| Summary: | In this paper, we study the implications of the Dark Large Mixing Angle (DLMA) solutions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mn>12</mn></msub></semantics></math></inline-formula> in the context of the IceCube data. We study the consequences in the measurement of the neutrino oscillation parameters, namely the octant of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mn>23</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>δ</mi><mi>CP</mi></msub></semantics></math></inline-formula> in light of both Large Mixing Angle (LMA) and DLMA solutions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mn>12</mn></msub></semantics></math></inline-formula>. We find that it will be impossible for IceCube to determine the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>δ</mi><mi>CP</mi></msub></semantics></math></inline-formula> and the true nature of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mn>12</mn></msub></semantics></math></inline-formula>, i.e., LMA or DLMA, at the same time. This is because of the existence of an intrinsic degeneracy at the Hamiltonian level between these parameters. Apart from that, we also identify a new degeneracy between <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mn>23</mn></msub></semantics></math></inline-formula> and two solutions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mn>12</mn></msub></semantics></math></inline-formula> for a fixed value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>δ</mi><mi>CP</mi></msub></semantics></math></inline-formula>. We perform a chi-square fit using three different astrophysical sources, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> source, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>π</mi></semantics></math></inline-formula> source, and <i>n</i> source, to find that both <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> source and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>π</mi></semantics></math></inline-formula> source are allowed within <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mi>σ</mi></mrow></semantics></math></inline-formula>, whereas the <i>n</i> source is excluded at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>σ</mi></mrow></semantics></math></inline-formula>. It is difficult to make any conclusion regarding the measurement of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mn>23</mn></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>δ</mi><mi>CP</mi></msub></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> source. However, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>π</mi></semantics></math></inline-formula> (<i>n</i>) source prefers the higher (lower) octant of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mn>23</mn></msub></semantics></math></inline-formula> for both LMA and DLMA solution of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mn>12</mn></msub></semantics></math></inline-formula>. The best-fit value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>δ</mi><mi>CP</mi></msub></semantics></math></inline-formula> is around <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>180</mn><mo>∘</mo></msup></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>0</mn><mo>∘</mo></msup><mo>/</mo><msup><mn>360</mn><mo>∘</mo></msup></mrow></semantics></math></inline-formula>) for the LMA (DLMA) solution of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mn>12</mn></msub></semantics></math></inline-formula>, whereas for the DLMA (LMA) solution of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mn>12</mn></msub></semantics></math></inline-formula>, the best-fit value is around <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>0</mn><mo>∘</mo></msup><mo>/</mo><msup><mn>360</mn><mo>∘</mo></msup></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>180</mn><mo>∘</mo></msup></semantics></math></inline-formula>) for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>π</mi></semantics></math></inline-formula> (<i>n</i>) source. If we assume the current best-fit values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mn>23</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>δ</mi><mi>CP</mi></msub></semantics></math></inline-formula> to be true, then the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>π</mi></semantics></math></inline-formula> sources prefer the LMA solution of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mn>12</mn></msub></semantics></math></inline-formula>, whereas the <i>n</i> source prefers the DLMA solution of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mn>12</mn></msub></semantics></math></inline-formula>.
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