = \frac{M N_c}{144} [1 + \text{sign}(\epsilon_v)] I_3 \left\{ \mp \int_0^\infty dp \int_{|Mx^\pm|} \frac{d\Omega_p}{4\pi} \tilde{\Psi}_v(p) \hat{p} \cdot \tau \tilde{\Psi}_v(p) + B_{\pm} \int_{|Mx^\pm|} \frac{d\Omega_p}{4\pi} \tilde{\Psi}_v(p) \hat{p} \cdot \tau \tilde{\Psi}_v(p) \right\},
\]
and that for the transverse polarized structure function as
\[
[g_{l1}^T(x)]^\pm = \frac{M N_c}{144} [1 + \text{sign}(\epsilon_v)] I_3 \left\{ \mp \int_0^\infty dp \int_{|Mx^\pm|} \frac{d\Omega_p}{4\pi} \tilde{\Psi}_v(p) \tau \cdot \sigma \tilde{\Psi}_v(p) \right\},
\]
The isovector unpolarized and isoscalar polarized structure functions are subleading in the $1/N_C$ counting. They are also more complicated to compute as they are quartic in the spinors and involve double sums over the basis states defined Eq. [18]. We refrain from presenting those lengthy expressions here and rather refer the interested reader to the Appendixes of Ref. [27].
VI. FORMAL DISCUSSION OF SUM RULES
In this section we discuss how the sum rules for the unpolarized and polarized structure functions work out when written explicitly in terms of the momentum space eigenspinors $\tilde{\Psi}_\alpha$. In this context it is important to note that we compute the structure functions for a localized configuration in its rest frame. Then the Bjorken variable has support on the half axis from zero to infinity. Lorentz covariance is regained by transforming to the infinite momentum frame, cf. Section VII.B
Sum rules relate integrated structure functions to static observables. In soliton models the latter are directly expressed in terms of the eigenspinors, Eq. [18] in coordinates space. Typically the sum rules can then be expressed as level-by-level identities. The only exception is the momentum (or energy) sum rule. For it to be obeyed it is compulsory that the soliton is an extremum of the energy functional, Eq. [20].
A. Momentum sum rule
For the momentum sum rule we require that $\frac{36}{M} \int dxx F_1(x)$ produces the quark contribution to the classical energy, i.e. all but the last integral in Eq. [20]. First we consider the scalar terms, $\pm \tilde{\Psi}_\alpha(p) \tilde{\Psi}_\alpha(p)$, from the vacuum contribution, Eq. [35].
\[
[M_{G0}] = \frac{M N_c}{4} \sum_{i=0}^2 c_i \int_0^\infty dx x \left\{ \left\langle \alpha | \alpha \right\rangle_{|Mx^+|} - \left\langle \alpha | \alpha \right\rangle_{|Mx^-|} \right\}
\]
\[
= \frac{M N_c}{4} \sum_{i=0}^2 c_i \int_0^\infty dy \left( y - \frac{\omega_\alpha}{M} \right) \left\langle \alpha | \alpha \right\rangle_{My} - \int_0^\infty dy \left( y + \frac{\omega_\alpha}{M} \right) \left\langle \alpha | \alpha \right\rangle_{MMy},
\]
\[
= -\frac{N_c}{2} \sum_{i=0}^2 c_i \int_0^\infty dy \left\langle \alpha | \alpha \right\rangle_{My} = \frac{N_c}{2} \sum_{i=0}^2 c_i \omega_\alpha \int_0^\infty dy \left\langle \alpha | \alpha \right\rangle_{MMy} \int_0^\infty dp \frac{d\Omega_p}{4\pi} \tilde{\Psi}_\alpha(p) \tilde{\Psi}_\alpha(p),
\]
\[
= -\frac{M N_c}{2} \sum_{i=0}^2 c_i \omega_\alpha \int_0^\infty dy \left[ \frac{d\Omega_p}{4\pi} \tilde{\Psi}_\alpha(p) \tilde{\Psi}_\alpha(p) \right] = -\frac{N_c}{2M} \sum_{i=0}^2 c_i \sqrt{c_\alpha^2 + \lambda_i^2},
\]
\[\tag{48}\]
We adopt the notation $\left\langle \alpha | \alpha \right\rangle_a = \int_0^\infty dp \frac{d\Omega_p}{4\pi} \tilde{\Psi}_\alpha(p) \tilde{\Psi}_\alpha(p)$.
\[\tag{49}\]
which is $1/M$ times the vacuum contribution to the classical energy. This contribution also includes subtraction of the trivial vacuum energy, when there is no soliton. Hence the isoscalar unpolarized structure function necessitates the analog subtraction, as indicated earlier. For the valence contribution the momentum sum rule gives
$$\left[\mathcal{M}_{G}^{0}\right]_{v} = \frac{36}{5} \int_{0}^{\infty} dx \varepsilon \left[F_{1}^{0}(x)\right]_{v} = \frac{N_{C}}{2M} \left[1 + \text{sign}(\varepsilon)\right] \varepsilon_{v}.$$ \hspace{1cm} (49)
Similarly, integrating the term with the operator $\hat{p} \cdot \alpha$ gives
$$\left[\mathcal{M}_{G}^{1}\right]_{s} = -\frac{M^{2}N_{C}}{2} \sum_{i=0}^{2} c_{i} \int_{0}^{\infty} dy \varepsilon \langle \alpha | \frac{\hat{p}}{\sqrt{\alpha}} | \alpha \rangle_{\text{My}} = -\frac{N_{C}}{6M} \sum_{i=0}^{2} c_{i} \frac{\varepsilon_{\alpha}}{\omega_{\alpha}} \langle \alpha | \alpha \rangle \varepsilon_{\alpha} \cdot p \alpha \rangle.$$ \hspace{1cm} (50)
Next we use the Dirac Hamiltonian, Eq. (13) to write
$$\varepsilon_{\alpha} \cdot p = \{ r \cdot p, h \} - m \beta \{ r \cdot p, U_{5}(r) \}$$
so that $\langle \alpha | \varepsilon_{\alpha} \cdot p | \alpha \rangle = \varepsilon_{\alpha} \langle \alpha | r \cdot p, U_{5}(r) | \alpha \rangle$. Since $r \cdot p$ is the dilatation operator this matrix element measures the change of the single particle energy when scaling the soliton extension by an infinitesimal amount. Furthermore
$$\frac{\omega_{\alpha}}{\omega_{\alpha}} = \frac{\partial}{\partial \omega_{\alpha}} \sqrt{\varepsilon_{\alpha}^{2} + \lambda^{2}}$$
so that $\left[\mathcal{M}_{G}^{1}\right]_{s}$ is the change of the vacuum energy when the soliton extension deviates slightly from its stationary point. Similarly, the valence quark adds $\left[\mathcal{M}_{G}^{2}\right]_{v} = i \frac{N_{C}}{12M} \left[1 + \text{sign}(\varepsilon)\right] \langle \psi | r \cdot p, U_{5}(r) | \psi \rangle$ to the sum rule. Then $\left[\mathcal{M}_{G}^{1}\right]_{s} + \left[\mathcal{M}_{G}^{2}\right]_{v}$ is the coefficient of $(\lambda - 1)$ term in the expansion
$$E[U(\lambda \psi)] = E_{0} + (\lambda - 1)E_{1} + \cdots (\lambda - 1)^{l}E_{l} + \cdots$$ \hspace{1cm} (51)
of the classical energy. Since $U(x)$ is a stationary point, $E_{1} = 0$ thus verifying the momentum sum rule \[13\]. Obviously the momentum sum rule is not saturated level by level; rather it requires summing all contributions to this isoscalar unpolarized structure function. Hence this sum rule will be a very sensitive test of the numerical simulation.
### B. Bjorken sum rule
Here we verify the Bjorken sum rule in our model, which relates the isovector polarized structure function $g_{1}^{T}$ to the axial charge \[23\]. First, we show that the term in Eq. (43) with the operator $\hat{p} \cdot \tau \gamma_{5}$ integrates to zero
$$\sum_{i=0}^{2} c_{i} \int_{0}^{\infty} dx \left\{ Mx^{-1} \langle \alpha | \frac{\hat{p}}{\sqrt{\alpha}} \tau \gamma_{5} | \alpha \rangle_{[Mx^{-1}]} - Mx^{+} \langle \alpha | \frac{\hat{p}}{\sqrt{\alpha}} \tau \gamma_{5} | \alpha \rangle_{[Mx^{+}]} \right\},$$
$$= \sum_{i=0}^{2} c_{i} \left\{ \int_{-\infty}^{0} \frac{dy}{\alpha} M y \langle \alpha | \frac{\hat{p}}{\sqrt{\alpha}} \tau \gamma_{5} | \alpha \rangle_{[M y]} - \int_{0}^{\infty} \frac{dy}{\alpha} M y \langle \alpha | \frac{\hat{p}}{\sqrt{\alpha}} \tau \gamma_{5} | \alpha \rangle_{[M y]} \right\},$$
$$= \sum_{i=0}^{2} c_{i} \int_{-\infty}^{\infty} \frac{dy}{\alpha} M y \langle \alpha | \frac{\hat{p}}{\sqrt{\alpha}} \tau \gamma_{5} | \alpha \rangle_{[M y]} = 0.$$ \hspace{1cm} (52)
There are two contributions without $\gamma_{5}$. The first one contributes
$$\frac{MN_{C}I_{3}}{72} \sum_{i=0}^{2} c_{i} \frac{\varepsilon_{\alpha}}{\omega_{\alpha}} \int_{0}^{\infty} dx \left[ \langle \alpha | \frac{pA_{+} \tau \cdot \sigma | \alpha \rangle_{[M x^{+}]} + \langle \alpha | \frac{pA_{-} \tau \cdot \sigma | \alpha \rangle_{[M x^{-}]} \right]$$
$$= \frac{MN_{C}I_{3}}{144} \sum_{i=0}^{2} c_{i} \frac{\varepsilon_{\alpha}}{\omega_{\alpha}} \left[ \int_{-\infty}^{\infty} \frac{dy}{\alpha} M y \langle \alpha | \tau \cdot \sigma \left( 1 - \frac{(M y)^{2}}{p^{2}} \right) | \alpha \rangle_{[M y]} + \int_{-\infty}^{\infty} \frac{dy}{\alpha} M y \langle \alpha | \tau \cdot \sigma \left( 1 - \frac{(M y)^{2}}{p^{2}} \right) | \alpha \rangle_{[M y]} \right]$$
$$= \frac{N_{C}I_{3}}{108} \sum_{i=0}^{2} c_{i} \frac{\varepsilon_{\alpha}}{\omega_{\alpha}} \langle \alpha | \tau \cdot \sigma | \alpha \rangle.$$ \hspace{1cm} (53)
The term with $\hat{p} \cdot \tau \cdot \sigma$ disappears because
$$
\int_0^\infty dy \int_{M_y}^\infty p^2 dp \int d\Omega_p \left( \frac{1}{p} - \frac{3}{y^2} \right) \bar{\Psi}_\alpha(p) \hat{p} \cdot \tau \cdot \sigma \Psi_\alpha(p) = \int_0^\infty dy \int_{M_y}^\infty dp \int d\Omega_p \frac{\partial}{\partial y} \left( py - \frac{M^2 y^3}{p} \right) \bar{\Psi}_\alpha(p) \hat{p} \cdot \tau \cdot \sigma \Psi_\alpha(p) = \int_0^\infty dy \left( \frac{M^2 y^3}{p} - py \right) \left[ \int d\Omega_p \bar{\Psi}_\alpha(p) \hat{p} \cdot \tau \cdot \sigma \Psi_\alpha(p) \right]_{p=M_y} = 0. \tag{54}
$$
Hence the Bjorken sum rule for the vacuum contribution of the longitudinal polarized structure function becomes
$$
\int dx [g_1^p(x) - g_1^n(x)]_v = \frac{N_C}{108} \sum_{i=0}^{2} c_i \frac{\epsilon_\alpha}{\omega_0} \langle \alpha | \tau \cdot \sigma | \alpha \rangle = \frac{1}{6} \left[ \frac{N_C}{18} \sum_{\alpha} \sum_{i=0}^{2} c_i \frac{\epsilon_\alpha}{\sqrt{\epsilon_\alpha^2 + \Lambda^2_5}} \langle \alpha | \gamma_3 \gamma_5 \tau_3 | \alpha \rangle \right]. \tag{55}
$$
The object in square brackets is the vacuum contribution to the axial charge \[29\]. Similar calculations from the valence contribution give
$$
\int dx [g_1^p(x) - g_1^n(x)]_v = -\frac{N_C}{54} \left[ 1 + \text{sign}(\epsilon_v) \right] \langle v | \tau \cdot \sigma | v \rangle = -\frac{1}{6} \left[ -\frac{N_C}{9} \left[ 1 + \text{sign}(\epsilon_v) \right] \langle v | \gamma_3 \gamma_5 \tau_3 | v \rangle \right] \tag{56}
$$
with the object in square brackets being the valence quark contribution to the axial charge. This indeed verifies the Bjorken sum rule for the total axial charge.
In an analog, yet much more tedious, calculation the sum rules for the subleading contributions in the $1/N_C$ expansion are also verified via level by level identities. Details may be found in Ref. \[27\]. We would like to mention however, that the Adler sum rule \[29\], which concerns a structure function from neutrino interactions and thus the exchange of a gauge boson (not considered here), measures the isospin of the nucleon. In that case the sum rule is not level by level; rather summing this integrated structure function over all levels reproduces the moment of inertia, Eq. \[24\] \[30\].
\[\text{VII. NUMERICAL RESULTS}\]
In this Section we present our numerical results for the structure functions. These results are obtained in a number of subsequent steps. First we construct the coordinate space eigenspinors of the self-consistent chiral soliton as described in Section \[IV\] for the parameters listed at the end of Section \[II\]. In the second step we evaluate the Fourier transform according to Eq. \[37\]. Details of this transformation are provided in Appendix \[A\]. Essentially the spinors in momentum space are combinations of spherical harmonic functions of the unit momentum vector and momentum space radial functions that are Bessel transforms of the radial functions in the coordinate space spinors from Section \[IV\]. In momentum space the spherical harmonic functions combine to the conserved grand spin just as do those in coordinate space. Hence we formally obtain matrix elements of operators as, for example $\alpha \cdot \hat{p}$, in the very same way as the matrix elements of $\alpha \cdot \hat{r}$ in coordinate space. In the third step the momentum space radial functions are numerically integrated to produce the structure functions in the nucleon rest frame. In the next step they are transformed to the infinite momentum frame \[31\] and subsequently the standard (perturbative QCD) evolution to the scale of the experimental data is performed to allow for a sensible comparison. We note that this evolution brings into the game a new model parameter, the scale at which the evolution commences. We take a single scale for all structure functions.
We test the outcome of our numerical simulations via the sum rules, that is, we compare the integrated functions with associated local quantity obtained from the coordinate space spinors. To gain acceptable agreement a very fine (equi-distant) grid for the radial variable in momentum space is required. Typically we take several thousand points on an interval between zero and ten times the physical cut-off, $\Lambda$. Needless to say that this consumes a large amount of CPU time and obtaining (in particular the subleading $1/N_C$ contributions to) the structure functions takes days or weeks on an ordinary desktop PC. Still, there are minor numerical inaccuracies as reflected by small oscillations of the structure functions around a central value at larger $x$, cf. figures below. Working in momentum space, rather than using the expansion coefficients $V_{\alpha\beta}$ introduced after Eq. \[18\] has, however, the advantage that no smearing \[14\] procedure is required.
FIG. 1: Isoscalar unpolarized structure function in the nucleon rest frame for a constituent quark mass of 400MeV. Dashed and dotted lines refer to the positive and negative frequency contributions, respectively.
FIG. 2: Isovector unpolarized structure function in the nucleon rest frame for a constituent quark mass of 400MeV. Note the logarithmic scale for the Bjorken variable. Dashed and dotted lines refer to the positive and negative frequency contributions, respectively.
A. Rest frame results
In Figures 1 and 2 we show typical results for the isoscalar and isovector components of the unpolarized structure functions respectively. In this case they have been obtained using the constituent quark mass of \( m = 400\text{MeV} \). We separately show the contributions of the discrete valence level, those of the vacuum contributions and their sums (labeled as total). For the vacuum contribution we find the unexpected result that it dominates the valence counterpart. Mainly this originates from the (additional) subtraction of the non-soliton piece mentioned after Eq. (40). Without that subtraction we would not get a finite result, of course. Neither would the momentum sum rule be fulfilled. However, this piece does not connect to the soliton rest frame and it is not clear at all whether or not transformation of the Bjorken variable should be performed before taking the difference between the soliton and non-soliton isoscalar unpolarized structure functions. Therefore we do not attach much relevance to this large vacuum contribution. As expected, the vacuum contribution is sub-dominant for the isovector unpolarized structure function.
In Figure 3 we present the unpolarized structure function that enters the Gottfried sum rule, i.e. \( F_{p}^{2}(x) - F_{n}^{2}(x) = 2x[F_{1}^{p}(x) - F_{1}^{n}(x)] \) as the Callan-Gross relation holds in the soliton rest frame. The vacuum contribution turns slightly negative at large \( x \) which persists when adding the dominating valence piece to form the total contribution of this structure function. In Table II we compare our results for the Gottfried sum rule, \( S_{G} = \int_{0}^{\infty} \frac{dx}{x} (F_{p}^{2} - F_{n}^{2}) \), for various constituent quark masses to the experimental data from the NMC Collaboration [32]. Under this integral, the vacuum part is even less significant as its positive and negative parts compensate. In total, the agreement for the Gottfried sum rule is surprisingly good since usually chiral soliton models reproduce empirical data with 30% accuracy [8].
In Figures 4 and 5 we show the isoscalar and isovector contributions to the longitudinal polarized structure functions \( g_{I=0,1} \), respectively. In both pictures we display the valence and vacuum contributions as well as their sums. Also the positive and negative frequency components of the valence and vacuum parts are shown. Obviously these structure functions are indeed dominated by their valence contributions and we thus a posteriori verify the valence
FIG. 3: Unpolarized structure function \( F_2^p(x) - F_2^n(x) \) in the nucleon rest frame for a constituent quark mass of 400MeV.
TABLE I: The Gottfried sum rule for various values of \( m \). The subscripts 'v' and 's' denote the valence and vacuum contributions, respectively. The third column contains their sums.
| \( m \) [MeV] | \( |S_G|_v \) | \( |S_G|_s \) | \( S_G \) | empirical value |
|-------------|-------------|-------------|-------|----------------|
| 400 | 0.214 | 0.000156 | 0.214 | |
| 450 | 0.225 | 0.000248 | 0.225 | 0.235 ± 0.026 [32] |
| 500 | 0.236 | 0.000356 | 0.237 | |
*only approximation* adopted in Ref. [12]. The only exception is the isoscalar structure function \( g_2 \) for which the valence contribution is small by its own due to large cancellations in \( g_2 = g_T - g_1 \). We also recognize some minor oscillations in the vacuum contributions at larger \( x \). These occur as remnants of numerical inaccuracies. We also remark that the present valence quark results do not exactly match those from Ref. [12] in which a soliton profile from the proper-time regularization scheme was employed.
We have computed the axial vector and singlet charges on one hand side via the respective sum rules, i.e. by integrating the structure functions \( g_{I=1,0}^I(x) \) and on the other hand via the coordinate space matrix elements of \( \gamma_3 \gamma_1 \gamma_5 \) and \( \gamma_3 \gamma_5 \) as e.g. in Eqs. (55) and (60). The comparison in table II serves as test for the numerical accuracy which works perfectly in the vector case while some minor discrepancies are observed for the axial singlet charge. This is understood as the latter is actually quartic in the quark wave-functions (two of which are Fourier transformed) and it is also a double sum over those wave-functions. So even tiny numerical errors are amplified.
As is typical for chiral soliton models, the axial vector charge falls short off the measured datum by about 30-40% [8]. It has been argued that this could be remedied by \( 1/N_C \) corrections arising from a particular handling of the collective coordinate quantization [33, 34]. However, these corrections do not emerge in the current approach and also lead to inconsistencies with PCAC [35]. On the other hand, the predicted axial singlet charge, which is linked to the
FIG. 4: Isoscalar longitudinal polarized structure functions in the nucleon rest frame for a constituent quark mass of 400MeV. For the valence and vacuum contributions we separately display the positive (dashed) and negative (dotted) frequency contributions. The full lines are their sums. Note the small vertical scale for the vacuum contribution.
proton spin problem\cite{36}, is well within the errors of the empirical value.
\textbf{B. Projection and evolution}
The soliton picture for baryons employs a localized field configuration which generally breaks translational invariance. This causes the structure functions not to vanish when $x > 1$ as would be demanded kinematically. This effect is obvious in the above figures. We note that it is not limited to soliton models but is observed, \textit{i.e.} in the bag model as well\cite{39}. Using light cone coordinates in the bag model in one space dimension a mapping of the structure functions from the localized field configuration was constructed that annihilated the structure functions for
TABLE II: The axial-vector and -singlet charges for various values of the constituent quark mass $m$. Subscripts are as in table 4. The data in parenthesis give the numerical results as obtained from the coordinate space representation.
| $m$ [MeV] | $g_A$ | $g_{A\perp}$ | empirical value | $m$ [MeV] | $g_A$ | $g_{A\perp}$ | empirical value |
|-----------|-------|--------------|-----------------|-----------|-------|--------------|-----------------|
| 400 | 0.734 | 0.065 | 0.799 (0.800) | 400 | 0.344 | 0.0016 | 0.345 (0.350) |
| 450 | 0.715 | 0.051 | 0.766 (0.765) | 450 | 0.327 | 0.0021 | 0.329 (0.332) |
| 500 | 0.704 | 0.029 | 0.733 (0.733) | 500 | 0.316 | 0.0028 | 0.318 (0.323) |
$x > 1$. Guided by that construction a Lorentz boost was applied transforming the rest frame structure functions to the infinite momentum frame.
$$f_{IMF}(x) = \frac{\Theta(1-x)}{1-x} f_{RF}(-\ln(1-x)),$$
(57)
where $f_{RF}$ refers to any of the structure functions computed in Section VII A. In what follows we will omit the label IMF for the boosted structure functions.
Even though we have adopted the high energy Bjorken limit in our kinematical analysis of the Compton tensor, it must be emphasized that the NJL model is (at best) an approximation to QCD at the low mass scale, $\mu^2$ which is thus a hidden parameter in the approach. To compare with experimental data that are taken at higher energy scales, $Q_{exp}^2$ we adopt Altarelli-Parisi (or DGLAP) equations (41–43), to evolve the model structure functions accordingly. To be precise, we integrate
$$f(x, t + \delta t) = f(x, t) + \delta t \frac{df(x, t)}{dt}.$$
(58)
with $t = \ln \left(\frac{Q^2}{\Lambda_{QCD}^2}\right)$ from $Q^2 = \mu^2$ to $Q^2 = Q_{exp}^2$. The structure functions from Eq. (57) are the initial values and we tune $\mu^2$ for best fit at $Q_{exp}^2$.
Since the isoscalar structure functions are associated with gluon type quantum numbers they mix under the evolution. We take this into account under the assumption that the gluon distributions vanish at $\mu^2$. We consider the leading order of the perturbative expansion sufficient to estimate the quality of our results. Then the evolution equations have the following structure
$$\frac{df^{(I=1)}(x, t)}{dt} = \frac{g_{QCD}(t)}{2\pi} C_R(F) \int_x^1 \frac{dy}{y} P_{qq}(y) f^{(I=1)} \left(\frac{x}{y}, t\right),$$
(59)
$$\frac{df^{(I=0)}(x, t)}{dt} = \frac{g_{QCD}(t)}{2\pi} C_R(F) \int_x^1 \frac{dy}{y} \left\{ P_{qq}(y) f^{(I=0)} \left(\frac{x}{y}, t\right) + P_{qq}(y) g \left(\frac{x}{y}, t\right) \right\},$$
(60)
$$\frac{dg(x, t)}{dt} = \frac{g_{QCD}(t)}{2\pi} C_R(F) \int_x^1 \frac{dy}{y} \left\{ P_{gg}(y) g \left(\frac{x}{y}, t\right) + P_{gg}(y) f^{(I=0)} \left(\frac{x}{y}, t\right) \right\},$$
(61)
where $C_R(F) = \frac{(N_f^2 - 1)}{2N_f}$ is the color factor for $N_f$ flavors. Furthermore $g_{QCD}(t) = \frac{4\pi}{\beta_0 t}$ with $\beta_0 = \frac{11}{3} N_C - \frac{2}{3} N_f$ is the leading order perturbative running coupling constant. Explicit expressions for the splitting functions $P_{qq}, \ldots, P_{gg}$, taken from Ref. (44) are listed in Appendix D for completeness. From the evolved isoscalar and isovector components we finally obtain the proton and neutron structure functions as sum and difference
$$f^{(p,n)}(x, Q^2) = \frac{1}{2} \left[ f^{I=1}(x, Q^2) \pm f^{I=0}(x, Q^2) \right].$$
(62)
We note that applying the perturbative QCD scheme to the model structure functions requires the identification of model and QCD degrees of freedom even though there is no definite reason for doing so other than the lack of any sensible alternative.
The second polarized structure function $g_2$ contains subleading, twist three, elements that undergo a different evolution procedure that is also described in Appendix D.
FIG. 8: Model prediction for the longitudinal polarized proton structure functions. Left panel: \( g_1^p(x) \); right panel: \( g_1^{3\text{He}}(x) \). These functions are “DGLAP” evolved from \( \mu^2 = 0.4\text{GeV}^2 \) to \( Q^2 = 3\text{GeV}^2 \) after projected to the infinite momentum frame “IMF”. Data are from Refs. [46, 47] for the proton and from Ref. [48] for helium. In the latter case \( E \) refers to the electron energy.
FIG. 9: Model prediction for the polarized proton structure functions \( g_{2W}^{WW(p)}(x) \) (left panel) and \( g_2^p(x) \) (right panel) the are the twist-2 and -3 pieces of \( g_2 \). These functions are “DGLAP” evolved from \( \mu^2 = 0.4\text{GeV}^2 \) to \( Q^2 = 5\text{GeV}^2 \) after projected to the infinite momentum frame “IMF”.
C. Comparison with experiment
As in previous calculations within the valence only approximation [11, 12, 45] we take \( \mu^2 = 0.4\text{GeV}^2 \) as initial value in the evolution differential equations. Smaller values contradict the perturbative nature of the evolution procedure as \( \Lambda^2_{\text{QCD}} \) becomes sizable in view of \( \Lambda^2_{\text{QCD}} = 0.2\text{GeV}^2 \). On the other hand, significantly larger \( \mu^2 \) values worsen the agreement with experimental data.
In the left panel of Figure 8 we show the numerical result for the polarized proton structure function \( g_1 \) obtained from the evolution equation at \( Q^2 = 3\text{GeV}^2 \). We compare our results to experimental results from the E143 Collaboration [46, 47]. At small \( x \) the model results are somewhat larger than the data, but definitely the gross features are predominantly reproduced.
For the neutron data are available in terms of the helium structure function [48]
\[
g_1^{3\text{He}}(x) \approx P_n g_1^n(x) + P_p g_1^p(x) - 0.014 \left[ g_1^p(x) - 4g_1^n(x) \right],
\]
with \( P_n \approx 0.86 \) and \( P_p \approx -0.028 \). From the right panel in Figure 8 we see that our model results reproduce the main features of the data: small positive values at large \( x \) turning negative at moderate \( x \), though the minimum is more pronounced by the model.
Next we discuss the results for the structure function \( g_2(x) \). As discussed in Appendix D the twist-2 and -3 pieces must be disentangled within the evolution whose result is shown in Figure 9. The effect of evolution is small for the twist-2 component but essential for the twist-3 counterpart. When the end point of evolution is reached, the two components are combined to \( g_2^p(x, Q^2) \). We display the model prediction in Figure 10 and see that the data are well produced. This shows that the higher twist contributions cannot be neglected. This suggests that the higher twist contributions cannot be neglected.
\[\text{In Ref. [48] direct neutron data are only given as the ratio } g_1^n(x)/F_1(x).\]
FIG. 10: Model prediction for the polarized proton structure functions $g_2^p(x)$. This function is “DGLAP” evolved from $\mu^2 = 0.4\, GeV^2$ to $Q^2 = 5\, GeV^2$ after projected to the infinite momentum frame “IMF”. Data is from Ref [40].
Recently data were reported for the neutron twist-3 moment
$$d_n^3(Q^2) = 3 \int_0^1 dx \, x^2 \, \bar{g}_n^2(x, Q^2)$$
et two different transferred momenta: $d_n^3(3.21 GeV^2) = -0.00421 \pm 0.00114$ and $d_n^3(4.32 GeV^2) = -0.00035 \pm 0.00104$, where we added the listed errors in quadrature. For $m = 400\, MeV$ the model calculation yields $-0.00426$ and $-0.00035$, respectively. While the lower $Q^2$ result nicely matches the observed value, the higher one differs by about three standard deviations. This discrepancy as a function of $Q^2$ indicates that the large $N_C$ approximation to evolve $g_2$ (cf. Appendix D) requires improvement.
Finally, in Figure 11 we display the unpolarized structure function that enters the Gottfried sum rule, i.e. $F_2^p(x) - F_2^n(x)$ using the evolution equation. Though the negative contribution from the Dirac vacuum (cf. Figure 2) around $x = 1$ is tiny in the rest frame, it gets amplified when transforming to the infinite momentum frame by the factor $1/(1-x)$ in Eq. (57) thereby worsening the agreement with the experimental data collected by the NMC [32].
VIII. CONCLUSION
We have presented the numerical simulation of nucleon structure functions within the NJL chiral soliton model. Central to this analysis has been the consistent implementation of the regularized vacuum contributions that arise from all quark spinors being distorted by the chiral soliton. Generally speaking, vacuum contributions should not be omitted in any quark model as no expansion scheme suppresses them. This is even more the case for the NJL model because the vacuum part significantly contributes to forming the soliton.
In our analysis we have only identified the symmetry currents of the model with those from QCD, not the quark distributions that are bilinear operators which are bilocal in the quark fields. Also, it is important to enforce the regularization on the action functional so that the regularization prescription for a given structure function is an unambiguous result. This increases the predictive power compared to previous similar studies that ”advocated” an ad hoc regularization of quark distributions [14, 16]. A first principle regularization is particularly important when
---
7 Schwinger’s proper time regularization scheme is popular in the context of the NJL chiral soliton [24]. Ref. [16] explicitly states that its
the sum rule for the structure function does not relate to coordinate space matrix elements of the quark fields. As an example we have seen that the isovector unpolarized structure functions are not subject to regularization (the explicit, lengthy formulas can be obtained from Ref. [27]). The prediction for the corresponding, so-called Gottfried, sum rule decently matches the empirical value. This is a very favorable case for the *valence only approximation* as the vacuum contribution essentially integrates to zero due to an unexpected negative contribution at large \( x \). For the isoscalar part we recognized that the subtraction of the zero-soliton vacuum contribution has a sizable effect at small \( x \). The emergence of this contribution is somewhat surprising as it suggests that the zero-soliton vacuum has structure. Yet it is required for convergence as well as fulfilling the momentum sum rule. We emphasize that we observe acceptable agreement for polarized proton structure functions between our model results and the experimental data. For the polarized structure functions our numerically expensive computation indeed showed that the vacuum contribution is sub-dominant, except maybe for the isoscalar part of \( g_2 \) where the valence part is tiny by itself. Nevertheless these results support the *valence only approximation* to a large extent.
In both, the unpolarized and polarized cases, the comparison with experiment required two additional operations on the model structure functions. As the soliton is a localized field configuration, translational invariance is lost and the rest frame structure functions must be Lorentz transformed to the infinite momentum frame. In turn the results from that transformation are subject to the perturbative QCD evolution scheme. This brings into the game the hidden parameter at which to commence the evolution. We took that to be the same as in the *valence only approximation*.
Even though we have separated positive and negative frequency contributions to the structure functions we stop short of identifying them as (anti-)quark distributions that parameterize semi-hard processes [50], like e.g. Drell-Yan [51]. The reason being that we avoid to identify model and QCD quark degrees of freedom at the model scale. There are many other nucleon matrix elements of bilocal, bilinear quark operators for which experimental results or lattice data are available. Examples are chiral odd distributions [52, 53] or quasi-distributions [54–56]. It is challenging to see whether quark distributions, or at least some of them, can also be formulated and computed with a first principle regularization scheme in the NJL chiral soliton model.
Acknowledgments
This project is is supported in part by the National Research Foundation of South Africa (NRF) by grant 109497. I. T. gratefully acknowledges a bursary from the *Stellenbosch Institute for Advanced Studies* (STIAS).
Appendix A: Soliton matrix elements
The Dirac Hamiltonian \( h \) of the hedgehog field configuration [19] commutes with the grand spin operator
\[
G = J + \frac{\tau}{2} = L + \frac{\sigma}{2} + \frac{\tau}{2},
\]
which is the operator sum of the total spin \( J \) and the isospin \( \tau/2 \). The total spin is the operator sum of the orbital angular momentum \( L \) and the intrinsic spin \( \sigma/2 \). Since \( h \) preserves \( G \), the eigenfunctions of the Dirac Hamiltonian are also eigenfunctions of \( G \). The quantum numbers of \( G \) are \( G^2 = G(G+1) \) and \( G_3 = M \). The respective eigenfunctions are tensor spherical harmonics associated with the grand spin
\[
[Y_{LMJG}(\hat{r})]_{is} = \sum_{m,s,s_3,i_3,j_3} C_{LMi,Lmi}^{GM} C^{J,j_3}_{Lm} Y_{LM}(\hat{r}) \chi_s(s_3) \chi_i(i_3)
\]
where \( C_{LMi,Lmi}^{GM} \) and \( C^{J,j_3}_{Lm} \) are \( SU(2) \) Clebsch-Gordon coefficients that describe the coupling of \( \chi_s \) and \( \chi_i \), which are two components spinors and isospinors, respectively, and the spherical harmonic functions \( Y_{LM} \).
For a prescribed profile function \( \Theta(r) \) the numerical diagonalization of the Dirac Hamiltonian [19] produces the application to the quark distributions is not yet known.
radial functions \( g_{α}^{(G;±i)} \) and \( f_{α}^{(G;±i)} \) that feature in eight component spinors \( 26 \):
\[
\Psi_{α}^{(+)}(r) = \left( \frac{ig_{α}^{(G;+;1)}(r)}{f_{α}^{(G;+;1)}(r)} \right) Y_{GG+\frac{1}{2}GM}(\hat{r}) + \left( \frac{-ig_{α}^{(G;+;2)}(r)}{f_{α}^{(G;+;2)}(r)} \right) Y_{GG-\frac{1}{2}GM}(\hat{r}) \tag{A3}
\]
\[
\Psi_{α}^{(-)}(r) = \left( \frac{-ig_{α}^{(G;-;1)}(r)}{f_{α}^{(G;-;1)}(r)} \right) Y_{GG+\frac{1}{2}GM}(\hat{r}) + \left( \frac{ig_{α}^{(G;-;2)}(r)}{f_{α}^{(G;-;2)}(r)} \right) Y_{GG-\frac{1}{2}GM}(\hat{r}) \tag{A4}
\]
Here, the second superscript \((±)\) denotes the intrinsic parity defined by the parity eigenvalue as \((-1)^{G} \times (±1)\). The radial functions are written as linear combinations of spherical Bessel functions that build the free spinors \( \Psi_{α}^{(0)} \). The order of these Bessel functions matches the angular momentum label \( G \) (first subscript) of the multiplying \( \mathcal{Y} \). The linear combination goes over momenta discretized by pertinent boundary conditions at a radius significantly larger than the extension of the profile function \( Θ(r) \). In Ref. \( 26 \) the condition that the radial function multiplying the \( G \) with equal orbital angular momentum and grand spin indexes vanished at that large distance. In contrast, we impose that condition on the radial function of the upper component. This avoids spurious contributions to the moment of inertia \( 18 \).
Writing
\[
e^{ip\tau} = 4π \sum_{LM} (i)^{L}j_{L}(pr)Y_{LM}(\hat{r})Y_{LM}(\hat{p}) \tag{A5}
\]
we find the Fourier transform, Eq. \( 37 \) of the spinors
\[
\bar{\Psi}_{α}^{(G,+)}(p) = (i)^{G+1} \left( \frac{g_{α}^{(G;+;1)}(p)}{f_{α}^{(G;+;1)}(p)} \right) Y_{GG+\frac{1}{2}GM}(\hat{p}) + (i)^{G-1} \left( \frac{-g_{α}^{(G;+;2)}(p)}{f_{α}^{(G;+;2)}(p)} \right) Y_{GG-\frac{1}{2}GM}(\hat{p}) \tag{A6}
\]
\[
\bar{\Psi}_{α}^{(G,-)}(p) = -(i)^{G} \left( \frac{g_{α}^{(G;-;1)}(p)}{f_{α}^{(G;-;1)}(p)} \right) Y_{GG+\frac{1}{2}GM}(\hat{p}) + (i)^{G+1} \left( \frac{-g_{α}^{(G;-;2)}(p)}{f_{α}^{(G;-;2)}(p)} \right) Y_{GG-\frac{1}{2}GM}(\hat{p}) \tag{A7}
\]
The radial functions in momentum space are the Fourier-Bessel transforms
\[
\bar{φ}_{α}(p) = \int_{0}^{∞} dr r^{2} j_{Lα}(pr)φ_{α}(r) \tag{A7}
\]
where \( Lα \) is the angular momentum associated with the coordinate space radial wave-function \( φ_{α}(r) \). Note that the grand spin spherical harmonic functions in momentum space are constructed precisely as in coordinate space, just that the argument is the momentum space solid angle. Note that the intrinsic parity is also conserved quantum number.
The valence quark carries \( G = 0 \), then only the components with \( J = +1/2 \) are allowed for the eigenspinor
\[
\Psi^{0,+}(r) = \Psi_{v}(r) = \left( \frac{ig_{v}(r)Y_{0,\frac{1}{2},0,0}(\hat{r})}{f_{v}(r)Y_{1,\frac{1}{2},0,0}(\hat{r})} \right) \tag{A8}
\]
here \( g_{v}(r) = g_{α}^{(0;+;1)}(r) \) etc, are the particular eigenvalue-functions. The cranking correction associated with the first order rotation \( \hat{H}_{α} \) dwells in the channel with \( G = 1 \) and negative intrinsic parity
\[
\Psi^{(-)}(r) = \left( \frac{ig_{α}^{(1)}(r)Y_{2,\frac{1}{2},1,0}(\hat{r})}{-f_{α}^{(1)}(r)Y_{1,\frac{1}{2},1,0}(\hat{r})} \right) + \left( \frac{ig_{α}^{(2)}(r)Y_{2,\frac{1}{2},1,0}(\hat{r})}{f_{α}^{(2)}(r)Y_{1,\frac{1}{2},1,0}(\hat{r})} \right) \tag{A9}
\]
for convenience we have written \( g_{α}^{(1;+;1)}(r) \) as \( g_{v}(r) \) etc. Taking the Fourier transform of equation \( 11 \) gives
\[
\bar{Ψ}_{v}(p) = \bar{Ψ}_{v}(p) + \sum_{α} \langle H_{α} | \bar{Ψ}_{α}(p) \rangle \tag{A10}
\]
where
\[
\bar{Ψ}_{v}(p) = \left( \frac{g_{v}(p)Y_{0,\frac{1}{2},0,0}(\hat{p})}{f_{v}(p)Y_{1,\frac{1}{2},0,0}(\hat{p})} \right) \tag{A11}
\]
TABLE III: Matrix elements $\int d\Omega p \bar{\psi}_{\alpha}(p) \gamma L' J' G M (p) \hat{p} \cdot \sigma \gamma L J G M (p)$.
\begin{tabular}{cccc}
\hline
$J' = G - \frac{i}{2}$ & $J' = G + \frac{i}{2}$ \\
$\frac{L'}{G} = G - 1$ & $\frac{L'}{G} = G$ & $\frac{L'}{G} = G$ & $\frac{L'}{G} = G + 1$
\hline
0 & 0 & 0 & $L = G - 1$
-1 & 0 & 0 & 0 & $L = G$
0 & 0 & 0 & -1 & $L = G$
0 & 0 & -1 & 0 & $L = G + 1$
\hline
\end{tabular}
and
$$\bar{\Psi}_\alpha(p) = -i \left( \bar{g}_{\alpha}^{(1)}(p) \gamma_{2,1,1,M}(p) - \bar{g}_{\alpha}^{(2)}(p) \gamma_{0,1,1,M}(p) \right) \left( \bar{f}_{\alpha}^{(1)}(p) \gamma_{1,1,1,M}(p) - \bar{f}_{\alpha}^{(2)}(p) \gamma_{1,1,1,M}(p) \right).$$ \hspace{1cm} \text{(A12)}$$
The “matrix element” $\langle H_\alpha \rangle$ arises from perturbatively treating the collective rotation
$$\langle H_\alpha \rangle = \frac{1}{2} \frac{\langle \sigma \cdot \Omega | v \rangle}{\epsilon_v - \epsilon_\alpha}.$$ \hspace{1cm} \text{(A13)}$$
Appendix B: Unpolarized Structure Functions at Leading Order
The level sums (over $\alpha$) as \textit{e.g.} in Eq. (63) concern the label of the radial function, grand spin ($G$) and its projection ($M$). As we average of the direction of the virtual photon \textit{[14]}, the matrix elements are degenerate in $M$. This produces the extra factor $2G + 1$ that we make explicit.
It is then straightforward to compute the matrix elements that appear in (51):
$$\int d\Omega p \bar{\Psi}_\alpha(p) \bar{\Psi}_\alpha(p)$$ \hspace{1cm} \text{(B1)}$$
and
$$\int d\Omega p \bar{\Psi}_\alpha(p) \hat{p} \cdot \alpha \bar{\Psi}_\alpha(p) = \int d\Omega p \bar{\Psi}_\alpha(p) \hat{p} \cdot \sigma \gamma_5 \bar{\Psi}_\alpha(p).$$ \hspace{1cm} \text{(B2)}$$
The positive intrinsic parity of the matrix element of (B2) is obtained as
$$\int d\Omega p \bar{\Psi}_\alpha^{(G,+)}(p) \bar{\Psi}_\alpha^{(G,+)}(p) = (2G + 1) \left( \bar{g}_{\alpha}^{(G,+,1)}(p)^2 + \bar{f}_{\alpha}^{(G,+,1)}(p)^2 + \bar{g}_{\alpha}^{(G,+,2)}(p)^2 + \bar{f}_{\alpha}^{(G,+,2)}(p)^2 \right),$$ \hspace{1cm} \text{(B3)}$$
and for the negative intrinsic parity as
$$\int d\Omega p \bar{\Psi}_\alpha^{(G,-)}(p) \bar{\Psi}_\alpha^{(G,-)}(p) = (2G + 1) \left( \bar{g}_{\alpha}^{(G,-,1)}(p)^2 + \bar{f}_{\alpha}^{(G,-,1)}(p)^2 + \bar{g}_{\alpha}^{(G,-,2)}(p)^2 + \bar{f}_{\alpha}^{(G,-,2)}(p)^2 \right).$$ \hspace{1cm} \text{(B4)}$$
Here the overall factor $(2G + 1)$ arises from summing the grand spin projection contained in $\sum_\alpha$. From Table (III) the positive intrinsic parity of the matrix element of (B2) is obtained as
$$\int d\Omega p \bar{\Psi}_\alpha^{(G,+)}(p) \hat{p} \cdot \sigma \gamma_5 \bar{\Psi}_\alpha^{(G,+)}(p) = -2(2G + 1) \left( \bar{g}_{\alpha}^{(G,+,1)}(p) \bar{f}_{\alpha}^{(G,+,1)}(p) + \bar{g}_{\alpha}^{(G,+,2)}(p) \bar{f}_{\alpha}^{(G,+,2)}(p) \right),$$ \hspace{1cm} \text{(B5)}$$
and that for the negative intrinsic parity as
$$\int d\Omega p \bar{\Psi}_\alpha^{(G,-)}(p) \hat{p} \cdot \sigma \gamma_5 \bar{\Psi}_\alpha^{(G,-)}(p) = -2(2G + 1) \left( \bar{g}_{\alpha}^{(G,-,1)}(p) \bar{f}_{\alpha}^{(G,-,1)}(p) + \bar{g}_{\alpha}^{(G,-,2)}(p) \bar{f}_{\alpha}^{(G,-,2)}(p) \right).$$
The matrix element from the valence contribution \textit{[12]} is easily obtained, using the definition of the decomposition of the valence wave function \textit{[11]}. They are given as
$$\int d\Omega p \bar{\Psi}_\alpha(p) \bar{\Psi}_\alpha(p) = \bar{g}_v(p)^2 + \bar{f}_v(p)^2$$ \hspace{1cm} \text{(B6)}$$
and
$$\int d\Omega p \bar{\Psi}_\alpha(p) \hat{p} \cdot \sigma \gamma_5 \bar{\Psi}_\alpha(p) = -2\bar{g}_v(p)\bar{f}_v(p)$$ \hspace{1cm} \text{(B7)}$$
at leading order $1/N_C$.
As contribution is given as
Also the matrix element (C2) is computed from the matrix elements from Table V: the positive intrinsic parity
Here we list the matrix elements that appear in the vacuum contribution of the polarized structure functions,
needs to be multiplied.
TABLE V: Matrix elements $\int d\Omega_{p} \Psi_{\alpha}^{(G,+)}(p) \hat{p} \cdot \tau \Psi_{\alpha}(p)$. The overall factor $1/(2G + 1)$
| $J = G - \frac{1}{2}$ | $J = G + \frac{1}{2}$ |
|------------------------|------------------------|
| $L = G - 1$ | $L = G - 1$ |
| $L' = G$ | $L' = G$ |
| $L' = G$ | $L' = G$ |
| $J = G - \frac{1}{2}$ | $J = G + \frac{1}{2}$ |
| -1 | 0 |
| 0 | -2/2G(G + 1) |
| -2/2G(G + 1) | 0 |
| 0 | -2/2G(G + 1) |
| 0 | 1 |
| 2/2G(G + 1) | 0 |
TABLE IV: Matrix elements $\int d\Omega_{p} \Psi_{\alpha}^{(G,+)}(p) \hat{p} \cdot \tau \Psi_{\alpha}(p)$. The overall factor $1/(2G + 1)$
$\int d\Omega_{p} \Psi_{\alpha}^{(G,+)}(p) \hat{p} \cdot \tau \Psi_{\alpha}(p)$ and $\int d\Omega_{p} \Psi_{\alpha}^{(G,+)}(p) \hat{p} \cdot \sigma \Psi_{\alpha}(p)$.
The matrix element (C1) is computed from the matrix elements from Table IV the positive intrinsic parity is obtained as
\[
\int d\Omega_{p} \Psi_{\alpha}^{(G,+)}(p) \hat{p} \cdot \tau \Psi_{\alpha}(p) = 2 \left( g_{\alpha}^{(G,+;+1)}(p) \tilde{f}_{\alpha}^{(G,+;+1)}(p) - g_{\alpha}^{(G,+;+2)}(p) \tilde{f}_{\alpha}^{(G,+;+2)}(p) \right) - 4 \sqrt{G(G + 1)} \left( g_{\alpha}^{(G,+;+1)}(p) \tilde{f}_{\alpha}^{(G,+;+2)}(p) + g_{\alpha}^{(G,+;+2)}(p) \tilde{f}_{\alpha}^{(G,+;+1)}(p) \right),
\]
and that for negative intrinsic parity as
\[
\int d\Omega_{p} \Psi_{\alpha}^{(G,-)}(p) \hat{p} \cdot \tau \Psi_{\alpha}(p) = 2 \left( g_{\alpha}^{(G,-;+1)}(p) \tilde{f}_{\alpha}^{(G,-;+1)}(p) - g_{\alpha}^{(G,-;+2)}(p) \tilde{f}_{\alpha}^{(G,-;+2)}(p) \right) + 4 \sqrt{G(G + 1)} \left( g_{\alpha}^{(G,-;+1)}(p) \tilde{f}_{\alpha}^{(G,-;+2)}(p) + g_{\alpha}^{(G,-;+2)}(p) \tilde{f}_{\alpha}^{(G,-;+1)}(p) \right).
\]
Also the matrix element (C2) is computed from the matrix elements from Table V the positive intrinsic parity contribution is given as
\[
\int d\Omega_{p} \Psi_{\alpha}^{(G,+)}(p) \hat{p} \cdot \tau \Psi_{\alpha}(p) = \left( -g_{\alpha}^{(G,+;+1)}(p)^2 - \tilde{f}_{\alpha}^{(G,+;+1)}(p)^2 + g_{\alpha}^{(G,+;+2)}(p)^2 + \tilde{f}_{\alpha}^{(G,+;+2)}(p)^2 \right) + 4 \sqrt{G(G + 1)} \left( g_{\alpha}^{(G,+;+1)}(p) \tilde{f}_{\alpha}^{(G,+;+2)}(p) + \tilde{f}_{\alpha}^{(G,+;+1)}(p) \tilde{f}_{\alpha}^{(G,+;+2)}(p) \right),
\]
Appendix C: Polarized Structure Functions at Leading Order
Here we list the matrix elements that appear in the vacuum contribution of the polarized structure functions, Eqs. (33) and (34). The matrix element to be considered are
\[
\int d\Omega_{p} \Psi_{\alpha}^{(G,+)}(p) \hat{p} \cdot \tau \Psi_{\alpha}(p),
\]
\[
\int d\Omega_{p} \Psi_{\alpha}^{(G,+)}(p) \hat{p} \cdot \sigma \Psi_{\alpha}(p)
\] and
\[
\int d\Omega_{p} \Psi_{\alpha}^{(G,+)}(p) \tau \cdot \sigma \Psi_{\alpha}(p).
\]
TABLE VI: Matrix elements \( \int d\Omega_p \mathcal{Y}_{L',J'GM}(p) \tau \cdot \sigma \mathcal{Y}_{LJGM}(p) \). The overall factor \( 1/(2G+1) \)
| \( J' = G - \frac{1}{2} \) | \( J' = G + \frac{1}{2} \) |
|-----------------|-----------------|
| \( L' = G - 1 \) | \( 0 \) | \( 0 \) | \( 0 \) | \( 2G+1 \) |
| \( L' = G \) | \( 0 \) | \( 0 \) | \( 0 \) | \( L = G \) |
| \( L' = G \) | \( 0 \) | \( 0 \) | \( 2G+1 \) | \( L = G + 1 \) |
| \( L' = G + 1 \) | \( J = G - \frac{1}{2} \) |
| \( J = G + \frac{1}{2} \) |
and that for the negative intrinsic parity as
\[
\int d\Omega_p \bar{\Psi}_a^{(G,-)}(p) \bar{\tau} \cdot \sigma \Psi_a^{(G,-)}(p) = \left( -\tilde{g}_a^{(G,-;1)}(p)^2 - \tilde{f}_a^{(G,-;2)}(p)^2 + \tilde{g}_a^{(G,-;2)}(p)^2 + \tilde{f}_a^{(G,-;2)}(p)^2 \right)
- 4\sqrt{G(G+1)} \left( \tilde{g}_a^{(G,-;1)}(p)\tilde{g}_a^{(G,-;2)}(p) + \tilde{f}_a^{(G,-;1)}(p)\tilde{f}_a^{(G,-;2)}(p) \right). \tag{C7}
\]
Furthermore the matrix element \( \text{(C3)} \) is computed from the matrix elements from Table VI, the positive intrinsic parity contribution becomes
\[
\int d\Omega_p \bar{\Psi}_a^{(G,+)}(p) \bar{\tau} \cdot \sigma \Psi_a^{(G,+)}(p) = (2G+1) \left( \tilde{f}_a^{(G,+;1)}(p)^2 + \tilde{f}_a^{(G,+;2)}(p)^2 \right) - (2G+3)\tilde{g}_a^{(G,+;1)}(p)^2
+ (2G-1)\tilde{g}_a^{(G,+;2)}(p)^2 + 8\sqrt{G(G+1)}\tilde{g}_a^{(G,+;1)}(p)\tilde{g}_a^{(G,+;2)}(p), \tag{C8}
\]
and for negative intrinsic parity becomes
\[
\int d\Omega_p \bar{\Psi}_a^{(G,-)}(p) \bar{\tau} \cdot \sigma \Psi_a^{(G,-)}(p) = (2G+1) \left( \tilde{g}_a^{(G,-;1)}(p)^2 + \tilde{g}_a^{(G,-;2)}(p)^2 \right) - (2G+3)\tilde{f}_a^{(G,-;1)}(p)^2
- (2G-1)\tilde{f}_a^{(G,-;2)}(p)^2 - 8\sqrt{G(G+1)}\tilde{f}_a^{(G,-;1)}(p)\tilde{f}_a^{(G,-;2)}(p). \tag{C9}
\]
Appendix D: Splitting functions
Here we list the splitting functions used in Eqs. (59), (60) and (61). They are different for the isovector, isosinglet and gluon contributions and are given as \([42, 57]\). They determine the probability for the parton \( m \) to emit a parton \( n \) such that the momentum of the parton \( m \) is reduced by the fraction \( z \). The regularized function \( (1-z)^{-1}_+ \) is defined under the integral by \([42]\)
\[
\int_0^1 dz \frac{f(z)}{(1-z)_+} = \int_0^1 dz \frac{f(z) - f(1)}{1-z}. \tag{D2}
\]
In the above \( C_H(F) = \frac{(N_f^2 - 1)}{2N_f} \) is the color factor for \( N_f \) flavors. Also the running coupling constant in the leading order is given by \( g_{QCD}(t) = \frac{4\pi}{\beta_0 t} \) with \( \beta_0 = \frac{11}{3} N_c - \frac{2}{3} N_f \) being the coefficient of the leading term of the QCD...
β-function. Using the “+” prescription, the evolution equations for the isovector, isosinglet and gluon contributions become \[57\]
\[
\frac{df^{(I=1)}(x,t)}{dt} = \frac{2C_R(F)}{9t} \left\{ \int_x^1 \frac{dy}{y} \left( \frac{1+y^2}{1-y} \right) \left[ \frac{1}{y} f^{(I=1)} \left( \frac{x}{y}, t \right) - f^{(I=1)}(x,t) \right] + \left[ x + \frac{x^2}{2} + 2 \ln(1-x) \right] f^{(I=1)}(x,t) \right\}.
\]
\[
\frac{df^{(I=0)}(x,t)}{dt} = \frac{2C_R(F)}{9t} \left\{ \int_x^1 \frac{dy}{y} \left( \frac{1+y^2}{1-y} \right) \left[ \frac{1}{y} f^{(I=0)} \left( \frac{x}{y}, t \right) - f^{(I=0)}(x,t) \right] + \frac{3}{4} \left( y^2 + (1-y^2) \right) g(x,t) + \left[ x + \frac{x^2}{2} + 2 \ln(1-x) \right] f^{(I=0)}(x,t) \right\}.
\]
\[
\frac{dg(x,t)}{dt} = \frac{2C_R(F)}{9t} \left\{ \int_x^1 \frac{dy}{y} \left( \frac{1+y^2}{1-y} \right) f^{(I=0)} \left( \frac{x}{y}, t \right) + \frac{9}{2} \left( \frac{1}{y} - y^2 (1-y) \right) g \left( \frac{x}{y}, t \right) + \frac{9}{2} \frac{g \left( \frac{x}{y}, t \right) - g(x,t)}{1-y} \right\} + \left[ \frac{3}{2} + \frac{9}{2} \ln(1-x) \right] g(x,t) \right\}.
\]
(D3)
Now, since our NJL model calculations do not account for any gluon content in the nucleon, we assume in our numerical calculations that, at the initial boundary scale \(\mu^2\), the gluon content, \(f(x,t_0) = 0\) for both the polarized and unpolarized structure functions.
Unlike the polarized spin structure function \(g_1(x)\) of the nucleon and the unpolarized structure functions, the nucleon’s second polarized spin structure function \(g_2\) involves contributions from quark-gluon iterations and quark masses \[58-60\]. According to the standard operator product expansion analysis, these contributions come from the twist-3 local operators. It, however, also receives contribution from twist-2 local operators under the impulse approximation. Thus, the structure function \(g_2\) can be written as the sum of the twist pieces
\[
g_2(x,Q^2) = g_{2W}(x,Q^2) + \bar{g}_2(x,Q^2), \quad (D4)
\]
where the twist-2 piece is given as \[61\]
\[
g_{2W}(x,Q^2) = -g_1(x,Q^2) + \int_0^1 \frac{1}{y} g_1(y,Q^2) dy, \quad (D5)
\]
while that of the twist-3 piece is
\[
\bar{g}_2(x,Q^2) = g_1(x,Q^2) + g_2(x,Q^2) - \int_0^1 \frac{1}{y} g_1(y,Q^2) dy. \quad (D6)
\]
The twist-2 part undergoes the ordinary evolution as in Eq. (D3) and the twist-3 piece is first parameterized by its moments
\[
M_j(\mu^2) = \int dx x^{j-1} \bar{g}_2(x,\mu^2) \quad (D7)
\]
that scale as \[58\]
\[
M_j(Q^2) = \left[ \ln(\mu^2) \right]^{\gamma_{j-1}/\gamma_0} \left\{ \ln(Q^2) \right\}^{\gamma_{j-1}/\gamma_0} \text{with} \quad \gamma_{j-1} = 2N_c \left[ \psi(j) + \frac{1}{2j} + \gamma_E - \frac{1}{4} \right]. \quad (D8)
\]
Here \(\psi(j)\) is the logarithmic derivative of the Γ-function. Then \(\bar{g}_2(x,Q^2)\) is obtained by expressing it in terms of the evolved moments, i.e. by inverting Eq. (D7).
[1] T. Muta, *Foundations of Quantum Chromodynamics* (World Scientific, Singapore, 1987).
[2] R. G. Roberts, *The Structure of the Proton* (Cambridge Monographs on Mathematical Physics, 1990).
| 2025-03-04T00:00:00 | olmocr | {
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} | Angular Dependence of Neutrino Flux in KM$^3$ Detectors in Low Scale Gravity Models
Pankaj Jain$^1$, Supriya Kar$^1$, Douglas W. McKay$^2$
Sukanta Panda$^1$ and John P. Ralston$^2$
$^1$Physics Department
I.I.T. Kanpur, India 208016
$^2$Department of Physics & Astronomy
University of Kansas
Lawrence, KS 66045
Abstract: Cubic kilometer neutrino telescopes are capable of probing fundamental questions of ultra-high energy neutrino interactions. There is currently great interest in neutrino interactions caused by low-scale, extra dimension models. Above 1 PeV the cross section in low scale gravity models rises well above the total Standard Model cross section. We assess the observability of this effect in the 1 PeV - 1000 PeV energy range of kilometer-scale detectors, emphasizing several new points that hinge on the enhancement of neutral current cross sections with respect to charged current cross sections. A major point is the importance of “feed-down” regeneration of upward neutrino flux, driven by new-physics neutral current interactions in the flux evolution equations. Feed-down is far from negligible, and it is essential to include its effect. We then find that the angular distribution of events has high discriminating value in separating models. In particular the “up-to-down” ratio between upward and downward-moving neutrino fluxes is a practical diagnostic tool which can discriminate between models in the near future. The slope of the angular distribution, in the region of maximum detected flux, is also substantially different in low-scale gravity and the Standard Model. These observables are only weakly dependent on astrophysical flux uncertainties. We conclude that angular distributions can reveal a breakdown of the Standard Model and probe the new physics beyond, as soon as data become available.
1 Introduction
The Ultra-high energy neutrino nucleon cross section $\sigma_{\nu N}$ is a topic of fundamental physical importance. Low scale gravity models \cite{1,2} predict enhancement of the neutrino-nucleon neutral current type cross section at center of mass energies above the fundamental gravity scale of about 1 TeV \cite{3,4,5}. Consequences for cosmic ray physics have been studied in application to the highest energy cosmic rays \cite{3,4,5,6,7,8,9,10,11}, where there is great interest in possible violation of the Greisen, Zatsepin, Kusmin (GZK) bound \cite{12}, at roughly 10 EeV ($10^{19}$eV). There is also great interest in application to an intermediate range \cite{5,13,14}, roughly 0.1 - 100 PeV ($10^{14} - 10^{17}$ eV). The history and future of the highest energy experiments \cite{15} and intermediate energy experiments \cite{16} provide a tremendous impetus for these studies pointing toward new physics.
Whatever the model, there are rich opportunities to study fundamental high energy interactions by focusing on (1) the ratio of neutral current type events to charged current events and (2) the angular distribution of events in upcoming experiments. The neutral-to-charged ratio and the angular distribution shape do not depend on the uncertainties of overall flux normalizations. The primary uncertainty in all cosmic ray comparisons with theory – the overall scale of the flux – drops right out. For a range of models with standard and reasonable flux spectral indices, the angular distributions are also remarkably insensitive to the details of the model. The strongest determining factor in the shape of the angular distribution is the fundamental physics of the interaction cross section itself. We emphasize and explore this fact, showing that arrays now planned or under construction could stringently test the Standard Model and proposals for new physics simply on the basis of the neutral-to-charge ratio and the slope of the angular distribution of neutrino-nucleon events. In the context of extra space-time dimensions, data could determine or bound such details as the scale and number of extra dimensions in the present models.
Here we continue earlier work \cite{5} that examined possible signatures of enhanced $\sigma_{\nu N}$ in kilometer scale detectors. Extensions of the currently operating AMANDA \cite{17} and RICE \cite{18,19} experiments to ICECUBE \cite{20}.
\footnote{In the present context, by neutral current type cross section we mean there is no leading, charged lepton produced. The shower is essentially hadronic, which includes the case of black hole final states.}
dimensions would certainly explore the 1 PeV - 100 PeV region. In the case of RICE, modest improvements even allow a reach above the EeV range. In [5] we used a linear extrapolation of the low scale gravity mediated neutral current $\sigma_{\nu N}$ from the low energy, $\sqrt{s} \ll 1$ TeV, region, to the $\sqrt{s} \geq 1$ TeV region and found a very sharp suppression of the up-to-down ratio compared to the standard model that set in at about 5 PeV for $M = 1$ TeV and at about 50 PeV for $M = 2$ TeV.
In that earlier study [5], we did not apply other extrapolations of the cross section to the up-to-down calculation, nor did we include the “feed down” effect [21, 22], which results from neutral current interactions degrading higher energy neutrinos as they travel through the earth and “feeding” the flux at lower energies. In the standard model, this effect is small above 1 PeV where the flux decrease steepens and the neutral current cross section is too small to compensate. In contrast, this effect turns out to be extremely important after including the gravity induced neutral current $\sigma_{\nu N}$, which rises rapidly with energy. This point has not been explicitly recognized previously in connection with gravity enhancement.
A number of new gravity effects relevant to energies above the fundamental scale, applicable to physics at the Large Hadron Collider (LHC), kilometer cubed detectors ($KM^3$) and GZK energies, have been proposed recently. In both Arkani-Hamed, Dimopoulos and Dvali (ADD) [1] and Randall and Sundrum (RS) [2] models, eikonal treatment of the effective low energy amplitude (used as the “Born term” input) has been studied [23] and applied to LHC [24] and GZK [8] energies. In string realizations of the ADD framework [25], “stringy” cross sections, relevant just below $M$, have been estimated [26, 27, 28], as has the black hole formation cross section [29, 7], which may be relevant above $M$, including the GZK energy region. We have investigated all of these options, and find that the eikonalized ADD low energy “Born amplitude” and “geometrical” black hole cross sections lead to the largest and least model dependent effects in our 1 PeV to 100 PeV $KM^3$ application. Our study goes beyond that reported recently in [13] in two respects: we emphasize higher energies and we include and analyze the consequences of the new, eikonalized graviton exchange component of the neutral current interaction. This latter point also distinguishes our work from a recent black hole detection study [14].
\footnote{We do not treat the possibility of brane production and decay here [30].}
We should note here that, though the neutral current does not produce a prompt electron shower or leading muon characteristic of charged current signatures, one expects that the hadronic shower, which is completely electromagnetic after several radiation lengths, will be an observable signature of neutral current interactions. For this reason we regard all of the current detection mechanisms to be relevant to our study, certainly including RICE, which can detect a radio pulse from any kind of shower, AMANDA and ICECUBE.
1.1 The Cross Section
At C.M. energies well above the Planck mass, the classical gravity Schwarzschild radius $R_s(\sqrt{s})$ is the dominant physical scale. The classical impact parameter, $b$, may make sense in this domain, and the eikonal approximation be valid, for values of $b$ larger than $R_s$. We will sketch the eikonal set-up shortly. At smaller impact parameters, the parton-level geometrical cross section
$$\hat{\sigma}_{BH} \approx \pi r_S^2$$
provides a classical, static estimate of the cross section to form black holes. In Eq. 1 $r_S$ is the Schwarzschild radius of a 4 + $n$ dimensional black hole of mass $M_{BH} = \sqrt{\hat{s}}$,
$$r_S = \frac{1}{M} \left( \frac{M_{BH}}{M} \right)^{1+n} \left[ \frac{2^n \pi^{(n-3)/2} \Gamma \left( \frac{3+n}{2} \right)}{2 + n} \right]^{1/n}$$
where $\sqrt{\hat{s}}$ is the parton-parton or, in our case, neutrino- parton C.M. energy, and $M$ is the 4+$n$-dimensional scale of quantum gravity. The black hole production process is expected to give a dominant contribution when $\sqrt{\hat{s}} >> M$. Black holes will form only if the impact parameter $b < r_S$. To convert Eq.1 into an estimate for the neutrino- nucleon cross-section, we fold it with the sum over parton distribution functions and integrate over $x$-values, where $\hat{s} = xs$, at a momentum transfer typical of the black hole production process:
$$\sigma_{\nu N \rightarrow BH}(s) = \Sigma_i \int_{x_{min}}^1 dx \hat{\sigma}_{BH}(xs) f_i(x, q).$$
3We use the mass scale convention discussed in [10], referred to as $M_D$ there.
Black hole formation requires $x > M^2/s$, so we take $x_{\text{min}} = M^2/s$. In addition, $q^{-1} = b < r_S$ is required. We adopt $q = \sqrt{s}$ up to $\sqrt{s} = 10\,\text{TeV}$, the maximum range in $q$ of the CTEQ parton distribution functions [31], the set we use, and $q = 10\,\text{TeV}$ when $\sqrt{s}$ is above this value. As remarked in [10], the dependence of $\sigma_{\nu N \rightarrow BH}(s)$ on the choice of $x_{\text{min}}$ and the treatment of $q$ is rather mild.
In the case of ADD model the black hole production cross sections can be large for $n > 2$, in which case the fundamental scale can be of the order of 1 TeV. A number of authors have adopted this estimate and applied it to ultra relativistic, parton level scattering. The approximation has been challenged on the basis that quantum corrections should lead to exponential suppression of individual channels, such as the black hole formation final state [32, 33], with several, independent arguments advanced in each case. In defense of the “black disk” approximation, several authors also point to success of internal consistency checks of the classical picture [28, 34, 24]. Recent phenomenological studies seem to be agnostic on this issue [9, 10, 11, 14, 35], treating the phenomenological consequences of both versions.
In a string picture with scale $M_s < M$, there is a range of energy $M_s \simeq \sqrt{s}$ where string resonances dominate [26], and a range $M_s < \sqrt{s} \leq M$, where stringball formation [28] could dominate. The cross section can be roughly expressed as [26]
$$\sigma_{\text{SR}}(\sqrt{s}) \sim g_s^2 \delta(s - M_{\text{SR}}^2), \sqrt{s} \simeq M_{\text{SR}},$$
(4)
for the string resonance case. Here $g_s$ is the (weak) string coupling constant and $M_{\text{SR}}$ is the mass of a string resonance state. Similarly, for the stringball case, estimates of the cross section give [28]
$$\sigma_{\text{SB}}(\sqrt{s}) \sim 1/M_s^2, M_s/g_s^2 < \sqrt{s} < M_s/g_s^2,$$
(5)
where $M$ is a few times less than $M_s/g_s^2$ for weak coupling. The impact of these various processes on the physics to be expected at the LHC, at a next linear collider (NLC) and very large hadron collider (VLHC) has been surveyed in a number of papers, summarized and referenced in the Snowmass 2001 report of the extra dimensions subgroup [36].
\[4\] In this discussion we suppress the $\hat{s}$ notation for convenience, though parton level processes are intended.
In our application to $KM^3$ physics in this paper, we mentioned above that the “classical” eikonal cross section [37, 23] and the geometric black hole formation cross section are the only cases where we find potentially observable effects. We outline our treatments of the eikonal model in the ADD [1] and RS1 [2] pictures next. The black hole cross section needs no further elaboration. The fundamental mass scale $M$ in the case of the ADD model for $n > 2$, can be of the order of 1 TeV, though new astrophysics analyses may constrain $n = 3$ more severely, as we comment below [38]. Similarly, in the RS picture the effective scale of gravity on the physical brane, the lowest K-K mode mass, can be arranged to be of the order of 1 TeV. In all of our quantitative work, we set the scale $M$ the same for every value of $n$ we use in our comparisons. As noted earlier, the scale $M$ is the same as $M_D$ defined in [24] and discussed in [10].
In the RS1 model with one extra dimension or in the ADD model with several, a possible choice for the input amplitude to the eikonal approximation, referred to as the Born amplitude, is given by,
$$iM_{\text{Born}} = \sum_i \frac{ics^2}{M^2 q^2 + m_i^2}$$
(6)
where $c$ is the gravitational coupling strength, which is effectively Newtonian for ADD and electroweak for RS. Here $q = \sqrt{-t}$ is the momentum transfer. In the Randall-Sundrum case, the sum runs over the massive K-K modes, constrained to start at or above the TeV scale when $c$ is of order electroweak strength. Their spacing is then also of TeV order. In the ADD case, the index $i$ must include the mass degeneracy for the $i$th K-K mode mass value. The spectrum is so nearly continuous that an integral evaluation of the sum is valid, but must be cut off at a scale generally taken to be of the order of $M$. Taking the transverse Fourier transform of the Born amplitude, we get the eikonal phase as a function of impact parameter, $b$,
$$\chi(s, b) = \frac{i}{2s} \int \frac{d^2q}{4\pi^2} \exp(iq \cdot b) iM_{\text{Born}}$$
(7)
For the ADD model, where $c = (M/\overline{M}_P)^2$ and $\overline{M}_P = 2.4 \times 10^{18}$ GeV is the reduced, four dimensional Planck mass,
$$\chi(s, b) = \frac{s(2^{2n-3} \pi^{\frac{4n}{3} - 1})}{M^{n+2} \Gamma(n/2)} 2 \int_0^\infty dmm^{n-1}K_0(mb)$$
6
\[ b^n_c = \frac{1}{2} (4\pi)^{\frac{n}{2}} \Gamma \left( \frac{n}{2} \right) \frac{s}{M^{2+n}}. \]
(9)
Because of the exponential decrease of \( K_0 \), the phase integral is actually ultraviolet finite. For the RS model, the corresponding expression is
\[ \chi(s, b) = \sum_i \frac{cs}{2M^2} K_0(m_i b). \]
(10)
For widely spaced KK modes in the Randall-Sundrum case, the lowest few modes dominate and contribute insignificantly in the 1 PeV < \( E_\nu < 100 \) PeV region. We do not discuss RS further.
The eikonal amplitude is then given in terms of the eikonal phase by
\[ M = -2is \int d^2 b \exp(iq \cdot b) \left[ \exp(i\chi) - 1 \right] \]
\[ = -i4\pi s \int db b J_0(qb) \left[ \exp(i\chi) - 1 \right], \]
(11)
The eikonal amplitude for the case of ADD model can be obtained analytically [23, 8, 24] in the strong coupling \( qb_c \gg 1 \) and weak coupling \( qb_c \ll 1 \) regimes.
In the strong coupling regime the eikonal amplitude can be computed using the stationary phase approximation and is given by,
\[ M = A_n e^{i\phi_n} \left[ \frac{s}{qM} \right]^{\frac{n+2}{2}} , \]
(12)
where
\[ A_n = \frac{(4\pi)^{\frac{3n}{2n+1}}} {\sqrt{n+1}} \left[ \Gamma \left( \frac{n}{2} + 1 \right) \right]^{\frac{1}{n}}, \]
(13)
\[ \phi_n = \frac{\pi}{2} + (n+1) \left[ \frac{b_c}{b_s} \right]^n \]
(14)
and \( b_s = b_c (qb_c/n)^{-1/(n+1)} \). In the weak coupling regime, \( q \to 0 \) the amplitude is given by
\[ M(q = 0) = 2\pi isb_c^2 \Gamma \left( 1 - \frac{2}{n} \right) e^{-\pi/n} . \]
(15)
As it turns out, the small q region contributes little to the cross section, and we use the simple rule that the amplitude is set to its value at $q = 1/b_c$ for values $q \leq 1/b_c$.
The parton-level cross section is calculated by assuming that it is given by the Born term as long as $\hat{s} < M^2$. For $\hat{s} > M^2$ the cross section is estimated by the eikonal amplitude. For $M = 1$ TeV, for example, the actual matching between the Born and eikonal amplitudes occurs in the range $\sqrt{\hat{s}} = 1 - 3$ TeV, depending on $n$ and the value of $y = q^2/\hat{s}$. In any case the region $\sqrt{\hat{s}} \sim M$ contributes negligibly to the cross-section, so the precise matching choice makes no difference in the final result. The eikonal calculation is not expected to be reliable if the momentum transfer $q > M$. At large momentum transfer we assume that the black hole production dominates the cross section. The eikonal cross section is, therefore, cut off once the momentum transfer $q > 1/r_S$. The neutrino-parton differential cross-section is folded with the CTEQ parton distributions and integrated over $x$ and $y$ variables, consistent with our momentum transfer restriction on the eikonal amplitude. The CTEQ limit at $x = 10^{-5}$ is exceeded only at the high end of the energy range we study. A standard power law extrapolation is used when $x$ does range below this value, though the $\hat{s}$ values are so low that the contribution of this range to the cross section is negligible. In Fig. 1 we plot the total neutrino-nucleon cross sections for several different values of the fundamental scale $M$ and the number of extra dimensions $n$, including both eikonal and black hole production contributions. The cross section is clearly more sensitive to the choice of the scale parameter, $M$, than to the number of dimensions, $n$. In fact the sensitivity to choice of $n$ comes primarily through the dependence of the differences in bounds on $M$ corresponding to different choices of $n$. The strongest bounds on $M$ come from astrophysical and cosmological considerations, which unavoidably require some degree of modeling.
A recent review of experimental and observational constraints is given in [38], where lower bounds on $M$ are quoted for $n = 3$ from various analyses such as supernova cooling, regarded as the least model dependent bound ($M \geq 2.5$ TeV), post-inflation re-heating ($M \geq 20$ TeV), and neutron star heat excess ($M \geq 60$ TeV). For $n = 4$, the most severe constraint is $M \geq 5$
\footnote{A discussion of the reliability of the eikonal amplitude in the $5$ TeV $\leq \sqrt{s} \leq 15$ TeV range is given in [24].}
Figure 1: The neutrino-proton cross section $\sigma_{\nu p}$ in the ADD model using the number of extra dimensions $n = 3, 4, 6$ as a function of the neutrino energy $E_{\nu}$ for $E_{\nu} < 10^8$ GeV. The geometric black hole production cross section is included. The solid, dotted and short dashed curves correspond to $n = 3, 4$ and 6 respectively. The long dashed curve represents the Standard Model prediction.
5TeV in the case of the post-inflation re-heating limit. There are essentially no constraints on the cases $n = 5$ and 6. Laboratory lower bounds are typically of order 1 TeV or less for all $n \geq 2$, with LEP II providing the strongest bound at 1.45 TeV for $n=2$.
2 Event Rates of Downward Neutrinos
In Fig. 2, we show the downward event rate, defined as the number of interactions from down- coming neutrinos, within a one kilometer cubed volume. We use two, quite different input flux assumptions to show the dependence of rate on the flux. A simple parameterization of an optically thick source model of flux above 1 PeV given in Ref. [39] (SDSS) is used, along with the flux bound for optically thin source environments of Ref. [40] (WB). The flux in [39] is about two orders of magnitude larger than the bound in [40], and it has roughly an $E^{-2}$ power law behavior from 0.1 PeV
to 10 PeV, and then it steepens to approximately $E^{-3}$. We parametrized the SDSS flux such that it falls as $E^{-2}$ for $10 \text{ TeV} < E < 10 \text{ PeV}$ and as $E^{-3}$ for $E > 10 \text{ PeV}$. The WB bound, on the other hand, falls as $E^{-2}$ over the whole energy range. The two flux curves cross at about $10^3 \text{ PeV}$.
Our estimate is made by taking the vertical flux result, multiplying it by $2\pi$ steradians and by the probability that a neutrino would interact within 1 km in ice, with density 0.93 gm/cc, given the cross section model in question. To get an actual event rate for a given detector, we would have to multiply by the acceptance of the detector. In the SDSS flux model, Fig. 2a, the number of interactions peaks at around 10 PeV for the $M = 1 \text{ TeV}$ case, with encouragingly large numbers of interactions, in the 350-650 range, induced by low-scale gravity. This is 10-20 times the SM interaction rate at the same energy. The number and the location of the peak rate depend upon the flux model of course [22]. This dependence is illustrated in Fig. 2b, where the event rate for the WB bound is shown. The event rates are lower by a factor of about 50 and the gravity induced events have a much broader peak, centered at about 15 PeV, compared to the SDSS case. The peak is broader for the WB flux than for SDSS because the WB limit falls as $E^{-2}$ throughout this energy region, while the SDSS flux falls as $E^{-3}$ above 10 PeV, cutting off the higher energy events more rapidly. The shape of the SM event rate is the same in both cases, since the flux shapes are the same below 10 PeV.
The excess above the SM is roughly half neutral current type events arising from eikonalized graviton exchange and half black hole events, which decay predominantly into hadrons. Therefore an optimal detection scheme requires sensitivity to the hadronic shower from the deposited energy in the ice. The ICECUBE and RICE detectors, for example would respond readily to hadron showers at these energies. The special role of enhanced neutral current type events, those with hadronic signatures like the graviton exchange and black hole events, prompts us to propose that the ratio of neutral current to charged current events provides a powerful tool to uncover new interactions. In particular, if the neutral current type component, distinguished by hadronic dominated showers and no leading charged lepton, dominates, as it does in the black hole and the eikonal regimes of the low scale gravity models, there would be no standard model explanation. The ratio of neutral current type events to charged current events, distinguished by the presence of a leading charged lepton, is shown in Fig. 3. The rapid rise that sets in above the threshold for new physics is remarkable, and would be so even without
the black hole contribution. Even scales $M > 2T eV$ are discernible with this observable. Putting together the information from several techniques, RICE and ICECUBE for example, one might well separate the “neutral” versus “charged” character of events and find a clear window on new physics.
3 The Regeneration and Angular Dependence of Neutrino Flux
We next calculate the up over down ratio of neutrino flux. The downward $\phi_0(E_\nu)$ is the flux of neutrinos incident on the surface of the earth from the sky. We will consider only the diffuse neutrino flux and assume that it is isotropic. Our calculations are easily generalized if the flux is found to be non-isotropic or if we are interested in individual sources. The upward flux $\phi_{up}(E_\nu)$ is defined as the flux of neutrinos coming upwards from the surface of the earth. This is the angular average
$$\phi_{up}(E_\nu) = \frac{1}{2\pi} \int_0^{2\pi} d\Phi \int_0^{\pi/2} \sin(\theta) d\theta \phi(E_\nu, \theta)$$
(16)
of the flux $\phi(E_\nu, \theta)$ emerging from the earth, where $\theta$ is the polar angle with respect to the nadir and $\Phi$ is the azimuthal angle. The ratio $R$ is essentially the ratio of up-to-down event rates. The event rates are given by the product of flux, cross section, number density, volume and acceptance, and to the extent that the latter four factors cancel in the ratio, only the up-to-down ratio of fluxes survives. This ratio is affected by the energy dependence of the flux, but not to its overall normalization. The upward flux depends on the cross sections of course, as we elaborate next.
In order to determine $\phi(E_\nu, \theta)$ we first need to solve the evolution equation for the neutrino propagating through the interior of the earth. In the case of the standard model, the neutrino cross section is dominated by the charged current, the neutral current does little to evolve, or “feed down”, the rapidly falling flux above 1 PeV, and neutrinos basically get lost after collision inside earth. In the present case, on the other hand, the eikonalized graviton exchange gives a large contribution to the feed down above 1 PeV, and we have to include this important enhancement of regeneration of lower
The large cross sections mean that neutrinos experience many interactions as they proceed through the earth, even at angles near the horizon. For instance, just 5 degrees below the horizon a 100 PeV neutrino traverses more than 20 interaction lengths of earth before reaching the detector region. This behavior supports our use of the continuous evolution model, summarized below in Eq.17, since fluctuations are small for UHE application, where the cross section is large. The neutrino loses little energy during any particular collision and hence it is reasonable to assume that it practically moves in a straight line path. The evolution equation for the neutrino is given by [21, 22]
\[
\frac{d\ln \phi}{dt}(E_\nu, \theta) = -\sigma_{W+BH}^{\nu N}(E_\nu) - \sigma_{Z+E}^{\nu N}(E_\nu) + \int_{E_\nu}^\infty dE'_\nu \frac{\phi(E'_\nu, \theta)}{\phi(E_\nu, \theta)} \frac{d\sigma_{Z+E}^{\nu N}(E'_\nu, E_\nu)}{dE'_\nu(E'_\nu, E_\nu)} \equiv -\sigma_{\text{eff}}^{\nu N}(E_\nu, \theta),
\]
where \(\sigma_{W+BH}^{\nu N}\) is the “neutrino absorbing” W-exchange + black hole cross section and \(\sigma_{Z+E}^{\nu N}\) is the “neutrino regenerating” Z-exchange + eikonalized graviton exchange cross section. In Eq.17, \(dt = n(r)dz\) and \(n(r)\) is the number density of nucleons at any distance \(r\) from the center of earth, radius \(R_e\). Expressing the flux \(\phi(E_\nu, \theta)\) as
\[
\phi(E_\nu, \theta) = \phi_0(E_\nu) \exp[-\sigma_{\text{eff}}(E_\nu, \theta)t(\theta)],
\]
where the column density at upcoming angle of entry \(\theta\), chord length \(2R_e\cos\theta\), is given by
\[
t(\theta) = \int_0^{2R_e\cos\theta} n(z, \theta)dz,
\]
we solve equations 17 and 18 numerically by iteratively improving the flux \(\phi(E_\nu, \theta)\). Using this solution we can determine the ratio \(R\) of up to down flux,
\[
R = \frac{\phi_{\text{up}}(E_\nu)}{\phi_0(E_\nu)}
\]
---
6 The black hole production component, in contrast, essentially leads to loss of neutrinos upon collision. In the present context, the black hole interaction acts like the charged current in the standard model. The eikonal component has the character of the usual neutral current, transferring a small fraction of the neutrino energy to the hadron and leaving a leading neutrino in the final state.
7 We do not treat the regeneration of \(\tau\) neutrinos through \(\tau\) production by \(\nu_\tau\) and subsequent \(\tau\) decay back to \(\nu_\tau\) [41].
By using $R$, one sharply reduces the effects of experimental systematics and flux normalization and isolates the dependence on cross sections and flux shape. Refinements of angular binning can add information according to the size of the data sample, as we discuss below.
In Fig. 4 we plot the ratio $R$ as a function of the neutrino energy. This figure illustrates two points: the ratio $R$ is insensitive to the normalization of the flux assumed and, given a flux, the feed-down from higher to lower energies as the neutrinos pass through the earth is a powerful effect. In this application, our two, quite different input flux assumptions, show the insensitivity of $R$ to the flux used. The long dashed curve gives the ratio for the larger flux with a “knee” from $[39]$, while the shorter dashed line gives the result of the smaller flux bound with uniform $E^{-2}$ fall-off of $[40]$. The solid curve refers to the SM cross section input, the two flux models give indistinguishable results in this case and are shown as one line. The lowest curve shows the ratio as a function of energy with pure absorption, in the sense that only the first two terms in $\sigma_{\text{eff}}$, Eq.17., are included. The two flux models give the same result, as they should. The sum of the eikonal and black hole cross section is used, the eikonal providing the neutral-current cross section and the black hole the purely absorptive cross section, as described in footnote 6. The cross section are computed in the ADD model, and the value $n = 3$ and the scale of quantum gravity $M = 1$ TeV. We see that the regeneration term gives a big effect for $M = 1$ TeV, producing factors of more than 10 above 20 PeV, and it is essential to include it in assessing the consequences of the low scale gravity models. Though radically different from each other, the two flux assumptions lead to nearly identical values of $R$ out to 50 PeV and then differ only weakly above that, where the effect of the steeper decrease of SDSS flux shows up.
In Fig. 5 we plot the results for the ratio $R$ in the ADD model as a function of the neutrino energy for the choice of the fundamental scale $M = 1, 2$ TeV and the number of extra dimensions equal to 3, 4 and 6. The result depends on both the number of extra dimensions and, especially strongly, on the fundamental scale $M$. If the fundamental scale is larger than about 2 TeV, the KM$^3$ neutrino detectors will not be able to distinguish the Standard model result from the predictions of quantum gravity by using $R$ as a diagnostic. However for $M \approx 1$ TeV, we find that the effect is very large and, given enough flux, could be seen in these detectors. For energies above 1 PeV, a noticeable difference in $R$ between the two cases is seen. With
sufficient data, a distinction between no deviation from the standard model and a deviation corresponding to $M = 1$ with $n = 3, 4$ or 6 may be drawn.
The angular distribution of upward flux is an important diagnostic tool, as we stressed in the Introduction. In Fig. 3 we show the integrated flux per square kilometer per year above 1.8 PeV as a function of nadir angle for $M = 1, 2$ and $n=3,4,6$. The (higher) 2 TeV curve is essentially the same as for the SM, while the (lower) curves for $M = 1$ show clear suppression at all nadir angles up to $\pi/2$. It is somewhat disguised by the log scale, but most of the contribution to the ratio $R$ comes from the highest event rates near the horizontal. With several bins of good statistics data above 1 radian in nadir angle, one can distinguish the slopes of the flux vs. nadir angle near the horizontal. The difference between these slopes for the SM and low scale gravity models when $M \approx 1$ TeV provides another new physics discriminator.
As in the case of the $R$ observable, the slope is insensitive to the flux value, enhancing its power at identifying the nature of the neutrino interactions.
Realistically, one needs enough upcoming events at a given energy to calculate a meaningful ratio. The situation for the two flux models we are discussing is shown in the table.
| n | M | 1-10 PeV | 5 - 10 PeV | > 10 PeV | > 15 PeV |
|---|---|---------|-----------|---------|---------|
| 3 | 1 | 83 | 7.2 | 1.3 | 0.22 |
| 4 | 1 | 79 | 5.4 | 0.62 | 0.07 |
| 6 | 1 | 74 | 2.6 | 0.09 | 0.004 |
| 3 | 2 | 80 | 8.9 | 4.3 | 1.8 |
| 4 | 2 | 80 | 8.9 | 4.2 | 1.7 |
| 6 | 2 | 80 | 9 | 4.1 | 1.6 |
| SM| | 80 | 8.8 | 4.2 | 1.7 |
Table 1: The numbers of upward events per year expected over the energy ranges shown, for models with different $n$ and $M$ choices and for the standard model. The flux used is our rough approximation to SDSS above 1 PeV.
For our two power law approximation to the flux of [39] (SDSS), we see from the table that there are enough events with $E_\nu > 10$ PeV in a 10 year run to determine the ratio $R$. Can one discriminate among the different physics models for the cross section with the events? Fig. 5 shows that
if there are enough upcoming events to calculate a meaningful value for $R$, there are clear distinctions among the models. For example, taking a naive, purely statistics estimate, the $M = 1$ TeV, $E_{\nu} > 10$ PeV case would produce $13 \pm 3, 6 \pm 2$ and $1 \pm 1$ events for $n = 3, 4$ and 6. Combining these values with the downward event rates (see Fig. 2), we see that the distinction between the $n = 3$ case and the $n = 6$ cases is significant. When $M = 2$ TeV, the numbers of upcoming events is essentially the same for the low scale gravity models and the SM. In 5 - 10 years of data, there would be enough events to determine an up-to-down ratio reasonably well. Figure 2 shows that the number of downward events is larger in the low scale gravity models than in the SM, because of their larger cross sections, so $R$ values are different for the different cases. Thus, even for $M = 2$, values of $R$ above 15 PeV are distinctly different in the two classes of models, as one sees in Fig. 5. In fact, within the low scale gravity models themselves there is more than a factor two difference between $n = 3$ and $n = 6$. Moreover, the low scale gravity models can all be easily distinguished from the SM with 5 years of data, with the number of events per year shown in the table. Even the cut $> 15$ PeV allows the meaningful distinction between the low scale models and the SM in 5 - 10 years of data. Using the WB bound on the optically thin source flux, we find that the upward events are too sparse to discriminate among models by use of the $R$ ratio. Flux values in between the two presented here offer various levels of discrimination, with useful information obtainable for the larger fluxes.
Putting together the rise in downward event rate above 1 PeV, the sensitivity of $R$ to the new physics cross sections, and the slope of the upward flux as a function of nadir angle, we see that a signal of low scale gravity models would stand out clearly.
4 Conclusions:
$KM^3$ detectors have been planned primarily as “neutrino telescopes”. However our study indicates that $KM^3$ will also provide vigorous new exploration of fundamental questions in ultra-high energy physics. Extra dimension, low scale gravity models show an observable impact on the signature of neutrino events in the $KM^3$ detectors such as the currently running RICE detector and the ICECUBE detector, which is in the R & D stage. Of the “strong
gravity” effects we looked at, the black hole formation cross section and the extrapolation of the small $q^2$, part of $\sigma^{\nu N}$ from the $s < M^2$ to $s \gg M^2$ within the ADD model yield the largest detectable effects.
While some observables are somewhat sensitive to the number of dimensions, $n$, most are quite sensitive to the value of the scale of gravity $M$. If $M$ is 1-2 TeV, the enhanced $\sigma^{\nu N}$ creates a clearly recognizable signature, with the “neutral-to-charged” event ratio still showing new physics effects at $M = 5$ TeV. The ratio $R$ of upcoming to down-going events is a powerful diagnostic, capable of discriminating between models if fluxes are large enough to produce a significant number of upcoming events. In the entire analysis, we emphasized that the regeneration of neutrino upcoming flux due to neutrinos scattering down from higher to lower energies is a crucial feature of the gravity-induced neutral current interactions. This is a general and important feature of our work presented here.
Considering only down-coming events, we propose that the fraction of neutral current type events, those with hadronic showers and no leading charged lepton, provides a probe of new interactions. In particular, event signatures arising from a dominant component of eikonalized graviton exchange or of black hole production as occur in low scale gravity models, would have no standard physics explanation, and would point the way toward physics beyond the Standard Model if observed. There is every reason to believe that the neutral-to-charged current ratio can be extracted from upcoming facilities well enough to make a practical signal. In particular, determining the ratio of events with a muon, to those making an isolated shower, is certainly feasible with a combination of AMANDA/ICECUBE and RICE technology. The striking behavior of the neutral current-to-charged current ratio is shown in Fig. 3.
Finally, the shape and slope of the angular distribution, Fig. 6, is found to have good discriminating power. The shape does not depend at all on the overall normalization of the flux. Moreover the slopes differ substantially right in the regime of maximum detectable flux, near $\pi/2$ nadir angle, and is ideal for comparing low scale gravity models to the Standard Model. Whether or not extra-dimension models as currently envisaged survive, the angular distribution can severely test the neutrino physics of the Standard Model, possibly strongly bounding or even discovering new physics, as soon as data becomes available.
**Acknowledgements:** P. Jain thanks the University of Kansas College of
Arts and Sciences and Department of Physics and Astronomy for hospitality and support in the course of this work. This research was supported in part by the U.S. Department of Energy under grant number DE-FG03-98ER41079. We used computational facilities of the Kansas Center for Advanced Scientific Computing for part of this work.
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Figure 2: The downwards event rate per cubic kilometer per year within the Standard Model (SM) and in the ADD model for the fundamental scale $M = 1, 2$ TeV and for the number of extra dimensions, $n = 3, 4, 6$ using (a) the SDSS flux model and (b) the WB flux model. The solid line shows the Standard Model prediction alone. The long dashed, dotted and short dashed curves show the predictions of the neutral current graviton exchange within the ADD model plus geometric, black hole cross sections, and including the SM contribution. The upper and lower histograms correspond to the $M = 1, 2$ TeV choices respectively. The new physics contributions are insignificant below 1 PeV, where their contribution above the SM is not visible on this scale.
Figure 3: The ratio of neutral current type events to SM charged current events as a function of neutrino energy for $n = 3, 4$ and $6$ and for $M = 1, 2,$ and $5$ TeV. Only the downward events are included in this plot. The neutral current type interactions, in the sense used here, are dominated by the black hole and eikonal components of the low scale gravity amplitude above the scale of gravity.
Figure 4: The contribution of the regeneration term to the ratio $R$ of the upwards and downwards flux within the ADD plus geometric black hole production model with the fundamental scale $M = 1$ TeV and for the number of extra dimensions equal to 3. The long dashed line (flux from [39]) and short dashed line (flux from [40]) predict nearly the same $R$ as a function of energy for the full calculation, including the regeneration term. Ignoring the regeneration of flux, one gets the lowest curve, identical for both flux models. The regeneration term has a profound effect on the up-to-down flux ratio $R$ due to the large, gravity driven neutral current cross section. Standard model prediction, the solid line, is also shown and it is the same for both flux assumptions.
Figure 5: The ratio $R$ as a function of neutrino energy for the fundamental scale $M = 1, 2$ TeV and for the number of extra dimensions 3 (solid curves), 4 (dotted curves) and 6 (dashed curves). The Standard model (SM) prediction is also shown.
Figure 6: The angular dependence of the upwards flux for the fundamental scale $M = 1, 2$ TeV and for the number of extra dimensions, $n = 3, 4, 6$. The Standard model (SM) prediction is also shown. Only neutrinos with energy $E_\nu > 1.8$ PeV are considered. The $M = 2$ TeV results are indistinguishable from the Standard Model, but the R value is different because downward event rate is larger in the M=2 low scale gravity model (see Fig.2). | 2025-03-05T00:00:00 | olmocr | {
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