Serveur d'exploration sur l'opéra

Attention, ce site est en cours de développement !
Attention, site généré par des moyens informatiques à partir de corpus bruts.
Les informations ne sont donc pas validées.

Superluminal velocity through near-maximal neutrino oscillations or by being off shell

Identifieur interne : 000104 ( Istex/Corpus ); précédent : 000103; suivant : 000105

Superluminal velocity through near-maximal neutrino oscillations or by being off shell

Auteurs : Tim R. Morris

Source :

RBID : ISTEX:018F1A8D7E43B5791309459D40767777B8531BBC

Abstract

Recently it was suggested that the observation of superluminal neutrinos by the OPERA collaboration may be due to group velocity effects resulting from close-to-maximal oscillation between neutrino mass eigenstates, in analogy to known effects in optics. We show that superluminal propagation does occur through this effect for a series of very narrow energy ranges, but this phenomenon cannot explain the OPERA measurement. Superluminal propagation can also occur if one of the neutrino masses is extremely small. However the effect only has appreciable amplitude at energies of order this mass and thus has negligible overlap with the multi-GeV scale of the experiment.

Url:
DOI: 10.1088/0954-3899/39/4/045010

Links to Exploration step

ISTEX:018F1A8D7E43B5791309459D40767777B8531BBC

Le document en format XML

<record>
<TEI wicri:istexFullTextTei="biblStruct">
<teiHeader>
<fileDesc>
<titleStmt>
<title>Superluminal velocity through near-maximal neutrino oscillations or by being off shell</title>
<author wicri:is="90%">
<name sortKey="Morris, Tim R" sort="Morris, Tim R" uniqKey="Morris T" first="Tim R" last="Morris">Tim R. Morris</name>
<affiliation>
<mods:affiliation>School of Physics and Astronomy, University of Southampton Highfield, Southampton, SO17 1BJ, UK</mods:affiliation>
</affiliation>
</author>
</titleStmt>
<publicationStmt>
<idno type="wicri:source">ISTEX</idno>
<idno type="RBID">ISTEX:018F1A8D7E43B5791309459D40767777B8531BBC</idno>
<date when="2012" year="2012">2012</date>
<idno type="doi">10.1088/0954-3899/39/4/045010</idno>
<idno type="url">https://api.istex.fr/document/018F1A8D7E43B5791309459D40767777B8531BBC/fulltext/pdf</idno>
<idno type="wicri:Area/Istex/Corpus">000104</idno>
</publicationStmt>
<sourceDesc>
<biblStruct>
<analytic>
<title level="a">Superluminal velocity through near-maximal neutrino oscillations or by being off shell</title>
<author wicri:is="90%">
<name sortKey="Morris, Tim R" sort="Morris, Tim R" uniqKey="Morris T" first="Tim R" last="Morris">Tim R. Morris</name>
<affiliation>
<mods:affiliation>School of Physics and Astronomy, University of Southampton Highfield, Southampton, SO17 1BJ, UK</mods:affiliation>
</affiliation>
</author>
</analytic>
<monogr></monogr>
<series>
<title level="j">Journal of Physics G Nuclear and Particle Physics</title>
<idno type="ISSN">0954-3899</idno>
<idno type="eISSN">1361-6471</idno>
<imprint>
<publisher>IOP Publishing</publisher>
<date type="published" when="2012-04">2012-04</date>
<biblScope unit="volume">39</biblScope>
<biblScope unit="issue">4</biblScope>
</imprint>
<idno type="ISSN">0954-3899</idno>
</series>
<idno type="istex">018F1A8D7E43B5791309459D40767777B8531BBC</idno>
<idno type="DOI">10.1088/0954-3899/39/4/045010</idno>
<idno type="href">http://stacks.iop.org/JPhysG/39/045010</idno>
<idno type="ArticleID">jpg417154</idno>
</biblStruct>
</sourceDesc>
<seriesStmt>
<idno type="ISSN">0954-3899</idno>
</seriesStmt>
</fileDesc>
<profileDesc>
<textClass></textClass>
<langUsage>
<language ident="en">en</language>
</langUsage>
</profileDesc>
</teiHeader>
<front>
<div type="abstract">Recently it was suggested that the observation of superluminal neutrinos by the OPERA collaboration may be due to group velocity effects resulting from close-to-maximal oscillation between neutrino mass eigenstates, in analogy to known effects in optics. We show that superluminal propagation does occur through this effect for a series of very narrow energy ranges, but this phenomenon cannot explain the OPERA measurement. Superluminal propagation can also occur if one of the neutrino masses is extremely small. However the effect only has appreciable amplitude at energies of order this mass and thus has negligible overlap with the multi-GeV scale of the experiment.</div>
</front>
</TEI>
<istex>
<corpusName>iop</corpusName>
<author>
<json:item>
<name>Tim R Morris</name>
<affiliations>
<json:string>School of Physics and Astronomy, University of Southampton Highfield, Southampton, SO17 1BJ, UK</json:string>
</affiliations>
</json:item>
</author>
<subject>
<json:item>
<lang>
<json:string>eng</json:string>
</lang>
<value>Particle physics</value>
</json:item>
<json:item>
<lang>
<json:string>eng</json:string>
</lang>
<value>14.60.Lm</value>
</json:item>
<json:item>
<lang>
<json:string>eng</json:string>
</lang>
<value>14.60.Pq</value>
</json:item>
<json:item>
<lang>
<json:string>eng</json:string>
</lang>
<value>Neutrinos OPERA</value>
</json:item>
</subject>
<language>
<json:string>eng</json:string>
</language>
<abstract>Recently it was suggested that the observation of superluminal neutrinos by the OPERA collaboration may be due to group velocity effects resulting from close-to-maximal oscillation between neutrino mass eigenstates, in analogy to known effects in optics. We show that superluminal propagation does occur through this effect for a series of very narrow energy ranges, but this phenomenon cannot explain the OPERA measurement. Superluminal propagation can also occur if one of the neutrino masses is extremely small. However the effect only has appreciable amplitude at energies of order this mass and thus has negligible overlap with the multi-GeV scale of the experiment.</abstract>
<qualityIndicators>
<score>6.712</score>
<pdfVersion>1.4</pdfVersion>
<pdfPageSize>595 x 842 pts (A4)</pdfPageSize>
<refBibsNative>true</refBibsNative>
<keywordCount>4</keywordCount>
<abstractCharCount>671</abstractCharCount>
<pdfWordCount>6407</pdfWordCount>
<pdfCharCount>31758</pdfCharCount>
<pdfPageCount>13</pdfPageCount>
<abstractWordCount>101</abstractWordCount>
</qualityIndicators>
<title>Superluminal velocity through near-maximal neutrino oscillations or by being off shell</title>
<genre>
<json:string>research-article</json:string>
</genre>
<host>
<volume>39</volume>
<issn>
<json:string>0954-3899</json:string>
</issn>
<issue>4</issue>
<genre></genre>
<language>
<json:string>unknown</json:string>
</language>
<eissn>
<json:string>1361-6471</json:string>
</eissn>
<title>Journal of Physics G Nuclear and Particle Physics</title>
</host>
<publicationDate>2012</publicationDate>
<copyrightDate>2012</copyrightDate>
<doi>
<json:string>10.1088/0954-3899/39/4/045010</json:string>
</doi>
<id>018F1A8D7E43B5791309459D40767777B8531BBC</id>
<fulltext>
<json:item>
<original>true</original>
<mimetype>application/pdf</mimetype>
<extension>pdf</extension>
<uri>https://api.istex.fr/document/018F1A8D7E43B5791309459D40767777B8531BBC/fulltext/pdf</uri>
</json:item>
<json:item>
<original>false</original>
<mimetype>application/zip</mimetype>
<extension>zip</extension>
<uri>https://api.istex.fr/document/018F1A8D7E43B5791309459D40767777B8531BBC/fulltext/zip</uri>
</json:item>
<istex:fulltextTEI uri="https://api.istex.fr/document/018F1A8D7E43B5791309459D40767777B8531BBC/fulltext/tei">
<teiHeader>
<fileDesc>
<titleStmt>
<title level="a">Superluminal velocity through near-maximal neutrino oscillations or by being off shell</title>
</titleStmt>
<publicationStmt>
<authority>ISTEX</authority>
<publisher>IOP Publishing</publisher>
<availability>
<p>IOP</p>
</availability>
<date>2012-03-16</date>
</publicationStmt>
<sourceDesc>
<biblStruct type="inbook">
<analytic>
<title level="a">Superluminal velocity through near-maximal neutrino oscillations or by being off shell</title>
<author>
<persName>
<forename type="first">Tim R</forename>
<surname>Morris</surname>
</persName>
<affiliation>School of Physics and Astronomy, University of Southampton Highfield, Southampton, SO17 1BJ, UK</affiliation>
</author>
</analytic>
<monogr>
<title level="j">Journal of Physics G Nuclear and Particle Physics</title>
<idno type="pISSN">0954-3899</idno>
<idno type="eISSN">1361-6471</idno>
<imprint>
<publisher>IOP Publishing</publisher>
<date type="published" when="2012-04"></date>
<biblScope unit="volume">39</biblScope>
<biblScope unit="issue">4</biblScope>
</imprint>
</monogr>
<idno type="istex">018F1A8D7E43B5791309459D40767777B8531BBC</idno>
<idno type="DOI">10.1088/0954-3899/39/4/045010</idno>
<idno type="href">http://stacks.iop.org/JPhysG/39/045010</idno>
<idno type="ArticleID">jpg417154</idno>
</biblStruct>
</sourceDesc>
</fileDesc>
<profileDesc>
<creation>
<date>2012-03-16</date>
</creation>
<langUsage>
<language ident="en">en</language>
</langUsage>
<abstract>
<p>Recently it was suggested that the observation of superluminal neutrinos by the OPERA collaboration may be due to group velocity effects resulting from close-to-maximal oscillation between neutrino mass eigenstates, in analogy to known effects in optics. We show that superluminal propagation does occur through this effect for a series of very narrow energy ranges, but this phenomenon cannot explain the OPERA measurement. Superluminal propagation can also occur if one of the neutrino masses is extremely small. However the effect only has appreciable amplitude at energies of order this mass and thus has negligible overlap with the multi-GeV scale of the experiment.</p>
</abstract>
<textClass>
<keywords scheme="keyword">
<list>
<head>article-type</head>
<item>
<term>Paper</term>
</item>
</list>
</keywords>
</textClass>
<textClass>
<keywords scheme="keyword">
<list>
<head>section</head>
<item>
<term>Particle physics</term>
</item>
</list>
</keywords>
</textClass>
<textClass>
<keywords scheme="keyword">
<list>
<head>author-pacs</head>
<item>
<term>14.60.Lm</term>
</item>
<item>
<term>14.60.Pq</term>
</item>
</list>
</keywords>
</textClass>
<textClass>
<keywords scheme="keyword">
<list>
<head>Keywords</head>
<item>
<term>Neutrinos OPERA</term>
</item>
</list>
</keywords>
</textClass>
</profileDesc>
<revisionDesc>
<change when="2012-03-16">Created</change>
<change when="2012-04">Published</change>
</revisionDesc>
</teiHeader>
</istex:fulltextTEI>
<json:item>
<original>false</original>
<mimetype>text/plain</mimetype>
<extension>txt</extension>
<uri>https://api.istex.fr/document/018F1A8D7E43B5791309459D40767777B8531BBC/fulltext/txt</uri>
</json:item>
</fulltext>
<metadata>
<istex:metadataXml wicri:clean="corpus iop not found" wicri:toSee="no header">
<istex:xmlDeclaration>version="1.0" encoding="ISO-8859-1" </istex:xmlDeclaration>
<istex:docType PUBLIC="-//NLM//DTD Journal Publishing DTD v3.0 20080202//EN" URI="http://ej.iop.org/dtd/nlm-3.0/journalpublishing3.dtd" name="istex:docType"></istex:docType>
<istex:document>
<article article-type="research-article">
<front>
<journal-meta>
<journal-id journal-id-type="publisher-id">jpg</journal-id>
<journal-id journal-id-type="coden">JPGPED</journal-id>
<journal-title-group>
<journal-title>Journal of Physics G: Nuclear and Particle Physics</journal-title>
<abbrev-journal-title abbrev-type="IOP">JPhysG</abbrev-journal-title>
<abbrev-journal-title abbrev-type="publisher">J. Phys. G: Nucl. Part. Phys.</abbrev-journal-title>
</journal-title-group>
<issn pub-type="ppub">0954-3899</issn>
<issn pub-type="epub">1361-6471</issn>
<publisher>
<publisher-name>IOP Publishing</publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="publisher-id">jpg417154</article-id>
<article-id pub-id-type="doi">10.1088/0954-3899/39/4/045010</article-id>
<article-id pub-id-type="manuscript">417154</article-id>
<article-categories>
<subj-group subj-group-type="article-type">
<subject>Paper</subject>
</subj-group>
<subj-group subj-group-type="section">
<subject>Particle physics</subject>
</subj-group>
</article-categories>
<title-group>
<article-title>Superluminal velocity through near-maximal neutrino oscillations or by being off shell</article-title>
<alt-title alt-title-type="ascii">Superluminal velocity through near-maximal neutrino oscillations or by being off shell</alt-title>
<alt-title alt-title-type="short">Superluminal velocity through near-maximal neutrino oscillations or by being off shell</alt-title>
<alt-title alt-title-type="short-ascii">Superluminal velocity through near-maximal neutrino oscillations or by being off shell</alt-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname>Morris</surname>
<given-names>Tim R</given-names>
</name>
<xref ref-type="aff" rid="jpg417154af1"></xref>
<xref ref-type="aff" rid="jpg417154em1"></xref>
</contrib>
<aff id="jpg417154af1">School of Physics and Astronomy,
<institution>University of Southampton Highfield</institution>
, Southampton, SO17 1BJ,
<country>UK</country>
</aff>
<ext-link ext-link-type="email" id="jpg417154em1">T.R.Morris@soton.ac.uk</ext-link>
<author-comment content-type="short-author-list">
<p>T R Morris</p>
</author-comment>
</contrib-group>
<pub-date pub-type="ppub">
<month>4</month>
<year>2012</year>
</pub-date>
<pub-date pub-type="epub">
<day>16</day>
<month>3</month>
<year>2012</year>
</pub-date>
<volume>39</volume>
<issue>4</issue>
<elocation-id content-type="artnum">045010</elocation-id>
<history>
<date date-type="received">
<day>19</day>
<month>12</month>
<year>2011</year>
</date>
</history>
<permissions>
<copyright-statement>© 2012 IOP Publishing Ltd</copyright-statement>
<copyright-year>2012</copyright-year>
</permissions>
<self-uri xlink:href="http://stacks.iop.org/JPhysG/39/045010"></self-uri>
<abstract>
<title>Abstract</title>
<p>Recently it was suggested that the observation of superluminal neutrinos by the OPERA collaboration may be due to group velocity effects resulting from close-to-maximal oscillation between neutrino mass eigenstates, in analogy to known effects in optics. We show that superluminal propagation does occur through this effect for a series of very narrow energy ranges, but this phenomenon cannot explain the OPERA measurement. Superluminal propagation can also occur if one of the neutrino masses is extremely small. However the effect only has appreciable amplitude at energies of order this mass and thus has negligible overlap with the multi-GeV scale of the experiment.</p>
</abstract>
<kwd-group kwd-group-type="author-pacs">
<kwd>14.60.Lm</kwd>
<kwd>14.60.Pq</kwd>
</kwd-group>
<kwd-group kwd-group-type="author">
<kwd>Neutrinos OPERA</kwd>
</kwd-group>
<counts>
<page-count count="13"></page-count>
</counts>
<custom-meta-group>
<custom-meta>
<meta-name>ccc</meta-name>
<meta-value>0954-3899/12/045010+13$33.00</meta-value>
</custom-meta>
<custom-meta>
<meta-name>printed</meta-name>
<meta-value>Printed in the UK & the USA</meta-value>
</custom-meta>
</custom-meta-group>
</article-meta>
</front>
<body>
<sec id="jpg417154s1">
<label>1.</label>
<title>Introduction</title>
<p>Recently the OPERA collaboration reported a measurement of the average time taken for neutrinos (ν
<sub>μ</sub>
up to % level contamination) created at CERN (CN) to arrive at the Gran Sasso Laboratory (GS) compared to the time taken travelling at the speed of light in vacuo (
<italic>c</italic>
). They found an early arrival time of approximately δ
<italic>t</italic>
= 60 ns, which corresponds, at a significance of 5.0σ, to faster-than-light travel by a positive fraction δ
<italic>v</italic>
=
<italic>v</italic>
− 1 ≈ 2.37 × 10
<sup>−5</sup>
[
<xref ref-type="bibr" rid="jpg417154bib01">1</xref>
]. (In this paper we set Planck's constant and the speed of light in vacuo ℏ =
<italic>c</italic>
= 1.)</p>
<p>The OPERA result has inspired many papers. For a selection see [
<xref ref-type="bibr" rid="jpg417154bib02">2</xref>
<xref ref-type="bibr" rid="jpg417154bib09">9</xref>
]. Here we rely on the fact that quantum mechanics, through Heisenberg's uncertainty principle, allows neutrinos to travel faster than light and demonstrate this through two different effects.</p>
<p>Mecozzi and Bellini [
<xref ref-type="bibr" rid="jpg417154bib03">3</xref>
] suggested an interpretation in analogy with established dispersion and interference effects in optics, where a superluminal group velocity has been both predicted and measured [
<xref ref-type="bibr" rid="jpg417154bib10">10</xref>
<xref ref-type="bibr" rid="jpg417154bib12">12</xref>
]. This effect does not imply that a signal is exchanged at faster than the speed of light, in violation with special relativity. Instead it arises from constructive and destructive interference deforming the leading and trailing edges of the pulse.</p>
<p>They do not analyse whether it is possible in practice in OPERA, and indeed in neutrino experiments in general. We show that remarkably such an effect does occur, causing superluminal ‘spikes’ in the neutrino velocity at various ‘critical’ values of the neutrino energy inside the neutrino beam energy spectrum. However, we will see that this behaviour is limited above by Heisenberg uncertainty:
<disp-formula id="jpg417154eqn01">
<label>1</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn01.gif"></graphic>
</disp-formula>
where the quantum mechanical uncertainty in position of the neutrino is Δ
<italic>x</italic>
and the time of flight is
<italic>t</italic>
.</p>
<p>We will see that this is too small to explain the OPERA result, and it also has the wrong energy dependence. The latter follows because the critical values of neutrino energy lie at 1.43 GeV and below. OPERA do not see dependence on such a low energy scale. They split the sample where reliable energies could be measured into bins with energy higher or lower than
<italic>E</italic>
<sub>med</sub>
= 20 GeV. With a significance of greater than 3σ, they still see a superluminal velocity in the higher energy sample of the same magnitude (within errors) [
<xref ref-type="bibr" rid="jpg417154bib01">1</xref>
].</p>
<p>Even though these spikes of superluminal group velocity cannot explain the OPERA measurements, it is an effect that is nevertheless interesting in its own right. We analyse carefully the size of the effect and the conditions that are required to realize it. In particular, we show how corrections neglected in [
<xref ref-type="bibr" rid="jpg417154bib03">3</xref>
], allow one to recover the intuitively expected result (
<xref ref-type="disp-formula" rid="jpg417154eqn01">1</xref>
).</p>
<p>We note that this effect has now been treated in two other papers. In [
<xref ref-type="bibr" rid="jpg417154bib04">4</xref>
], it was noted that the OPERA experiment might be a manifestation of weak measurement. This is in fact another way of interpreting the same effect however here also there is no attempt to determine whether this is realized for neutrinos in practice.</p>
<p>Independently, the weak measurement interpretation was also treated in a paper that appeared after ours [
<xref ref-type="bibr" rid="jpg417154bib05">5</xref>
], where also a numerical estimate is provided. This is in agreement with our more detailed calculations, however the limiting corrections, multiple peaks and energy spectrum are not treated. On the other hand, the treatment of the effect is performed in position space with Gaussian wave packets, which exposes particularly clearly the root cause as being through the small relative displacement of the mass-eigenstate wave packets and their near-cancellation.</p>
<p>We will see that in order to maximize the effect we will need a mixing angle such that sin 
<sup>2</sup>
(2θ) is as close to 1 as possible. It is known that θ
<sub>12</sub>
≈ 0.86, however the effective value of sin 
<sup>2</sup>
(2θ
<sub>12</sub>
) in rock is close to zero [
<xref ref-type="bibr" rid="jpg417154bib13">13</xref>
]. θ
<sub>13</sub>
is tiny [
<xref ref-type="bibr" rid="jpg417154bib14">14</xref>
], so we are therefore left with θ
<sub>23</sub>
for which only the limits 0.92 ≲ sin 
<sup>2</sup>
(2θ
<sub>23</sub>
) ⩽ 1 are so far known [
<xref ref-type="bibr" rid="jpg417154bib13">13</xref>
]. θ
<sub>23</sub>
drives ν
<sub>μ</sub>
↔ν
<sub>τ</sub>
mixing and takes place practically as in vacuum [
<xref ref-type="bibr" rid="jpg417154bib13">13</xref>
]. Clearly then we can work effectively with 2-neutrino mixing between ν
<sub>μ</sub>
and ν
<sub>τ</sub>
. (Actually, this is not quite true: we know that the other mixing matrix parameters will supply corrections in exceptional circumstances. This will be discussed in the conclusions.)</p>
<p>In the second example, we note that effective superluminal propagation of ν
<sub>μ</sub>
(both for individual events and, as we will later see, on average), is possible if the mass
<italic>m</italic>
of one of the mass-eigenstates is extremely small, so that space-like propagation can take place with appreciable probability even over the
<italic>L</italic>
= 730 km distance between CN and GS. Consider a neutrino of such a mass created at CN at time
<italic>x</italic>
<sup>0</sup>
= 0 and position
<italic>x</italic>
= 0, and arriving at GS with space-time coordinates (
<italic>y</italic>
<sup>0</sup>
,
<italic>y</italic>
) = (
<italic>L</italic>
− δ
<italic>t</italic>
,
<italic>L</italic>
). The neutrino beam has average energy ⟨
<italic>E</italic>
⟩ = 17 GeV [
<xref ref-type="bibr" rid="jpg417154bib01">1</xref>
]. The neutrinos are created left handed, and being ultra-relativistic, will stay that way to very good approximation throughout their flight. Therefore we need only the left-handed component, which effectively reduces the propagator to that of a scalar particle:
<disp-formula id="jpg417154eqn02">
<label>2</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn02.gif"></graphic>
</disp-formula>
(Here
<italic>p</italic>
<sup>μ</sup>
is the 4-momentum.) When the interval
<italic>s</italic>
≔ (
<italic>x</italic>
<sup>0</sup>
<italic>y</italic>
<sup>0</sup>
)
<sup>2</sup>
− (
<italic>x</italic>
<italic>y</italic>
)
<sup>2</sup>
is negative as is measured by OPERA, i.e. is space-like, then we have that
<disp-formula id="jpg417154eqn03">
<label>3</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn03.gif"></graphic>
</disp-formula>
where
<italic>K</italic>
<sub>1</sub>
is a modified Bessel function. For large value of its argument it decays exponentially as
<disp-formula id="jpg417154eqn04">
<label>4</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn04.gif"></graphic>
</disp-formula>
On the other hand, for small values,
<italic>K</italic>
<sub>1</sub>
diverges
<disp-formula id="jpg417154eqn05">
<label>5</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn05.gif"></graphic>
</disp-formula>
We see that if we choose
<disp-formula id="jpg417154eqn06">
<label>6</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn06.gif"></graphic>
</disp-formula>
then a substantial fraction of the neutrinos can propagate superluminally.</p>
<p>Of course it is not true that such neutrinos are really tachyonic. Classic arguments due to Feynman, which will not be repeated here, show that this can be interpreted in a way that is consistent with Lorentz invariance. Since
<italic>s</italic>
is negative, there is an inertial frame in which the emission at CN and absorption at GS happen simultaneously. In this frame,
<inline-formula>
<tex-math></tex-math>
<inline-graphic xlink:href="jpg417154ieqn1.gif"></inline-graphic>
</inline-formula>
is the spatial separation of CN and GS, and (
<xref ref-type="disp-formula" rid="jpg417154eqn06">6</xref>
) is just an expression of Heisenberg's uncertainty relation. In the conclusions we will develop this to show that issues with causality are resolved through this observation.</p>
<p>For this effect, we can ignore neutrino mixing. The known mass-squared differences [
<xref ref-type="bibr" rid="jpg417154bib13">13</xref>
] show that only one mass eigenstate can be as light as (
<xref ref-type="disp-formula" rid="jpg417154eqn06">6</xref>
). Therefore the effect we are looking for is entirely due to evanescence in this mass eigenstate. The only effect of mixing is the multiplicative inclusion of mixing angles at the beginning where the CN ν
<sub>μ</sub>
converts to this state, and at the end where it reverts to ν
<sub>μ</sub>
, as detected in GS.</p>
<p>While this paper was being prepared, [
<xref ref-type="bibr" rid="jpg417154bib08">8</xref>
] appeared, where similar ideas are proposed as an explanation for the OPERA measurement. In fact, if we adapt this observation to the setup in the OPERA experiment, we can see that the effect vanishes or at best is far too small to explain the measurement.</p>
<p>For a sample of papers that have developed an approach to neutrino oscillations based on virtual neutrinos and wave packets see [
<xref ref-type="bibr" rid="jpg417154bib09">9</xref>
,
<xref ref-type="bibr" rid="jpg417154bib17">17</xref>
,
<xref ref-type="bibr" rid="jpg417154bib18">18</xref>
,
<xref ref-type="bibr" rid="jpg417154bib21">21</xref>
].</p>
<p>In section
<xref ref-type="sec" rid="jpg417154s2">2</xref>
we argue that the spread in energy for each neutrino's wave packet is
<italic>w</italic>
<italic>E</italic>
<sub>0</sub>
. In the ensemble of neutrinos that make up the neutrino bunches,
<italic>E</italic>
<sub>0</sub>
is spread over a wide range [
<xref ref-type="bibr" rid="jpg417154bib01">1</xref>
,
<xref ref-type="bibr" rid="jpg417154bib19">19</xref>
] but they are at least GeV. From (
<xref ref-type="disp-formula" rid="jpg417154eqn04">4</xref>
), the evanescent part of the propagator appears dominated by energies of order (
<xref ref-type="disp-formula" rid="jpg417154eqn06">6</xref>
)—indeed we will see that it is a manifestation of quantum mechanical tunnelling, requiring energy to be less than
<italic>m</italic>
; this mismatch with the energies in the wavepacket ∼
<italic>E</italic>
<sub>0</sub>
±
<italic>w</italic>
, ensures that any remaining effect is consigned to any small tail in the wavepacket that reaches down to these small energies. For example, if we assume that the wavepacket is a Gaussian then this supplies a suppression factor exp −
<italic>E</italic>
<sup>2</sup>
<sub>0</sub>
/
<italic>w</italic>
<sup>2</sup>
. Alternatively, a wave packet with a lower cutoff >
<italic>m</italic>
on the neutrino energy would eliminate the effect entirely.</p>
<p>One might try to resuscitate the proposal by bringing
<italic>w</italic>
and
<italic>E</italic>
<sub>0</sub>
closer to each other. The relevant
<italic>E</italic>
<sub>0</sub>
could be as low as
<italic>E</italic>
<sub>min</sub>
, where this is the threshold energy of 112 MeV for detection at GS via the process ν
<sub>μ</sub>
<italic>n</italic>
→ μ
<sup></sup>
<italic>p</italic>
, or more realistically the resolution of the GS detector
<italic>E</italic>
<sub>min</sub>
≈ 1 GeV [
<xref ref-type="bibr" rid="jpg417154bib16">16</xref>
]. We still cannot fit the data since, as we have already remarked, OPERA do not see dependence on a low energy scale.</p>
<p>Another way one might try nevertheless to use this effect to explain the OPERA measurement is to boost (
<xref ref-type="disp-formula" rid="jpg417154eqn02">2</xref>
). In other words, we note that the neutrinos are neither created at an exact time nor at an exact location; in reality we need to integrate over position space terms that supply the neutrino with the appropriate ultra relativistic momentum. One could then hope that the behaviour (
<xref ref-type="disp-formula" rid="jpg417154eqn04">4</xref>
) would be the correct one for directions transverse to the neutrino's momentum, corresponding to small deficits in the energy required by the ultra relativistic on-shell energy–momentum relation. The amplitude would effectively take the form of the kilometres-wide wave function assumed
<xref ref-type="fn" rid="jpg417154fn1">
<sup>1</sup>
</xref>
<fn id="jpg417154fn1">
<label>1</label>
<p>The assumption is made to be consistent with technical requirements [
<xref ref-type="bibr" rid="jpg417154bib17">17</xref>
] which ensure that wave packet dispersion can be neglected. It is not explained however why nature should obey these requirements.</p>
</fn>
in [
<xref ref-type="bibr" rid="jpg417154bib09">9</xref>
], where the OPERA result is then explained without any adjustable parameters by the off-centre detection of these wave packets. However we will see that this set-up does not result in such transverse evanescence.</p>
<p>The rest of the paper is organized as follows. In the next section we review the derivation in [
<xref ref-type="bibr" rid="jpg417154bib03">3</xref>
], including the notation and adaptations we will need to match the OPERA experiment. In particular we take care to include corrections neglected in [
<xref ref-type="bibr" rid="jpg417154bib03">3</xref>
] but which are crucial in limiting the maximum size of the effect. To carry through the derivation, we will need to make some assumptions about the coherence of individual neutrino wave packets. In section
<xref ref-type="sec" rid="jpg417154s3">3</xref>
we show these are correct. In section
<xref ref-type="sec" rid="jpg417154s4">4</xref>
we show that while there is an effect, it is far too small and confirm that it has the wrong energy dependence to agree with the data. In section
<xref ref-type="sec" rid="jpg417154s5">5</xref>
, we turn to the second idea, constructing the initial wavefunction and the amplitude for neutrinos as seen at GS, and draw out the off-shell piece relevant for this effect, confirming and extending the arguments given above. Finally, in section
<xref ref-type="sec" rid="jpg417154s6">6</xref>
, we present our conclusions, underlining why these results also provide the reason why there is no violation of causality.</p>
</sec>
<sec id="jpg417154s2">
<label>2.</label>
<title>Review and development of [
<xref ref-type="bibr" rid="jpg417154bib03">3</xref>
]</title>
<p>Our treatment follows closely [
<xref ref-type="bibr" rid="jpg417154bib03">3</xref>
] but with adaptations and corrections. We write the mass eigenstates in standard convention [
<xref ref-type="bibr" rid="jpg417154bib13">13</xref>
] as
<disp-formula id="jpg417154eqn07">
<label>7</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn07.gif"></graphic>
</disp-formula>
<disp-formula id="jpg417154eqn08">
<label>8</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn08.gif"></graphic>
</disp-formula>
where
<italic>p</italic>
is the momentum of the neutrino in the beam. For momenta much larger than the masses, which is the case here (since the masses are known from tritium decay to be ≲ 2 eV [
<xref ref-type="bibr" rid="jpg417154bib13">13</xref>
]), energies are given by
<disp-formula id="jpg417154eqn09">
<label>9</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn09.gif"></graphic>
</disp-formula>
where
<disp-formula id="jpg417154eqn10">
<label>10</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn10.gif"></graphic>
</disp-formula>
We start at CERN with a normalized pure ν
<sub>μ</sub>
wave function |ψ
<sub>0</sub>
⟩ at time
<italic>t</italic>
= 0, and position
<italic>x</italic>
<sub>0</sub>
= ⟨ψ
<sub>0</sub>
|
<italic>x</italic>
<sub>0</sub>
⟩. Projecting on mass eigenstates using ∑
<sub>
<italic>i</italic>
</sub>
|
<italic>p</italic>
′, ν
<sub>
<italic>i</italic>
</sub>
⟩⟨
<italic>p</italic>
′, ν
<sub>
<italic>i</italic>
</sub>
|, we have at later times
<disp-formula id="jpg417154eqn11">
<label>11</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn11.gif"></graphic>
</disp-formula>
where ψ
<sub>0</sub>
(
<italic>k</italic>
) = ⟨
<italic>k</italic>
<sub>0</sub>
⟩ describes the spread of momenta in the initial neutrino wave packet.</p>
<p>To find the ν
<sub>μ</sub>
wave function when measured at Gran Sasso, we collapse |ψ
<sub>
<italic>t</italic>
</sub>
⟩ using ⟨
<italic>p</italic>
′, ν
<sub>μ</sub>
| to obtain
<disp-formula id="jpg417154eqn12">
<label>12</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn12.gif"></graphic>
</disp-formula>
where
<disp-formula id="jpg417154eqn13">
<label>13</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn13.gif"></graphic>
</disp-formula>
and we have replaced the initial ψ
<sub>0</sub>
(
<italic>p</italic>
′) with an effective measured ψ
<sub>1</sub>
(
<italic>p</italic>
′) to allow for further decoherence in the momentum space of the particle during its measurement.</p>
<p>We need to normalize (
<xref ref-type="disp-formula" rid="jpg417154eqn12">12</xref>
), which requires dividing by ∫d
<italic>p</italic>
′|ψ′(
<italic>p</italic>
′)|
<sup>2</sup>
. We now assume, as effectively was done in [
<xref ref-type="bibr" rid="jpg417154bib03">3</xref>
], that ψ
<sub>1</sub>
(
<italic>p</italic>
′) is strongly peaked around the momentum
<italic>p</italic>
′ =
<italic>p</italic>
, and that the variation in |
<italic>F</italic>
(
<italic>p</italic>
′)| is gradual in comparison. We will confirm these assumptions in the next section. We note that in this case ∫d
<italic>p</italic>
′|ψ′(
<italic>p</italic>
′)|
<sup>2</sup>
≈ |
<italic>F</italic>
(
<italic>p</italic>
)|
<sup>2</sup>
and so the Gran Sasso wave function is typically normalized to good approximation by dividing by |
<italic>F</italic>
(
<italic>p</italic>
)|. This approximation breaks down if
<italic>p</italic>
takes a value such that
<italic>F</italic>
(
<italic>p</italic>
) is close to vanishing (as is clear because ∫d
<italic>p</italic>
′|ψ′(
<italic>p</italic>
′)|
<sup>2</sup>
is positive definite since
<italic>F</italic>
cannot vanish for all momenta). We will return to this in the next section.</p>
<p>For now we take the expected position of the ν
<sub>μ</sub>
at Gran Sasso, to thus be given by
<disp-formula id="jpg417154eqn14">
<label>14</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn14.gif"></graphic>
</disp-formula>
which depends on the time of arrival through
<italic>F</italic>
. Substituting (
<xref ref-type="disp-formula" rid="jpg417154eqn12">12</xref>
), we have that the integral is given by
<italic>N</italic>
<sub>1</sub>
+
<italic>N</italic>
<sub>2</sub>
where
<disp-formula id="jpg417154eqn15">
<label>15</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn15.gif"></graphic>
</disp-formula>
Using again the fact that ψ
<sub>1</sub>
is strongly peaked we see that
<disp-formula id="jpg417154eqn16">
<label>16</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn16.gif"></graphic>
</disp-formula>
Since (
<xref ref-type="disp-formula" rid="jpg417154eqn14">14</xref>
) is the expectation of an Hermitian operator it must give a real answer. Therefore the imaginary part of the above must get cancelled. This can be seen to be true by integrating
<italic>N</italic>
<sub>1</sub>
by parts to get:
<disp-formula id="jpg417154eqn17">
<label>17</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn17.gif"></graphic>
</disp-formula>
and thus extract the imaginary part. The remaining real part of
<italic>N</italic>
<sub>1</sub>
is approximately
<disp-formula id="jpg417154eqn18">
<label>18</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn18.gif"></graphic>
</disp-formula>
Putting it all together we see that
<disp-formula id="jpg417154eqn19">
<label>19</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn19.gif"></graphic>
</disp-formula>
where we have identified the second term as
<italic>t</italic>
times the group velocity. Finally, substituting (
<xref ref-type="disp-formula" rid="jpg417154eqn13">13</xref>
) and using (
<xref ref-type="disp-formula" rid="jpg417154eqn10">10</xref>
), we have
<disp-formula id="jpg417154eqn20">
<label>20</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn20.gif"></graphic>
</disp-formula>
and evaluating the numerator:
<disp-formula id="jpg417154eqn21">
<label>21</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn21.gif"></graphic>
</disp-formula>
where
<disp-formula id="jpg417154eqn22">
<label>22</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn22.gif"></graphic>
</disp-formula>
is the expected negative ‘classical’ correction to the speed of light as a consequence of the average squared mass of the two mass eigenstates, and the last term is a correction resulting from interference between the two mass eigenstates:
<disp-formula id="jpg417154eqn23">
<label>23</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn23.gif"></graphic>
</disp-formula>
This is the main result of [
<xref ref-type="bibr" rid="jpg417154bib03">3</xref>
] (up to correction of the power of sin 2θ
<sub>23</sub>
and the explicit statement of approximation).</p>
</sec>
<sec id="jpg417154s3">
<label>3.</label>
<title>Limitations to coherence and the strongly-peaked wave packet approximation</title>
<p>Let us revisit the assumption that the momentum of the measured neutrino wave packet ψ
<sub>1</sub>
(
<italic>p</italic>
′) is strongly peaked about
<italic>p</italic>
′ ≈
<italic>p</italic>
. This was used to derive the key formulae in section
<xref ref-type="sec" rid="jpg417154s2">2</xref>
below (
<xref ref-type="disp-formula" rid="jpg417154eqn13">13</xref>
), by assuming that the variation in |
<italic>F</italic>
(
<italic>p</italic>
′)| was by comparison more gradual. It should be clear that here we are not discussing the momentum spectrum of the ensemble of neutrinos in the beam, which is very broad—as we will describe in the next section, but rather the inherent quantum mechanical uncertainty in the momentum of a given neutrino as it is measured at GS.</p>
<p>We start with the neutrino wave function ψ
<sub>0</sub>
(
<italic>p</italic>
′) produced in the CNGS decay tunnel at CERN. The momentum uncertainty can certainly be no smaller than that set by Δ
<italic>t</italic>
≈ 5 ns, the smallest time features in the proton bunch [
<xref ref-type="bibr" rid="jpg417154bib01">1</xref>
].
<xref ref-type="fn" rid="jpg417154fn2">
<sup>2</sup>
</xref>
<fn id="jpg417154fn2">
<label>2</label>
<p>In the new version of the OPERA experiment Δ
<italic>t</italic>
≈ 3 ns, as set by the width of the bunch.</p>
</fn>
This corresponds to
<italic>c</italic>
Δ
<italic>p</italic>
= 1/Δ
<italic>t</italic>
≈ 1.3 × 10
<sup>−7</sup>
eV. However there are many other larger potential sources of decoherence. Thermalization in the hot graphite target limits energy–momentum resolution to
<italic>k
<sub>B</sub>
T</italic>
, where
<italic>T</italic>
∼ 300 °C [
<xref ref-type="bibr" rid="jpg417154bib01">1</xref>
]. One could argue that the pion wavefunction is initially localized on a nuclear scale, where it would also be subject to decoherence due to nuclear Fermi motion. The produced neutrino could be subject to decoherence from the internal dynamics of the pion, and finally the neutrinos are subject to decoherence via quantum entanglement to the associated muon in the decay π
<sup>+</sup>
→ μ
<sup>+</sup>
ν
<sub>μ</sub>
. This is because the muon is then localized by its (in principle) detection at the μm level. The same effect happens when the neutrino converts back to a muon which is detected at GS [
<xref ref-type="bibr" rid="jpg417154bib16">16</xref>
].</p>
<p>Without delving into these sources of decoherence in any further detail, we can conclude conservatively that each neutrino's wave packet has a momentum uncertainty lying somewhere in the range
<disp-formula id="jpg417154eqn24">
<label>24</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn24.gif"></graphic>
</disp-formula>
The equivalent uncertainty in position is Δ
<italic>x</italic>
= 1/
<italic>w</italic>
lying somewhere between a Fermi and a metre.</p>
<p>We now contrast this with the variation in |
<italic>F</italic>
(
<italic>p</italic>
′)| from (
<xref ref-type="disp-formula" rid="jpg417154eqn13">13</xref>
). We see that this is controlled by the phase
<disp-formula id="jpg417154eqn25">
<label>25</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn25.gif"></graphic>
</disp-formula>
using (
<xref ref-type="disp-formula" rid="jpg417154eqn10">10</xref>
), where
<italic>t</italic>
can be taken to be
<italic>L</italic>
/
<italic>c</italic>
. At the average energy 17 GeV, and using Δ
<italic>m</italic>
<sup>2</sup>
<sub>23</sub>
≈ 2.43 × 10
<sup>−3</sup>
eV
<sup>2</sup>
[
<xref ref-type="bibr" rid="jpg417154bib13">13</xref>
], we have that φ has magnitude 0.26, and thus we see that |
<italic>F</italic>
(
<italic>p</italic>
′)| is indeed slowly varying even on changes of order Δ
<italic>p</italic>
′ ∼ GeV.</p>
<p>Although this establishes that |
<italic>F</italic>
(
<italic>p</italic>
)| is slowly varying compared to the fundamental uncertainty
<italic>w</italic>
, as we mentioned in section
<xref ref-type="sec" rid="jpg417154s2">2</xref>
we cannot trust (
<xref ref-type="disp-formula" rid="jpg417154eqn23">23</xref>
) if
<italic>F</italic>
(
<italic>p</italic>
) is close to vanishing. This leads to extra conditions that must be satisfied if we are to trust the approximation. From (
<xref ref-type="disp-formula" rid="jpg417154eqn20">20</xref>
), we see that |
<italic>F</italic>
(
<italic>p</italic>
)| takes its minimum value, cos 
<sup>2</sup>
<sub>23</sub>
, when φ(
<italic>p</italic>
) = π + 2π
<italic>a</italic>
, where
<italic>a</italic>
= 0, 1, 2, … . This corresponds to ‘critical’ values of momentum
<disp-formula id="jpg417154eqn26">
<label>26</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn26.gif"></graphic>
</disp-formula>
Taylor expanding about these values we have
<disp-formula id="jpg417154eqn27">
<label>27</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn27.gif"></graphic>
</disp-formula>
where Δ
<italic>p</italic>
′ =
<italic>p</italic>
′ −
<italic>p</italic>
<sub>crit</sub>
. Thus from (
<xref ref-type="disp-formula" rid="jpg417154eqn12">12</xref>
), at
<italic>p</italic>
=
<italic>p</italic>
<sub>crit</sub>
, we can compute the first correction to ∫d
<italic>p</italic>
′|ψ′(
<italic>p</italic>
′)|
<sup>2</sup>
≈ |
<italic>F</italic>
(
<italic>p</italic>
)|
<sup>2</sup>
, from the finite size of the wave packet. It is independent of the details other than through the RMS width
<italic>w</italic>
<sup>2</sup>
= ⟨ψ
<sub>1</sub>
|[Δ
<italic>p</italic>
′]
<sup>2</sup>
<sub>1</sub>
⟩:
<disp-formula id="jpg417154eqn28">
<label>28</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn28.gif"></graphic>
</disp-formula>
Thus if
<disp-formula id="jpg417154eqn29">
<label>29</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn29.gif"></graphic>
</disp-formula>
the first term in (
<xref ref-type="disp-formula" rid="jpg417154eqn28">28</xref>
) dominates and ∫d
<italic>p</italic>
′|ψ′(
<italic>p</italic>
′)|
<sup>2</sup>
≈ |
<italic>F</italic>
(
<italic>p</italic>
)|
<sup>2</sup>
is always a good approximation. If (
<xref ref-type="disp-formula" rid="jpg417154eqn29">29</xref>
) is not satisfied then
<italic>p</italic>
cannot be taken too close to
<italic>p</italic>
=
<italic>p</italic>
<sub>crit</sub>
(
<italic>a</italic>
): we can only trust the expression (
<xref ref-type="disp-formula" rid="jpg417154eqn23">23</xref>
) for δ
<italic>v</italic>
<sub>sup</sub>
providing
<disp-formula id="jpg417154eqn30">
<label>30</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn30.gif"></graphic>
</disp-formula>
</p>
</sec>
<sec id="jpg417154s4">
<label>4.</label>
<title>Confronting the experiment</title>
<p>Given the average 17 GeV energy of the beam, the classical correction to the speed of light (
<xref ref-type="disp-formula" rid="jpg417154eqn22">22</xref>
) is negligible and can be neglected. For most values of
<italic>p</italic>
, δ
<italic>v</italic>
<sub>sup</sub>
is also clearly negligible.</p>
<p>However, (
<xref ref-type="disp-formula" rid="jpg417154eqn23">23</xref>
) provides a correction that is maximal at the critical values of momenta (
<xref ref-type="disp-formula" rid="jpg417154eqn26">26</xref>
). Note that the requirement of detection at Gran Sasso sets the upper limit to
<italic>a</italic>
as
<disp-formula id="jpg417154eqn31">
<label>31</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn31.gif"></graphic>
</disp-formula>
This implies
<italic>a</italic>
can only be zero if
<italic>E</italic>
<sub>min</sub>
= 1 GeV, or
<italic>a</italic>
⩽ 6 for
<italic>E</italic>
<sub>min</sub>
= 112 MeV.</p>
<p>If θ
<sub>23</sub>
is infinitesimally close to the maximal mixing value of π/4, then (
<xref ref-type="disp-formula" rid="jpg417154eqn23">23</xref>
) actually diverges at the points
<italic>p</italic>
=
<italic>p</italic>
<sub>crit</sub>
(
<italic>a</italic>
). To see this more clearly, let
<disp-formula id="jpg417154eqn32">
<label>32</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn32.gif"></graphic>
</disp-formula>
where the sign is opposite to that of Δ
<italic>m</italic>
<sup>2</sup>
<sub>23</sub>
, δθ > 0 is a small shift from maximal mixing, and ε is a small deviation in energy from the critical value. Then (
<xref ref-type="disp-formula" rid="jpg417154eqn23">23</xref>
) becomes
<disp-formula id="jpg417154eqn33">
<label>33</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn33.gif"></graphic>
</disp-formula>
However, at ε = 0, from (
<xref ref-type="disp-formula" rid="jpg417154eqn29">29</xref>
), this is only trustworthy if
<disp-formula id="jpg417154eqn34">
<label>34</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn34.gif"></graphic>
</disp-formula>
Substituting this in (
<xref ref-type="disp-formula" rid="jpg417154eqn33">33</xref>
) and using (
<xref ref-type="disp-formula" rid="jpg417154eqn26">26</xref>
), we see that the maximum superluminal effect that can be predicted is nothing but the intuitive bound (
<xref ref-type="disp-formula" rid="jpg417154eqn01">1</xref>
):
<disp-formula id="jpg417154eqn35">
<label>35</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn35.gif"></graphic>
</disp-formula>
We see that in order to explain the OPERA measurement we would have to assume that the width of the neutrino wave packet created at CERN is Δ
<italic>x</italic>
<italic>c</italic>
δ
<italic>t</italic>
= 18 m, which is not compatible with the analysis in the previous section.</p>
<p>We have already noted in the introduction that the energy dependence of the effect is also in conflict with OPERA. Finally let us show that, taking into account the spread in energies in the neutrino beam, the superluminal contribution from this effect is in fact washed out by many orders of magnitude. Only a narrow spread in energies in the neutrino beam contributes significantly to (
<xref ref-type="disp-formula" rid="jpg417154eqn33">33</xref>
). We see that the energies contributing must lie in the range
<disp-formula id="jpg417154eqn36">
<label>36</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn36.gif"></graphic>
</disp-formula>
In fact, from the talk given at CERN [
<xref ref-type="bibr" rid="jpg417154bib19">19</xref>
] and earlier OPERA analysis [
<xref ref-type="bibr" rid="jpg417154bib20">20</xref>
], we can see that the energy spectrum is broad, rising approximately linearly from
<italic>E</italic>
<sub>min</sub>
to a maximum at
<italic>E</italic>
=
<italic>E</italic>
<sub>med</sub>
, and then falling with a large tail reaching energies of ∼ 200 GeV. Let the fraction of neutrino events measured between momenta
<italic>p</italic>
and
<italic>p</italic>
+ d
<italic>p</italic>
be
<italic>n</italic>
(
<italic>p</italic>
). If we model
<italic>n</italic>
(
<italic>p</italic>
) = ρ
<italic>p</italic>
for
<italic>p</italic>
<
<italic>E</italic>
<sub>med</sub>
, where ρ is a constant, then, since about half the events are found below
<italic>E</italic>
<sub>med</sub>
, we have by ρ ≈ 1/
<italic>E</italic>
<sup>2</sup>
<sub>med</sub>
. It follows that
<italic>n</italic>
(
<italic>p</italic>
<sub>crit</sub>
) ≈
<italic>p</italic>
<sub>crit</sub>
/
<italic>E</italic>
<sup>2</sup>
<sub>med</sub>
= 3.6 (1 + 2
<italic>a</italic>
)
<sup>−1</sup>
× 10
<sup>−12</sup>
eV
<sup>−1</sup>
. Since (
<xref ref-type="disp-formula" rid="jpg417154eqn23">23</xref>
) is sharply peaked around
<italic>p</italic>
=
<italic>p</italic>
<sub>crit</sub>
, the average contribution from (
<xref ref-type="disp-formula" rid="jpg417154eqn33">33</xref>
) is to good approximation
<disp-formula id="jpg417154eqn37">
<label>37</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn37.gif"></graphic>
</disp-formula>
where
<italic>A</italic>
is the number of peaks (1 or 7 depending on
<italic>E</italic>
<sub>min</sub>
). This is numerically 3
<italic>A</italic>
× 10
<sup>−24</sup>
, which is of course too small to measure. It is also competitive with the classical term (
<xref ref-type="disp-formula" rid="jpg417154eqn22">22</xref>
). Using the same model for
<italic>n</italic>
(
<italic>p</italic>
) we have for the classical term
<disp-formula id="jpg417154eqn38">
<label>38</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn38.gif"></graphic>
</disp-formula>
where the inequality results from neglecting the contribution of energies greater than
<italic>E</italic>
<sub>med</sub>
. This term will dominate unless there is a large hierarchy between the two masses, in which case this is of the same order, making the detected average group velocity slower (faster) than the speed of light, depending on whether the larger (smaller) value of
<italic>E</italic>
<sub>min</sub>
is used.</p>
</sec>
<sec id="jpg417154s5">
<label>5.</label>
<title>Tunnelling to Gran Sasso</title>
<p>Finally we turn to the second effect. From hereon we can ignore neutrino mixing. We start by regarding our neutrino as being created at position
<italic>x</italic>
= 0 at an uncertain time centred around
<italic>t</italic>
= 0, with energy localized to
<italic>E</italic>
=
<italic>E</italic>
<sub>0</sub>
with an accuracy
<italic>w</italic>
. Let us model the shape as a Gaussian. Then we have for the initial wavefunction:
<disp-formula id="jpg417154eqn39">
<label>39</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn39.gif"></graphic>
</disp-formula>
The amplitude to find the neutrino at Gran Sasso at time
<italic>t</italic>
<sub>2</sub>
=
<italic>y</italic>
<sup>0</sup>
is then
<disp-formula id="jpg417154eqn40">
<label>40</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn40.gif"></graphic>
</disp-formula>
where
<italic>S</italic>
is the Feynman propagator (
<xref ref-type="disp-formula" rid="jpg417154eqn02">2</xref>
). Since we are dealing with a situation where we know the neutrino arrives at Gran Sasso, we will choose the normalization factor so that the probability distribution reflects this:
<disp-formula id="jpg417154eqn41">
<label>41</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn41.gif"></graphic>
</disp-formula>
We can now make a further simplification. We can drop the transverse momentum integrations in (
<xref ref-type="disp-formula" rid="jpg417154eqn02">2</xref>
) since these will only result in ∼1/
<italic>L</italic>
<sup>2</sup>
losses that are scaled away when we normalize. Therefore, substituting (
<xref ref-type="disp-formula" rid="jpg417154eqn02">2</xref>
) and using (
<xref ref-type="disp-formula" rid="jpg417154eqn39">39</xref>
), we have
<disp-formula id="jpg417154eqn42">
<label>42</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn42.gif"></graphic>
</disp-formula>
where Δ
<italic>E</italic>
=
<italic>E</italic>
<italic>E</italic>
<sub>0</sub>
, and we have performed the
<italic>t</italic>
integration. Now we do the momentum integration. Since we know that
<italic>L</italic>
> 0, the iε prescription tells us to close the contour over the top. We obtain
<disp-formula id="jpg417154eqn43">
<label>43</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn43.gif"></graphic>
</disp-formula>
where
<italic>N</italic>
is the normalization constant, where
<inline-formula>
<tex-math></tex-math>
<inline-graphic xlink:href="jpg417154ieqn2.gif"></inline-graphic>
</inline-formula>
has either a vanishing positive imaginary part (the iε) for |
<italic>E</italic>
| >
<italic>m</italic>
, or
<inline-formula>
<tex-math></tex-math>
<inline-graphic xlink:href="jpg417154ieqn3.gif"></inline-graphic>
</inline-formula>
for |
<italic>E</italic>
| <
<italic>m</italic>
.</p>
<p>We see that the exponential decay in (
<xref ref-type="disp-formula" rid="jpg417154eqn04">4</xref>
) indeed arises from energies of order
<italic>m</italic>
, as we claimed. We also confirm that the exponential decay component is suppressed into the tail of the probability distribution by ∼exp −
<italic>E</italic>
<sup>2</sup>
<sub>0</sub>
/
<italic>w</italic>
<sup>2</sup>
. The amount of evanescent component thus depends crucially on the unknown shape of this tail. If we had chosen a wave packet with a sharp cutoff at some realistic minimum energy, we would eliminate the evanescent contribution completely.</p>
<p>For completeness we note that (
<xref ref-type="disp-formula" rid="jpg417154eqn43">43</xref>
) can be evaluated by the method of steepest descents. The dominant term comes from
<italic>E</italic>
<italic>E</italic>
<sub>0</sub>
gives:
<disp-formula id="jpg417154eqn44">
<label>44</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn44.gif"></graphic>
</disp-formula>
(The approximation is valid providing |
<italic>t</italic>
<sub>2</sub>
<italic>L</italic>
/
<italic>v</italic>
<sub>0</sub>
| ≪
<italic>p</italic>
<sub>0</sub>
/
<italic>w</italic>
<sup>2</sup>
.) Here
<inline-formula>
<tex-math></tex-math>
<inline-graphic xlink:href="jpg417154ieqn4.gif"></inline-graphic>
</inline-formula>
is the central momentum and
<italic>v</italic>
<sub>0</sub>
=
<italic>p</italic>
<sub>0</sub>
/
<italic>E</italic>
<sub>0</sub>
is the classical velocity. We recognize this as nothing but the expected result of propagating the wave packet to Gran Sasso without dispersion. The evanescent part can also be evaluated. Writing
<italic>E</italic>
=
<italic>m</italic>
(1 −
<italic>z</italic>
) for small positive
<italic>z</italic>
at the top boundary, the integral can again be evaluated by steepest descents. Adding to this the term from the bottom boundary
<italic>E</italic>
= −
<italic>m</italic>
(1 −
<italic>z</italic>
), and similar size pieces from
<italic>E</italic>
= ±
<italic>m</italic>
(1 +
<italic>z</italic>
) (which can again be evaluated by steepest descents), we get for the evanescent part
<disp-formula id="jpg417154eqn45">
<label>45</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn45.gif"></graphic>
</disp-formula>
This approximation is valid providing
<italic>mt</italic>
<sub>2</sub>
≫ 1. We see again the damping by the tail of the wave function. We can now carry this through to a computation of the superluminal component of velocity. The evanescent term will provide one, but we do not present the computation since we have seen that it necessarily depends on a vanishingly small unknown quantity.</p>
<p>Finally, we address the question raised in the introduction: whether, after taking into account appropriate spatial dependence of ψ
<sub>0</sub>
and spatial integration in (
<xref ref-type="disp-formula" rid="jpg417154eqn40">40</xref>
), so as to incorporate the fact that the neutrino has some momentum
<inline-formula>
<tex-math></tex-math>
<inline-graphic xlink:href="jpg417154ieqn5.gif"></inline-graphic>
</inline-formula>
(with associated small uncertainty) which is slightly larger in magnitude than the energetically allowed
<inline-formula>
<tex-math></tex-math>
<inline-graphic xlink:href="jpg417154ieqn6.gif"></inline-graphic>
</inline-formula>
, the resulting amplitude could take the form similar to (
<xref ref-type="disp-formula" rid="jpg417154eqn44">44</xref>
) in the direction of
<inline-formula>
<tex-math></tex-math>
<inline-graphic xlink:href="jpg417154ieqn7.gif"></inline-graphic>
</inline-formula>
but with large transverse evanescent tails corresponding to imaginary transverse momenta? Recall that such an amplitude would be very similar to the wave packet envisaged in [
<xref ref-type="bibr" rid="jpg417154bib09">9</xref>
] where results consistent with the OPERA measurement were derived as a result of off-centre measurement of these wave packets.</p>
<p>Clearly in order to investigate this we now need to keep the transverse momentum integrations in (
<xref ref-type="disp-formula" rid="jpg417154eqn02">2</xref>
). We start with (
<xref ref-type="disp-formula" rid="jpg417154eqn40">40</xref>
) which now reads:
<disp-formula id="jpg417154eqn46">
<label>46</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn46.gif"></graphic>
</disp-formula>
For ψ
<sub>0</sub>
we could, as in (
<xref ref-type="disp-formula" rid="jpg417154eqn39">39</xref>
), write ψ
<sub>0</sub>
as a Gaussian, now multiplied by a Gaussian in position space of width 1/
<italic>w</italic>
and exponential
<inline-formula>
<tex-math></tex-math>
<inline-graphic xlink:href="jpg417154ieqn8.gif"></inline-graphic>
</inline-formula>
. However, our conclusions are independent of the exact form of ψ
<sub>0</sub>
. Instead we notice that on performing the space-time integral above, we get the analogous formula to (
<xref ref-type="disp-formula" rid="jpg417154eqn42">42</xref>
) which we can write as
<disp-formula id="jpg417154eqn47">
<label>47</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn47.gif"></graphic>
</disp-formula>
where
<disp-formula id="jpg417154eqn48">
<label>48</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn48.gif"></graphic>
</disp-formula>
and Φ, strongly peaked about
<inline-formula>
<tex-math></tex-math>
<inline-graphic xlink:href="jpg417154ieqn9.gif"></inline-graphic>
</inline-formula>
and
<italic>E</italic>
=
<italic>E</italic>
<sub>0</sub>
, incorporates both the momentum and energy dependence induced by the initial wave function.</p>
<p>Now we appeal to the Grimus–Stockinger theorem [
<xref ref-type="bibr" rid="jpg417154bib21">21</xref>
] which states that
<xref ref-type="fn" rid="jpg417154fn3">
<sup>3</sup>
</xref>
<fn id="jpg417154fn3">
<label>3</label>
<p>Providing Φ satisfies the reasonable conditions that it is three-times continuously differentiable and first and second derivatives decrease at least as
<inline-formula>
<tex-math></tex-math>
<inline-graphic xlink:href="jpg417154ieqn10.gif"></inline-graphic>
</inline-formula>
for
<inline-formula>
<tex-math></tex-math>
<inline-graphic xlink:href="jpg417154ieqn11.gif"></inline-graphic>
</inline-formula>
.
<inline-formula>
<tex-math></tex-math>
<inline-graphic xlink:href="jpg417154ieqn12.gif"></inline-graphic>
</inline-formula>
is the unit vector
<inline-formula>
<tex-math></tex-math>
<inline-graphic xlink:href="jpg417154ieqn13.gif"></inline-graphic>
</inline-formula>
.</p>
</fn>
<disp-formula id="jpg417154eqn49">
<label>49</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn49.gif"></graphic>
</disp-formula>
if
<inline-formula>
<tex-math></tex-math>
<inline-graphic xlink:href="jpg417154ieqn14.gif"></inline-graphic>
</inline-formula>
is real, and is
<italic>O</italic>
(1/
<italic>L</italic>
<sup>2</sup>
) otherwise.</p>
<p>We see that in the domain |
<italic>E</italic>
| >
<italic>m</italic>
, the remaining integral (
<xref ref-type="disp-formula" rid="jpg417154eqn47">47</xref>
) generalizes the case where
<italic>p</italic>
is real in (
<xref ref-type="disp-formula" rid="jpg417154eqn43">43</xref>
), and thus gives only the analogous dependence to (
<xref ref-type="disp-formula" rid="jpg417154eqn44">44</xref>
). The difference is that
<inline-formula>
<tex-math></tex-math>
<inline-graphic xlink:href="jpg417154ieqn15.gif"></inline-graphic>
</inline-formula>
attempts simultaneously to constrain
<inline-formula>
<tex-math></tex-math>
<inline-graphic xlink:href="jpg417154ieqn16.gif"></inline-graphic>
</inline-formula>
through dependence on its first argument and
<italic>p</italic>
<italic>p</italic>
<sub>0</sub>
through dependence on its second argument. The result is suppression through the partial overlap of these two peaks. Note however that we do not generate evanescent tails as a result of this mismatch.</p>
<p>Therefore it is
<italic>not</italic>
the case that giving the neutrino some momentum
<inline-formula>
<tex-math></tex-math>
<inline-graphic xlink:href="jpg417154ieqn17.gif"></inline-graphic>
</inline-formula>
slightly larger in magnitude than the energetically allowed value
<italic>p</italic>
<sub>0</sub>
, results in a wave packet that has form (
<xref ref-type="disp-formula" rid="jpg417154eqn44">44</xref>
) in the direction
<inline-formula>
<tex-math></tex-math>
<inline-graphic xlink:href="jpg417154ieqn18.gif"></inline-graphic>
</inline-formula>
but with large evanescent transverse tails.</p>
<p>On the other hand, for |
<italic>E</italic>
| <
<italic>m</italic>
,
<italic>p</italic>
is imaginary, and
<italic>J</italic>
decays faster in
<italic>L</italic>
. This is the expected evanescence. We see that even with this ‘boosted’ wave packet, it is still the case that the evanescent behaviour responsible for superluminal propagation, is restricted to the regime
<italic>E</italic>
<
<italic>m</italic>
.</p>
</sec>
<sec id="jpg417154s6">
<label>6.</label>
<title>Conclusions</title>
<p>We have seen that if the mixing angle θ
<sub>23</sub>
is close to maximal then thanks to interference between the two mass eigenstates, the group velocity for the muon neutrino wave packet, (
<xref ref-type="disp-formula" rid="jpg417154eqn21">21</xref>
) using (
<xref ref-type="disp-formula" rid="jpg417154eqn22">22</xref>
) and (
<xref ref-type="disp-formula" rid="jpg417154eqn23">23</xref>
), develops a series of sharp peaks at critical momenta
<disp-formula id="jpg417154eqn50">
<label>50</label>
<tex-math></tex-math>
<graphic xlink:href="jpg417154eqn50.gif"></graphic>
</disp-formula>
where
<italic>a</italic>
is a non-negative integer. It is surely interesting in its own right that superluminal propagation of neutrinos is possible in principle through such an effect. In the OPERA experiment this corresponds to energies 1.43/(1 + 2
<italic>a</italic>
) GeV and in practice
<italic>a</italic>
is bounded above by the fact that for sufficiently large
<italic>a</italic>
it is no longer possible to detect the neutrino, definitely
<italic>a</italic>
⩽ 6.</p>
<p>However from (
<xref ref-type="disp-formula" rid="jpg417154eqn35">35</xref>
) we have seen that the maximum extra displacement caused by this is limited above by the width of the wave packet Δ
<italic>x</italic>
. This translates into a maximum increase in group velocity of Δ
<italic>x</italic>
/
<italic>t</italic>
, where
<italic>t</italic>
is the naïvely expected time of flight. Therefore we see that the effect is not in conflict with relativity and causality, being limited to no more than that caused by the inherent quantum mechanical uncertainty in position.</p>
<p>We have pointed to a number of potential sources of decoherence which could contribute to Δ
<italic>x</italic>
. The maximum possible coherence length is set by features in the proton beam, which would give Δ
<italic>x</italic>
∼ 1m. In practice the other sources of decoherence will contribute, so realistically Δ
<italic>x</italic>
is order 1 μm or less. The maximum superluminal correction is therefore at most δ
<italic>v</italic>
<sub>sup</sub>
≈ 10
<sup>−11</sup>
<italic>c</italic>
, which is far too small to correspond to the effect seen by OPERA.</p>
<p>We note that this also potentially provides a severe constraint on proposals to explain the OPERA measurement via superluminal propagation of only certain mass eigenstates (e.g. a sterile neutrino in extra dimensions [
<xref ref-type="bibr" rid="jpg417154bib06">6</xref>
]) since the OPERA measurement corresponds to an extra displacement by approximately 20 m, whereas a relative shift between mass eigenstate wave packets of more than a few μm is enough to destroy the coherence of the observed neutrino oscillations.</p>
<p>We do not reach the upper bound, Δ
<italic>x</italic>
/
<italic>t</italic>
, on the maximum extra group velocity unless the mixing angle θ
<sub>23</sub>
is closer to maximal than a term of order
<italic>w</italic>
/
<italic>p</italic>
, cf equation (
<xref ref-type="disp-formula" rid="jpg417154eqn34">34</xref>
). Such finite-width corrections are small everywhere except in this case, and then only when close to the group velocity peaks, where they matter because of they correct the near vanishing of the denominator term in (
<xref ref-type="disp-formula" rid="jpg417154eqn23">23</xref>
). There are some corrections to the numerator that become important similarly since it is also close to vanishing, although we did not need them for our analysis. In practice the neglected small corrections due to the other neutrino mixing parameters (δ, θ
<sub>13</sub>
and the effective value of θ
<sub>12</sub>
in rock) could also have an effect when close to these peaks for the same reason. The exact behaviour close to the peaks is therefore the result of a competition between these corrections, and will depend on which one dominates.</p>
<p>None of this matters for the measurements performed in the OPERA experiment however since they measure an effect that is averaged over the broad energy spectrum of the neutrinos. The average effect would give a strong dependence on energy, which is not seen in the OPERA experiment, if it weren't for the fact that the result is anyway too small to measure. Furthermore we have seen that, once averaged over the energy spectrum of the neutrino beam, the resulting net correction (
<xref ref-type="disp-formula" rid="jpg417154eqn37">37</xref>
) is ∼10
<sup>−23</sup>
, at best of the same order as the negative (i.e. subluminal) classical correction.</p>
<p>It seems very difficult to imagine a scenario where this effect could explain the OPERA measurement. We note that the neutrinos with superluminal group velocity would be found preferentially at the leading edge of the neutrino bunch. But from the plots [
<xref ref-type="bibr" rid="jpg417154bib01">1</xref>
,
<xref ref-type="bibr" rid="jpg417154bib19">19</xref>
], it is clear that there is no tail of early arrivers: the shift due to an excess of superluminal neutrinos at the leading edge cannot be much more than the measured δ
<italic>v</italic>
, and besides we have too few of them by this effect in the energy spectrum. Similar comments apply to the superluminal-depleted population at the trailing edge. With the new measurements using shorter pulses [
<xref ref-type="bibr" rid="jpg417154bib01">1</xref>
] all such types of explanation are ruled out.</p>
<p>We have also seen that although superluminal propagation from CN to GS is possible if the mass of the lowest neutrino mass eigenstate is so small that it remains off shell, cf (
<xref ref-type="disp-formula" rid="jpg417154eqn06">6</xref>
), the effect does not survive projection on the relevant energy scales, being killed typically by a huge suppression. This projection is required because one must integrate over the initial neutrino wave function which carries a very rapidly oscillating exponential, set by the multi-GeV energy of the created neutrino. One could reduce the suppression (i.e. increase the overlap with energy scale
<italic>m</italic>
) by concentrating on muon neutrinos with energies only just higher than the threshold energy of 112 MeV. This is still not enough: one would have to assume an estimate of
<italic>w</italic>
at the larger end and even if one does this, the effect is still concentrated at energies of order
<italic>m</italic>
, resulting in the wrong energy dependence.</p>
<p>One could reduce the time for which the neutrino has to remain off shell by considering cases where it propagates on shell to a point near the detector and, due to interaction with the rock, is then kicked off shell for the remaining part of its journey. Note that an off-shell neutrino can be passed from this point to the detector in principle almost instantaneously. In this way we achieve the smallest negative interval
<italic>s</italic>
from the instantaneous jump (0, δ
<italic>t</italic>
) from (
<italic>L</italic>
− δ
<italic>t</italic>
,
<italic>L</italic>
− δ
<italic>t</italic>
) to the detector. In (
<xref ref-type="disp-formula" rid="jpg417154eqn06">6</xref>
) we then get the larger mass
<italic>m</italic>
∼ 1/δ
<italic>t</italic>
∼ 10
<sup>−8</sup>
eV [
<xref ref-type="bibr" rid="jpg417154bib08">8</xref>
]. This is still far too small to bridge the gap to the energy scales set by the experiment.</p>
<p>Although the OPERA measurements cannot be explained by assuming very low mass off-shell neutrinos, it is still the case that such neutrinos can propagate superluminally. How can this be reconciled with causality, in particular why does this not lead to faster-than-light communication? Note that we have shown that these neutrinos must carry energy
<italic>E</italic>
less than
<italic>m</italic>
. Leaving aside practical issues involved in detecting such neutrinos, we note that even in principle, a detector for such neutrinos would have to be restricted to measuring distances Δ
<italic>x</italic>
larger than or of order their Compton wavelength λ = 1/
<italic>m</italic>
, and similarly periods Δ
<italic>t</italic>
> 1/
<italic>m</italic>
, otherwise the observation itself would disturb the system too much—for example by pair creating the very neutrinos it was trying to measure. It follows that the uncertainty in the speed measurement from measuring the interval (
<italic>t</italic>
,
<italic>x</italic>
) ≈ (
<italic>L</italic>
,
<italic>L</italic>
) is
<disp-formula id="jpg417154ueq01">
<tex-math></tex-math>
<graphic xlink:href="jpg417154ueq01.gif"></graphic>
</disp-formula>
Here we have used (
<xref ref-type="disp-formula" rid="jpg417154eqn06">6</xref>
) and the fact that the faster than light travel
<italic>v</italic>
− 1 = δ
<italic>t</italic>
/
<italic>L</italic>
≪ 1. Thus we see that the restriction to
<italic>E</italic>
<
<italic>m</italic>
for superluminal neutrinos, which comes out from the detailed analysis, ensures that they cannot be detected with sufficient spatial resolution to allow faster than light signals.</p>
</sec>
</body>
<back>
<ack>
<title>Acknowledgments</title>
<p>The author thanks the STFC for financial support and Konstantin Kuzmin and Bartolome Alles Salom for useful conversations.</p>
</ack>
<ref-list content-type="numerical">
<title>References</title>
<ref id="jpg417154bib01">
<label>1</label>
<element-citation publication-type="preprint">
<person-group person-group-type="author">
<name>
<surname>Adam</surname>
<given-names>T</given-names>
</name>
<etal></etal>
<collab>(OPERA Collaboration)</collab>
</person-group>
<year>2011</year>
<comment>arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1109.4897v2">1109.4897v2</ext-link>
[hep-ex]</comment>
</element-citation>
</ref>
<ref id="jpg417154bib02">
<label>2</label>
<element-citation publication-type="preprint" id="jpg417154bib02a">
<person-group person-group-type="author">
<name>
<surname>Fargion</surname>
<given-names>D</given-names>
</name>
<name>
<surname>D'Armiento</surname>
<given-names>D</given-names>
</name>
</person-group>
<year>2011</year>
<comment>arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1109.5368">1109.5368</ext-link>
[astro-ph.HE]</comment>
</element-citation>
<element-citation publication-type="preprint" id="jpg417154bib02b">
<person-group person-group-type="author">
<name>
<surname>Giudice</surname>
<given-names>G F</given-names>
</name>
<name>
<surname>Sibiryakov</surname>
<given-names>S</given-names>
</name>
<name>
<surname>Strumia</surname>
<given-names>A</given-names>
</name>
</person-group>
<year>2011</year>
<comment>arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1109.5682">1109.5682</ext-link>
[hep-ph]</comment>
</element-citation>
<element-citation publication-type="preprint" id="jpg417154bib02c">
<person-group person-group-type="author">
<name>
<surname>Oikonomou</surname>
<given-names>V K</given-names>
</name>
</person-group>
<year>2011</year>
<comment>arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1109.6170">1109.6170</ext-link>
[hep-th]</comment>
</element-citation>
<element-citation publication-type="preprint" id="jpg417154bib02d">
<person-group person-group-type="author">
<name>
<surname>Wang</surname>
<given-names>P</given-names>
</name>
<name>
<surname>Wu</surname>
<given-names>H</given-names>
</name>
<name>
<surname>Yang</surname>
<given-names>H</given-names>
</name>
</person-group>
<year>2011</year>
<comment>arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1109.6930">1109.6930</ext-link>
[hep-ph]</comment>
</element-citation>
<element-citation publication-type="preprint" id="jpg417154bib02e">
<person-group person-group-type="author">
<name>
<surname>Anacleto</surname>
<given-names>M A</given-names>
</name>
<name>
<surname>Brito</surname>
<given-names>F A</given-names>
</name>
<name>
<surname>Passos</surname>
<given-names>E</given-names>
</name>
</person-group>
<year>2011</year>
<comment>arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1109.6298">1109.6298</ext-link>
[hep-th]</comment>
</element-citation>
<element-citation publication-type="preprint" id="jpg417154bib02f">
<person-group person-group-type="author">
<name>
<surname>Giacosa</surname>
<given-names>F</given-names>
</name>
<name>
<surname>Kovacs</surname>
<given-names>P</given-names>
</name>
<name>
<surname>Lottini</surname>
<given-names>S</given-names>
</name>
</person-group>
<year>2011</year>
<comment>arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1110.3642">1110.3642</ext-link>
[hep-ph]</comment>
</element-citation>
<element-citation publication-type="preprint" id="jpg417154bib02g">
<person-group person-group-type="author">
<name>
<surname>Bramante</surname>
<given-names>J</given-names>
</name>
</person-group>
<year>2011</year>
<comment>arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1110.4871">1110.4871</ext-link>
[hep-ph]</comment>
</element-citation>
</ref>
<ref id="jpg417154bib03">
<label>3</label>
<element-citation publication-type="preprint">
<person-group person-group-type="author">
<name>
<surname>Mecozzi</surname>
<given-names>A</given-names>
</name>
<name>
<surname>Bellini</surname>
<given-names>M</given-names>
</name>
</person-group>
<year>2011</year>
<comment>arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1110.1253">1110.1253</ext-link>
[hep-ph]</comment>
</element-citation>
</ref>
<ref id="jpg417154bib04">
<label>4</label>
<element-citation publication-type="preprint">
<person-group person-group-type="author">
<name>
<surname>Tanimura</surname>
<given-names>S</given-names>
</name>
</person-group>
<year>2011</year>
<comment>arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1110.1790">1110.1790</ext-link>
[hep-ph]</comment>
</element-citation>
</ref>
<ref id="jpg417154bib05">
<label>5</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Berry</surname>
<given-names>M V</given-names>
</name>
<name>
<surname>Brunner</surname>
<given-names>N</given-names>
</name>
<name>
<surname>Popescu</surname>
<given-names>S</given-names>
</name>
<name>
<surname>Shukla</surname>
<given-names>P</given-names>
</name>
</person-group>
<year>2011</year>
<source>J. Phys. A: Math. Theor.</source>
<volume>44</volume>
<elocation-id content-type="artnum">492001</elocation-id>
<comment>(arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1110.2832">1110.2832</ext-link>
[hep-ph])</comment>
<pub-id pub-id-type="doi">10.1088/1751-8113/44/49/492001</pub-id>
</element-citation>
</ref>
<ref id="jpg417154bib06">
<label>6</label>
<element-citation publication-type="preprint">
<person-group person-group-type="author">
<name>
<surname>Nicolaidis</surname>
<given-names>A</given-names>
</name>
</person-group>
<year>2011</year>
<comment>arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1109.6354">1109.6354</ext-link>
[hep-ph]</comment>
</element-citation>
</ref>
<ref id="jpg417154bib07">
<label>7</label>
<element-citation publication-type="preprint" id="jpg417154bib07a">
<person-group person-group-type="author">
<name>
<surname>Bi</surname>
<given-names>X J</given-names>
</name>
<name>
<surname>Yin</surname>
<given-names>P F</given-names>
</name>
<name>
<surname>Yu</surname>
<given-names>Z H</given-names>
</name>
<name>
<surname>Yuan</surname>
<given-names>Q</given-names>
</name>
</person-group>
<year>2011</year>
<comment>arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1109.6667">1109.6667</ext-link>
[hep-ph]</comment>
</element-citation>
<element-citation publication-type="preprint" id="jpg417154bib07b">
<person-group person-group-type="author">
<name>
<surname>Cohen</surname>
<given-names>A G</given-names>
</name>
<name>
<surname>Glashow</surname>
<given-names>S L</given-names>
</name>
</person-group>
<year>2011</year>
<comment>arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1109.6562">1109.6562</ext-link>
[hep-ph]</comment>
</element-citation>
</ref>
<ref id="jpg417154bib08">
<label>8</label>
<element-citation publication-type="preprint">
<person-group person-group-type="author">
<name>
<surname>Ahluwalia</surname>
<given-names>D V</given-names>
</name>
<name>
<surname>Horvath</surname>
<given-names>S P</given-names>
</name>
<name>
<surname>Schritt</surname>
<given-names>D</given-names>
</name>
</person-group>
<year>2011</year>
<comment>arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1110.1162">1110.1162</ext-link>
[hep-ph]</comment>
</element-citation>
</ref>
<ref id="jpg417154bib09">
<label>9</label>
<element-citation publication-type="preprint">
<person-group person-group-type="author">
<name>
<surname>Naumov</surname>
<given-names>D V</given-names>
</name>
<name>
<surname>Naumov</surname>
<given-names>V A</given-names>
</name>
</person-group>
<year>2011</year>
<comment>arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1110.0989">1110.0989</ext-link>
[hep-ph]</comment>
</element-citation>
</ref>
<ref id="jpg417154bib10">
<label>10</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Wang</surname>
<given-names>Y-P</given-names>
</name>
<name>
<surname>Zhang</surname>
<given-names>D-L</given-names>
</name>
</person-group>
<year>1995</year>
<source>Phys. Rev.
<named-content content-type="jnl-part">A</named-content>
</source>
<volume>52</volume>
<fpage>2597</fpage>
<lpage>2600</lpage>
<page-range>2597–600</page-range>
<pub-id pub-id-type="doi">10.1103/PhysRevA.52.2597</pub-id>
</element-citation>
</ref>
<ref id="jpg417154bib11">
<label>11</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Sánchez-López</surname>
<given-names>M M</given-names>
</name>
<name>
<surname>Sánchez-Meroño</surname>
<given-names>A</given-names>
</name>
<name>
<surname>Arias</surname>
<given-names>J</given-names>
</name>
<name>
<surname>Davis</surname>
<given-names>J A</given-names>
</name>
<name>
<surname>Moreno</surname>
<given-names>I</given-names>
</name>
</person-group>
<year>2008</year>
<source>Appl. Phys. Lett.</source>
<volume>93</volume>
<elocation-id content-type="artnum">074102</elocation-id>
<pub-id pub-id-type="doi">10.1063/1.2969407</pub-id>
</element-citation>
</ref>
<ref id="jpg417154bib12">
<label>12</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Brunner</surname>
<given-names>N</given-names>
</name>
<name>
<surname>Scarani</surname>
<given-names>V</given-names>
</name>
<name>
<surname>Wegmüller</surname>
<given-names>M</given-names>
</name>
<name>
<surname>Legré</surname>
<given-names>M</given-names>
</name>
<name>
<surname>Gisin</surname>
<given-names>N</given-names>
</name>
</person-group>
<year>2004</year>
<source>Phys. Rev. Lett.</source>
<volume>93</volume>
<elocation-id content-type="artnum">203902</elocation-id>
<pub-id pub-id-type="doi">10.1103/PhysRevLett.93.203902</pub-id>
</element-citation>
</ref>
<ref id="jpg417154bib13">
<label>13</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Nakamura</surname>
<given-names>K</given-names>
</name>
<etal></etal>
<collab>(Particle Data Group)</collab>
</person-group>
<year>2010</year>
<source>J. Phys. G: Nucl. Part. Phys.</source>
<volume>37</volume>
<elocation-id content-type="artnum">075021</elocation-id>
<comment>and 2011 partial update for the 2012 edition</comment>
<pub-id pub-id-type="doi">10.1088/0954-3899/37/7A/075021</pub-id>
</element-citation>
</ref>
<ref id="jpg417154bib14">
<label>14</label>
<element-citation publication-type="preprint">
<person-group person-group-type="author">
<name>
<surname>Fogli</surname>
<given-names>G L</given-names>
</name>
<name>
<surname>Lisi</surname>
<given-names>E</given-names>
</name>
<name>
<surname>Marrone</surname>
<given-names>A</given-names>
</name>
<name>
<surname>Palazzo</surname>
<given-names>A</given-names>
</name>
<name>
<surname>Rotunno</surname>
<given-names>A M</given-names>
</name>
</person-group>
<year>2011</year>
<comment>arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1106.6028">1106.6028</ext-link>
[hep-ph]</comment>
</element-citation>
</ref>
<ref id="jpg417154bib15">
<label>15</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Ball</surname>
<given-names>A E</given-names>
</name>
<etal></etal>
</person-group>
<year>2002</year>
<article-title>CERN SL Division</article-title>
<source>Technical Note</source>
<comment>SL-Note-2002-040 EA EMDS No. 36258
<ext-link ext-link-type="uri" xlink:href="http://cdsweb.cern.ch/record/702721">http://cdsweb.cern.ch/record/702721</ext-link>
</comment>
</element-citation>
</ref>
<ref id="jpg417154bib16">
<label>16</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Agafonova</surname>
<given-names>N</given-names>
</name>
<etal></etal>
</person-group>
<year>2009</year>
<source>J. Instrum.</source>
<volume>4</volume>
<elocation-id content-type="artnum">P06020</elocation-id>
<comment>(arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/0903.2973">0903.2973</ext-link>
[hep-ex])</comment>
<pub-id pub-id-type="doi">10.1088/1748-0221/4/06/P06020</pub-id>
</element-citation>
</ref>
<ref id="jpg417154bib17">
<label>17</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Naumov</surname>
<given-names>D V</given-names>
</name>
<name>
<surname>Naumov</surname>
<given-names>V A</given-names>
</name>
</person-group>
<year>2010</year>
<source>J. Phys. G: Nucl. Part. Phys.</source>
<volume>37</volume>
<elocation-id content-type="artnum">105014</elocation-id>
<comment>(arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1008.0306">1008.0306</ext-link>
[hep-ph])</comment>
<pub-id pub-id-type="doi">10.1088/0954-3899/37/10/105014</pub-id>
</element-citation>
</ref>
<ref id="jpg417154bib18">
<label>18</label>
<element-citation publication-type="journal" id="jpg417154bib18a">
<person-group person-group-type="author">
<name>
<surname>Bilenky</surname>
<given-names>S M</given-names>
</name>
<name>
<surname>von Feilitzsch</surname>
<given-names>F</given-names>
</name>
<name>
<surname>Potzel</surname>
<given-names>W</given-names>
</name>
</person-group>
<year>2011</year>
<source>J. Phys. G: Nucl. Part. Phys.</source>
<volume>38</volume>
<elocation-id content-type="artnum">115002</elocation-id>
<comment>(arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1102.2770">1102.2770</ext-link>
[hep-ph])</comment>
<pub-id pub-id-type="doi">10.1088/0954-3899/38/11/115002</pub-id>
</element-citation>
<element-citation publication-type="journal" id="jpg417154bib18b">
<person-group person-group-type="author">
<name>
<surname>Beuthe</surname>
<given-names>M</given-names>
</name>
</person-group>
<year>2003</year>
<source>Phys. Rep.</source>
<volume>375</volume>
<fpage>105</fpage>
<comment>(arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/hep-ph/0109119">hep-ph/0109119</ext-link>
)</comment>
<pub-id pub-id-type="doi">10.1016/S0370-1573(02)00538-0</pub-id>
</element-citation>
<element-citation publication-type="journal" id="jpg417154bib18c">
<person-group person-group-type="author">
<name>
<surname>Giunti</surname>
<given-names>C</given-names>
</name>
</person-group>
<year>2004</year>
<source>Found. Phys. Lett.</source>
<volume>17</volume>
<fpage>103</fpage>
<comment>(arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/hep-ph/0302026">hep-ph/0302026</ext-link>
)</comment>
<pub-id pub-id-type="doi">10.1023/B:FOPL.0000019651.53280.31</pub-id>
</element-citation>
<element-citation publication-type="journal" id="jpg417154bib18d">
<person-group person-group-type="author">
<name>
<surname>Giunti</surname>
<given-names>C</given-names>
</name>
</person-group>
<year>2008</year>
<source>AIP Conf. Proc.</source>
<volume>1026</volume>
<fpage>3</fpage>
<lpage>119</lpage>
<page-range>3–19</page-range>
<comment>(arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/0801.0653">0801.0653</ext-link>
)</comment>
<pub-id pub-id-type="doi">10.1063/1.2965075</pub-id>
</element-citation>
<element-citation publication-type="journal" id="jpg417154bib18e">
<person-group person-group-type="author">
<name>
<surname>Akhmedov</surname>
<given-names>E</given-names>
</name>
<name>
<surname>Kopp</surname>
<given-names>J</given-names>
</name>
</person-group>
<year>2010</year>
<source>J. High Energy Phys.</source>
<elocation-id content-type="artnum">JHEP04(2010)008</elocation-id>
<comment>(arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1001.4815v2">1001.4815v2</ext-link>
)</comment>
<pub-id pub-id-type="doi">10.1007/JHEP04(2010)008</pub-id>
</element-citation>
<element-citation publication-type="preprint" id="jpg417154bib18f">
<person-group person-group-type="author">
<name>
<surname>Akhmedov</surname>
<given-names>E Kh</given-names>
</name>
<name>
<surname>Smirnov</surname>
<given-names>A Yu</given-names>
</name>
</person-group>
<year>2011</year>
<comment>arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1008.2077">1008.2077</ext-link>
</comment>
</element-citation>
<element-citation publication-type="preprint" id="jpg417154bib18g">
<person-group person-group-type="author">
<name>
<surname>Wu</surname>
<given-names>J</given-names>
</name>
<name>
<surname>Hutasoit</surname>
<given-names>J A</given-names>
</name>
<name>
<surname>Boyanovsky</surname>
<given-names>D</given-names>
</name>
<name>
<surname>Holman</surname>
<given-names>R</given-names>
</name>
</person-group>
<year>2010</year>
<comment>arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1002.2649">1002.2649</ext-link>
</comment>
</element-citation>
</ref>
<ref id="jpg417154bib19">
<label>19</label>
<element-citation publication-type="webpage">
<person-group person-group-type="author">
<name>
<surname>Autiero</surname>
<given-names>D</given-names>
</name>
</person-group>
<year>2011</year>
<article-title>New results from OPERA on neutrino properties</article-title>
<comment>
<ext-link ext-link-type="uri" xlink:href="http://cdsweb.cern.ch/record/1384486">http://cdsweb.cern.ch/record/1384486</ext-link>
</comment>
</element-citation>
</ref>
<ref id="jpg417154bib20">
<label>20</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Agafonova</surname>
<given-names>N</given-names>
</name>
<etal></etal>
<collab>(OPERA Collaboration)</collab>
</person-group>
<year>2011</year>
<source>New J. Phys.</source>
<volume>13</volume>
<elocation-id content-type="artnum">053051</elocation-id>
<comment>(arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/1102.1882">1102.1882</ext-link>
[hep-ex])</comment>
<pub-id pub-id-type="doi">10.1088/1367-2630/13/5/053051</pub-id>
</element-citation>
</ref>
<ref id="jpg417154bib21">
<label>21</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Grimus</surname>
<given-names>W</given-names>
</name>
<name>
<surname>Stockinger</surname>
<given-names>P</given-names>
</name>
</person-group>
<year>1996</year>
<source>Phys. Rev.
<named-content content-type="jnl-part">D</named-content>
</source>
<volume>54</volume>
<fpage>3414</fpage>
<comment>(arXiv:
<ext-link ext-link-type="arxiv" xlink:href="http://arxiv.org/abs/hep-ph/9603430">hep-ph/9603430</ext-link>
)</comment>
<pub-id pub-id-type="doi">10.1103/PhysRevD.54.3414</pub-id>
</element-citation>
</ref>
</ref-list>
</back>
</article>
</istex:document>
</istex:metadataXml>
<mods version="3.6">
<titleInfo>
<title>Superluminal velocity through near-maximal neutrino oscillations or by being off shell</title>
</titleInfo>
<titleInfo type="alternative" contentType="CDATA">
<title>Superluminal velocity through near-maximal neutrino oscillations or by being off shell</title>
</titleInfo>
<name type="personal">
<namePart type="given">Tim R</namePart>
<namePart type="family">Morris</namePart>
<affiliation>School of Physics and Astronomy, University of Southampton Highfield, Southampton, SO17 1BJ, UK</affiliation>
</name>
<typeOfResource>text</typeOfResource>
<genre type="research-article" displayLabel="research-article"></genre>
<subject>
<genre>article-type</genre>
<topic>Paper</topic>
</subject>
<subject>
<genre>section</genre>
<topic>Particle physics</topic>
</subject>
<originInfo>
<publisher>IOP Publishing</publisher>
<dateIssued encoding="w3cdtf">2012-04</dateIssued>
<dateCreated encoding="w3cdtf">2012-03-16</dateCreated>
<copyrightDate encoding="w3cdtf">2012</copyrightDate>
</originInfo>
<language>
<languageTerm type="code" authority="iso639-2b">eng</languageTerm>
<languageTerm type="code" authority="rfc3066">en</languageTerm>
</language>
<physicalDescription>
<internetMediaType>text/html</internetMediaType>
</physicalDescription>
<abstract>Recently it was suggested that the observation of superluminal neutrinos by the OPERA collaboration may be due to group velocity effects resulting from close-to-maximal oscillation between neutrino mass eigenstates, in analogy to known effects in optics. We show that superluminal propagation does occur through this effect for a series of very narrow energy ranges, but this phenomenon cannot explain the OPERA measurement. Superluminal propagation can also occur if one of the neutrino masses is extremely small. However the effect only has appreciable amplitude at energies of order this mass and thus has negligible overlap with the multi-GeV scale of the experiment.</abstract>
<subject>
<genre>author-pacs</genre>
<topic>14.60.Lm</topic>
<topic>14.60.Pq</topic>
</subject>
<subject>
<genre>Keywords</genre>
<topic>Neutrinos OPERA</topic>
</subject>
<relatedItem type="host">
<titleInfo>
<title>Journal of Physics G Nuclear and Particle Physics</title>
</titleInfo>
<genre type="Journal">journal</genre>
<identifier type="ISSN">0954-3899</identifier>
<identifier type="eISSN">1361-6471</identifier>
<identifier type="PublisherID">jpg</identifier>
<part>
<date>2012</date>
<detail type="volume">
<caption>vol.</caption>
<number>39</number>
</detail>
<detail type="issue">
<caption>no.</caption>
<number>4</number>
</detail>
<extent unit="pages">
<total>13</total>
</extent>
</part>
</relatedItem>
<identifier type="istex">018F1A8D7E43B5791309459D40767777B8531BBC</identifier>
<identifier type="DOI">10.1088/0954-3899/39/4/045010</identifier>
<identifier type="href">http://stacks.iop.org/JPhysG/39/045010</identifier>
<identifier type="ArticleID">jpg417154</identifier>
<accessCondition type="use and reproduction" contentType="copyright">2012 IOP Publishing Ltd</accessCondition>
<recordInfo>
<recordContentSource>IOP</recordContentSource>
</recordInfo>
</mods>
</metadata>
<serie></serie>
</istex>
</record>

Pour manipuler ce document sous Unix (Dilib)

EXPLOR_STEP=$WICRI_ROOT/Wicri/Musique/explor/OperaV1/Data/Istex/Corpus
HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 000104 | SxmlIndent | more

Ou

HfdSelect -h $EXPLOR_AREA/Data/Istex/Corpus/biblio.hfd -nk 000104 | SxmlIndent | more

Pour mettre un lien sur cette page dans le réseau Wicri

{{Explor lien
   |wiki=    Wicri/Musique
   |area=    OperaV1
   |flux=    Istex
   |étape=   Corpus
   |type=    RBID
   |clé=     ISTEX:018F1A8D7E43B5791309459D40767777B8531BBC
   |texte=   Superluminal velocity through near-maximal neutrino oscillations or by being off shell
}}

Wicri

This area was generated with Dilib version V0.6.21.
Data generation: Thu Apr 14 14:59:05 2016. Site generation: Thu Oct 8 06:48:41 2020