The KLOE collaboration recently reported bounds on the directional dependence
of the lifetime of the short-lived neutral kaon K_S with respect to the cosmic
microwave background dipole anisotropy. We interpret their results in a general
framework developed to probe Lorentz violation in the weak interaction. In this
approach a Lorentz-violating tensor \chi_{\mu\nu} is added to the standard
propagator of the W boson. We derive the K_S decay rate in a naive tree-level
model and calculate the asymmetry for the lifetime. By using the KLOE data the
real vector part of \chi_{\mu\nu} is found to be smaller than 10^-2. We briefly
discuss the theoretical challenges concerning nonleptonic decays.Comment: Presented at the Sixth Meeting on CPT and Lorentz Symmetry,
  Bloomington, Indiana, June 17-21, 2013

Timmermans, R. G. E.

Vos, K. K.

Wilschut, H. W.

English

arXiv

Vos, K.K.

University of Groningen Research Database

  
 University of Groningen
Limits on Lorentz violation in neutral-Kaon decay
Vos, K.K.; Wilschut, Henricus; Timmermans, Robertus
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):
Vos, K. K., Wilschut, H. W., & Timmermans, R. G. E. (2013). Limits on Lorentz violation in neutral-Kaon
decay.
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.
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Download date: 11-02-2018
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Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)
1
LIMITS ON LORENTZ VIOLATION IN
NEUTRAL-KAON DECAY
K.K. VOS,∗ H.W. WILSCHUT and R.G.E. TIMMERMANS
Kernfysisch Versneller Instituut, University of Groningen
9747 AA Groningen, The Netherlands
∗E-mail: K.K.Vos@rug.nl
The KLOE collaboration recently reported bounds on the directional depen-
dence of the lifetime of the short-lived neutral kaon KS with respect to the
cosmic microwave background dipole anisotropy. We interpret their results in
a general framework developed to probe Lorentz violation in the weak interac-
tion. In this approach a Lorentz-violating tensor χµν is added to the standard
propagator of the W boson. We derive the KS decay rate in a naive tree-level
model and calculate the asymmetry for the lifetime. By using the KLOE data
the real vector part of χµν is found to be smaller than 10
−2. We briefly discuss
the theoretical challenges concerning nonleptonic decays.
1. Testing Lorentz violation in the weak interaction
Recently Lorentz violation in the weak interaction has been studied in
allowed1,2 and forbidden3 β-decay. A general theoretical framework was
developed for these tests,1 in which the standard W -boson propagator is
modified by a general Lorentz-violating tensor χµν . In the low-energy limit
the modified propagator is
〈
Wµ+(q)W ν−(−q)
〉
= −i
gµν + χµν
M2W
. (1)
To constrain χµν we derive the KS decay rate with Lorentz violation and
use data from the KLOE collaboration,4 which searched for the directional
dependence of the lifetime of KS mesons.
2. KLOE lifetime asymmetry
The KLOE collaboration measured the lifetime asymmetry of the neutral
KS meson, by transforming the KS momenta event-by-event to galactic
coordinates {ℓ, b}, where ℓ is the galactic longitude and b is the galactic
Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)
2
latitude. The lifetime asymmetry is defined as
A =
τ+ − τ−
τ+ + τ−
, (2)
where τ+(−) is the lifetime parallel (antiparallel) to a specific direction in
space. Three directions were studied: CMB0, the direction of the dipole
anisotropy in the Cosmic Microwave Background (CMB), and two perpen-
dicular directions, CMB1 and CMB2. Only events inside a cone of 30◦
opening angle were used. The measured asymmetries are given in Table 1,
where the errors are mainly statistical.
Table 1. Observed KS lifetime asymmetry.
{ℓ, b} A × 103 Ref.
CMB0 = {264◦, 48◦} −0.2± 1.0 4
CMB0 = {264◦, 48◦} −0.13± 0.4 5
CMB1 = {174◦, 0◦} 0.2± 1.0 4
CMB2 = {264◦,−42◦} 0.0± 0.9 4
3. ∆I = 1/2 rule
The description of nonleptonic processes is somewhat more involved than
leptonic and semileptonic processes, due to the importance of gluon loops.
These QCD corrections can be of the same order as tree-level contribu-
tions. Within the Standard Model, nonleptonic ∆S = 1 decays are usually
described by an effective lagrangian,6 which contains different operators and
their coefficients. An important consequence of the gluon loops is that they
generate a so-called penguin diagram, depicted in Fig. 1, which can effec-
tively be described by an operator that also contains right-handed quarks.
For KS decays there are two isospin I final states, I = 0 and I = 2. Exper-
imentally it is found that the former, a ∆I = 1/2 transition, is enhanced
compared to the latter, a ∆I = 3/2 transition. This enhancement is an order
of magnitude larger than expected from theoretical calculations with an ef-
fective lagrangian, and is known as the ∆I = 1/2 rule. The penguin diagram
generates a ∆I = 1/2 operator O5 that explains part of the enhancement
due to the coupling to right-handed quarks, but the enhancement is not
sufficient to explain the ∆I = 1/2 rule.7
The derivation of the complete Lorentz violating lagrangian is more
involved, due to mixing between operators, and lies beyond the scope of
Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)
3
g
W
s d
q q
c c
Fig. 1. The penguin diagram for an s to d quark transition
this work. The penguin diagram does not mix and we are able to derive the
Lorentz violating operator associated with this diagram by integrating out
the W boson, following Ref. 8. We find,9
OLV+SM5 = −
1
2
d¯Lt
a [−2gµν + χµν + χνµ + χ˜µν ] γ
νsL (q¯Rt
aγµqR) + h.c.,
(3)
where the gµν part is the Standard Model part and χ˜µν ≡ iǫ
αβµνχαβ. Fur-
ther discussion and implications of O5 are given in Ref. 9. The outstanding
problem of the ∆I = 1/2 rule makes our calculation with Lorentz violation
challenging. We therefore first explore the possibilities of the KLOE data
with a naive tree-level model.
4. Tree-level model
To explore the possible bounds from the KS decay we first use a tree-level
model, in which we assume that the dominant process for KS decay is the
tree-level decay. We thus treat this process naively as being semileptonic.
For the tree-level amplitude with the modified W -boson propagator we find
M∼
〈
π+|u¯Lγ
µdL|0
〉
(gµν + χ
∗
µν)
〈
π−|s¯Lγ
νuL|K
0
〉
. (4)
After integrating, we find for the asymmetry,
A~n = −
4
3 +
2
3
m2
pi
m2
K
(1 − β2K)
(
1−
m2
pi
m2
K
) (χri0 + χr0i)βiK = −0.343(χri0 + χr0i) βˆiK , (5)
where 2χrµν = χµν +χ
∗
µν and βˆK is the normalized kaon velocity, where for
the last part we used β = 0.2. The asymmetry thus gets a γ2 enhancement
and is sensitive to the symmetric real vector part of χµν .
Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)
4
For our final results we transform χµν to the Sun-centered reference
frame10 and find at 95% confidence level,9
|Xr10 +X
r
01| < 3× 10
−3,
|Xr20 +X
r
02| < 6× 10
−3,
|Xr30 +X
r
03| < 6× 10
−3. (6)
5. Conclusion and outlook
We investigated Lorentz violation in neutral KS decays by modifying the
W -boson propagator. This nonleptonic decay is theoretically more challeng-
ing than semileptonic or leptonic decays. Using a tree-level model we are
able to put bounds on the real vector part of χµν of the order of 10
−2. These
bounds complement bounds from allowed2 and forbidden3 β-decay, but are
several orders of magnitude less strict than bounds in other sectors.10 In
order to significantly improve these bounds a possibility would be to use
the γ2 enhancement that occurs in decay asymmetries. From the theoretical
point of view we argued that nonleptonic decays are much more involved
due to QCD corrections, making experiments with leptonic or semileptonic
decays preferable.
Acknowledgments
This research was supported by the Dutch Stichting voor Fundamenteel
Onderzoek der Materie (FOM) under Programmes 104 and 114 and project
08PR2636.
References
1. J.P. Noordmans, H.W. Wilschut, and R.G.E. Timmermans, Phys. Rev. C
87, 055502 (2013).
2. H.W. Wilschut et al., arXiv:1303.6419; S.E. Mu¨ller et al., in preparation.
3. J.P. Noordmans et al., arXiv:1308.5829.
4. F. Ambrosino et al., Eur. Phys. J. C 71, 1604 (2011).
5. A. De Angelis et al., Nuov. Cim. C 034N3, 323 (2011).
6. M. Shifman et al., Nucl. Phys. B 120, 316 (1977).
7. J.F. Donoghue, Phys. Rev. D 30, 1499 (1984).
8. M.B. Wise and E. Witten, Phys. Rev. D 20, 1216 (1979).
9. K.K. Vos et al., in preparation.
10. Data Tables for Lorentz and CPT Violation, V.A. Kostelecky´ and N. Russell,
2013 edition, arXiv:0801.0287v6.


Limits on Lorentz violation in neutral-Kaon decay

The KLOE collaboration recently reported bounds on the directional dependence of the lifetime of the short-lived neutral kaon K_S with respect to the cosmic microwave background dipole anisotropy. We interpret their results in a general framework developed to probe Lorentz violation in the weak interaction. In this approach a Lorentz-violating tensor \chi_{\mu\nu} is added to the standard propagator of the W boson. We derive the K_S decay rate in a naive tree-level model and calculate the asymmetry for the lifetime. By using the KLOE data the real vector part of \chi_{\mu\nu} is found to be smaller than 10^-2. We briefly discuss the theoretical challenges concerning nonleptonic decays

NARCIS 

University of Groningen

   University of GroningenLimits on Lorentz violation in neutral-Kaon decayVos, K.K.; Wilschut, H. W.; Timmermans, R. G. E.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):Vos, K. K., Wilschut, H. W., & Timmermans, R. G. E. (2013). Limits on Lorentz violation in neutral-Kaondecay.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: 26-12-2020arXiv:1308.6468v1  [hep-ph]  29 Aug 2013Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)1LIMITS ON LORENTZ VIOLATION INNEUTRAL-KAON DECAYK.K. VOS,∗ H.W. WILSCHUT and R.G.E. TIMMERMANSKernfysisch Versneller Instituut, University of Groningen9747 AA Groningen, The Netherlands∗E-mail: K.K.Vos@rug.nlThe KLOE collaboration recently reported bounds on the directional depen-dence of the lifetime of the short-lived neutral kaon KS with respect to thecosmic microwave background dipole anisotropy. We interpret their results ina general framework developed to probe Lorentz violation in the weak interac-tion. In this approach a Lorentz-violating tensor χµν is added to the standardpropagator of the W boson. We derive the KS decay rate in a naive tree-levelmodel and calculate the asymmetry for the lifetime. By using the KLOE datathe real vector part of χµν is found to be smaller than 10−2. We briefly discussthe theoretical challenges concerning nonleptonic decays.1. Testing Lorentz violation in the weak interactionRecently Lorentz violation in the weak interaction has been studied inallowed1,2 and forbidden3 β-decay. A general theoretical framework wasdeveloped for these tests,1 in which the standard W -boson propagator ismodified by a general Lorentz-violating tensor χµν . In the low-energy limitthe modified propagator is〈Wµ+(q)W ν−(−q)〉= −igµν + χµνM2W. (1)To constrain χµν we derive the KS decay rate with Lorentz violation anduse data from the KLOE collaboration,4 which searched for the directionaldependence of the lifetime of KS mesons.2. KLOE lifetime asymmetryThe KLOE collaboration measured the lifetime asymmetry of the neutralKS meson, by transforming the KS momenta event-by-event to galacticcoordinates {ℓ, b}, where ℓ is the galactic longitude and b is the galacticProceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)2latitude. The lifetime asymmetry is defined asA =τ+ − τ−τ+ + τ−, (2)where τ+(−) is the lifetime parallel (antiparallel) to a specific direction inspace. Three directions were studied: CMB0, the direction of the dipoleanisotropy in the Cosmic Microwave Background (CMB), and two perpen-dicular directions, CMB1 and CMB2. Only events inside a cone of 30◦opening angle were used. The measured asymmetries are given in Table 1,where the errors are mainly statistical.Table 1. Observed KS lifetime asymmetry.{ℓ, b} A × 103 Ref.CMB0 = {264◦, 48◦} −0.2± 1.0 4CMB0 = {264◦, 48◦} −0.13± 0.4 5CMB1 = {174◦, 0◦} 0.2± 1.0 4CMB2 = {264◦,−42◦} 0.0± 0.9 43. ∆I = 1/2 ruleThe description of nonleptonic processes is somewhat more involved thanleptonic and semileptonic processes, due to the importance of gluon loops.These QCD corrections can be of the same order as tree-level contribu-tions. Within the Standard Model, nonleptonic ∆S = 1 decays are usuallydescribed by an effective lagrangian,6 which contains different operators andtheir coefficients. An important consequence of the gluon loops is that theygenerate a so-called penguin diagram, depicted in Fig. 1, which can effec-tively be described by an operator that also contains right-handed quarks.For KS decays there are two isospin I final states, I = 0 and I = 2. Exper-imentally it is found that the former, a ∆I = 1/2 transition, is enhancedcompared to the latter, a ∆I = 3/2 transition. This enhancement is an orderof magnitude larger than expected from theoretical calculations with an ef-fective lagrangian, and is known as the ∆I = 1/2 rule. The penguin diagramgenerates a ∆I = 1/2 operator O5 that explains part of the enhancementdue to the coupling to right-handed quarks, but the enhancement is notsufficient to explain the ∆I = 1/2 rule.7The derivation of the complete Lorentz violating lagrangian is moreinvolved, due to mixing between operators, and lies beyond the scope ofProceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)3gWs dq qc cFig. 1. The penguin diagram for an s to d quark transitionthis work. The penguin diagram does not mix and we are able to derive theLorentz violating operator associated with this diagram by integrating outthe W boson, following Ref. 8. We find,9OLV +SM5 = −12d̄Lta [−2gµν + χµν + χνµ + χ̃µν ] γνsL (q̄RtaγµqR) + h.c.,(3)where the gµν part is the Standard Model part and χ̃µν ≡ iǫαβµνχαβ. Fur-ther discussion and implications of O5 are given in Ref. 9. The outstandingproblem of the ∆I = 1/2 rule makes our calculation with Lorentz violationchallenging. We therefore first explore the possibilities of the KLOE datawith a naive tree-level model.4. Tree-level modelTo explore the possible bounds from the KS decay we first use a tree-levelmodel, in which we assume that the dominant process for KS decay is thetree-level decay. We thus treat this process naively as being semileptonic.For the tree-level amplitude with the modified W -boson propagator we findM ∼〈π+|ūLγµdL|0〉(gµν + χ∗µν)〈π−|s̄LγνuL|K0〉. (4)After integrating, we find for the asymmetry,A~n = −43 +23m2πm2K(1 − β2K)(1−m2πm2K) (χri0 + χr0i)βiK = −0.343(χri0 + χr0i) β̂iK , (5)where 2χrµν = χµν +χ∗µν and β̂K is the normalized kaon velocity, where forthe last part we used β = 0.2. The asymmetry thus gets a γ2 enhancementand is sensitive to the symmetric real vector part of χµν .Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)4For our final results we transform χµν to the Sun-centered referenceframe10 and find at 95% confidence level,9|Xr10 +Xr01| < 3× 10−3,|Xr20 +Xr02| < 6× 10−3,|Xr30 +Xr03| < 6× 10−3. (6)5. Conclusion and outlookWe investigated Lorentz violation in neutral KS decays by modifying theW -boson propagator. This nonleptonic decay is theoretically more challeng-ing than semileptonic or leptonic decays. Using a tree-level model we areable to put bounds on the real vector part of χµν of the order of 10−2. Thesebounds complement bounds from allowed2 and forbidden3 β-decay, but areseveral orders of magnitude less strict than bounds in other sectors.10 Inorder to significantly improve these bounds a possibility would be to usethe γ2 enhancement that occurs in decay asymmetries. From the theoreticalpoint of view we argued that nonleptonic decays are much more involveddue to QCD corrections, making experiments with leptonic or semileptonicdecays preferable.AcknowledgmentsThis research was supported by the Dutch Stichting voor FundamenteelOnderzoek der Materie (FOM) under Programmes 104 and 114 and project08PR2636.References1. J.P. Noordmans, H.W. Wilschut, and R.G.E. Timmermans, Phys. Rev. C87, 055502 (2013).2. H.W. Wilschut et al., arXiv:1303.6419; S.E. Müller et al., in preparation.3. J.P. Noordmans et al., arXiv:1308.5829.4. F. Ambrosino et al., Eur. Phys. J. C 71, 1604 (2011).5. A. De Angelis et al., Nuov. Cim. C 034N3, 323 (2011).6. M. Shifman et al., Nucl. Phys. B 120, 316 (1977).7. J.F. Donoghue, Phys. Rev. D 30, 1499 (1984).8. M.B. Wise and E. Witten, Phys. Rev. D 20, 1216 (1979).9. K.K. Vos et al., in preparation.10. Data Tables for Lorentz and CPT Violation, V.A. Kostelecký and N. Russell,2013 edition, arXiv:0801.0287v6.

ARTS repository - University of Groningen

   University of GroningenLimits on Lorentz violation in neutral-Kaon decayVos, K.K.; Wilschut, H. W.; Timmermans, R. G. E.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):Vos, K. K., Wilschut, H. W., & Timmermans, R. G. E. (2013). Limits on Lorentz violation in neutral-Kaondecay.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:1308.6468v1  [hep-ph]  29 Aug 2013Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)1LIMITS ON LORENTZ VIOLATION INNEUTRAL-KAON DECAYK.K. VOS,∗ H.W. WILSCHUT and R.G.E. TIMMERMANSKernfysisch Versneller Instituut, University of Groningen9747 AA Groningen, The Netherlands∗E-mail: K.K.Vos@rug.nlThe KLOE collaboration recently reported bounds on the directional depen-dence of the lifetime of the short-lived neutral kaon KS with respect to thecosmic microwave background dipole anisotropy. We interpret their results ina general framework developed to probe Lorentz violation in the weak interac-tion. In this approach a Lorentz-violating tensor χµν is added to the standardpropagator of the W boson. We derive the KS decay rate in a naive tree-levelmodel and calculate the asymmetry for the lifetime. By using the KLOE datathe real vector part of χµν is found to be smaller than 10−2. We briefly discussthe theoretical challenges concerning nonleptonic decays.1. Testing Lorentz violation in the weak interactionRecently Lorentz violation in the weak interaction has been studied inallowed1,2 and forbidden3 β-decay. A general theoretical framework wasdeveloped for these tests,1 in which the standard W -boson propagator ismodified by a general Lorentz-violating tensor χµν . In the low-energy limitthe modified propagator is〈Wµ+(q)W ν−(−q)〉= −igµν + χµνM2W. (1)To constrain χµν we derive the KS decay rate with Lorentz violation anduse data from the KLOE collaboration,4 which searched for the directionaldependence of the lifetime of KS mesons.2. KLOE lifetime asymmetryThe KLOE collaboration measured the lifetime asymmetry of the neutralKS meson, by transforming the KS momenta event-by-event to galacticcoordinates {ℓ, b}, where ℓ is the galactic longitude and b is the galacticProceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)2latitude. The lifetime asymmetry is defined asA =τ+ − τ−τ+ + τ−, (2)where τ+(−) is the lifetime parallel (antiparallel) to a specific direction inspace. Three directions were studied: CMB0, the direction of the dipoleanisotropy in the Cosmic Microwave Background (CMB), and two perpen-dicular directions, CMB1 and CMB2. Only events inside a cone of 30◦opening angle were used. The measured asymmetries are given in Table 1,where the errors are mainly statistical.Table 1. Observed KS lifetime asymmetry.{ℓ, b} A × 103 Ref.CMB0 = {264◦, 48◦} −0.2± 1.0 4CMB0 = {264◦, 48◦} −0.13± 0.4 5CMB1 = {174◦, 0◦} 0.2± 1.0 4CMB2 = {264◦,−42◦} 0.0± 0.9 43. ∆I = 1/2 ruleThe description of nonleptonic processes is somewhat more involved thanleptonic and semileptonic processes, due to the importance of gluon loops.These QCD corrections can be of the same order as tree-level contribu-tions. Within the Standard Model, nonleptonic ∆S = 1 decays are usuallydescribed by an effective lagrangian,6 which contains different operators andtheir coefficients. An important consequence of the gluon loops is that theygenerate a so-called penguin diagram, depicted in Fig. 1, which can effec-tively be described by an operator that also contains right-handed quarks.For KS decays there are two isospin I final states, I = 0 and I = 2. Exper-imentally it is found that the former, a ∆I = 1/2 transition, is enhancedcompared to the latter, a ∆I = 3/2 transition. This enhancement is an orderof magnitude larger than expected from theoretical calculations with an ef-fective lagrangian, and is known as the ∆I = 1/2 rule. The penguin diagramgenerates a ∆I = 1/2 operator O5 that explains part of the enhancementdue to the coupling to right-handed quarks, but the enhancement is notsufficient to explain the ∆I = 1/2 rule.7The derivation of the complete Lorentz violating lagrangian is moreinvolved, due to mixing between operators, and lies beyond the scope ofProceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)3gWs dq qc cFig. 1. The penguin diagram for an s to d quark transitionthis work. The penguin diagram does not mix and we are able to derive theLorentz violating operator associated with this diagram by integrating outthe W boson, following Ref. 8. We find,9OLV +SM5 = −12d̄Lta [−2gµν + χµν + χνµ + χ̃µν ] γνsL (q̄RtaγµqR) + h.c.,(3)where the gµν part is the Standard Model part and χ̃µν ≡ iǫαβµνχαβ. Fur-ther discussion and implications of O5 are given in Ref. 9. The outstandingproblem of the ∆I = 1/2 rule makes our calculation with Lorentz violationchallenging. We therefore first explore the possibilities of the KLOE datawith a naive tree-level model.4. Tree-level modelTo explore the possible bounds from the KS decay we first use a tree-levelmodel, in which we assume that the dominant process for KS decay is thetree-level decay. We thus treat this process naively as being semileptonic.For the tree-level amplitude with the modified W -boson propagator we findM ∼〈π+|ūLγµdL|0〉(gµν + χ∗µν)〈π−|s̄LγνuL|K0〉. (4)After integrating, we find for the asymmetry,A~n = −43 +23m2πm2K(1 − β2K)(1−m2πm2K) (χri0 + χr0i)βiK = −0.343(χri0 + χr0i) β̂iK , (5)where 2χrµν = χµν +χ∗µν and β̂K is the normalized kaon velocity, where forthe last part we used β = 0.2. The asymmetry thus gets a γ2 enhancementand is sensitive to the symmetric real vector part of χµν .Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)4For our final results we transform χµν to the Sun-centered referenceframe10 and find at 95% confidence level,9|Xr10 +Xr01| < 3× 10−3,|Xr20 +Xr02| < 6× 10−3,|Xr30 +Xr03| < 6× 10−3. (6)5. Conclusion and outlookWe investigated Lorentz violation in neutral KS decays by modifying theW -boson propagator. This nonleptonic decay is theoretically more challeng-ing than semileptonic or leptonic decays. Using a tree-level model we areable to put bounds on the real vector part of χµν of the order of 10−2. Thesebounds complement bounds from allowed2 and forbidden3 β-decay, but areseveral orders of magnitude less strict than bounds in other sectors.10 Inorder to significantly improve these bounds a possibility would be to usethe γ2 enhancement that occurs in decay asymmetries. From the theoreticalpoint of view we argued that nonleptonic decays are much more involveddue to QCD corrections, making experiments with leptonic or semileptonicdecays preferable.AcknowledgmentsThis research was supported by the Dutch Stichting voor FundamenteelOnderzoek der Materie (FOM) under Programmes 104 and 114 and project08PR2636.References1. J.P. Noordmans, H.W. Wilschut, and R.G.E. Timmermans, Phys. Rev. C87, 055502 (2013).2. H.W. Wilschut et al., arXiv:1303.6419; S.E. Müller et al., in preparation.3. J.P. Noordmans et al., arXiv:1308.5829.4. F. Ambrosino et al., Eur. Phys. J. C 71, 1604 (2011).5. A. De Angelis et al., Nuov. Cim. C 034N3, 323 (2011).6. M. Shifman et al., Nucl. Phys. B 120, 316 (1977).7. J.F. Donoghue, Phys. Rev. D 30, 1499 (1984).8. M.B. Wise and E. Witten, Phys. Rev. D 20, 1216 (1979).9. K.K. Vos et al., in preparation.10. Data Tables for Lorentz and CPT Violation, V.A. Kostelecký and N. Russell,2013 edition, arXiv:0801.0287v6.

   University of GroningenLimits on Lorentz violation in neutral-Kaon decayVos, K.K.; Wilschut, H. W.; Timmermans, R. G. E.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):Vos, K. K., Wilschut, H. W., & Timmermans, R. G. E. (2013). Limits on Lorentz violation in neutral-Kaondecay.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:1308.6468v1  [hep-ph]  29 Aug 2013Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)1LIMITS ON LORENTZ VIOLATION INNEUTRAL-KAON DECAYK.K. VOS,∗ H.W. WILSCHUT and R.G.E. TIMMERMANSKernfysisch Versneller Instituut, University of Groningen9747 AA Groningen, The Netherlands∗E-mail: K.K.Vos@rug.nlThe KLOE collaboration recently reported bounds on the directional depen-dence of the lifetime of the short-lived neutral kaon KS with respect to thecosmic microwave background dipole anisotropy. We interpret their results ina general framework developed to probe Lorentz violation in the weak interac-tion. In this approach a Lorentz-violating tensor χµν is added to the standardpropagator of the W boson. We derive the KS decay rate in a naive tree-levelmodel and calculate the asymmetry for the lifetime. By using the KLOE datathe real vector part of χµν is found to be smaller than 10−2. We briefly discussthe theoretical challenges concerning nonleptonic decays.1. Testing Lorentz violation in the weak interactionRecently Lorentz violation in the weak interaction has been studied inallowed1,2 and forbidden3 β-decay. A general theoretical framework wasdeveloped for these tests,1 in which the standard W -boson propagator ismodified by a general Lorentz-violating tensor χµν . In the low-energy limitthe modified propagator is〈Wµ+(q)W ν−(−q)〉= −igµν + χµνM2W. (1)To constrain χµν we derive the KS decay rate with Lorentz violation anduse data from the KLOE collaboration,4 which searched for the directionaldependence of the lifetime of KS mesons.2. KLOE lifetime asymmetryThe KLOE collaboration measured the lifetime asymmetry of the neutralKS meson, by transforming the KS momenta event-by-event to galacticcoordinates {ℓ, b}, where ℓ is the galactic longitude and b is the galacticProceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)2latitude. The lifetime asymmetry is defined asA =τ+ − τ−τ+ + τ−, (2)where τ+(−) is the lifetime parallel (antiparallel) to a specific direction inspace. Three directions were studied: CMB0, the direction of the dipoleanisotropy in the Cosmic Microwave Background (CMB), and two perpen-dicular directions, CMB1 and CMB2. Only events inside a cone of 30◦opening angle were used. The measured asymmetries are given in Table 1,where the errors are mainly statistical.Table 1. Observed KS lifetime asymmetry.{ℓ, b} A × 103 Ref.CMB0 = {264◦, 48◦} −0.2± 1.0 4CMB0 = {264◦, 48◦} −0.13± 0.4 5CMB1 = {174◦, 0◦} 0.2± 1.0 4CMB2 = {264◦,−42◦} 0.0± 0.9 43. ∆I = 1/2 ruleThe description of nonleptonic processes is somewhat more involved thanleptonic and semileptonic processes, due to the importance of gluon loops.These QCD corrections can be of the same order as tree-level contribu-tions. Within the Standard Model, nonleptonic ∆S = 1 decays are usuallydescribed by an effective lagrangian,6 which contains different operators andtheir coefficients. An important consequence of the gluon loops is that theygenerate a so-called penguin diagram, depicted in Fig. 1, which can effec-tively be described by an operator that also contains right-handed quarks.For KS decays there are two isospin I final states, I = 0 and I = 2. Exper-imentally it is found that the former, a ∆I = 1/2 transition, is enhancedcompared to the latter, a ∆I = 3/2 transition. This enhancement is an orderof magnitude larger than expected from theoretical calculations with an ef-fective lagrangian, and is known as the ∆I = 1/2 rule. The penguin diagramgenerates a ∆I = 1/2 operator O5 that explains part of the enhancementdue to the coupling to right-handed quarks, but the enhancement is notsufficient to explain the ∆I = 1/2 rule.7The derivation of the complete Lorentz violating lagrangian is moreinvolved, due to mixing between operators, and lies beyond the scope ofProceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)3gWs dq qc cFig. 1. The penguin diagram for an s to d quark transitionthis work. The penguin diagram does not mix and we are able to derive theLorentz violating operator associated with this diagram by integrating outthe W boson, following Ref. 8. We find,9OLV +SM5 = −12d̄Lta [−2gµν + χµν + χνµ + χ̃µν ] γνsL (q̄RtaγµqR) + h.c.,(3)where the gµν part is the Standard Model part and χ̃µν ≡ iǫαβµνχαβ. Fur-ther discussion and implications of O5 are given in Ref. 9. The outstandingproblem of the ∆I = 1/2 rule makes our calculation with Lorentz violationchallenging. We therefore first explore the possibilities of the KLOE datawith a naive tree-level model.4. Tree-level modelTo explore the possible bounds from the KS decay we first use a tree-levelmodel, in which we assume that the dominant process for KS decay is thetree-level decay. We thus treat this process naively as being semileptonic.For the tree-level amplitude with the modified W -boson propagator we findM ∼〈π+|ūLγµdL|0〉(gµν + χ∗µν)〈π−|s̄LγνuL|K0〉. (4)After integrating, we find for the asymmetry,A~n = −43 +23m2πm2K(1 − β2K)(1−m2πm2K) (χri0 + χr0i)βiK = −0.343(χri0 + χr0i) β̂iK , (5)where 2χrµν = χµν +χ∗µν and β̂K is the normalized kaon velocity, where forthe last part we used β = 0.2. The asymmetry thus gets a γ2 enhancementand is sensitive to the symmetric real vector part of χµν .Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)4For our final results we transform χµν to the Sun-centered referenceframe10 and find at 95% confidence level,9|Xr10 +Xr01| < 3× 10−3,|Xr20 +Xr02| < 6× 10−3,|Xr30 +Xr03| < 6× 10−3. (6)5. Conclusion and outlookWe investigated Lorentz violation in neutral KS decays by modifying theW -boson propagator. This nonleptonic decay is theoretically more challeng-ing than semileptonic or leptonic decays. Using a tree-level model we areable to put bounds on the real vector part of χµν of the order of 10−2. Thesebounds complement bounds from allowed2 and forbidden3 β-decay, but areseveral orders of magnitude less strict than bounds in other sectors.10 Inorder to significantly improve these bounds a possibility would be to usethe γ2 enhancement that occurs in decay asymmetries. From the theoreticalpoint of view we argued that nonleptonic decays are much more involveddue to QCD corrections, making experiments with leptonic or semileptonicdecays preferable.AcknowledgmentsThis research was supported by the Dutch Stichting voor FundamenteelOnderzoek der Materie (FOM) under Programmes 104 and 114 and project08PR2636.References1. J.P. Noordmans, H.W. Wilschut, and R.G.E. Timmermans, Phys. Rev. C87, 055502 (2013).2. H.W. Wilschut et al., arXiv:1303.6419; S.E. Müller et al., in preparation.3. J.P. Noordmans et al., arXiv:1308.5829.4. F. Ambrosino et al., Eur. Phys. J. C 71, 1604 (2011).5. A. De Angelis et al., Nuov. Cim. C 034N3, 323 (2011).6. M. Shifman et al., Nucl. Phys. B 120, 316 (1977).7. J.F. Donoghue, Phys. Rev. D 30, 1499 (1984).8. M.B. Wise and E. Witten, Phys. Rev. D 20, 1216 (1979).9. K.K. Vos et al., in preparation.10. Data Tables for Lorentz and CPT Violation, V.A. Kostelecký and N. Russell,2013 edition, arXiv:0801.0287v6.

Limits on Lorentz violation in neutral-Kaon decay

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