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README.md

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 # fritzsch-manuscript
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+Dynamic neutrino flavor mixing through transition magnetic moments
+
+## abstract
+We show that Majorana neutrino flavor mixing can be driven by transition dipole moments in the presence of external electromagnetic fields. We demonstrate the sensitivity of the rotation mixing matrix to strong fields obtaining effective mass eigenstates in the two-flavor model. Contribution to the book edited by Gerhard Buchalla, Dieter L\"ust
+and Zhi-Zhong Xing \textbf{dedicated to memory of Harald Fritzsch
+## authors
+<a
+id="cy-effective-orcid-url"
+class="underline"
+href="https://orcid.org/0000-0001-8217-1484"
+target="orcid.widget"
+rel="me noopener noreferrer"
+style="vertical-align: top"><img
+src="https://orcid.org/sites/default/files/images/orcid_16x16.png"
+style="width: 1em; margin-inline-start: 0.5em"
+alt="ORCID iD icon"/> Johann Rafelski</a>, <a
+id="cy-effective-orcid-url"
+class="underline"
+href="https://orcid.org/0000-0001-5474-2649"
+target="orcid.widget"
+rel="me noopener noreferrer"
+style="vertical-align: top"><img
+src="https://orcid.org/sites/default/files/images/orcid_16x16.png"
+style="width: 1em; margin-inline-start: 0.5em"
+alt="ORCID iD icon"/> Andrew Steinmetz</a>, and <a
+id="cy-effective-orcid-url"
+class="underline"
+href="https://orcid.org/0000-0001-5038-8427"
+target="orcid.widget"
+rel="me noopener noreferrer"
+style="vertical-align: top"><img
+src="https://orcid.org/sites/default/files/images/orcid_16x16.png"
+style="width: 1em; margin-inline-start: 0.5em"
+alt="ORCID iD icon"/> Cheng Tao Yang</a>
+
+## cite as
+Rafelski, J., Steinmetz, A., & Yang, C. T. Dynamic neutrino flavor mixing through transition magnetic moments. arXiv preprint arXiv:XXXX.XXXXX [hep-ph] (2023).
+
+## doi/arXiv id
+- TBD

Rafelski_Steinmetz_for_Harald.tex

@@ -55,7 +55,7 @@ \section{Introduction}
 
 Advancement of the understanding of neutrino physical properties attracts much interest today: Neutrinos are very abundant in the Universe, they were a dominant form of energy density in the Universe for much of its history, and they influence stellar and supernova evolution. As neutrinos are naturally massless in the standard model, the observed flavor oscillation signal non-vanishing neutrino mass. This suggests that neutrinos could provide a window to explore Beyond Standard Model (BSM) physics. This also motivates intense efforts to determine whether they are Dirac-type or Majorana-type fermions with the latter serving as their own antiparticle.
 
-We study the connection between Majorana neutrino transition magnetic dipole moments~\cite{Fujikawa:1980yx,Shrock:1980vy,Shrock:1982sc} and neutrino flavor oscillation. Neutrino electromagnetic (EM) properties have been considered before~\cite{Schechter:1981hw,Giunti:2014ixa,Chukhnova:2019oum,Popov:2019nkr} including the effect of oscillation~\cite{Pal:1991pm,Elizalde:2004mw}. The case of transition moments has the mathematical characteristics of an off-diagonal mass which is distinct from normal direct dipole moment behavior. EM field effects are also distinct from weak interaction remixing within matter, {\it i.e.\/} the Mikheyev-Smirnov-Wolfenstein effect~\citep{Wolfenstein:1977ue,Mikheyev:1985zog,Smirnov:2003da}.
+We study the connection between Majorana neutrino transition magnetic dipole moments~\cite{Fujikawa:1980yx,Shrock:1980vy,Shrock:1982sc} and neutrino flavor oscillation. Neutrino electromagnetic (EM) properties have been considered before~\cite{Schechter:1981hw,Giunti:2014ixa,Chukhnova:2019oum,Popov:2019nkr} including the effect of oscillation~\cite{Pal:1991pm,Elizalde:2004mw}. The case of transition moments has the mathematical characteristics of an off-diagonal mass which is distinct from normal direct dipole moment behavior. EM field effects are also distinct from weak interaction remixing within matter, {\it i.e.\/} the Mikheyev-Smirnov-Wolfenstein effect~\cite{Wolfenstein:1977ue,Mikheyev:1985zog,Smirnov:2003da}.
 
 We study for the case of two nearly degenerate neutrinos and their eigenstates in the presence of EM fields. We obtain an EM-mass basis, distinct from flavor, and free-particle mass basis, which mixes flavors as a function of EM fields. Moreover we show solutions relating to full EM field tensor which result in covariant expressions allowing for both magnetic and electrical fields we have not seen considered before. As neutrinos are electrically neutral, they have no intrinsic magnetic moment due to their spin, therefore any nonzero dipole moment is anomalous. 
 
@@ -460,6 +460,8 @@ \section{Toy model: EM-flavor mixing for two generations with a real Hermitian m
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Conclusions}
+\label{sec:conclusions}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 We have incorporated electromagnetic effects in the Majorana neutrino mixing matrix by introducing an anomalous transition magnetic dipole moment. We have described the formalism for three generations of neutrinos and explicitly explored the two generation case as a toy model. 
 
 In the two generation case, we determined the effect of electric and magnetic fields on flavor rotation in \req{zrot:2} by introducing an electromagnetic flavor unitary rotation $Z_{kj}^\mathrm{ext}$. We presented remixed mass eigenstates $\widetilde m(E,B)$ in \req{poly:3} which are the propagating mass-states in a background electromagnetic field $F^{\alpha\beta}_\mathrm{ext}(x^{\mu})$. These EM-mass eigenstates were also further split by spin aligned and anti-aligned states relative to the external field momentum density. There is much left to do to explore further the nascent connection between spin and flavor via transition magnetic moments. 
@@ -470,6 +472,8 @@ \section{Conclusions}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section*{Acknowledgements (by Johann Rafelski)}
+\label{sec:acknowledgements}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 Harald Fritzsch enjoyed the American Southwest. On the way between Caltech and the Santa Fe Institute, he sometimes made a stop at the half-way point: Tucson. On one such occasion he explored Arizona's mountains and deserts. Figure\,\ref{Fig:AZcolloq2007} shows our outing the morning after he gave a physics colloquium at the University of Arizona on March 23rd, 2007. Harald was an avid observer and photographer of the desert fauna and flora. 
 
 \begin{figure}%[hb]
@@ -483,8 +487,6 @@ \section*{Acknowledgements (by Johann Rafelski)}
 
 In Figure\,\ref{Fig:RANP2004} we see Harald's first transparency on ``Time Dependence of QCD and Experimental Tests'' which he presented at the 9th Hadron Physics and 7th Relativistic Aspects of Nuclear Physics (HADRON-RANP 2004) meeting. This meeting, which we both attended, was a joint event on QCD and Quark-Gluon Plasma in Rio de Janeiro, Brazil on March 28th - April 3rd, 2004~\cite{Fritzsch:2004civ}. Harald used a typed prepared lecture which arose from Ref.~(\citeauthor{Calmet:2001nu}). However, his actual talk included transparencies that were annotated with handwritten notes of his latest thoughts and insights.
 
-
-
 \begin{figure}%[ht]
 \centerline{\includegraphics[width=0.99\columnwidth]{04RANPHarald1Ed.jpg}}
 \caption{Harald begins his presentation in Rio de Janeiro 2004 about time dependence of QCD, see text for details. Picture by Johann Rafelski
@@ -493,11 +495,9 @@ \section*{Acknowledgements (by Johann Rafelski)}
 \end{figure}
 
 With Harald's passing I lost a friend of more than 40 years and, equally importantly, a colleague whose fast mind, willingness to listen, and clarity of thought, helped in some of my own challenges. I had very much wished to hear his opinion on this work.
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