We consider the possibility of Lorentz-invariance violation in weak-decay
processes. We present a general approach that entails modifying the W-boson
propagator by adding a Lorentz-violating tensor to it. We describe the effects
of Lorentz violation on nuclear beta decay in this scenario. In particular we
show the expression for a first-forbidden transition with a spin change of two.
Using data from an old experiment on the rotational invariance of yttrium-90,
we derive several bounds on the Lorentz-violating parameters of the order of
10^(-6)-10^(-8).Comment: 4 pages; presented at the Sixth Meeting on CPT and Lorentz Symmetry,
  Bloomington, Indiana, June 17-21, 2013; Added reference

Noordmans, J. P.

Timmermans, R. G. E.

Wilschut, H. W.

English

arXiv

Noordmans, J.P.

University of Groningen Research Database

  
 University of Groningen
Nuclear beta decay with Lorentz violation
Noordmans, J.P.; Wilschut, Henricus; Timmermans, Robertus
Published in:
Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)
IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from
it. Please check the document version below.
Publication date:
2013
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Noordmans, J. P., Wilschut, H. W., & Timmermans, R. G. E. (2013). Nuclear beta decay with Lorentz
violation. In Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)
Copyright
Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the
author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).
Take-down policy
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately
and investigate your claim.
Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the
number of authors shown on this cover page is limited to 10 maximum.
Download date: 11-02-2018
ar
X
iv
:1
30
8.
58
29
v2
  [
he
p-
ph
]  
7 N
ov
 20
13
Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)
1
NUCLEAR β DECAY WITH LORENTZ VIOLATION
J.P. NOORDMANS,∗ H.W. WILSCHUT, and R.G.E. TIMMERMANS
KVI, University of Groningen
Zernikelaan 25, NL-9747 AA Groningen, The Netherlands
∗E-mail: J.P.Noordmans@rug.nl
We consider the possibility of Lorentz-invariance violation in weak-decay pro-
cesses. We present a general approach that entails modifying the W -boson
propagator by adding a Lorentz-violating tensor to it. We describe the effects
of Lorentz violation on nuclear β decay in this scenario. In particular we show
the expression for a first-forbidden transition with a spin change of two. Us-
ing data from an old experiment on the rotational invariance of yttrium-90,
we derive several bounds on the Lorentz-violating parameters of the order of
10−6-10−8.
1. Introduction
There has been considerable scientific interest in the possibility of the viola-
tion of Lorentz symmetry, in the context of searches for a theory of quantum
gravity. Its phenomenological consequences have been studied extensively,
in particular in the context of the Standard-Model Extension (SME).1 Ex-
tensive experimental efforts have been made to bound the coefficients in all
sectors of the SME.2
Missing in these efforts is the study of weak decay. Until recently, there
were, to our knowledge, only two dedicated experiments that addressed ro-
tational invariance violation in β decay.3,4 This effort is now supplemented
by an experiment at KVI.5
Theoretically, the issue of calculating cross sections and decay rates is
also far from settled, although some amount of work has already gone into
this.6
We started a joint experimental and theoretical effort at KVI, addressing
the issue of Lorentz violation in nuclear β decay. We incorporate Lorentz
violation in weak decay by modifying the gauge boson propagator to〈
Wµ+(p)W ν−(−p)
〉
=
−i(gµν + χµν)
M2W
, (1)
Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)
2
where gµν is the usual Minkowski metric and χµν a general complex (possi-
bly momentum-dependent) tensor, that parametrizes the Lorentz violation.
What we miss in this way are mainly kinematic properties of the external
particles (see Ref. 7 for more discussion).
2. β-decay transitions and forbiddenness
β-decay transitions are classified by the spin change from parent to daughter
nucleus (∆I) and the parities of the parent and daughter nuclei, denoted
by πi and πf , respectively.
One distinguishes different β-decay transitions also by their forbidden-
ness, or the degree of suppression of the quantum mechanical amplitude
that describes the process. This suppression is determined by factors of
three small quantities: R/λ, vN and αZ, which are the ratio of the nuclear
radius and the de Broglie wavelength of the leptons, the velocity of the
decaying nucleon in units of c, and the fine-structure constant times the
charge of the daughter nucleus, respectively. Factors of R/λ come from the
leptons carrying away orbital angular momentum (corresponding to higher
order terms in the multipole expansion of the lepton wavefunctions). The
quantities vN and αZ originate from relativistic and Coulomb effects, re-
spectively.
If we assume these three quantities to vanish, we limit ourselves to so-
called allowed β-decay transitions. The leptons do not carry away orbital
angular momentum in this case. Consequently, the total angular momentum
of the lepton pair can be either zero or one. In the Lorentz-symmetric
case this translates directly into the selection rules for allowed β decay,
which can have spin change zero or one while the relative parity must
be πiπf = +1. With Lorentz violation this connection is not that direct
anymore, as explained later.
When allowing for factors of R/λ, vN and αZ, the leptons can carry
off orbital angular momentum and the total angular momentum of the
lepton pair can be J = 0, 1, 2, . . ., which allows for a larger spin change in
the transition, as well as the possibility of a relative parity of πiπf = −1.
For a complete classification of transitions and their forbiddenness in the
Lorentz-symmetric and Lorentz-violating cases, see Ref. 8.
3. The effects of Lorentz violation
Using the propagator in Eq. (1), we have calculated the rate of general
β-decay transitions. The expression for allowed β decay can be found in
Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)
3
Ref. 7, and the one for forbidden transitions can be found in Ref. 9.
In Refs. 8 and 9 we found that there is a relative enhancement of Lorentz-
violating effects in transitions with ∆I ≥ 2 by a factor of αZ/pR. From the
very schematic expression
(1,σ)µhadron(gµν + χµν)(1,σ)
ν
lepton , (2)
that shows the hadron and lepton current, connected by theW -boson prop-
agator, we see that the effect of the tensor χµν is to connect the singlet and
triplet parts of the lepton and hadron current in an unconventional way.
For example the spacetime part of χµν connects the triplet of the hadron
current to the singlet of the lepton current. This means that there can be
a unit of spin change in the nucleus, while the lepton pair does not carry
off the corresponding unit of angular momentum. This is the cause of the
enhancement factor that occurs in some Lorentz-violating terms of the am-
plitude of transitions with ∆I ≥ 2. For a more elaborate discussion of this
effect, see Refs. 8 and 9.
4. Bounds on Lorentz violation
Using our results for a general β-decay transition we calculated the Lorentz-
violating transition rate of a first forbidden transition with ∆I = 2 and
πiπf = −1, dependent on the emission direction of the outgoing β particle.
It is given by
dλ
dΩdE
∝ R2
{
p2 + q2
+
αZ
pR
[
3
10
p3
E
(
χijr pˆ
ipˆj − 1
3
χ00r
)
∓
1
2
p2χ˜lipˆ
l ± p2χl0r pˆ
l
]}
, (3)
where p, q, R are the β-particle momentum, the neutrino momentum and
the nuclear radius respectively, χ˜i = ǫijkχjk, Latin indices run over spa-
tial components only, the subscripts r, i denote real and imaginary parts
respectively, and the upper (lower) sign corresponds to β− (β+) decay. We
see the relative enhancement factor αZ/pR of the Lorentz-violating effects.
Using the expression in Eq. (3) we reanalyzed results from an experi-
ment published in 1976 and described in Ref. 3. The experiment was done
using a 10 Ci strontium-90 source, which decays to yttrium-90 and subse-
quently to zirconium-90. The relevant decay is the one from 90Y to 90Z,
which is a first-forbidden transition with ∆I = 2 and πiπf = −1. In the ex-
periment the source and the detector were mounted on a turntable, which
Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)
4
rotated with a frequency of 0.75 Hz. Meanwhile the current of electrons
that originated from the source was measured. Using the data of 232 good
2-h runs, asymmetries were defined in terms of the emission direction of the
electrons. These asymmetries were fitted to distributions in sidereal time,
given by
δ = a0 + a1 sin(ωt+ φ1) + a2(2ωt+ φ2) . (4)
Bounds of the order of 10−8 were obtained on the constants a0, a1, and a2.
Using Eq. (3) we calculated the theoretical prediction for the asymmetries
defined in Ref. 3 and extracted the expressions for a0, a1, and a2 in terms
of χµν . In this way we were able to put the following bounds at a 95%
confidence level:9
|2X30r − X˜
3
i | < 2× 10
−8 , (5a)
|3X33r −X
00
r | < 3× 10
−6 , (5b)[
(X12r +X
21
r )
2 + (X22r −X
11
r )
2
]1/2
< 1× 10−6 , (5c)[
(X13r +X
31
r )
2 + (X23r +X
32
r )
2
]1/2
< 1× 10−6 , (5d)[
(2X20r − X˜
2
i )
2 + (2X10r − X˜
1
i )
2
]1/2
< 4× 10−8 , (5e)
where the Lorentz-violating parameter Xµν is now given in the standard
Sun-centered intertial reference frame.2 These bounds are the best direct
bounds on the parameter χµν of which we are aware.
References
1. D. Colladay and V.A. Kostelecky´, Phys. Rev. D 55, 6760 (1997); D. Colladay
and V.A. Kostelecky´, Phys. Rev. D 58, 116002 (1998).
2. Data Tables for Lorentz and CPT Violation, V.A. Kostelecky´ and N. Russell,
2013 edition, arXiv:0801.0287v6.
3. R. Newman and S. Wiesner, Phys. Rev. D 14, 1 (1976).
4. J.D. Ullman, Phys. Rev. D 17, 1750 (1978).
5. S.E. Mu¨ller, Hyperfine Int. 210, 33 (2012); H.W. Wilschut et al.,
arXiv:1303.6419; S.E. Mu¨ller et al., in preparation.
6. D. Colladay and V.A. Kostelecky´, Phys. Lett. B 511, 209 (2001); V.A. Kost-
elecky´ and R. Lehnert, Phys. Rev. D 63, 065008 (2001); R. Lehnert, J. Math.
Phys. 45, 3399 (2004); R. Potting, Phys. Rev. D 85, 045033 (2012);
7. J.P. Noordmans, H.W. Wilschut, and R.G.E. Timmermans, Phys. Rev. C 87,
055502 (2013).
8. J.P. Noordmans, H.W. Wilschut, and R.G.E. Timmermans, in preparation.
9. J.P. Noordmans, H.W. Wilschut, and R.G.E. Timmermans, Phys. Rev. Lett.
111, 171601 (2013).


Nuclear beta decay with Lorentz violation

University of Groningen

   University of GroningenNuclear beta decay with Lorentz violationNoordmans, J.P.; Wilschut, H. W.; Timmermans, R. G. E.Published in:Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Publication date:2013Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Noordmans, J. P., Wilschut, H. W., & Timmermans, R. G. E. (2013). Nuclear beta decay with Lorentzviolation. In Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019arXiv:1308.5829v2  [hep-ph]  7 Nov 2013Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)1NUCLEAR β DECAY WITH LORENTZ VIOLATIONJ.P. NOORDMANS,∗ H.W. WILSCHUT, and R.G.E. TIMMERMANSKVI, University of GroningenZernikelaan 25, NL-9747 AA Groningen, The Netherlands∗E-mail: J.P.Noordmans@rug.nlWe consider the possibility of Lorentz-invariance violation in weak-decay pro-cesses. We present a general approach that entails modifying the W -bosonpropagator by adding a Lorentz-violating tensor to it. We describe the effectsof Lorentz violation on nuclear β decay in this scenario. In particular we showthe expression for a first-forbidden transition with a spin change of two. Us-ing data from an old experiment on the rotational invariance of yttrium-90,we derive several bounds on the Lorentz-violating parameters of the order of10−6-10−8.1. IntroductionThere has been considerable scientific interest in the possibility of the viola-tion of Lorentz symmetry, in the context of searches for a theory of quantumgravity. Its phenomenological consequences have been studied extensively,in particular in the context of the Standard-Model Extension (SME).1 Ex-tensive experimental efforts have been made to bound the coefficients in allsectors of the SME.2Missing in these efforts is the study of weak decay. Until recently, therewere, to our knowledge, only two dedicated experiments that addressed ro-tational invariance violation in β decay.3,4 This effort is now supplementedby an experiment at KVI.5Theoretically, the issue of calculating cross sections and decay rates isalso far from settled, although some amount of work has already gone intothis.6We started a joint experimental and theoretical effort at KVI, addressingthe issue of Lorentz violation in nuclear β decay. We incorporate Lorentzviolation in weak decay by modifying the gauge boson propagator to〈Wµ+(p)W ν−(−p)〉=−i(gµν + χµν)M2W, (1)Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)2where gµν is the usual Minkowski metric and χµν a general complex (possi-bly momentum-dependent) tensor, that parametrizes the Lorentz violation.What we miss in this way are mainly kinematic properties of the externalparticles (see Ref. 7 for more discussion).2. β-decay transitions and forbiddennessβ-decay transitions are classified by the spin change from parent to daughternucleus (∆I) and the parities of the parent and daughter nuclei, denotedby πi and πf , respectively.One distinguishes different β-decay transitions also by their forbidden-ness, or the degree of suppression of the quantum mechanical amplitudethat describes the process. This suppression is determined by factors ofthree small quantities: R/λ, vN and αZ, which are the ratio of the nuclearradius and the de Broglie wavelength of the leptons, the velocity of thedecaying nucleon in units of c, and the fine-structure constant times thecharge of the daughter nucleus, respectively. Factors of R/λ come from theleptons carrying away orbital angular momentum (corresponding to higherorder terms in the multipole expansion of the lepton wavefunctions). Thequantities vN and αZ originate from relativistic and Coulomb effects, re-spectively.If we assume these three quantities to vanish, we limit ourselves to so-called allowed β-decay transitions. The leptons do not carry away orbitalangular momentum in this case. Consequently, the total angular momentumof the lepton pair can be either zero or one. In the Lorentz-symmetriccase this translates directly into the selection rules for allowed β decay,which can have spin change zero or one while the relative parity mustbe πiπf = +1. With Lorentz violation this connection is not that directanymore, as explained later.When allowing for factors of R/λ, vN and αZ, the leptons can carryoff orbital angular momentum and the total angular momentum of thelepton pair can be J = 0, 1, 2, . . ., which allows for a larger spin change inthe transition, as well as the possibility of a relative parity of πiπf = −1.For a complete classification of transitions and their forbiddenness in theLorentz-symmetric and Lorentz-violating cases, see Ref. 8.3. The effects of Lorentz violationUsing the propagator in Eq. (1), we have calculated the rate of generalβ-decay transitions. The expression for allowed β decay can be found inProceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)3Ref. 7, and the one for forbidden transitions can be found in Ref. 9.In Refs. 8 and 9 we found that there is a relative enhancement of Lorentz-violating effects in transitions with ∆I ≥ 2 by a factor of αZ/pR. From thevery schematic expression(1,σ)µhadron(gµν + χµν)(1,σ)νlepton , (2)that shows the hadron and lepton current, connected by theW -boson prop-agator, we see that the effect of the tensor χµν is to connect the singlet andtriplet parts of the lepton and hadron current in an unconventional way.For example the spacetime part of χµν connects the triplet of the hadroncurrent to the singlet of the lepton current. This means that there can bea unit of spin change in the nucleus, while the lepton pair does not carryoff the corresponding unit of angular momentum. This is the cause of theenhancement factor that occurs in some Lorentz-violating terms of the am-plitude of transitions with ∆I ≥ 2. For a more elaborate discussion of thiseffect, see Refs. 8 and 9.4. Bounds on Lorentz violationUsing our results for a general β-decay transition we calculated the Lorentz-violating transition rate of a first forbidden transition with ∆I = 2 andπiπf = −1, dependent on the emission direction of the outgoing β particle.It is given bydλdΩdE∝ R2{p2 + q2+αZpR[310p3E(χijr pˆipˆj − 13χ00r)∓12p2χ˜lipˆl ± p2χl0r pˆl]}, (3)where p, q, R are the β-particle momentum, the neutrino momentum andthe nuclear radius respectively, χ˜i = ǫijkχjk, Latin indices run over spa-tial components only, the subscripts r, i denote real and imaginary partsrespectively, and the upper (lower) sign corresponds to β− (β+) decay. Wesee the relative enhancement factor αZ/pR of the Lorentz-violating effects.Using the expression in Eq. (3) we reanalyzed results from an experi-ment published in 1976 and described in Ref. 3. The experiment was doneusing a 10 Ci strontium-90 source, which decays to yttrium-90 and subse-quently to zirconium-90. The relevant decay is the one from 90Y to 90Z,which is a first-forbidden transition with ∆I = 2 and πiπf = −1. In the ex-periment the source and the detector were mounted on a turntable, whichProceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)4rotated with a frequency of 0.75 Hz. Meanwhile the current of electronsthat originated from the source was measured. Using the data of 232 good2-h runs, asymmetries were defined in terms of the emission direction of theelectrons. These asymmetries were fitted to distributions in sidereal time,given byδ = a0 + a1 sin(ωt+ φ1) + a2(2ωt+ φ2) . (4)Bounds of the order of 10−8 were obtained on the constants a0, a1, and a2.Using Eq. (3) we calculated the theoretical prediction for the asymmetriesdefined in Ref. 3 and extracted the expressions for a0, a1, and a2 in termsof χµν . In this way we were able to put the following bounds at a 95%confidence level:9|2X30r − X˜3i | < 2× 10−8 , (5a)|3X33r −X00r | < 3× 10−6 , (5b)[(X12r +X21r )2 + (X22r −X11r )2]1/2< 1× 10−6 , (5c)[(X13r +X31r )2 + (X23r +X32r )2]1/2< 1× 10−6 , (5d)[(2X20r − X˜2i )2 + (2X10r − X˜1i )2]1/2< 4× 10−8 , (5e)where the Lorentz-violating parameter Xµν is now given in the standardSun-centered intertial reference frame.2 These bounds are the best directbounds on the parameter χµν of which we are aware.References1. D. Colladay and V.A. Kostelecky´, Phys. Rev. D 55, 6760 (1997); D. Colladayand V.A. Kostelecky´, Phys. Rev. D 58, 116002 (1998).2. Data Tables for Lorentz and CPT Violation, V.A. Kostelecky´ and N. Russell,2013 edition, arXiv:0801.0287v6.3. R. Newman and S. Wiesner, Phys. Rev. D 14, 1 (1976).4. J.D. Ullman, Phys. Rev. D 17, 1750 (1978).5. S.E. Mu¨ller, Hyperfine Int. 210, 33 (2012); H.W. Wilschut et al.,arXiv:1303.6419; S.E. Mu¨ller et al., in preparation.6. D. Colladay and V.A. Kostelecky´, Phys. Lett. B 511, 209 (2001); V.A. Kost-elecky´ and R. Lehnert, Phys. Rev. D 63, 065008 (2001); R. Lehnert, J. Math.Phys. 45, 3399 (2004); R. Potting, Phys. Rev. D 85, 045033 (2012);7. J.P. Noordmans, H.W. Wilschut, and R.G.E. Timmermans, Phys. Rev. C 87,055502 (2013).8. J.P. Noordmans, H.W. Wilschut, and R.G.E. Timmermans, in preparation.9. J.P. Noordmans, H.W. Wilschut, and R.G.E. Timmermans, Phys. Rev. Lett.111, 171601 (2013).

We consider the possibility of Lorentz-invariance violation in weak-decay processes. We present a general approach that entails modifying the W-boson propagator by adding a Lorentz-violating tensor to it. We describe the effects of Lorentz violation on nuclear beta decay in this scenario. In particular we show the expression for a first-forbidden transition with a spin change of two. Using data from an old experiment on the rotational invariance of yttrium-90, we derive several bounds on the Lorentz-violating parameters of the order of 10^(-6)-10^(-8)

ARTS repository - University of Groningen

   University of GroningenNuclear beta decay with Lorentz violationNoordmans, J.P.; Wilschut, H. W.; Timmermans, R. G. E.Published in:Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Publication date:2013Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Noordmans, J. P., Wilschut, H. W., & Timmermans, R. G. E. (2013). Nuclear beta decay with Lorentzviolation. In Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 08-06-2022arXiv:1308.5829v2  [hep-ph]  7 Nov 2013Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)1NUCLEAR β DECAY WITH LORENTZ VIOLATIONJ.P. NOORDMANS,∗ H.W. WILSCHUT, and R.G.E. TIMMERMANSKVI, University of GroningenZernikelaan 25, NL-9747 AA Groningen, The Netherlands∗E-mail: J.P.Noordmans@rug.nlWe consider the possibility of Lorentz-invariance violation in weak-decay pro-cesses. We present a general approach that entails modifying the W -bosonpropagator by adding a Lorentz-violating tensor to it. We describe the effectsof Lorentz violation on nuclear β decay in this scenario. In particular we showthe expression for a first-forbidden transition with a spin change of two. Us-ing data from an old experiment on the rotational invariance of yttrium-90,we derive several bounds on the Lorentz-violating parameters of the order of10−6-10−8.1. IntroductionThere has been considerable scientific interest in the possibility of the viola-tion of Lorentz symmetry, in the context of searches for a theory of quantumgravity. Its phenomenological consequences have been studied extensively,in particular in the context of the Standard-Model Extension (SME).1 Ex-tensive experimental efforts have been made to bound the coefficients in allsectors of the SME.2Missing in these efforts is the study of weak decay. Until recently, therewere, to our knowledge, only two dedicated experiments that addressed ro-tational invariance violation in β decay.3,4 This effort is now supplementedby an experiment at KVI.5Theoretically, the issue of calculating cross sections and decay rates isalso far from settled, although some amount of work has already gone intothis.6We started a joint experimental and theoretical effort at KVI, addressingthe issue of Lorentz violation in nuclear β decay. We incorporate Lorentzviolation in weak decay by modifying the gauge boson propagator to〈Wµ+(p)W ν−(−p)〉=−i(gµν + χµν)M2W, (1)Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)2where gµν is the usual Minkowski metric and χµν a general complex (possi-bly momentum-dependent) tensor, that parametrizes the Lorentz violation.What we miss in this way are mainly kinematic properties of the externalparticles (see Ref. 7 for more discussion).2. β-decay transitions and forbiddennessβ-decay transitions are classified by the spin change from parent to daughternucleus (∆I) and the parities of the parent and daughter nuclei, denotedby πi and πf , respectively.One distinguishes different β-decay transitions also by their forbidden-ness, or the degree of suppression of the quantum mechanical amplitudethat describes the process. This suppression is determined by factors ofthree small quantities: R/λ, vN and αZ, which are the ratio of the nuclearradius and the de Broglie wavelength of the leptons, the velocity of thedecaying nucleon in units of c, and the fine-structure constant times thecharge of the daughter nucleus, respectively. Factors of R/λ come from theleptons carrying away orbital angular momentum (corresponding to higherorder terms in the multipole expansion of the lepton wavefunctions). Thequantities vN and αZ originate from relativistic and Coulomb effects, re-spectively.If we assume these three quantities to vanish, we limit ourselves to so-called allowed β-decay transitions. The leptons do not carry away orbitalangular momentum in this case. Consequently, the total angular momentumof the lepton pair can be either zero or one. In the Lorentz-symmetriccase this translates directly into the selection rules for allowed β decay,which can have spin change zero or one while the relative parity mustbe πiπf = +1. With Lorentz violation this connection is not that directanymore, as explained later.When allowing for factors of R/λ, vN and αZ, the leptons can carryoff orbital angular momentum and the total angular momentum of thelepton pair can be J = 0, 1, 2, . . ., which allows for a larger spin change inthe transition, as well as the possibility of a relative parity of πiπf = −1.For a complete classification of transitions and their forbiddenness in theLorentz-symmetric and Lorentz-violating cases, see Ref. 8.3. The effects of Lorentz violationUsing the propagator in Eq. (1), we have calculated the rate of generalβ-decay transitions. The expression for allowed β decay can be found inProceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)3Ref. 7, and the one for forbidden transitions can be found in Ref. 9.In Refs. 8 and 9 we found that there is a relative enhancement of Lorentz-violating effects in transitions with ∆I ≥ 2 by a factor of αZ/pR. From thevery schematic expression(1,σ)µhadron(gµν + χµν)(1,σ)νlepton , (2)that shows the hadron and lepton current, connected by the W -boson prop-agator, we see that the effect of the tensor χµν is to connect the singlet andtriplet parts of the lepton and hadron current in an unconventional way.For example the spacetime part of χµν connects the triplet of the hadroncurrent to the singlet of the lepton current. This means that there can bea unit of spin change in the nucleus, while the lepton pair does not carryoff the corresponding unit of angular momentum. This is the cause of theenhancement factor that occurs in some Lorentz-violating terms of the am-plitude of transitions with ∆I ≥ 2. For a more elaborate discussion of thiseffect, see Refs. 8 and 9.4. Bounds on Lorentz violationUsing our results for a general β-decay transition we calculated the Lorentz-violating transition rate of a first forbidden transition with ∆I = 2 andπiπf = −1, dependent on the emission direction of the outgoing β particle.It is given bydλdΩdE∝ R2{p2 + q2+αZpR[310p3E(χijr p̂ip̂j − 13χ00r)∓12p2χ̃lip̂l ± p2χl0r p̂l]}, (3)where p, q, R are the β-particle momentum, the neutrino momentum andthe nuclear radius respectively, χ̃i = ǫijkχjk, Latin indices run over spa-tial components only, the subscripts r, i denote real and imaginary partsrespectively, and the upper (lower) sign corresponds to β− (β+) decay. Wesee the relative enhancement factor αZ/pR of the Lorentz-violating effects.Using the expression in Eq. (3) we reanalyzed results from an experi-ment published in 1976 and described in Ref. 3. The experiment was doneusing a 10 Ci strontium-90 source, which decays to yttrium-90 and subse-quently to zirconium-90. The relevant decay is the one from 90Y to 90Z,which is a first-forbidden transition with ∆I = 2 and πiπf = −1. In the ex-periment the source and the detector were mounted on a turntable, whichProceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)4rotated with a frequency of 0.75 Hz. Meanwhile the current of electronsthat originated from the source was measured. Using the data of 232 good2-h runs, asymmetries were defined in terms of the emission direction of theelectrons. These asymmetries were fitted to distributions in sidereal time,given byδ = a0 + a1 sin(ωt+ φ1) + a2(2ωt+ φ2) . (4)Bounds of the order of 10−8 were obtained on the constants a0, a1, and a2.Using Eq. (3) we calculated the theoretical prediction for the asymmetriesdefined in Ref. 3 and extracted the expressions for a0, a1, and a2 in termsof χµν . In this way we were able to put the following bounds at a 95%confidence level:9|2X30r − X̃3i | < 2× 10−8 , (5a)|3X33r −X00r | < 3× 10−6 , (5b)[(X12r +X21r )2 + (X22r −X11r )2]1/2< 1× 10−6 , (5c)[(X13r +X31r )2 + (X23r +X32r )2]1/2< 1× 10−6 , (5d)[(2X20r − X̃2i )2 + (2X10r − X̃1i )2]1/2< 4× 10−8 , (5e)where the Lorentz-violating parameter Xµν is now given in the standardSun-centered intertial reference frame.2 These bounds are the best directbounds on the parameter χµν of which we are aware.References1. D. Colladay and V.A. Kostelecký, Phys. Rev. D 55, 6760 (1997); D. Colladayand V.A. Kostelecký, Phys. Rev. D 58, 116002 (1998).2. Data Tables for Lorentz and CPT Violation, V.A. Kostelecký and N. Russell,2013 edition, arXiv:0801.0287v6.3. R. Newman and S. Wiesner, Phys. Rev. D 14, 1 (1976).4. J.D. Ullman, Phys. Rev. D 17, 1750 (1978).5. S.E. Müller, Hyperfine Int. 210, 33 (2012); H.W. Wilschut et al.,arXiv:1303.6419; S.E. Müller et al., in preparation.6. D. Colladay and V.A. Kostelecký, Phys. Lett. B 511, 209 (2001); V.A. Kost-elecký and R. Lehnert, Phys. Rev. D 63, 065008 (2001); R. Lehnert, J. Math.Phys. 45, 3399 (2004); R. Potting, Phys. Rev. D 85, 045033 (2012);7. J.P. Noordmans, H.W. Wilschut, and R.G.E. Timmermans, Phys. Rev. C 87,055502 (2013).8. J.P. Noordmans, H.W. Wilschut, and R.G.E. Timmermans, in preparation.9. J.P. Noordmans, H.W. Wilschut, and R.G.E. Timmermans, Phys. Rev. Lett.111, 171601 (2013).

   University of GroningenNuclear beta decay with Lorentz violationNoordmans, J.P.; Wilschut, H. W.; Timmermans, R. G. E.Published in:Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Publication date:2013Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Noordmans, J. P., Wilschut, H. W., & Timmermans, R. G. E. (2013). Nuclear beta decay with Lorentzviolation. In Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 31-10-2023arXiv:1308.5829v2  [hep-ph]  7 Nov 2013Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)1NUCLEAR β DECAY WITH LORENTZ VIOLATIONJ.P. NOORDMANS,∗ H.W. WILSCHUT, and R.G.E. TIMMERMANSKVI, University of GroningenZernikelaan 25, NL-9747 AA Groningen, The Netherlands∗E-mail: J.P.Noordmans@rug.nlWe consider the possibility of Lorentz-invariance violation in weak-decay pro-cesses. We present a general approach that entails modifying the W -bosonpropagator by adding a Lorentz-violating tensor to it. We describe the effectsof Lorentz violation on nuclear β decay in this scenario. In particular we showthe expression for a first-forbidden transition with a spin change of two. Us-ing data from an old experiment on the rotational invariance of yttrium-90,we derive several bounds on the Lorentz-violating parameters of the order of10−6-10−8.1. IntroductionThere has been considerable scientific interest in the possibility of the viola-tion of Lorentz symmetry, in the context of searches for a theory of quantumgravity. Its phenomenological consequences have been studied extensively,in particular in the context of the Standard-Model Extension (SME).1 Ex-tensive experimental efforts have been made to bound the coefficients in allsectors of the SME.2Missing in these efforts is the study of weak decay. Until recently, therewere, to our knowledge, only two dedicated experiments that addressed ro-tational invariance violation in β decay.3,4 This effort is now supplementedby an experiment at KVI.5Theoretically, the issue of calculating cross sections and decay rates isalso far from settled, although some amount of work has already gone intothis.6We started a joint experimental and theoretical effort at KVI, addressingthe issue of Lorentz violation in nuclear β decay. We incorporate Lorentzviolation in weak decay by modifying the gauge boson propagator to〈Wµ+(p)W ν−(−p)〉=−i(gµν + χµν)M2W, (1)Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)2where gµν is the usual Minkowski metric and χµν a general complex (possi-bly momentum-dependent) tensor, that parametrizes the Lorentz violation.What we miss in this way are mainly kinematic properties of the externalparticles (see Ref. 7 for more discussion).2. β-decay transitions and forbiddennessβ-decay transitions are classified by the spin change from parent to daughternucleus (∆I) and the parities of the parent and daughter nuclei, denotedby πi and πf , respectively.One distinguishes different β-decay transitions also by their forbidden-ness, or the degree of suppression of the quantum mechanical amplitudethat describes the process. This suppression is determined by factors ofthree small quantities: R/λ, vN and αZ, which are the ratio of the nuclearradius and the de Broglie wavelength of the leptons, the velocity of thedecaying nucleon in units of c, and the fine-structure constant times thecharge of the daughter nucleus, respectively. Factors of R/λ come from theleptons carrying away orbital angular momentum (corresponding to higherorder terms in the multipole expansion of the lepton wavefunctions). Thequantities vN and αZ originate from relativistic and Coulomb effects, re-spectively.If we assume these three quantities to vanish, we limit ourselves to so-called allowed β-decay transitions. The leptons do not carry away orbitalangular momentum in this case. Consequently, the total angular momentumof the lepton pair can be either zero or one. In the Lorentz-symmetriccase this translates directly into the selection rules for allowed β decay,which can have spin change zero or one while the relative parity mustbe πiπf = +1. With Lorentz violation this connection is not that directanymore, as explained later.When allowing for factors of R/λ, vN and αZ, the leptons can carryoff orbital angular momentum and the total angular momentum of thelepton pair can be J = 0, 1, 2, . . ., which allows for a larger spin change inthe transition, as well as the possibility of a relative parity of πiπf = −1.For a complete classification of transitions and their forbiddenness in theLorentz-symmetric and Lorentz-violating cases, see Ref. 8.3. The effects of Lorentz violationUsing the propagator in Eq. (1), we have calculated the rate of generalβ-decay transitions. The expression for allowed β decay can be found inProceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)3Ref. 7, and the one for forbidden transitions can be found in Ref. 9.In Refs. 8 and 9 we found that there is a relative enhancement of Lorentz-violating effects in transitions with ∆I ≥ 2 by a factor of αZ/pR. From thevery schematic expression(1,σ)µhadron(gµν + χµν)(1,σ)νlepton , (2)that shows the hadron and lepton current, connected by the W -boson prop-agator, we see that the effect of the tensor χµν is to connect the singlet andtriplet parts of the lepton and hadron current in an unconventional way.For example the spacetime part of χµν connects the triplet of the hadroncurrent to the singlet of the lepton current. This means that there can bea unit of spin change in the nucleus, while the lepton pair does not carryoff the corresponding unit of angular momentum. This is the cause of theenhancement factor that occurs in some Lorentz-violating terms of the am-plitude of transitions with ∆I ≥ 2. For a more elaborate discussion of thiseffect, see Refs. 8 and 9.4. Bounds on Lorentz violationUsing our results for a general β-decay transition we calculated the Lorentz-violating transition rate of a first forbidden transition with ∆I = 2 andπiπf = −1, dependent on the emission direction of the outgoing β particle.It is given bydλdΩdE∝ R2{p2 + q2+αZpR[310p3E(χijr p̂ip̂j − 13χ00r)∓12p2χ̃lip̂l ± p2χl0r p̂l]}, (3)where p, q, R are the β-particle momentum, the neutrino momentum andthe nuclear radius respectively, χ̃i = ǫijkχjk, Latin indices run over spa-tial components only, the subscripts r, i denote real and imaginary partsrespectively, and the upper (lower) sign corresponds to β− (β+) decay. Wesee the relative enhancement factor αZ/pR of the Lorentz-violating effects.Using the expression in Eq. (3) we reanalyzed results from an experi-ment published in 1976 and described in Ref. 3. The experiment was doneusing a 10 Ci strontium-90 source, which decays to yttrium-90 and subse-quently to zirconium-90. The relevant decay is the one from 90Y to 90Z,which is a first-forbidden transition with ∆I = 2 and πiπf = −1. In the ex-periment the source and the detector were mounted on a turntable, whichProceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)4rotated with a frequency of 0.75 Hz. Meanwhile the current of electronsthat originated from the source was measured. Using the data of 232 good2-h runs, asymmetries were defined in terms of the emission direction of theelectrons. These asymmetries were fitted to distributions in sidereal time,given byδ = a0 + a1 sin(ωt+ φ1) + a2(2ωt+ φ2) . (4)Bounds of the order of 10−8 were obtained on the constants a0, a1, and a2.Using Eq. (3) we calculated the theoretical prediction for the asymmetriesdefined in Ref. 3 and extracted the expressions for a0, a1, and a2 in termsof χµν . In this way we were able to put the following bounds at a 95%confidence level:9|2X30r − X̃3i | < 2× 10−8 , (5a)|3X33r −X00r | < 3× 10−6 , (5b)[(X12r +X21r )2 + (X22r −X11r )2]1/2< 1× 10−6 , (5c)[(X13r +X31r )2 + (X23r +X32r )2]1/2< 1× 10−6 , (5d)[(2X20r − X̃2i )2 + (2X10r − X̃1i )2]1/2< 4× 10−8 , (5e)where the Lorentz-violating parameter Xµν is now given in the standardSun-centered intertial reference frame.2 These bounds are the best directbounds on the parameter χµν of which we are aware.References1. D. Colladay and V.A. Kostelecký, Phys. Rev. D 55, 6760 (1997); D. Colladayand V.A. Kostelecký, Phys. Rev. D 58, 116002 (1998).2. Data Tables for Lorentz and CPT Violation, V.A. Kostelecký and N. Russell,2013 edition, arXiv:0801.0287v6.3. R. Newman and S. Wiesner, Phys. Rev. D 14, 1 (1976).4. J.D. Ullman, Phys. Rev. D 17, 1750 (1978).5. S.E. Müller, Hyperfine Int. 210, 33 (2012); H.W. Wilschut et al.,arXiv:1303.6419; S.E. Müller et al., in preparation.6. D. Colladay and V.A. Kostelecký, Phys. Lett. B 511, 209 (2001); V.A. Kost-elecký and R. Lehnert, Phys. Rev. D 63, 065008 (2001); R. Lehnert, J. Math.Phys. 45, 3399 (2004); R. Potting, Phys. Rev. D 85, 045033 (2012);7. J.P. Noordmans, H.W. Wilschut, and R.G.E. Timmermans, Phys. Rev. C 87,055502 (2013).8. J.P. Noordmans, H.W. Wilschut, and R.G.E. Timmermans, in preparation.9. J.P. Noordmans, H.W. Wilschut, and R.G.E. Timmermans, Phys. Rev. Lett.111, 171601 (2013).

https://pure.rug.nl/ws/files/31340762/1308.5829v2

Nuclear beta decay with Lorentz violation

Abstract

Similar works

Full text

Available Versions

University of Groningen Research Database

University of Groningen

ARTS repository - University of Groningen

ARTS repository - University of Groningen