tiny [table]font=tiny

# A scenario for the anisotropy of galactic cosmic rays related to nearby
source and local regular magnetic field

Ai-feng Li  Qiang Yuan  Wei Liu,11footnotetext: Corresponding author. and Yi-
qing Guo 22footnotetext: Corresponding author.

###### Abstract

In our recent work, we build a propagation scenario to simultaneously explain
the spectra and anisotropy of cosmic rays (CRs) by considering spatially
dependent propagation (SDP) model and nearby Geminga supernova remnant (SNR)
source. But the phase of anisotropy is still inconsistent with the
experimental data. Recent observations of CR anisotropy show that the phase is
consistent with local regular magnetic field (LRMF) observed by Interstellar
Boundary Explorer (IBEX) below 100 TeV, which indicates that diffusion along
LRMF is important. In this work, we further introduce the LRMF and take into
account the effect of corresponding anisotropic diffusion to explain the
anisotropy of CRs. We find that when the diffusion coefficient perpendicular
to the LRMF is much smaller than the parallel one, the phase of anisotropy
points to $\sim R.A.=3^{h}$, which accords with experimental observation below
100 TeV. We also analyze the influence of the ratio of perpendicular and
parallel diffusion coefficient on the anisotropy and the energy dependence of
the ratio. The results illustrate that with the decrease of perpendicular
diffusion, the anisotropic phase changes from the direction of nearby source
to the LRMF below 100 TeV, meanwhile it changes from the galactic center (GC)
to opposite direction of LRMF above 100 TeV. When the perpendicular diffusion
coefficient grows faster than the parallel one with energy, the diffusion
approaches to the isotropic at high energy, the phase of anisotropy shifts
from the LRMF to the GC above 100 TeV. This could be helpful to ascertain the
energy dependence of diffusion coefficients.

##  1 Introduction

Owing to the deflection of the Galactic magnetic field (GMF), CRs lose their
original direction and become nearly isotropic. However, small CR anisotropy
with relative amplitudes at the order of $10^{-4}\sim 10^{-3}$ is still
observed at a wide energy range from 100 GeV to hundreds of TeV. Numerous
experiments such as Tibet AS$\gamma$ [1, 2, 3], Super-Kamiokande [4], Milagro
[5, 6], IceCube/Ice-Top [7, 8, 9, 10, 11], ARGO-YBJ [12, 13], HAWC [14]
provided the high-precision and two dimensional observations of CR anisotropy,
which show a complex relationship with energy. Less than 100 TeV, the
amplitude increases first and then decreases, but increases again above 100
TeV. At the same time the phase reverses from $\sim R.A.=3^{h}$ to the GC at
an energy of $100\sim 300$ TeV. The CR anisotropy may arise due to several
causes. It may result from the uneven distribution of CR sources such as SNRs
[15, 16] or be due to the LRMF which governs the propagation of CRs nearby the
solar system [17, 18, 19]. The anisotropy can also be a pure kinetic effect
called the Compton-Getting effect arising from the relative motion between the
observer and the CR plasma [20, 21].

Conventional CR propagation model has been successfully explained many
observed phenomena, such as broken power-law energy spectra, secondary-to-
primary ratios, the diffuse gamma-ray distribution, etc. However it fails to
explain the anisotropy of CRs [22, 23]. it is expected that the amplitude of
anisotropy is proportional to the diffusion coefficient $D({\cal
R})\propto{\cal R}^{\delta}$, with $\delta\sim 0.3-0.6$ . Thus, one can expect
that the anisotropy amplitude will increase with energy. But the expected
anisotropy amplitude is larger than the observation above hundreds of TeV
[22]. Furthermore, conventional CR propagation model expects the anisotropic
phase point to GC, because SNRs are mostly in the inner galactic disk region.
Nevertheless, the observed phase of anisotropy varies with the energy,
especially the phase below 100 TeV is obviously deviated from the GC.

Based on the conventional propagation model, various improvements have been
explored to explain the anisotropy problem, including nearby source [15, 16,
24, 23], spatially dependent diffusion process [25, 26, 27], ensemble
fluctuations of CR sources, etc. Our recent work attempted to explain
anisotropy of CRs by involve the nearby source under SDP model[28, 29]. We
found that the anisotropic phase of CRs is sensitive to the direction of the
nearby source. Because Geminga SNR is located close to the direction of anti
GC and far from the galactic disk and it may suppress the expected large
amplitude, so it is a prime candidate [30]. Involve the nearby Geminga SNR
source can explain the spectra and amplitude of anisotropy very well, however
it couldn’t account for the phase of anisotropy which is inconsistent with the
observation below 100 TeV [28].

In galaxy, there is GMF with the order of $\sim 3\mu G$, that have two
components including regular and the turbulent part. A PeV CR particle with a
nucleon number of $Z$ move in a circular motion in GMF with a Larmor radius
$r_{L}\simeq 0.4E_{PeV}/(ZB_{3\mu G})$ pc [19, 31, 32]. The Larmor radius of
PeV CRs is much smaller than their scattering length. The propagation of CRs
is anisotropic when the regular magnetic field strength is comparable to or
beyond that of the turbulent one. Therefore, the anisotropy may be induced by
LRMF nearby solar system below PeV[33]. Recently Interstellar Boundary
Explorer (IBEX) performed observations of neutral particles which unveiled
LRMF [34]. The LRMF is along $(l,b)=(210.5^{\circ},-57.1^{\circ})$ with an
uncertainty of $\sim 1.5^{\circ}$ within $20$ pc. The observation of
anisotropic phase of CRs is coincident with the direction of LRMF below 100
TeV. Several literatures [17, 18, 19, 35] studied the relationship between
anisotropy and LRMF.

The deflecting effect of the magnetic field on the CR particles combined with
nearby sources may explain the amplitude and phase of the anisotropy. To
demonstrate this view, we introduce LRMF and it’s corresponding anisotropic
diffusion simultaneously considering the contribution of nearby Geminga SNR
source under SDP model to account for anisotropy of CRs. In this work, we
first calculate the proton, helium and all-particle energy spectra to match
the experimental data. Then we analyze the anisotropy of CRs and compare the
results with recent work [28]. We find that when the diffusion coefficient
perpendicular to the LRMF is much smaller than the parallel one, the phase of
anisotropy clearly points to LRMF, which is consistent with the experimental
data. We also investigate the influence of the ratio of perpendicular and
parallel diffusion coefficient on the anisotropy and the energy dependence of
the ratio. This could be helpful to ascertain the ratio and energy dependence
of diffusion coefficients.

##  2  Model

###  2.1 Spatially dependent diffusion

The region where CRs diffuse in the Milky Way is called a magnetic halo, which
is usually approximated as a cylinder with its radial boundary equal to the
Galactic radius, i.e. $R=20$ kpc and its half thickness $z_{h}$ is about a few
kpc. $z_{h}$ is usually determined by fitting the B/C ratio along with
diffusion coefficient [36]. Both CR sources and the interstellar medium are
usually assumed to be concentrated near the Galactic disk, whose average
thickness $z_{s}$ is roughly $200$ pc.

HAWC observations have recently that the diffusion coefficient of CRs near the
galactic disk is at least two orders of magnitude smaller than the
conventional one [30]. Therefore, we work in aSDP frame [26, 27, 37], whose
diffusion coefficient are different in inner halo and outer halo. The
parameterized diffusion coefficient we adopt is [38, 39].

$\displaystyle\ D_{xx}(r,z,R)$ $\displaystyle=D_{0}F(r,z)({\dfrac{\cal
R}{{\cal R}_{0}}})^{{\delta_{0}}F(r,z)}$ (2.1)

$F(r,z)$ is parameterized as

$\displaystyle\begin{split}F(r,z)=\left\\{\begin{array}[]{ll}g(r,z)+[1-g(r,z)]{(\dfrac{z}{\xi
z_{0}})}^{n},&|z|\leq\xi z_{0}\\\ 1,&|z|>\xi z_{0}\\\
\end{array}\right.\end{split}$ (2.2)

where $g(r,z)=N_{m}/[1+f(r,z)]$, and $f(r,z)$ is the source density
distribution. The spatial distribution of sources takes the form of SNR
distribution [40].
$f(r,z)\propto(r/r_{\odot})^{1.69}\exp[-3.33(r-r_{\odot})/r_{\odot}]\exp(-|z|/z_{s})$,
where $r_{\odot}=8.5$ kpc and $z_{s}=0.2$ kpc. In this work, we adopt
numerical package DRAGON to solve the transport equation [41]. The
corresponding transport parameters are shown in Table 1.

Table 1: parameters of SDP model. | $D_{0}$ | $\delta_{0}$ | $N_{\rm m}$ | $\xi$ | $n$ | $v_{A}$ | $z_{h}$ |
---|---|---|---|---|---|---|---|---
| [${\rm cm}^{2}\cdot{\rm s}^{-1}$] |  |  |  |  | [${\rm km}\cdot{\rm s}^{-1}$] | [kpc] |
| $4.87\times 10^{28}$ | 0.58 | 0.62 | 0.1 | 4 | 6 | 5 |

The injection spectrum of background sources is assumed to be a power-law of
rigidity with a high-energy exponential cutoff, $q({\cal R})\propto{\cal
R}^{-\nu}\exp(-{\cal R}/{\cal R}_{\rm c})$. The cutoff rigidity of each
element could be either $Z$\- or $A$-dependent.

###  2.2 Local source

Green’s function method is adopted to calculate the propagation of particles
from the local source, assuming a spherical geometry with infinite boundary
conditions [22, 24]. The CR density of local source as a function of space,
rigidity, and time is described as

$\phi(r,{\cal R},t)=\frac{q_{\rm inj}({\cal
R})}{(\sqrt{2\pi}\sigma)^{3}}\exp\left(-\frac{r^{2}}{2\sigma^{2}}\right),$
(2.3)

where $q_{\rm inj}({\cal R})\delta(t)\delta({\bf r})$ is the instantaneous
injection spectrum of a point source, $\sigma({\cal R},t)=\sqrt{2D({\cal
R})t}$ is the effective diffusion length within time $t$, $D({\cal R})$ is the
diffusion coefficient which was adopted as the value nearby the solar system.
The injection spectrum is aslo parameterized as a cutoff power-law form,
$q_{\rm inj}({\cal R})=q_{0}{\cal R}^{-\alpha}\exp(-{\cal R}/{\cal
R}^{\prime}_{\rm c})$. The normalization $q_{0}$ is obtained through fitting
to the CR energy spectra. In this work, we selected the Geminga SNR source
$(l,b=194.3^{\circ},-13^{\circ})$, whose distance and age are $d=330$ pc and
$\tau=3.4\times 10^{5}$ years respectively [30, 42]. The corresponding
injection parameters are given in table 2.

| Background | Local source
---|---|---
Element | Normalization† |  $\nu$ |  $\mathcal{R}_{c}$ |  $q_{0}$ |  $\alpha$ |  ${\cal R}^{\prime}_{c}$
| $[({\rm m}^{2}\cdot{\rm sr}\cdot{\rm s}\cdot{\rm GeV})^{-1}]$ |  | [PV] | [GeV-1] |  | [TV]
p | $1.91\times 10^{-2}$ | 2.34 | 7 | $8.28\times 10^{52}$ | 2.16 | 25
He | $1.43\times 10^{-3}$ | 2.27 | 7 | $2.35\times 10^{52}$ | 2.08 | 25
C | $6.15\times 10^{-5}$ | 2.31 | 7 | $7.2\times 10^{50}$ | 2.13 | 25
N | $7.67\times 10^{-6}$ | 2.34 | 7 | $1.13\times 10^{50}$ | 2.13 | 25
O | $8.20\times 10^{-5}$ | 2.36 | 7 | $1.11\times 10^{51}$ | 2.13 | 25
Ne | $8.05\times 10^{-6}$ | 2.28 | 7 | $1.13\times 10^{50}$ | 2.13 | 25
Mg | $1.62\times 10^{-5}$ | 2.39 | 7 | $1.08\times 10^{50}$ | 2.13 | 25
Si | $1.28\times 10^{-5}$ | 2.37 | 7 | $1.05\times 10^{50}$ | 2.13 | 25
Fe | $1.23\times 10^{-5}$ | 2.29 | 7 | $2.20\times 10^{50}$ | 2.13 | 25

†The normalization is set at total energy $E=100$ GeV.

Table 2: Injection parameters of the background and local source.

###  2.3  Anisotropic diffusion

Under the anisotropic diffusion model, diffusion of CRs consists of two
components: parallel and perpendicular to the magnetic field. The diffusion
tensor $D_{ij}$ associated with the magnetic field is written as

$D_{ij}\,\equiv\,D_{\perp}\delta_{ij}\,+\,\big{(}D_{\|}-D_{\perp}\big{)}b_{i}b_{j}~{},~{}~{}b_{i}=\dfrac{B_{i}}{|\boldsymbol{B}|}$
(2.4)

Where $b_{i}$ is the $i$-th component of the unit vector [43]. Here
$D_{\parallel}$ and $D_{\perp}$ are the diffusion coefficients aligned
parallel and perpendicular to the regular magnetic field, respectively.

CR particles propagate in a magnetic field, $B=\delta B+B_{0}$, that is the
sum of a regular $B_{0}$ field and a turbulent $\delta B$ field. Mean free
path of the CR particles in the TeV region is $\sim 20$ pc which is larger
than the Lamore radius. According to the quasi-linear theory [44, 45], $\delta
B(k)^{2}=\int{d^{3}\delta B(k)^{2}\ll|B_{0}|^{2}}$ at this scale. The ratio
between the perpendicular and parallel diffusion coefficients is expected to
be as

$\dfrac{D_{\perp}}{D_{\parallel}}\sim\mathcal{F}(k)\sim\dfrac{\delta
B(k)^{2}}{B_{0}^{2}}\ll 1~{},$ (2.5)

where $\mathcal{F}(k)$ is the normalized power of the turbulent modes with
wave number $k$ [46, 18, 47]. However, if $\delta B/B_{0}\sim 1$,
$D_{\perp}\simeq D_{\|}$, then the diffusion of CRs for higher energies is
isotropic. This ratio need to be calculated with the numerical simulation.

In this work, we adopt two different rigidity dependents $D_{\parallel}$ and
$D_{\perp}$ following [46],

$\displaystyle D_{\parallel}$ $\displaystyle=D_{0\parallel}\left(\frac{\cal
R}{{\cal R}_{0}}\right)^{\delta_{\|}}~{},$ (2.6) $\displaystyle D_{\perp}$
$\displaystyle=\,D_{0\perp}\left(\frac{\cal R}{{\cal
R}_{0}}\right)^{\delta_{\perp}}=\varepsilon D_{0\parallel}\left(\frac{\cal
R}{{\cal R}_{0}}\right)^{\delta_{\perp}}~{},$ (2.7)

which $\varepsilon=\dfrac{D_{0\perp}}{D_{0\parallel}}$ is the ratio between
perpendicular and parallel diffusion coefficient at the reference rigidity
${\cal R}_{0}$. The parameters $\delta_{\parallel}$, $\delta_{\perp}$, ${\cal
R}_{0}$, and $\varepsilon$ need to be carefully chosen to keep
$D_{\perp}\lesssim D_{\parallel}$ within the concerned energy region(GeV -
PeV).

###  2.4  Large-scale dipole anisotropy

In a simple model of isotropic diffusion, the amplitude of the dipole
anisotropy is usually defined with observed relative difference between the
fluxes in the maximum and minimum directions $\phi_{\rm max}$ and $\phi_{\rm
max}$ as [18]

$|\delta|=\dfrac{(\phi_{\rm max}-\phi_{\rm min})}{(\phi_{\rm max}+\phi_{\rm
min})}=\dfrac{3D\cdot\nabla\psi}{v\psi}~{},$ (2.8)

It can be seen that the amplitude of the dipole anisotropy is proportional to
the spatial gradient of the CR density $\nabla\psi$ and the diffusion
coefficient $D$.

In anisotropic diffusion model, its vector form can be written as [19, 35]

${\delta}=\dfrac{3}{v\psi}D_{ij}\dfrac{\partial\psi}{\partial x_{j}}~{}.$
(2.9)

The rigidity dependence of the dipole amplitude results from both diffusion
tensor $D_{ij}$ and $\dfrac{\nabla\psi}{\psi}$. For GeV–PeV CRs, parallel
diffusion predominates, the anisotropic phase tends to the direction of
magnetic field. In order to study the influence of two diffusion coefficients
($D_{\perp}$ and $D_{\parallel}$ ) on anisotropy, we would choose several sets
of parameters ($\varepsilon,~{}\delta_{\parallel},~{}\delta_{\perp}$) to
calculate the anisotropy.

##  3 Results and Discussion

Figure 1: Energy spectra of protons (left) and helium nuclei (right). The data
points are taken from DAMPE[48, 49], AMS-02 [50, 51], CREAM-III [52], NUCLEON
[53], KASCADE [54] and KASCADE-Grande [55] respectively. The blue lines are
the background fluxes, and the red lines are the fluxes from a nearby Geminga
SNR source respectively. The black lines represent the sum contributions of
the background and nearby Geminga SNR source.  Figure 2: The all-particle
spectra multiplied by $E^{2.6}$. The data points are taken from ref [56]. The
solid lines with different colors are the model predictions of different mass
groups, and the black solid line is the total contribution.

The LRMF within 20 pc has been measured accurately by IBEX experiment, which
is much smaller than the size of the Milky Way [34, 46]. Therefore, when we
simulate the CRs background using DRARGON public numerical packages [41], the
CRs is approximately isotropic diffusion in the Milky Way. First, we calculate
the proton and helium spectra. Figure 1 shows the energy spectra of protons
and helium obtained for the model calculation, with the red, blue and black
lines representing the contributions from the local source, the background
sources and the sum of them, respectively. It can be seen that the
contribution of nearby Geminga SNR source can simultaneously account for the
spectral hardening features at $\sim$ 200 GeV, and softening features at
$\sim$ 10 TeV. Through adding different compositions together, we get the all-
particle spectrum as shown in Figure 2, which is well consistent with the
observational data.

Figure 3: When $\varepsilon=0.01$, $\Delta\delta=0.32$, energy dependences of
the amplitude (left) and phase (right) of the dipole anisotropies together
with the contribution from nearby Geminga SNR source and LRMF. The red broken
line indicates the result [28].

The anisotropy of CRs depends on the sum of the CR flows from the background
and the local source. In this work, the parameters of parallel diffusion
coefficient $D_{\parallel}$ are set as those shown in Table 1. The CRs in the
100 GeV - PeV energy region require $D_{\parallel}>D_{\perp}$, therefore we
set $\varepsilon=0.01$ and
$\Delta\delta=\delta_{\perp}-\delta_{\parallel}=0.32$, where $\Delta\delta$ is
the difference between $\delta_{\perp}$ and $\delta_{\parallel}$. Figure 3
shows the comparison of the anisotropy of CRs with and without LRMF [28],
where the solid black line presents anisotropy of the introduced LRMF and it’s
anisotropic diffusion effect combined with nearby Geminga SNR source, and red
broken line is the result of the work [28], respectively. For $E<100$ TeV,
anisotropic phase with the LRMF clearly points to LRMF ($\sim R.A.=3^{h}$ ),
which is better in accordance with the experimental observation than the
result without magnetic field. The nearby source dominates the observed
anisotropies, although its flux is sub-dominant. The position of nearby source
and the direction of LRMF together determine the anisotropic phase. The phase
flips at 100 TeV from $\sim R.A.=3^{h}$ to GC. For $E\gtrsim 100$ TeV, the
contribution from the local source and LRMF decrease significantly, and
background become dominant instead. The phase of the anisotropy turns to the
GC, since CR sources are more abundant in the inner Galaxy.

Figure 4:  Energy dependences of the amplitude (left) and phase (right) of the
anisotropies for $\Delta\delta(\delta_{\perp}-\delta_{\parallel})=0.32$
together with the contribution from nearby Geminga SNR source and local
regular magnetic field. The three black lines correspond to the results for
three different $\varepsilon(=0.1,~{}0.01,~{}0.001)$.

Anisotropy of CRs is related to the diffusion coefficient, which is determined
by the parameters $\varepsilon$, $\delta_{\parallel}$, $\delta_{\perp}$, and
${\cal R}_{0}$. To study the induced anisotropy changes of CRs by the ratio of
perpendicular and parallel diffusion coefficients, we calculate the anisotropy
of different $\varepsilon$ (0.1, 0.01, 0.001) at $\Delta\delta=0.32$. Figure 4
shows the anisotropies of three different $\varepsilon$ and demonstrates the
amplitude and phase of anisotropy are significantly related to $\varepsilon$.
At the same energy, the amplitude of anisotropy increases with increasing of
$\varepsilon$, which is due to the amplitude is proportional to the diffusion
coefficient. As $\varepsilon$ decreases from 0.1 to 0.001, the phase of
anisotropy shifts from the position of the nearby source to the LRMF below 100
TeV, that results from the parallel diffusion gradually increases and the
perpendicular diffusion weakens. At the same time, the phase of anisotropy
shifts from the GC to opposite direction of LRMF above 100 TeV.

Figure 5:  Energy dependences of the amplitude (left) and phase (right) of the
anisotropies for $\varepsilon=0.01$ together with the contribution from nearby
Geminga SNR source and LRMF. The three black lines correspond to the results
for three different
$\Delta\delta(\delta_{\perp}-\delta_{\parallel})(=0,~{}0.3,~{}0.8)$.

In addition to $\varepsilon$, we also study the relationship between
anisotropy and $\Delta\delta$. Figure 5 shows the amplitude and phase of the
anisotropy with different $\Delta\delta$ ( 0, 0.3 and 0.8) at
$\varepsilon$=0.01. From Figure 5 (left), we can see that with the increase of
$\Delta\delta$, the anisotropic amplitude of CRs rises, which is due to the
diffusion coefficient become larger. Figure 5 (right) illustrates that with
the increase of $\Delta\delta$, anisotropic phase of CRs changes from the LRMF
to the direction of the nearby source below 100 TeV, while it changes from the
opposite direction of LRMF to GC above 100 TeV. It can be attributed to that
the diffusion gradually transforms from anisotropic to the isotropic, when
perpendicular diffusion coefficient grows faster than the parallel one. The
current experimental observation around PeV energy range are still in dispute.
The AS$\gamma$ measurement points to the GC, while the ICECUBE observation
points to the LRMF. Further experimental measurements of anisotropic phase,
such as LHAASO, will help to understand the energy dependence of the diffusion
coefficient.

##  4 Summary

The observation of CR anisotropy is hard to explain for a long time. Our
recent work attempted to account for the anisotropy of CRs through the
contribution of nearby Geminga SNR source under the SDP model. However the
anisotropic phase is about $\sim R.A.=5^{h}$ below 100 TeV, which disagrees
with the experimental data. LRMF observed by IBEX is coincident with
observational phase of anisotropy, which indicates that diffusion along LRMF
is important. Here, we further take into account the LRMF and the
corresponding effect of anisotropic diffusion on CR anisotropy.

We find that when the diffusion coefficient perpendicular to the LRMF is much
smaller than the parallel one, i.e. the CR propagation is principally along
the magnetic field, the phase of anisotropy changes from the direction of
local source to the LRMF, which conforms with the observations below 100 TeV.

We further study the influence of the ratio of the perpendicular and parallel
diffusion coefficients on anisotropy. When the perpendicular diffusion
coefficient becomes smaller, the phase of anisotropy gradually shifts from the
local source to the LRMF below 100 TeV, while it shifts from the GC to the
opposite direction of LRMF above 100 TeV.

We also investigate the energy dependence of the ratio. When the perpendicular
diffusion coefficient grows faster than the parallel one with energy, the
diffusion approaches to the isotropic at high energy, the phase of anisotropy
changes from the LRMF to the GC above 100 TeV. However the current
measurements at that energy range are still in dispute. The Tibet AS$\gamma$
measurement points to the GC, while the ICECUBE observation points to the
LRMF. We hope that the CR experiments, for example LHAASO, could measure the
anisotropy phase precisely, which could be helpful to ascertain the energy
dependence of diffusion coefficients.

Overall, we provide a good scenario for anisotropy of CRs related to nearby
Geminga SNR source and LRMF inferred from IBEX, in the energy region from 100
GeV to PeV.

## Acknowledgements

This work is supported by the National Key $R\&D$ Program of China grant
(2018YFA0404202) and the National Natural Science Foundation of China
(11963004, 11635011, 11875264, U1831208, U1738205, U2031110) and Shandong
Province Natural Science Foundation ( ZR2020MA095).

## References

  * [1] M. Amenomori, S. Ayabe, X. J. Bi, et al.  Anisotropy and Corotation of Galactic Cosmic Rays.  Science, 314(5798):439–443, October 2006.
  * [2] M. Amenomori et al.  On Temporal Variations of the Multi-TeV Cosmic Ray Anisotropy using the Tibet III Air Shower Array.  Astrophys. J., 711:119–124, 2010.
  * [3] M. Amenomori.  Northern sky Galactic Cosmic Ray anisotropy between 10-1000 TeV with the Tibet Air Shower Array.  Astrophys. J., 836(2):153, 2017.
  * [4] G. Guillian, J. Hosaka, K. Ishihara, et al.  Observation of the anisotropy of 10TeV primary cosmic ray nuclei flux with the Super-Kamiokande-I detector.  Phys. Rev. D, 75(6):062003, March 2007.
  * [5] A. A. Abdo, B. Allen, T. Aune, et al.  Discovery of Localized Regions of Excess 10-TeV Cosmic Rays.  Physical Review Letters, 101(22):221101, November 2008.
  * [6] A. A. Abdo, B. T. Allen, T. Aune, et al.  The Large-Scale Cosmic-Ray Anisotropy as Observed with Milagro.  apj, 698:2121–2130, June 2009.
  * [7] R. Abbasi, Y. Abdou, T. Abu-Zayyad, et al.  Measurement of the Anisotropy of Cosmic-ray Arrival Directions with IceCube.  apj, 718:L194–L198, August 2010.
  * [8] R. Abbasi, Y. Abdou, T. Abu-Zayyad, et al.  Observation of Anisotropy in the Arrival Directions of Galactic Cosmic Rays at Multiple Angular Scales with IceCube.  apj, 740:16, October 2011.
  * [9] R. Abbasi, Y. Abdou, T. Abu-Zayyad, et al.  Observation of Anisotropy in the Galactic Cosmic-Ray Arrival Directions at 400 TeV with IceCube.  apj, 746:33, February 2012.
  * [10] M. G. Aartsen, R. Abbasi, Y. Abdou, et al.  Observation of Cosmic-Ray Anisotropy with the IceTop Air Shower Array.  apj, 765:55, March 2013.
  * [11] M. G. Aartsen, K. Abraham, M. Ackermann, et al.  Anisotropy in Cosmic-Ray Arrival Directions in the Southern Hemisphere Based on Six Years of Data from the IceCube Detector.  apj, 826:220, August 2016.
  * [12] B. Bartoli, P. Bernardini, X. J. Bi, et al.  Medium scale anisotropy in the TeV cosmic ray flux observed by ARGO-YBJ.  prd, 88(8):082001, October 2013.
  * [13] B. Bartoli, P. Bernardini, X. J. Bi, et al.  ARGO-YBJ Observation of the Large-scale Cosmic Ray Anisotropy During the Solar Minimum between Cycles 23 and 24.  apj, 809:90, August 2015.
  * [14] A. U. Abeysekara, R. Alfaro, C. Alvarez, et al.  Observation of Small-scale Anisotropy in the Arrival Direction Distribution of TeV Cosmic Rays with HAWC.  apj, 796:108, December 2014.
  * [15] A. D. Erlykin and A. W. Wolfendale.  The anisotropy of galactic cosmic rays as a product of stochastic supernova explosions.  Astroparticle Physics, 25:183–194, April 2006.
  * [16] P. Blasi and E. Amato.  Diffusive propagation of cosmic rays from supernova remnants in the Galaxy. I: spectrum and chemical composition.  jcap, 1:10, January 2012.
  * [17] N. A. Schwadron, F. C. Adams, E. R. Christian, et al.  Global Anisotropies in TeV Cosmic Rays Related to the Sun rsquos Local Galactic Environment from IBEX.  Science, 343:988–990, February 2014.
  * [18] P. Mertsch and S. Funk.  Solution to the Cosmic Ray Anisotropy Problem.  Physical Review Letters, 114(2):021101, January 2015.
  * [19] M. Ahlers.  Deciphering the Dipole Anisotropy of Galactic Cosmic Rays.  Physical Review Letters, 117(15):151103, October 2016.
  * [20] Arthur H. Compton and Ivan A. Getting.  An Apparent Effect of Galactic Rotation on the Intensity of Cosmic Rays.  Physical Review, 47(11):817–821, June 1935.
  * [21] L. J. Gleeson and W. I. Axford.  The Compton-Getting Effect.  Ap&SS, 2(4):431–437, December 1968.
  * [22] P. Blasi and E. Amato.  Diffusive propagation of cosmic rays from supernova remnants in the Galaxy. II: anisotropy.  jcap, 1:11, January 2012.
  * [23] Wei Liu, Xiao-Jun Bi, Su-Jie Lin, Bing-Bing Wang, and Peng-Fei Yin.  Excesses of cosmic ray spectra from a single nearby source.  prd, 96(2):023006, July 2017.
  * [24] L. G. Sveshnikova, O. N. Strelnikova, and V. S. Ptuskin.  Spectrum and anisotropy of cosmic rays at TeV-PeV-energies and contribution of nearby sources.  Astroparticle Physics, 50:33–46, December 2013.
  * [25] C. Evoli, D. Gaggero, D. Grasso, and L. Maccione.  Common Solution to the Cosmic Ray Anisotropy and Gradient Problems.  Physical Review Letters, 108(21):211102, May 2012.
  * [26] N. Tomassetti.  Origin of the Cosmic-Ray Spectral Hardening.  apj, 752:L13, June 2012.
  * [27] Y.-Q. Guo, Z. Tian, and C. Jin.  Spatial-dependent Propagation of Cosmic Rays Results in the Spectrum of Proton, Ratios of P/P, and B/C, and Anisotropy of Nuclei.  ApJ, 819:54, March 2016.
  * [28] Wei Liu, Yi-Qing Guo, and Qiang Yuan.  Indication of nearby source signatures of cosmic rays from energy spectra and anisotropies.  jcap, 2019(10):010, October 2019.
  * [29] Bing-Qiang Qiao, Wei Liu, Yi-Qing Guo, and Qiang Yuan.  Anisotropies of different mass compositions of cosmic rays.  jcap, 2019(12):007, December 2019.
  * [30] A. U. Abeysekara, A. Albert, R. Alfaro, et al.  Extended gamma-ray sources around pulsars constrain the origin of the positron flux at Earth.  Science, 358(6365):911–914, November 2017.
  * [31] Markus Ahlers and Philipp Mertsch.  Origin of small-scale anisotropies in Galactic cosmic rays.  Progress in Particle and Nuclear Physics, 94:184–216, May 2017\.
  * [32] A. P. Snodin, A. Shukurov, G. R. Sarson, P. J. Bushby, and L. F. S. Rodrigues.  Global diffusion of cosmic rays in random magnetic fields.  MNRAS, 457(4):3975–3987, April 2016.
  * [33] Eduardo Battaner, Joaquín Castellano, and Manuel Masip.  Galactic Magnetic Fields and the Large-Scale Anisotropy at Milagro.  ApJ, 703(1):L90–L93, September 2009.
  * [34] H. O. Funsten, R. DeMajistre, P. C. Frisch, et al.  Circularity of the Interstellar Boundary Explorer Ribbon of Enhanced Energetic Neutral Atom (ENA) Flux.  ApJ, 776(1):30, Oct 2013.
  * [35] Wei Liu, Su-jie Lin, Hong-bo Hu, Yi-qing Guo, and Ai-feng Li.  Two Numerical Methods for the 3D Anisotropic Propagation of Galactic Cosmic Rays.  ApJ, 892(1):6, March 2020.
  * [36] Andrew W. Strong, Igor V. Moskalenko, and Vladimir S. Ptuskin.  Cosmic-Ray Propagation and Interactions in the Galaxy.  Annual Review of Nuclear and Particle Science, 57(1):285–327, November 2007.
  * [37] Chao Jin, Yi-Qing Guo, and Hong-Bo Hu.  Spatial dependent diffusion of cosmic rays and the excess of primary electrons derived from high precision measurements by AMS-02.  Chinese Physics C, 40(1):015101, January 2016.
  * [38] Yi-Qing Guo and Qiang Yuan.  Understanding the spectral hardenings and radial distribution of Galactic cosmic rays and Fermi diffuse $\gamma$ rays with spatially-dependent propagation.  Phys. Rev. D, 97(6):063008, March 2018.
  * [39] Wei Liu, Yu-hua Yao, and Yi-Qing Guo.  Revisiting the Spatially Dependent Propagation Model with the Latest Observations of Cosmic-Ray Nuclei.  ApJ, 869(2):176, December 2018.
  * [40] G. Case and D. Bhattacharya.  Revisiting the galactic supernova remnant distribution.  A&AS, 120:437–440, December 1996.
  * [41] Carmelo Evoli, Daniele Gaggero, Dario Grasso, and Luca Maccione.  Cosmic Ray propagation in the Galaxy and diffuse gamma-ray emission.  In Felix A. Aharonian, Werner Hofmann, and Frank Rieger, editors, American Institute of Physics Conference Series, volume 1085 of American Institute of Physics Conference Series, pages 380–383, December 2008.
  * [42] Jacqueline Faherty, Frederick M. Walter, and Jay Anderson.  The trigonometric parallax of the neutron star Geminga.  Ap&SS, 308(1-4):225–230, April 2007.
  * [43] J. Giacalone and J. R. Jokipii.  The Transport of Cosmic Rays across a Turbulent Magnetic Field.  ApJ, 520:204–214, July 1999.
  * [44] J. R. Jokipii.  Cosmic-Ray Propagation. I. Charged Particles in a Random Magnetic Field.  ApJ, 146:480, November 1966.
  * [45] J. R. Jokipii and E. N. Parker.  Random Walk of Magnetic Lines of Force in Astrophysics.  Phys. Rev. Lett., 21(1):44–47, July 1968.
  * [46] S. S. Cerri, D. Gaggero, A. Vittino, C. Evoli, and D. Grasso.  A signature of anisotropic cosmic-ray transport in the gamma-ray sky.  jcap, 10:019, October 2017.
  * [47] Eduardo Battaner, Joaquín Castellano, and Manuel Masip.  Magnetic Fields and Cosmic-Ray Anisotropies at TeV Energies.  ApJ, 799(2):157, February 2015.
  * [48] Q. An, R. Asfandiyarov, P. Azzarello, et al.  Measurement of the cosmic ray proton spectrum from 40 GeV to 100 TeV with the DAMPE satellite.  Science Advances, 5(9):eaax3793, September 2019.
  * [49] F. Alemanno, Q. An, P. Azzarello, et al.  Measurement of the Cosmic Ray Helium Energy Spectrum from 70 GeV to 80 TeV with the DAMPE Space Mission.  Phys. Rev. Lett., 126(20):201102, May 2021.
  * [50] M. Aguilar, D. Aisa, B. Alpat, et al.  Precision Measurement of the Proton Flux in Primary Cosmic Rays from Rigidity 1 GV to 1.8 TV with the Alpha Magnetic Spectrometer on the International Space Station.  Phys. Rev. Lett., 114(17):171103, May 2015.
  * [51] M. Aguilar, L. Ali Cavasonza, B. Alpat, et al.  Observation of the Identical Rigidity Dependence of He, C, and O Cosmic Rays at High Rigidities by the Alpha Magnetic Spectrometer on the International Space Station.  Phys. Rev. Lett., 119(25):251101, December 2017.
  * [52] Y. S. Yoon, T. Anderson, A. Barrau, et al.  Proton and Helium Spectra from the CREAM-III Flight.  ApJ, 839(1):5, April 2017.
  * [53] E. Atkin, V. Bulatov, V. Dorokhov, et al.  First results of the cosmic ray NUCLEON experiment.  J. Cosmology Astropart. Phys, 2017(7):020, July 2017.
  * [54] T. Antoni, W. D. Apel, A. F. Badea, et al.  KASCADE measurements of energy spectra for elemental groups of cosmic rays: Results and open problems.  Astroparticle Physics, 24(1-2):1–25, September 2005.
  * [55] W. D. Apel, J. C. Arteaga-Velázquez, K. Bekk, et al.  KASCADE-Grande measurements of energy spectra for elemental groups of cosmic rays.  Astroparticle Physics, 47:54–66, July 2013.
  * [56] Jörg R. Hörandel.  On the knee in the energy spectrum of cosmic rays.  Astroparticle Physics, 19(2):193–220, May 2003.


# RNS model from a new angle for strings charged under the Maximal Gauge
Symmetry of the Standard model

J. S. Bhattacharyya
Department of Physics, Kanchrapara College, Kanchrapara 743145,
North 24-PGS, West Bengal, India Electronic address<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Dated: )

###### Abstract

We consider the RNS model from a new angle. The longitudinal and time
components of the world-sheet fermions add a $U(1)$ charge to a state. Unlike
the gauginos, the ground state fermions in the open string sector are complex;
spinor representations of $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y_{W}}$.

In the supersymmetric formulation of $N=1,D=10$ string theory, the GSO
conditions are in-built to respect modular invariance and the odd $G$ parity
states like tachyons are mapped out. Thus, the ground states in the bosonic
sector and the fermionic sector of the open strings are identified as the
gauge bosons and the gauginos of a particular chirality. They are connected by
a $N=1$ local supersymmetry and both of them transform in the adjoint
representation of the gauge group. We argue that the longitudinal and the time
components of the world-sheet fermions can add a $U(1)$ charge to the ground
state fermions. Since the representations become complex after the addition of
the $U(1)$ charge, the gauge group should be unitary and the three generation
Standard Model seems to be a good choice.

The light-cone gauge condition: $\psi^{+}=0$ cannot be taken as an operator
identity. It then conflicts with the anti-commutation relation:
$\\{\psi^{+},\psi^{-}\\}=\eta^{+-}$. The physical states can only be
annihilated by the positive frequency parts of $\psi^{+}$ according to Gupta-
Bleuler formalism. Let $Y$ be the boson we get after bosonization of the pair
of longitudinal and time components of the world-sheet fermions. We interpret
$Y$ as the curled up coordinate to define the $U(1)$ gauge symmetry.

Since the spinor representation is complex, the gauge group should be unitary.
We assume that the six dimensional internal space is $CP^{2}\otimes S^{2}\cong
SU(3)_{C}\otimes SU(2)_{L}$.

Though $CP^{2}$ does not admit spinors, $CP^{2}\otimes S^{2}$ does [1]. It is
a well known fact that the index of the Dirac operator on a six dimensional
space of Euclidean signature is $\frac{\chi}{2}$ [2], where $\chi$ is the
Euler number of the manifold. Since the Euler number of a product manifold is
the product of its Euler numbers, the index of the Dirac operator for the
product space $CP^{2}\otimes S^{2}$ is

$\displaystyle\nu_{+}-\nu_{-}$ $\displaystyle=$
$\displaystyle\frac{\chi_{CP^{2}}.\chi_{S^{2}}}{2}$ $\displaystyle=$
$\displaystyle\frac{3.2}{2}$ $\displaystyle=$ $\displaystyle 3$ 111The Betti-
Hodge numbers for $CP^{n}$ are given by  $\displaystyle b_{pq}$
$\displaystyle=$ $\displaystyle\delta_{p,q}\hskip 13.0ptp\leq n$
$\displaystyle=$ $\displaystyle 0\hskip 13.0ptp>n$ Hence, for $S^{2}$ or
$CP^{1}$ they are given by the matrix
$b_{pq}(S^{2})=\left[{\begin{array}[]{cc}1&0\\\ 0&1\\\ \end{array}}\right]$
and those for $CP^{2}$ are given by the matrix
$b_{pq}(CP^{2})=\left[{\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&1\\\
\end{array}}\right].$ The same for the product manifold, $M\otimes N$, is
given by the Kunneth formula:  $\displaystyle b_{pq}(M\otimes
N)=\sum_{r,m,n,s}b_{rs}(M).b_{mn}(N),$ (3) subject to the constraints $r+m=p$
and $s+n=q$. Thus, we can write the Betti-Hodge numbers for $CP^{2}\otimes
S^{2}$ as  $b_{pq}(CP^{2}\otimes S^{2})=\left[{\begin{array}[]{cccc}1&0&0&0\\\
0&2&0&0\\\ 0&0&2&0\\\ 0&0&0&1\\\ \end{array}}\right].$ (4) This matrix is
homeomorphic to the Hodge matrix of the six dimensional Calabi-Yau Manifold:
$\left[{\begin{array}[]{cccc}1&0&0&1\\\ 0&3&0&0\\\ 0&0&3&0\\\ 1&0&0&1\\\
\end{array}}\right].$ (5) corresponding to three generations of fermions. We
have added 1 to each of $b_{11}$ and $b_{22}$ going from (4) to (5). It
corresponds to adding one simplex of dimension two and one simplex of
dimension four. To keep the Euler characteristics the same, we have added 1 to
each of $b_{03}$ and $b_{30}$. It corresponds to the addition of two simplices
of dimension three to the manifold.

Since $\chi$ vanishes if the internal space is a product space of the form
$S^{1}\otimes X$ by the product rule for $\chi s$, there is perhaps no other
choice than $CP^{2}\otimes S^{2}\neq S^{1}\otimes X$ for the six dimensional
internal space if the gauge group is unitary.

We can write an arbitrary spinor as

$\eta=\omega^{(0,0)}\zeta+\omega^{(0,2)}_{\bar{\mu}\bar{\nu}}\gamma^{\bar{\mu}\bar{\nu}}\zeta+\omega^{(0,1)}_{\bar{\mu}}\gamma^{\bar{\mu}}\zeta+\omega^{(0,3)}_{\bar{\mu}\bar{\nu}\bar{\rho}}\gamma^{\bar{\mu}\bar{\nu}\bar{\rho}}\zeta,$
(6)

where $\gamma^{\mu}$-matrices with complex indices $\mu$=1,2 or 3 for the six
dimensional internal manifold, can be used as creation and annihilation
operators [3]. The eight components of the spinor break up under $SU(3)$
subgroup of $SU(4)\cong SO(6)$ as $8=4+\bar{4}=1+3+\bar{3}+\bar{1}$. The four
states of the four-component Weyl spinor of $CP^{2}\otimes S^{2}$ can be
identified with the singlet and the triplet representations (1+3) of the left-
handed leptons and quarks of a particular generation. For isoscalar right-
handed fermions we should consider the conjugate pair ($\bar{1}+\bar{3}$) of
opposite helicity contributing 2.(1+3)=8 degrees of freedom to the fermionic
sector. Left-handed fermions are isodoublets. Hence, the corresponding number
is sixteen. Therefore, the total number of degrees of freedom of the fermions
is twenty-four. The total number of degrees of freedom of the gluons, W
bosons, Z bosons and photons is also 2.(8+3+1)=24. So there is an exact
matching between the number of degrees of freedom of bosons and fermions at
the ground level. This will be so for other states also if the GSO conditions
are respected in both the fermionic and the bosonic sectors. Hence, the zero-
point energy vanishes and there is no Casimir force between two parallel
D-branes [4, 5, 6].

The $SU(3)$ quantum numbers are part of the dynamics of the system, but the
Chan-Paton factors for the isospin can only be added to the states through the
inclusion of Wilson lines and therefore the $SU(2)$ gauge symmetry can break
spontaneously.

We can ignore the group indices locally. Hence, from the triality symmetry of
$SO(8)$, the open string ground states in the fermionic and the bosonic
sectors can be identified with copies of the eight primitive states of the
Majorana-Weyl fermion and the eight states of the vector meson of the D = 10 N
= 1 local supersymmetry.

D-branes cannot be wrapped in the compact internal space in a non-trivial way
to add Chan-Paton factors to the states through Wilson loops. Every loop can
be shrunk to a point in this case, because $CP^{2}\otimes S^{2}\neq
S^{1}\otimes X$. Hence, they should be wrapped in the four dimensional
Minkowski space only, making the theory four dimensional in effect. It
corresponds to the gauge symmetry

$\displaystyle SO(2^{\frac{4}{2}})$ $\displaystyle=$ $\displaystyle SO(4)$
$\displaystyle\cong$ $\displaystyle SU(2)_{L}\otimes SU(2)_{R}.$

The states are $SU(2)_{R}$ singlets in this case. It confirms that only the
$SU(2)$ Chan-Paton factor can be added to the states through D-branes.

Type-I theory has odd-branes, D3 and D1 branes in this case. A D3-brane fills
the whole Minkowski space and should correspond to the ordinary Chan-Paton
factor of $SU(3)_{C}$. The Chan-Paton factors of $SU(2)$ introduced on Wilson
lines correspond to D1-branes. The magnetic flux enclosed by a Wilson loop
through the two dimensional plane normal to the D1-branes is quantized,
because $\pi_{1}(U(1))=\mathbb{Z}$. The size of the loop is a measure of the
energy scale. When it is small, the size of the loop is sufficiently large to
enclose a magnetic flux, breaking the $SU(2)$ gauge symmetry, which is
restored when the energy scale is high.

## References

  * [1] B. Dolan, Annales de l’I.H.P. Physique th´eorique, 42(1985)375
  * [2] M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory, Volume 2, Cambridge University Press 1987
  * [3] S.P. Misra, Introduction to Supersymmetry and Supergravity, SERC School Series, Department of Science and Technology, Wiley Eastern Limited 1992
  * [4] S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888
  * [5] J. Polchinski, Comm. Math. Phys. 104 (1986) 37
  * [6] J. Polchinski, arXiv:hep-th/9611050v2 23 Apr 1997