# A Corpus-based Study of Temporal Signals

###### Abstract

Automatic temporal ordering of events described in discourse has been of great
interest in recent years. Event orderings are conveyed in text via various
linguistic mechanisms including the use of expressions such as “before”,
“after” or “during” that explicitly assert a temporal relation – temporal
signals. In this paper, we investigate the role of temporal signals in
temporal relation extraction and provide a quantitative analysis of these
expressions in the TimeBank annotated corpus.

A Corpus-based Study of Temporal Signals

Leon Derczynski and Robert Gaizauskas
---
University of Sheffield
211 Portobello, S1 4DP, UK
<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract content

##  1\. Introduction

The task of automatically determining the temporal relations that hold between
events described in a text is a research challenge that has increasingly
occupied researchers in computational language processing [Setzer and
Gaizauskas, 2000, Pustejovsky et al., 2004, Verhagen et al., 2009, Verhagen et
al., 2010]. The mechanisms used to convey temporal relational information in
text are complex and include tense, textual ordering, as well as specific
lexical cues; and of course readers and writers bring to bear lexical and
world knowledge, informing them of likely event sequences and inter-
relationships.

Of the mechanisms that play a part in conveying temporal relational
information, one that has been under-investigated is the use of expressions,
typically adverbials or conjunctions, which overtly signal temporal relations
– words or phrases such as after, during and as soon as. Very few of the teams
participating in the recent TempEval challenges [Verhagen et al., 2009,
Verhagen et al., 2010] exploited these words as features in their automated
temporal relation classification systems. Certainly no detailed study of these
words and their potential contribution to the task of temporal relation
detection has been carried out to date, despite their demonstrable utility
[Derczynski and Gaizauskas, 2010b]. This paper begins to address this
deficiency. Using the TimeBank corpus, a corpus of news wire texts annotated
with TimeML [Pustejovsky et al., 2003], in which a class of expressions
referred to as temporal signals is explicitly annotated, we set out to answer
the following questions:

  1. 1.

What proportion of temporal relations annotated in TimeBank have an associated
temporal signal? That is, are explicitly signalled using a signal word or
phrase?

  2. 2.

Of the expressions which can function as temporal signals, what proportion of
their usage in the TimeBank corpus is as a temporal signal? E.g. how ambiguous
are these expressions in terms of their role as temporal signals?

  3. 3.

Of the occurrences of these expressions as temporal signals, how ambiguous are
they with respect to the temporal relation they convey?

The following paper provides provisional answers to these questions –
provisional as one of the difficulties we encountered was significant under-
annotation of temporal signals in TimeBank. We have addressed this to some
extent, but more work remains to be done. Nonetheless we believe the current
study provides important insights into the behaviour of temporal signals and
how they may be exploited by computational systems carrying out the temporal
relation detection task.

The remainder of the paper is divided into three parts. In section two we give
a more detailed characterisation of temporal signals, further describe
TimeBank and TimeML and discuss prior related work. In section three we
describe the additional annotation work we have done on TimeBank and present
the quantitative analysis that provides answers to the questions framed above.
The fourth section considers, on a case by case basis, specific examples of
expressions which are highly ambiguous as regards their role as temporal
signals and discusses their behaviour in detail.

##  2\. Temporal Signals

###  2.1. Linguistic Characterisation

Signal expressions explicitly indicate the existence and nature of a temporal
relation between two events or states or between an event or state and a time
point or interval. Hence a temporal signal has two arguments, which are the
temporal ”entities” that are related. One of these arguments may be deictic
instead of directly attached to an event or time; anaphoric temporal
references are also permitted. For example, the temporal function and
arguments of _after_ in _He slept after a long day at work_ are clear and
available in the immediately surrounding text. With After that, he swiftly
finished his meal and left we must look back to the antecedent of that to
locate the second argument.

Sometimes a signal will appear to be missing an argument; for example,
sentence-initial signals with only one event in the sentence (_“Later, they
subsided.”_). These relate an event in their sentence with the discourse’s
current temporal focus – for example, document creation time, the previous
sentence’s main event, or reference time [Reichenbach, 1947, Dowty, 1979]. In
a more complex case, such as Example 2.1., we suggest that two temporal links
are present. First, _Later_ is attached to the current focus, as is
_surveyed_. Secondly, _after_ describes the relation between _the storm_ and
_surveyed_.

_It rained heavily. Later, after the storm, we surveyed the damage._

Sometimes a signal may appear to only take one argument, when the other is
(implicitly) reference time. For example, _afterwards_ and _after that_ are
temporally equivalent, though _afterwards_ only takes one extra argument.

Signal surface forms have a compound structure consisting of a head and an
optional qualifier. The head describes the temporal operation of the signal
phrase and the qualifier modifies or clarifies this operation. An example of
an unqualified signal expression is _after_ , which provides information about
the nature of a temporal link, but does not say anything about the absolute or
relative magnitude of the temporal separation of its arguments. We can
elaborate on this with phrases which give qualitative information about the
relative size of temporal separation between events (such as _very shortly
after_), or which give a specific separation between events using a duration
as a modifying phrase (e.g. _two weeks after_).

###  2.2. TimeML and TimeBank

TimeML [Pustejovsky et al., 2004] is a temporal annotation language. It may be
used to annotate events, time expressions or timex’s (times, dates,
durations), temporal relations between events and times (such as before or
during), and signal expressions – words or phrases (such as conjunctions,
adverbials) that provide information about temporal relations. TimeBank
[Pustejovsky et al., 2003] is currently the largest TimeML-annotated gold
standard corpus available, including over 6 000 temporal relation annotations,
as well as events, times and signals. It consists of around 65 000 tokens of
English newswire text.

TimeML offers the following definition of temporal signal. From the annotation
guidelines111See http://timeml.org/site/publications/timeMLdocs
/annguide_1.2.1.pdf .:

> A signal is a textual element that makes explicit the relation holding
> between two entities (timex and event, timex and timex, or event and event).
> Signals are generally:
>
>   * •
>
> Temporal prepositions: on, in, at, from, to, before, after, during, etc.
>
>   * •
>
> Temporal conjunctions: before, after, while, when, etc.
>
>   * •
>
> Prepositions signaling modality: to.
>
>   * •
>
> Special characters: “-” and “/”, in temporal expressions denoting ranges.
>
>

In cases where a specific duration occurs as part of a complex qualifier-head
temporal signal, e.g. two weeks after, TimeBank has followed the convention
that the signal head alone is annotated as a signal and the qualifier is
annotated as a TIMEX of type duration.

###  2.3. Previous Work

Signals help create well-structured discourse. Temporal signals can provide
context shifts and orderings [Hitzeman, 1997]. These signal expressions
therefore work as discourse segmentation markers [Ho-Dac and Péry-Woodley,
2008]. It has been shown that correctly including such explicit markers make
texts easier for human readers to process [Bestgen and Vonk, 1999].

Some prior work has approached linguistic characterisation of signals. Brée et
al. [Brée and Smit, 1986] performed a study of temporal conjunctions and
prepositions and suggested rules for discriminating temporal from non-temporal
uses of signal expressions that fall into these classes. However, this work is
purely theoretical and not a corpus-based study. Schlüter [Schlüter, 2001]
identifies signal expressions used with the present perfect and compares their
frequency in British and US English. Vlach [Vlach, 1993] presents a semantic
framework that deals with duratives when used as signal modifiers (see Section
2.1.). Brée et al. [Brée et al., 1993] later describe the ambiguity of nine
temporal prepositions in terms of their roles as temporal signals. Our work
differs from the literature in that is it the first to be based on gold
standard annotations of temporal semantics and that it encompasses all
temporal signal expressions, not just those of a particular grammatical class.

Intuitively, signal expressions contain temporal ordering information that
human readers can access easily. Once temporal conjunctions are identified,
existing semantic formalisms may be applied to discourse semantics [Dowty,
1979]. It is however ambiguous which temporal expression they attempt to
convey [Hitzeman, 2005]. Our work quantifies this ambiguity for a subset of
expressions.

Previous work applying temporal signals has been related to the labeling of
temporal links [Min et al., 2007] and question answering [Pustejovsky et al.,
2005, Saquete et al., 2009]. In particular, Lapata and Lascarides [Lapata and
Lascarides, 2006] remove the temporal signal from sentences containing two
temporally connected clauses and attempt to learn sentence-level temporal
relations using the orderings suggested by the removed signal as training
data. Directly applying signals to the temporal relation identification task,
Derczynski and Gaizauskas [Derczynski and Gaizauskas, 2010b] halved the error
rate of TLINK classification for TLINKs that have a signal by adding features
describing signals. This raised classification accuracy from 62% to 82%.

##  3\. Signals in TimeBank

In this section, we give a detailed profiling of temporal signals in the
TimeBank corpus. Statistics are generated using the CAVaT [Derczynski and
Gaizauskas, 2010a] tool for TimeML-annotated corpus analysis.

First, we note that in TimeML signals may be divided into three classes based
on the type of relation they signal: temporal (tlink), sub-ordinating (slink)
or aspectual (alink). The distribution of signals by class in Timebank is
shown in Table 1. For the rest of the paper we discuss temporal signals only.

Annotated SIGNAL elements | 758
---|---
Signals used by a TLINK | 721
Signals used by an ALINK | 1
Signals used by a SLINK | 39
TLINKs that use a SIGNAL | 787
Signals used by more than one TLINK | 54
Table 1: How <SIGNAL> elements are used in TimeBank. TLINKs per signal | Number of signals
---|---
1 | 597
2 | 41
3 | 12
5 | 1
Table 2: The number of TLINKs associated with each temporal signal word/phrase, in TimeBank. Signals not used on TLINKs (e.g. those used on aspectual or subordinate links, or for event cardinality) are excluded. The distribution is Zipfian. Part of speech | Frequency | Proportion
---|---|---
IN | 521 | 77.3%
RB | 73 | 10.8%
WRB | 53 | 7.9%
JJ | 14 | 2.1%
RBR | 5 | 0.7%
VBG | 4 | 0.6%
CC | 2 | 0.3%
RP | 1 | 0.1%
JJR | 1 | 0.1%
Table 3: Distribution of part-of-speech in signals and the first word of multiword signals, using the Penn Treebank tag set. Expression | Count in corpus | As signal | Proportion as signals | After curation | Proportion
---|---|---|---|---|---
in | 1214 | 161 | 13.3% |  |
after | 72 | 56 | 77.8% | 66 | 91.7%
for | 621 | 52 | 8.4% |  |
if | 65 | 37 | 56.9% |  |
when | 62 | 35 | 56.5% | 56 | 90.3%
on | 344 | 33 | 9.6% |  |
until | 36 | 25 | 69.4% | 36 | 100.0%
before | 33 | 23 | 69.7% | 30 | 90.9%
by | 356 | 20 | 5.6% |  |
from | 366 | 19 | 5.2% |  |
since | 31 | 17 | 54.8% | 18 | 58.1%
through | 69 | 15 | 21.7% |  |
as | 271 | 14 | 5.2% |  |
over | 59 | 14 | 23.7% |  |
already | 32 | 13 | 40.6% | 13 | 40.6%
ended | 21 | 13 | 61.9% |  |
during | 19 | 13 | 68.4% |  |
at | 311 | 11 | 3.5% |  |
previously | 19 | 11 | 57.9% | 16 | 84.2%
within | 23 | 8 | 34.8% |  |
s | 10 | 8 | 80.0% |  |
later | 15 | 7 | 46.7% |  |
earlier | 50 | 6 | 12.0% |  |
while | 39 | 6 | 15.4% | 9 | 23.1%
then | 23 | 5 | 21.7% |  |
once | 15 | 5 | 33.3% |  |
still | 35 | 4 | 11.4% |  |
following | 15 | 4 | 26.7% |  |
meanwhile | 14 | 4 | 28.6% | 9 | 64.3%
at the same time | 6 | 4 | 66.7% |  |
to | 1600 | 3 | 0.2% |  |
into | 63 | 3 | 4.8% |  |
follows | 4 | 3 | 75.0% |  |
subsequently | 3 | 3 | 100.0% |  |
followed | 10 | 2 | 20.0% | 4 | 40.0%
former | 16 | 0 | 0.0% | 12 | 75.0%
Table 4: Frequency of candidate signal expressions in TimeBank. We include counts of how often these occur as signal expressions both before and after manual curation. Signal Expression | TLINK count |  after |  before |  begins |  begun_by |  during |  ended_by |  ends |  iafter |  ibefore |  includes |  is_included |  simultaneous
---|---|---|---|---|---|---|---|---|---|---|---|---|---
after | 76 | 62 | 3 | 4 |  |  |  | 5 | 2 |  |  |  |
when | 57 | 16 | 3 | 1 | 2 | 1 |  |  | 1 | 1 | 9 | 9 | 14
until | 37 | 4 | 7 | 1 |  |  | 21 | 1 | 1 | 2 |  |  |
before | 36 | 1 | 28 | 2 |  |  | 1 | 2 | 1 | 1 |  |  |
since | 19 | 9 | 1 | 2 | 7 |  |  |  |  |  |  |  |
already | 13 |  | 6 |  |  |  |  |  |  |  | 4 | 3 |
previously | 18 | 6 | 12 |  |  |  |  |  |  |  |  |  |
while | 9 |  |  |  |  |  |  |  |  |  |  |  | 9
meanwhile | 9 |  | 1 |  |  | 2 |  |  |  |  |  | 1 | 5
followed | 4 | 2 | 2 |  |  |  |  |  |  |  |  |  |
former | 12 |  | 12 |  |  |  |  |  |  |  |  |  |
Table 5: Signal expressions and the TimeML relations that they can denote,
ordered as per Table 4 for comparison. Counts do not match because a single
signal expression can support more than one temporal link.

###  3.1. Additional Annotation

Upon examination of the non-annotated instances of words that often occur as a
temporal signal (such as _after_) it became evident that TimeBank’s signals
are under-annotated. As we are certain of some annotation errors in the source
data, we revisited the original annotations. A subset of signal words was
selected for re-annotation. This set consisted of signals that were ambiguous
(occurred temporally close to 50% of the time) or that we expected contained,
based on informal observations, would yield a number of missed temporal
annotations. All temporal instances of these words were re-annotated with
TimeML, adding EVENTs, TIMEX3s and TLINKs where necessary to create a
signalled TLINK.

A single annotator checked the source documents and annotated 70 extra
signals, as well as adding 34 events, 1 temporal expression and 49 extra
temporal links.

###  3.2. Proportion of Temporal Relations with Signals

TimeBank contains 6 418 TLINKs (6 467 after re-annotation) of which 718 (787)
are explicitly indicated by a temporal signal – 11.2% (12.2%). This provides
an answer to the first question we posed in Section 1. Thus while ability to
successfully detect temporal signals will not solve the problem of assigning
temporal relations, it is likely to make a noticeable difference (see
Derczynski and Gaizauskas [Derczynski and Gaizauskas, 2010b]). Perhaps of more
interest is that so few temporal relations are explicitly signalled – we must
look elsewhere for an explanations of how temporal relations are conveyed in
natural language.

While many TLINKs do not have any associated temporal signal it is also the
case that some temporal signals are associated with more than one TLINK. Table
2 shows details of just how signals are being used by TLINKs.

###  3.3. Temporal vs Non-temporal Uses

The semantic function that a temporal signal expression performs is that of
relating two temporal entities. However, the words that can function as
temporal signals also play other roles.

Table 4 details the distribution of expressions that are found as temporal
signals more than twice (after re-annotation) in TimeBank. The most frequent
signal word was “in”, accounting for 24.8% of all signal-using TLINKs.
However, only 13.3% of occurrences of the word “in” have a temporal sense. The
word “after” is far more likely (91.7% of all occurrences) to have a temporal
sense. In total TimeBank contains 62 unique signal words and phrases (ignoring
case) and of these, over half (36) are also found in Table 4 .

As an aside, note that any thought that temporal signals might be easily
picked out based on word class may be dispelled by examining the distribution
of parts-of-speech possessed by temporal signals – see Table 3.

###  3.4. Relation Ambiguity

The nature of the temporal relation described by a signal is not constant,
though each signal tends to describe a particular relation type most often.
Table 5 gives an excerpt of data showing which temporal relations are made
explicit by each signal expression. The variation in relation type associated
with a signal is not as great as it might appear as the assignment of temporal
relation type has an element of arbitrariness – one may choose to annotate a
before or after relation for the same event pair by simply reversing the
temporal link’s argument order, for example. Nevertheless, it is possible to
draw useful information from the table; for example, one can see that
_meanwhile_ is much more likely to suggest some sort of temporal overlap
between events than an ordering where arguments occur discretely.

##  4\. Per-expression details

We chose to curate signal annotations in TimeBank for a subset of candidate
signal expressions (as described in Section 3.1.). During this curation, we
attempted to determine distinguishing features that could aid automatic
discrimination of temporal from non-temporal sense of the expressions. Details
of our findings are given below.

#### Previously

TimeBank contains eight instances of the word that were not annotated as a
signal. Of these, all were being used as temporal signals. The word only takes
one event or time as its direct argument, which is placed temporally before an
event or time that is in focus. For example:

_“X reported a third-quarter loss, citing a previously announced capital
restructuring program”_

In this sentence, the second argument of _previously_ is _“announced”_ , which
is temporally situated before its first argument (_“reported”_). When
_previously_ occurs at the top of a section, the temporal element that has
focus is either document creation time or, if one has been specified in
previous discourse, the time currently in focus.

#### After

Of the nineteen instances of this word not annotated as temporal, only three
were actually non-temporal. The cases that were non-temporal were a different
sense of the word. The temporal signals are adverbial, with a temporal
function. Two non-temporal cases used a positional sense. The last case was in
a phrasal verb _to go after_ ; _“whether we would go after attorney’s fees”_.

#### When

There are 35 annotated and 27 non-annotated occurrences of this phrase. It
indicates either an overlap between intervals, or a point relation that
matches an interval’s start. Twenty-three of the twenty-seven non-annotated
occurrences are used as temporal signals. Two of the remaining four are in
negated phrases and not used to link an interval pair. for example, _“did not
say when the reported attempt occurred”_. The other two are used in context
setting phrases, e.g. _“we think he is someone who is capable of rational
judgements when it comes to power”_, which are not temporal in nature.

#### While

The cases of _while_ that have not been annotated as a signal – the majority
class, 33 to 6 – are often used in a contrastive sense. This does suggest that
the connected events have some overlap, often between statives. For example,
_“But while the two Slavic neighbours see themselves as natural partners,
their relations since the breakup of the Soviet Union have been bedeviled”_.
As two states described in the same sentences are likely to temporally overlap
and any events or times outside or bounding these states will be related to
the state, it is unlikely that any contribution to TLINK annotation would be
made by linking the two states with a “roughly simultaneous” relation; the
closest suitable label is TempEval’s overlap relation [Verhagen et al., 2010].

_“nor can the government easily back down on promised protection for a
privatized company while it proceeds with …”_

The cases of _while_ that were not of this sense were easier to annotate.
Sometimes it was used as a temporal expression; _“for a while”_. Other times,
it was not used in a contrastive sense, but instead as irrealis – see Example
4.. The four cases of non-contrastive usage were annotated as temporal
signals.

[.PP [.IN before ] [.NP [.Det the ] [.NNS wars ] ] ]

Figure 1: An example of the common syntactic surroundings of a _before_
signal.

#### Before

Three of the ten negative examples are correctly annotated. They are _before_
in the spatial sense of “in front of” (as in _“The procedures are to go before
the Security Council next week”_) and also a logical before that does not link
instantiated or specific events (_“ before taxes”_). The remaining seven
unannotated examples of the word are all temporal signals. These directly
precede either an NP describing a nominalised event, or directly precede a
subordinate clause (e.g. [IN before, S] – see Figure 1).

#### Until

All eleven non-annotated instances of _until_ should have been annotated as
temporal signals. This word suggests a TimeML ibefore relation, unless
qualified otherwise by something like “not until” or “at least until”.

#### Already

There were thirteen positive examples of _already_. All of the non-annotated
examples had a non-temporal sense as per our description of temporal signals.
The word tends to be used for emphasis, but can also suggest a broad “before
DCT” position, which goes without saying for any past and present tensed
events. As _already_ can be removed without changing the temporal links
present in a sentence, we have not annotated any more examples of this beyond
the thirteen present in TimeBank.

#### Meanwhile

This word tends to refer to a reference or event time introduced earlier in
discourse, often from the same sentence. As well as a temporal sense, it can
have a contrastive “despite”-like meaning. _Meanwhile_ tends to refer more to
previous actions, instead of states specified in immediately prior sentences.
Sometimes _meanwhile_ is used with no previous temporal reference. In these
cases, the implicit argument is DCT. Five of the ten non-annotated _meanwhile_
s were temporal signals.

#### Again

This word shows recurrence and is always used for this purpose where it occurs
in TimeBank not annotated as a temporal signal. No instances of _“again”_ were
annotated.

[.NX [.NX [.NN founder ] ] [.CC and ] [.NX [.JJ former ] [.NN chairman ] ] ]

Figure 2: Example of a non-annotated signal (_former_) from TimeBank’s
wsj_0778.tml.

#### Former

This word indicates a state that persisted before DCT or current speech time
and has now finished. Generally the construction that is found is an NP, which
contains an optional determiner, followed by _former_ and then a substituent
NP which may be annotated as an EVENT of class state. This configuration
suggests a TLINK that places the event before the state’s utterance.

_“The San Francisco sewage plant was named in honour of former President
Bush.”_

In Example 4., there is a state-class event – _President_ – that at one time
has applied to the named entity _Bush_. The signal expression _former_
indicates that this state terminated before the time of the sentence’s
utterance.

Three-quarters of the non-annotated instances of _former_ in TimeBank are
temporal signals.

#### Recently

Although _recently_ is a temporal adverb, it can only be to applied simple or
anterior tensed verbs (using Reichenbach’s tense nomenclature). In the corpus,
these are only seen in reported speech or of verbal events that happened
before DCT. _Recently_ adds a qualitative distance between event and utterance
time, but is of reduced use when we can already use tense information. The
phrase “Until recently” appears awkward when cast as a temporal signal but can
be interpreted as “before DCT”, with the interval’s endpoint being close to
DCT. In this case, recently functions as a temporal expression, not a signal.
Only one of the non-annotated _recently_ s in TimeBank is a temporal signals.
The exception, _“More recently”_ , includes a comparative and is annotated as
a TIMEX3; both this phrase and, e.g., _“less recently”_ suggest a relation to
a previously-mentioned (and in-focus) past event. As a result, we posit that
_recently_ behaves as an abstract temporal point (as seen in the behaviour of
_“until recently”_). Structures such as _[comparative] recently_ may be
interpreted as a qualified temporal signal, as they convey information about
the relative ordering of the event that they dominate vent compared with a
previously mentioned interval.

##  5\. Conclusion

We have provided a characterisation of temporal signal expressions. In an
analysis of the TimeBank corpus we have shown what proportion of temporal
relations are explicitly signalled by these expressions and have given
quantitative descriptions of how ambiguous these phrases are, both regarding
their temporal/non-temporal senses and the type of temporal relation that they
convey.

###  5.1. Acknowledgments

The first author would like to acknowledge the UK Engineering and Physical
Science Research Council’s support in the form of a doctoral studentship.

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# Relic Neutrino Helicity Evolution in Galactic Magnetic Field and Its
Implications

Kuo(i:to).Liao<EMAIL_ADDRESS>Center for Cosmology and Particle Physics,
Department of Physics, New York University  Glennys R. Farrar<EMAIL_ADDRESS>Center for Cosmology and Particle Physics, Department of Physics, New York
University

###### Abstract

We simulate the evolution of the helicity of relic neutrinos as they propagate
to Earth through a realistic model of the Galactic magnetic field, improving
upon the rough estimates in the pioneering work of Baym and Peng. We find that
with magnetic moments consistent with experimental bounds and even several
orders of magnitude smaller, the helicity of relic neutrinos rotates with a
substantial directional anisotropy. Averaged over directions this would simply
reduce the apparent flux; if the direction of the incident neutrino could be
measured, the directional anisotropy in the interaction probability could
become a powerful diagnostic. We study the effects of $\nu$ spin rotation on
C$\nu$B detection through the inverse tritium decay process.

Suggested keywords

††preprint: APS/123-QED

##  I Introduction

Relic neutrinos carry information from the early Universe. They decoupled from
the hot plasma about 1 second after the big bang, much earlier than the cosmic
microwave background decoupling. They are decoupled in chirality eigenstates
at temperatures $\simeq$ MeV. When decoupled, they were highly relativistic,
and their helicity eigenstates coincided with chiral eigenstates. Previous
research [1][2] studied the effects of astrophysical magnetic field and
gravitational inhomogeneities on the helicities of present-day relic
neutrinos. The helicity eigenstates of both Dirac and Majorana neutrinos are
modified by gravitational effects, with finite mass, and for Dirac neutrinos
helicity modifications are also induced by magnetic moments during propagation
through astrophysical magnetic fields.

Inverse tritium beta decay [3] (ITBD) is a promising experimental tool [4] to
observe relic neutrinos. Since the ITBD cross-section depends on helicity, in
principle ITBD measurements can provide a tool to detect helicity
modification. Moreover, with a polarized target, directional anisotropies are
in principle accessible [5] thus detection of a helicity modification which
depends on the neutrino arrival direction – due in our case to the Galactic
magnetic field having a non-trivial structure – may one day be possible to
explore observationally.

Our understanding of the Galactic magnetic field developed over the decades.
Today the Jasson and Farrar model (JF12) [6] fits the largest range of
observations and should be a good semi-realistic model to study neutrino
helicity evolution. Here we focus on Dirac neutrinos with non-zero diagonal
magnetic moments.

##  II Theoretical Background

###  II.1 Relic Neutrino Spin Rotation in Astrophysical Magnetic Field

Neutrinos are produced in flavor eigenstates, but they arrive at Earth in
well-separated mass packets. The three flavors are superpositions of mass
eigenstates via the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix:

$\displaystyle\begin{bmatrix}\nu_{e}\\\ \nu_{\mu}\\\
\nu_{\tau}\end{bmatrix}=\begin{bmatrix}U_{e1}&U_{e2}&U_{e3}\\\ U_{\mu
1}&U_{\mu 2}&U_{\mu 3}\\\ U_{\tau 1}&U_{\tau 2}&U_{\tau
3}\end{bmatrix}\begin{bmatrix}\nu_{1}\\\ \nu_{2}\\\ \nu_{3}\end{bmatrix}.$ (1)

While coupled to ambient plasma, relic neutrinos had a relativistic thermal
distribution at temperature $T$. As momenta and temperature redshift
identically with the expansion of the Universe, the Fermi-Dirac distribution
is preserved even after the decoupling process, so the present momenta
$\vec{p}_{0}$ and temperature $T_{0}$, for flavor $\alpha$:

$f_{\alpha}(\vec{p_{0}},T_{0})=\frac{1}{e^{|\vec{p_{0}}|/T_{0}}+1},$ (2)

where present cosmic neutrino background temperature $T_{0}$ $\simeq$ 1.67
$\times$ $10^{-4}$ eV = $(\frac{4}{11})^{1/3}$ $T_{CMB}$.

Propagating in a magnetic field, the neutrino spin vector rotates in its rest
frame [1][2]:

$\frac{d\vec{S}}{d\tau}=2{\mathbf{\mu}}_{\nu}(\vec{S}\times{\vec{B}}),$ (3)

where $\vec{B}$ represents the magnetic field vector in its rest frame, $\tau$
is the neutrino proper time, ${\mathbf{\mu}}_{\nu}$ is the diagonal magnetic
moment of the Dirac neutrino, expressed in Bohr magnetons,
$\mu_{B}=1.4$MHz/Gauss. In the (extended) standard model [7]:

$\mu^{SM}_{\nu}\approx\frac{3G_{F}}{4\sqrt{2}\pi^{2}}m_{\nu}m_{e}\mu_{B}\approx
3\times 10^{-21}\frac{m_{\nu}}{0.01eV}\mu_{B},$ (4)

this is approximately $10^{-9}$ lower than the present experimental upper
limit result from the GEMMA [8] Reactor and Borexino Phase-II [9] $\simeq$
2.9$\times$$10^{-11}$ $\times$ $\mu_{B}$ 111We also revisited the
supersymmetric contribution to the neutrino magnetic moment of massive
neutrino. Based on the updated LHC limit and the supersymmetry pattern [10],
we found that the SUSY value cannot be larger than the contribution of the
standard model result.. The neutrino lab frame time $dt$ = $\gamma$ d$\tau$,
with the relativistic factor $\gamma$ $\approx$1 for a relic neutrino with a
velocity much smaller than the speed of light. $\vec{S}$ interacts with the
magnetic field and deviates with respect to its initial orientation. The spin
vector $\vec{S}$ will precess around the local $\vec{B}$ as the neutrino
propagates through an astrophysical magnetic field in the lab frame:

$\Delta\vec{S}=2{\mathbf{\mu}}_{\nu}\int{\vec{S}\times\vec{B}}\frac{dt}{\gamma}.$
(5)

The $\vec{S}$ evolution in terms of path dependence is

$\frac{\partial\vec{S}}{\partial\vec{r}}=\frac{1}{|\vec{v}|}\frac{\partial\vec{S}}{\partial{t}}=\frac{2}{\gamma}\frac{{\mathbf{\mu}}_{\nu}}{|\vec{v}|}{\vec{S}\times\vec{B}}.$
(6)

$\vec{S}_{final}$ can be calculated by integrating over the total propagation
path length. Assuming that the $\vec{P}$ direction does not change during
propagation through the Galaxy, the $\nu$ helicity evolution is well-justified
according to [1]. The probability of helicity flipping can be expressed as a
function of the rotation angle $\theta$: $P_{f}=sin^{2}(\theta/2)$.

In the lab frame, with respect to the neutrino’s lab frame velocity
$\vec{B}_{\perp}^{restframe}$=$\gamma$$\vec{B}_{\perp}^{labframe}$ and
$\vec{B}_{\parallel}^{labframe}$=$\vec{B}_{\parallel}^{restframe}$, so in
terms of lab frame magnetic field and time [1]:

$\frac{d\vec{S}_{\perp}}{dt}=(\vec{S}_{\parallel}\times\vec{B}_{\perp}^{labframe}+\frac{1}{\gamma}\vec{S}_{\perp}\times\vec{B}_{\parallel}^{labframe})2{\mu_{\nu}}.$
(7)

The cumulative rotation angle $\theta$ at a specific moment compared to
neutrino’s initial $\vec{S}$ state is
$\theta=sin^{-1}({\vec{S}_{\perp}/\vec{S}})$. When $\theta\ll 1$,
$\vec{S}_{\perp}\approx 0$, the second term can be neglected in the
computation process: Baym-Peng makes this appoximation but we do not.

###  II.2 Simulation Method

In this section, we summarize the essential elements of the simulation
procedure.

Following the thermal distribution and its relationship with the neutrino
total number density $n$= $\frac{1}{(2\pi)^{3}}$$\int$
f($\vec{p_{0}}$,$T_{0}$) $d^{3}\vec{p_{0}}$ $\approx$ 56.25$cm^{-3}$ we
generate velocity samples for non-relativistic cases based on:

$1=\frac{1}{n}\frac{1}{(2\pi)^{3}}\int\frac{1}{e^{m|\vec{v}|/T_{0}}+1}4\pi
m^{3}v^{2}dv.$ (8)

The $Astropy.units$ function is imported to track the conversion of units
better.

Throughout the simulation of each neutrino trajectory, we assume that the
$\nu$ momentum direction is fixed in the laboratory frame. Neutrino’s helicity
change depends purely on the $\vec{S}$ variation during propagation through
the astrophysical magnetic field. The JF12 Coherent Galactic Magnetic Field
Model and the coherent model with random components are used in the simulation
[6]. The coherent model includes the striated components, poloidal component,
toroidal halo, and spiral arm disk components. Along the galactic disk, $>$5
kpc are 8 spiral arms. Between 3-5kpc is the purely azimuthal molecular ring.
The magnetic field strength is set as zero at r$<$1kpc and r$>$20kpc. Field
data are presented in Cartesian Coordinates ranging [-20Kpc, 20Kpc], in units
of $[\mu G]$, where the galaxy center coordinate is [0,0,0].

Given the magnetic field model, we apply the Riemann sum to compute the spin
vector evolution

$\vec{S}_{N}-\vec{S}_{N-1}=2\frac{\mu_{\nu}}{|\vec{v}|}\times(\vec{S}_{N-1}\times\vec{B}_{N-1})\delta
r,$ (9)

where $\delta{r}$ = $R_{total}$/$N_{sample}$; $R_{total}$ is the propagation
path length of a test neutrino coming from a known direction and N is the
number of steps sampled.

The incoming directions of relic neutrinos are generated as pixelated data.
Using the Hierarchical Equal Area isoLatitude Pixelization (HEALPix) library
and the Python package $healpy$, we plot values on a skymap with high
resolution. The HEALPix resolution parameter (scalar integer) is set to 12288
pixels =$12\times N_{side}$; function $healpy.nside2npix$ from the package
returns pixel number at $N_{side}$=32. Every pixel on the map represents a
specific direction with latitude and longitude. For our purpose, each
neutrino’s incoming direction in the simulation is identical to each pixel on
the mollweide view projection map. The area of each pixel is
$\Omega_{pixel}$=$\pi/3(N_{side})^{2}$ and angular resolution is
$(\Omega_{pixel})^{1/2}$ = 1.83 degrees.

We propagate $\nu$ at each pixel outward from Earth to a distance where the
local field vanishes and far away from the galactic center, then we reverse
the neutrino momentum vector and propagate the neutrino inward starting with
relic helicity eigenstate -1. Along the propagation inward we track the spin
rotation of $\nu$. The Riemann sum process yields the final direction of
$\vec{S}$; the angle difference between $\vec{S}_{final}$ and
$\vec{S}_{initial}$ is calculated and the expectation values of rotation for
each pixel are computed based on simulation results at different velocity
samples. Attempted magnetic moment values $\mu_{\nu}$ are the GEMMA reactor
experimental upper limit $\mu_{\nu}$$<$ 2.9 $\times$ $10^{-11}$ $\mu_{B}$ and
$10^{-1}$, $10^{-2}$ and $10^{-3}$ times smaller cases. The $\nu$ mass is
assigned to be 0.1eV during the simulation; for this mass, refer to Equation
(2) to have the $\nu$ velocity.

We note that for non-relativistic neutrinos, $v/c<<1$, the spin rotation is
linear in $m_{\nu}$. This is because the spin rotation scales linearly in the
time spent traversing the field, hence inversely with the velocity. But as
noted above, the relic neutrino momentum distribution is fixed by cosmology so
$v_{\nu}\sim p_{0}/m_{\nu}$.

##  III Discussion of the results

We use Healpy Projview and molleweide sky-view visualization to show the all-
sky directional anisotropy of relic neutrino spin rotation and helicity-flip.
In both FIG.1 and FIG. 2 Galactic center is $\sim$ 8kpc away from the earth
with coordinates being longitude = 0 and colatitude = 0.

FIG. 1 shows the spin rotation angles (modulo $\pi$) of a 0.1 eV neutrino in
the JF12 Galactic magnetic field model as a function of arrival direction, for
four different choices of magnetic moment, scaling the GEMMA Reactor
experimental bound by 1, 0.1, 0.01 and 0.001. For an incoming neutrino at the
upper bound value of magnetic moment from the GEMMA Reactor experiment, the
total deviation angles between $\vec{S}_{final}$ and $\vec{v}$
($\vec{S}_{initial}$) range from $\simeq$ 0.792 to $\simeq$ 2.061 radians in
the coherent field model and $\simeq$ 1.27 to $\simeq$ 1.82 radians in the
field model including the random field. In fact, the spin rotates multiple
times for many arrival directions when the moment is within two orders of
magnitude of the GEMMA limit. However, at $10^{-3}$ $\times$
$\mu^{GEMMA}_{\nu}$, the rotation angles are in general less than $\pi/2$.

FIG. 2 shows the helicity flip probability, in the Molleweide view. A large-
scale directional anisotropy pattern is visible in the coherent field model
for magnetic moments $10^{-1}$ or less than the experimental magnetic moment
upper bound, with the high helicity-flip probability directions pointing
toward the galactic center. However, for the more realistic GMF model
including the random components, large-scale regions with higher flip
probability start to be observable at $10^{-2}$ $\times$ $\mu_{\nu}^{GEMMA}$.
For larger magnetic moments the rotation is so large that the neutrino is in a
nearly equal superposition of helicity eigenstates over most of the sky. The
angular power spectrum is shown in FIG. 3 to quantify how the angular scale of
the flip-probability anisotropy depends on magnetic moment.

Both field model results confirm that regions with longer propagation paths
and traversing regions of the stronger fields have a higher helicity-flip
probability – hence the noticeable difference in the Galactic and antigalactic
center directions, which could be depicted in FIG.1, the center of Molleweide
plots represents the direction of Galactic direction

At the same time, the variation of the magnetic field intensity and the
direction of the magnetic field vector $\vec{B}$ among different regions along
the trajectory affects the helicity-flip probability results.

(a)  (b)  (c)  (d)  (e)  (f)  (g)  (h)

Figure 1: Spin vector rotation angle of a 0.1 eV relic neutrino traversing
(first row) the JF12 coherent Galactic magnetic field and (second row) the
full coherent plus random field. From left to right the columns are for a
magnetic moment equal to the GEMMA Reactor experiment bound
$\mu_{\nu}^{GEMMA}\approx 2.9\times 10^{-11}\mu_{B}$ and factor-10 smaller
values: 0.1$\mu_{\nu}^{GEMMA}$, 0.01$\mu_{\nu}^{GEMMA}$, and
0.001$\mu_{\nu}^{GEMMA}$. $Healpy$ $Projview$ polar view has been used for
sky-view mapping, with the coordinates being longitude from 0 to 2$\pi$ and
colatitude from -$\pi/2$ to $\pi$/2.

(a)  (b)  (c)  (d)  (e)  (f)  (g)  (h)

Figure 2: Relic Neutrino Helicity-Flip Probability in Molleweide view, for the
same cases shown in FIG. 1. (first row) the JF12 coherent Galactic magnetic
field and (second row) the full coherent plus random field.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 3: Relic Neutrino Helicity-Flip Probability angular power spectrum for
scales of GEMMA Reactor experiment bounds $\mu_{\nu}^{GEMMA}\approx 2.9\times
10^{-11}\mu_{B}$, 0.1$\mu_{\nu}^{GEMMA}$, 0.01$\mu_{\nu}^{GEMMA}$,
0.001$\mu_{\nu}^{GEMMA}$(a)-(d) simulation results in JF-12 Coherent Galactic
Magnetic Filed Model, (e)-(f) simulation results in JF12 Coherent Galactic
Magnetic Filed Plus random components Model; magnetic moment applied are
$10^{-1}$ smaller in sequence for each model Figure 4: The sky-averaged
helicity flip probability in the JF12 Coherent and Coherent + Random Magnetic
Field models, for $m_{\nu}$=0.1 eV (green and red data points, respectively,
with RMS spread on the sky shown as error bar), overlaid on the prediction
using Eq. 10. For $m_{\mu}$ $\gtrsim$ $10^{-4}$ times the GEMMA limit, the
oscillation in the Baym-Peng model is so rapidly varying, we show only the
average value $\frac{1}{2}$.

##  IV Comparison with earlier work

The Baym-Peng treatment [2] took the Galactic magnetic field to be entirely
random with magnitude $B_{g}$ $\simeq$ 10 $\mu$Gauss, with a coherence length
$\Lambda_{g}$ $\simeq$ kpc within a mean crossing distance of the galaxy of
$l_{g}$$\simeq$ 16 kpc. In a uniform random field the spin vector of a
neutrino undergoes a random walk in the field during propagation, generating a
mean-square spin rotation [2]

$<\theta^{2}>\simeq(2\mu_{\nu}B_{g}\frac{\Lambda_{g}}{v})^{2}\frac{l_{g}}{\Lambda_{g}}$
(10)

For ($\mu_{\nu}$$B_{g}$$\frac{\Lambda_{g}}{v}$) of order unity and
sufficiently large magnetic moment, this predicts cumulative rotation angles
large enough for significant $\nu$ helicity-flip. For $m_{\nu}=0.1$ eV, their
predicted spin rotation is generically large and the flip probability is of
order one, as long as $\mu_{\nu}\gtrsim 2\times 10^{-6}$ $\mu^{GEMMA}_{\nu}$,
as shown in FIG. 4. By contrast, we find the maximum sky-averaged flip
probability is about 20%, and becomes very small for $\mu_{\nu}\leq 10^{-3}$
$\mu^{GEMMA}_{\nu}$.

Much of the discrepancy is due to the unrealistic field parameters assumed in
[2]. Averaging over the 20 kpc sphere centered on the Galactic center in which
the Jansson and Farrar (2012) model is given, the average path length to Earth
is 24.4 kpc and the average value of $|B_{\perp}|$ is $0.104\,\mu$G.
(Histograms of $|B_{\perp}|$ and $l_{g}$ for all the lines of sight are shown
in Fig. 5.) Moreover, the coherence length of the random component of the GMF
is typically estimated to be 10 to 100 pc; for our calculations we took it to
be 30 pc. Using our more realistic magnetic field properties in the Baym-Peng
expression Eq. (10), decreases the estimated $<\theta^{2}>$ by a factor $\sim
5\times 10^{-6}$.

(a)

(b)

Figure 5: Histograms of (a) neutrino propagation length $l_{g}$ and (b)
$|B_{\perp}|$ for all the lines of sight in JF 12 coherent plus random
magnetic field model.

##  V Effects of helicity flip on the detection of Relic Neutrinos through
Inverse Tritium Beta Decay

A proposed experimental method for detecting relic neutrinos is based on
neutrino capture through the inverse tritium beta decay process [3] $\nu_{j}$
+${}^{3}H$ $\rightarrow$ ${}^{3}He$ $+$ $e^{-}$.

The total cross-section multiplied by the neutrino velocity for capturing
$\nu_{e}$ [11] is

$\displaystyle\sigma^{h}(E_{\nu})v={}$
$\displaystyle\frac{G_{F}^{2}}{2\pi}|V_{ud}|^{2}F(Z,E_{e})\frac{m_{{}^{3}He}}{m_{{}^{3}H}}E_{e}P_{e}\times$
(11) $\displaystyle(\langle f_{F}\rangle^{2}+(g_{A}/g_{V})^{2}\langle
g_{GT}\rangle^{2})A^{h},$

where $v$ is the neutrino velocity, $V_{ud}$ is the up-down quark element of
the CKM matrix, $F(Z,E_{e})$ is the Fermi Coulomb correction, and $h$ labels
the helicity. We define a polarization-averaged cross-section value
$\overline{\sigma}$ = $\sum_{j=1,2,3}$$\sigma_{j}({\pm})$ $v_{j}$ $\simeq$
3.834 $\times$$10^{-45}$ cm2 for later use.

The helicity-dependent factor $A^{h}$ is:

$A^{\pm}=1\mp\beta,$ (12)

$+$ for right-hand helicity eigenstate, $-$ for left-hand helicity eigenstate;
$\beta$=$v/c$, where $v$ is the $\nu$ velocity. For massive neutrinos such
that $\beta$ $\ll$ 1, the cross-sections for different helicity states are
thus practically speaking indistinguishable and the impact of spin rotation by
the GMF is not observable in the total rate. A significant impact of neutrino
helicity flip in ITBD requires the cross-section to be sensitive to helicity.

References [5][12] discuss the detection of a cosmic neutrino background
arrival direction anisotropy. Such an anisotropy can appear due to density
perturbations as for the CMB, or in our case, due to anisotropic impacts of
helicity rotation in the GMF. Equation (4.3) of Ref. [12] gives the
differential cross section depending on the direction of ${}^{3}H$
polarization, $\hat{s}_{H}$, the direction of the outgoing electron, the $\nu$
velocity, and $\nu$ spin direction. Neglecting terms giving small
contributions:

$\frac{d\sigma(\hat{s}_{H},\hat{v}_{e})}{d\Omega_{e}}\approx\frac{\overline{\sigma}}{4\pi}[(1-\hat{s}_{\nu}\cdot\vec{v}_{\nu})+B\,\hat{s}_{H}\cdot(\vec{v}_{\nu}-\hat{s}_{\nu})],$
(13)

in units with $c=1$, where $B$ $=$ $\frac{2g_{A}(1+g_{A})}{1+3g^{2}_{A}}$
$\simeq$ 0.99 is an asymmetry parameter from the amplitude calculation of
$\nu$ to be captured on the neutron, and term
(1-$\hat{s}_{\nu}$$\cdot$$\vec{v}_{\nu})$ is identical to the $A^{\pm}$ in
Eq.(12), the expression of helicity here is in terms of spin operators. This
expression must be summed over all neutrinos contributing to the measurement,
i.e., summed over all arrival directions in the sky.

For unpolarized ${}^{3}H$, only the $<\hat{s}_{\nu}$$\cdot$$\vec{v}_{\nu}>$
term is of interest because the second term averages to zero. One might think
since $v_{\nu}\sim p_{0}/m$ for non-relativistic neutrinos, that a lower
neutrino mass would lead to a larger $<v>$ and hence larger sensitivity to the
spin rotation via $<\hat{s}_{\nu}$$\cdot$$\vec{v}_{\nu}>$. However, the spin
rotation is proportional to the time spent traversing the GMF, and is thus
inversely proportional to the mass, so that
$<\hat{s}_{\nu}$$\cdot$$\vec{v}_{\nu}>$ is independent of neutrino mass. We
show in FIG. 6 the variation of $<\hat{s}_{\nu}$$\cdot$$\vec{v}_{\nu}>$ as
$\mu_{\nu}$ ranges from standard model magnetic moment to experimental upper
bound taking $m_{\nu}$= 0.1eV.

Neutrino spin rotation not only impacts the total event rate but, as can be
seen in Eq. (13), it generates a sensitivity of the event rate to the
direction of polarization of the tritium target. Using the results from our
simulations, we calculate the fractional difference of ITBD event rate for
$\hat{s}_{H}$ pointing to a given Galactic coordinate, summing over the
neutrino arrival directions, due to the term
$\hat{s}_{H}\cdot(\vec{v}_{\nu}-\hat{s}_{\nu})$. Repeating this for each
direction $\hat{s}_{H}$ on the sky, and dividing by the average at the given
$\mu_{\nu}$, yields the “dipole anisotropy” maps for the ITBD signal
fractional difference as a function of its polarization direction, shown in
Fig. 7 taking $m_{\nu}=0.1$ eV, for the suite of values of $\mu_{\nu}$ and
assuming the flux of cosmic background neutrinos is isotropic.

Figure 6:  Arrival-direction-averaged value of
$<$$\hat{s}_{\nu}$$\cdot$$\vec{v}_{\nu}$$>$ versus $\mu_{\nu}$ ranging from
the standard model magnetic moment $\mu_{\nu}^{SM}$ $\sim$ $10^{-18}$
$\mu_{B}$ to the experimental upper bound
$\mu^{GEMMA}_{\nu}$$\approx$$2.9\times 10^{-11}$$\mu_{B}$, in the JF12
coherent GMF. As shown in the text,
$<$$\hat{s}_{\nu}$$\cdot$$\vec{v}_{\nu}$$>$ is independent of neutrino mass
for non-relativistic neutrinos.

(a)

(b)

(c)

(d)

Figure 7: The ITBD event rate as a function of the tritium polarization
direction, expressed as the fractional difference relative to the mean, in
Galactic coordinates and for $m_{\nu}=0.1$eV and $\nu$ spin rotation in JF12
Coherent Galactic Magnetic Filed Model, for $\mu_{\nu}$= $\mu^{GEMMA}_{\nu}$,
$10^{-1}$ $\times$ $\mu^{GEMMA}_{\nu}$, $10^{-2}$ $\times$ $\mu^{GEMMA}_{\nu}$
and $10^{-3}$ $\times$ $\mu^{GEMMA}_{\nu}$ in sequence from left to right

##  VI  Spin Rotation of solar neutrinos

In this section, we comment on another possible effect of a neutrino magnetic
moment large enough to produce observable effects. Consider a solar neutrino
traveling to Earth, at mass scale $\sim$ $10^{-2}$ eV and energy $\sim$ 100
keV (proton-proton reaction case). Helioseismological data provide an upper
bound on the field strength in the convective zone of the Sun, between 0.71-1
$R_{\odot}$, of $B_{CZ}\lesssim 300$ kG. [13]. With these values, even a
magnetic moment $\sim$ 2.9$\times$$10^{-11}$$\mu_{B}$ saturating the GEMMA
upper bound value yields a negligible spin rotation in the convective zone. In
the radiative zone, 0.2$\times$$R_{\odot}$ to 0.7$\times$$R_{\odot}$, an upper
bound to $B_{RZ}$ of $\approx$ 7 MG is provided by estimation in the
axisymmetric toroidal model [14]. Here, with a GEMMA or smaller value of the
magnetic moment, the magnetic field strength could modify the solar neutrino
helicity. Such spin-precession would in general result in a reduced solar
neutrino flux observed and induce periodic modulations in solar neutrino
signals over the course of a year as the angle of propagation of solar
neutrinos reaching a detector on Earth changes with respect to the solar
magnetic field. The idea of a relic magnetic field with small effects of solar
evolution in the radiative zone was proposed in previous research [14] in an
axis-symmetric toroidal field. Following the standard solar model BP2000,
eigenmodes of the magnetic field with lifetime $\approx$ 5 Gyr do exist with
an upper bound of an order of $\simeq$ MGauss.

Unlike the magnetic fields, gravitational inhomogeneities affect both Dirac
$\nu$ and Majorana $\nu$ helicity. Effects of solar neutrino spin rotation
caused by gravitational fields of the sun itself are also estimated by Baym
and Peng [2], who show that estimated spin rotation is $\simeq$ $10^{-6}$
radians for $\nu$ with $\beta$ = $v_{\nu}$/c $\simeq$ 1, a smaller effect
compared to the possible rotation caused by $\mu_{\nu}$ $\times$
$\vec{B}_{\odot}$ discussed above.

Future detectors and event constructions with better precision will possibly
be able to test such predictions. This could result in a better constraint on
magnetic moment coupling with the magnetic field in the radiative zone, and at
the same time contribute to our understanding of the solar magnetic field
structure.

##  VII Discussion

We have explored the effects of a neutrino magnetic moment on the evolution of
the helicity of relic neutrinos propagating through the Galaxy to a detector
on Earth, by tracking through a realistic model of the Galactic magnetic field
with numerical simulations. Our results show that for magnetic moments
consistent with experimental bounds and even several orders of magnitude
smaller, a relic neutrinos’ helicity has a significant rotation, which depends
on the arrival direction. Using these results, we studied the effects of a
neutrino magnetic moment on relic neutrino detection through inverse tritium
beta decay and calculated the dipole anisotropy in the neutrino detection rate
as a function of tritium polarization direction. We find that the absolute
reduction in the detection rate due to spin rotation is small, on account of
the low velocity of cosmic neutrinos. However, the fractional directional
anisotropy is up to a factor-10 larger. Many difficulties still remain,
however, in reaching the required levels of sensitivity or indeed even
detecting a relic neutrino signal.

Though neutrino helicity modification due to the astrophysical magnetic field
is still not measurable by current experiments, the topic itself could
eventually be a significant probe to complete our understanding of the
astrophysical magnetic field, cosmic neutrino background studies, constraints
for non-zero neutrino magnetic moment value, and whether the neutrino mass is
Dirac or Majorana since only Dirac neutrinos experience spin rotation in a
magnetic field.

###### Acknowledgements.

We appreciate helpful discussions with G. Baym, C. Tully, A. Gruzinov, and X.
Xu. The research of GRF is supported by National Science Foundation Grant NSF-
PHY-2013199 and by the Simons Foundation.

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