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Dahaka14

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In summary: Can someone tell me why this is?In summary, when an expression is derived using an "expansion in small parameters," it is generally a method for calculating a probability. However, in the case of neutrino oscillation probabilities, only functions of the small parameters are left after a Taylor expansion. This may be due to the fact that the approximation is based on the inner product of the states, which does not involve multiplication by the small parameter.

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Dahaka14

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ansgar

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Dahaka14

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[tex]P_{\nu_{e}\rightarrow\nu_{\mu}}=\sin^{2}\theta_{23}\sin^{2}2\theta_{13}\sin^{2}\left(\frac{\Delta_{13}L}{2}\right)+\cos^{2}\theta_{23}\sin^{2}2\theta_{12}\sin^{2}\left(\Delta_{12}L}{2}\right)+\tilde{J}\cos\left(\pm\delta-\frac{\Delta_{13}L}{2}\right)\frac{\Delta_{12}L}{2}\sin\left(\frac{\Delta_{13}L}{2}\right)[/tex]

where [tex]\tilde{J}\equiv\cos\theta_{13}\sin2\theta_{12}\sin2\theta_{23}\sin2\theta_{13}[/tex].

How is it that only functions of these parameters are left after Taylor expansion, and no linear or quadratic terms are left of say [tex]\theta_{13}[/tex]?

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Norman

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Dahaka14 said:

[tex]P_{\nu_{e}\rightarrow\nu_{\mu}}=\sin^{2}\theta_{23}\sin^{2}2\theta_{13}\sin^{2}\left(\frac{\Delta_{13}L}{2}\right)+\cos^{2}\theta_{23}\sin^{2}2\theta_{12}\sin^{2}\left(\Delta_{12}L}{2}\right)+\tilde{J}\cos\left(\pm\delta-\frac{\Delta_{13}L}{2}\right)\frac{\Delta_{12}L}{2}\sin\left(\frac{\Delta_{13}L}{2}\right)[/tex]

where [tex]\tilde{J}\equiv\cos\theta_{13}\sin2\theta_{12}\sin2\theta_{23}\sin2\theta_{13}[/tex].

How is it that only functions of these parameters are left after Taylor expansion, and no linear or quadratic terms are left of say [tex]\theta_{13}[/tex]?

Correct me if I am wrong, but is the first term in your equation for P, not quadratic in [tex]\theta_{13}[/tex]?

Have you worked through the derivation yourself? The devil is always in the details. Unless someone familiar with the problem can argue about the probability without doing the derivation.

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Dahaka14

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I double checked the expression, and it should be quadratic. Take a look at equation 7 on page 7 in the arxiv version http://arxiv.org/PS_cache/hep-ph/pdf/0002/0002108v3.pdf" . I took the inner product of the states and took absolute square, which has given me a large mess of terms. When I try to expand parts in [tex]\theta_{13}[/tex], I always end up with factors of [tex]\theta_{13}[/tex] and [tex]\sin^{2}2\theta_{13}[/tex], but there only appear to be factors of [tex]\sin^{2}2\theta_{13}[/tex] in their answer. There are other papers that perform similar approximations, and yet they never end up with things being multiplied by [tex]\theta_{13}[/tex].

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Expansion in a small parameter is a mathematical technique used to approximate complicated functions by breaking them down into simpler components. It involves expanding a function into a series of terms, where each term is multiplied by a small parameter that is assumed to be close to zero.

This technique is useful because it allows us to approximate complex functions with a much simpler form, making them easier to analyze and manipulate. It is commonly used in physics, engineering, and other fields to solve problems that would otherwise be too difficult to solve directly.

An example would be using Taylor series expansion to approximate a function in terms of a small variable, such as expanding *sin(x)* around *x=0* into *x* and higher order terms.

One limitation is that it only provides an approximation of the function, which may not be accurate for large values of the small parameter. Additionally, it may not be applicable to all functions, as some may not have a convergent series expansion.

Perturbation theory is a specific application of expansion in a small parameter, where the small parameter represents a small deviation from a known solution. It is commonly used to approximate solutions to equations that cannot be solved exactly, such as in quantum mechanics and celestial mechanics.

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