2018 Vol. 42, No. 9
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2018, 42(9): 093101. doi: 10.1088/1674-1137/42/9/093101
Abstract:
In this work, we analyse semi-leptonic and non-leptonic weak decays of the heavy baryons:∧b,Ξb,Ωb and ∧c,Ξc,Ωc. For non-leptonic decay modes, we study only the factorizable channels induced by the external W-emission. The two spectator quarks in the baryonic transitions are treated as a diquark and form factors are calculated in the light-front approach. Using the results for form factors, we also calculate some corresponding semi-leptonic and non-leptonic decay widths. We find that our results are comparable with the available experimental data and other theoretical predictions. Decay branching fractions for many channels are found to reach the level 10-3~10-2, which is promising for discovery in future measurements at BESⅢ, LHCb and Belle Ⅱ. The SU(3) symmetry in semi-leptonic decays is examined and sources of symmetry breaking are discussed.
In this work, we analyse semi-leptonic and non-leptonic weak decays of the heavy baryons:∧b,Ξb,Ωb and ∧c,Ξc,Ωc. For non-leptonic decay modes, we study only the factorizable channels induced by the external W-emission. The two spectator quarks in the baryonic transitions are treated as a diquark and form factors are calculated in the light-front approach. Using the results for form factors, we also calculate some corresponding semi-leptonic and non-leptonic decay widths. We find that our results are comparable with the available experimental data and other theoretical predictions. Decay branching fractions for many channels are found to reach the level 10-3~10-2, which is promising for discovery in future measurements at BESⅢ, LHCb and Belle Ⅱ. The SU(3) symmetry in semi-leptonic decays is examined and sources of symmetry breaking are discussed.
2018, 42(9): 093102. doi: 10.1088/1674-1137/42/9/093102
Abstract:
In light of the developments of the chiral constituent quark model (χCQM) in studying low energy hadronic matrix elements of the ground-state baryons, we extend this model to investigate their transition properties. The magnetic moments of transitions from the JP=(3/2)+ decuplet to JP=(1/2)+ octet baryons are calculated with explicit valence quark spin, sea quark spin and sea quark orbital angular momentum contributions. Since the experimental data is available for only a few transitions, we compare our results with the results of other available models. The implications of other complicated effects such as chiral symmetry breaking and SU(3) symmetry breaking arising due to confinement of quarks are also discussed.
In light of the developments of the chiral constituent quark model (χCQM) in studying low energy hadronic matrix elements of the ground-state baryons, we extend this model to investigate their transition properties. The magnetic moments of transitions from the JP=(3/2)+ decuplet to JP=(1/2)+ octet baryons are calculated with explicit valence quark spin, sea quark spin and sea quark orbital angular momentum contributions. Since the experimental data is available for only a few transitions, we compare our results with the results of other available models. The implications of other complicated effects such as chiral symmetry breaking and SU(3) symmetry breaking arising due to confinement of quarks are also discussed.
2018, 42(9): 093103. doi: 10.1088/1674-1137/42/9/093103
Abstract:
The lowest-lying glueballs are investigated in lattice QCD using Nf=2 clover Wilson fermions on anisotropic lattices. We simulate at two different and relatively heavy quark masses, corresponding to physical pion masses of mπ~938 MeV and 650 MeV. The quark mass dependence of the glueball masses has not been investigated in the present study. Only the gluonic operators built from Wilson loops are utilized in calculating the corresponding correlation functions. In the tensor channel, we obtain the ground state mass to be 2.363(39) GeV and 2.384(67) GeV at mπ~938 MeV and 650 MeV, respectively. In the pseudoscalar channel, when using the gluonic operator whose continuum limit has the form of εijkTrBiDjBk, we obtain the ground state mass to be 2.573(55) GeV and 2.585(65) GeV at the two pion masses. These results are compatible with the corresponding results in the quenched approximation. In contrast, if we use the topological charge density as field operators for the pseudoscalar, the masses of the lowest state are much lighter (around 1 GeV) and compatible with the expected masses of the flavor singlet qq meson. This indicates that the operator εijkTrBiDjBk and the topological charge density couple rather differently to the glueball states and qq mesons. The observation of the light flavor singlet pseudoscalar meson can be viewed as the manifestation of effects of dynamical quarks. In the scalar channel, the ground state masses extracted from the correlation functions of gluonic operators are determined to be around 1.4-1.5 GeV, which is close to the ground state masses from the correlation functions of the quark bilinear operators. In all cases, the mixing between glueballs and conventional mesons remains to be further clarified in the future.
The lowest-lying glueballs are investigated in lattice QCD using Nf=2 clover Wilson fermions on anisotropic lattices. We simulate at two different and relatively heavy quark masses, corresponding to physical pion masses of mπ~938 MeV and 650 MeV. The quark mass dependence of the glueball masses has not been investigated in the present study. Only the gluonic operators built from Wilson loops are utilized in calculating the corresponding correlation functions. In the tensor channel, we obtain the ground state mass to be 2.363(39) GeV and 2.384(67) GeV at mπ~938 MeV and 650 MeV, respectively. In the pseudoscalar channel, when using the gluonic operator whose continuum limit has the form of εijkTrBiDjBk, we obtain the ground state mass to be 2.573(55) GeV and 2.585(65) GeV at the two pion masses. These results are compatible with the corresponding results in the quenched approximation. In contrast, if we use the topological charge density as field operators for the pseudoscalar, the masses of the lowest state are much lighter (around 1 GeV) and compatible with the expected masses of the flavor singlet qq meson. This indicates that the operator εijkTrBiDjBk and the topological charge density couple rather differently to the glueball states and qq mesons. The observation of the light flavor singlet pseudoscalar meson can be viewed as the manifestation of effects of dynamical quarks. In the scalar channel, the ground state masses extracted from the correlation functions of gluonic operators are determined to be around 1.4-1.5 GeV, which is close to the ground state masses from the correlation functions of the quark bilinear operators. In all cases, the mixing between glueballs and conventional mesons remains to be further clarified in the future.
2018, 42(9): 093104. doi: 10.1088/1674-1137/42/9/093104
Abstract:
In this paper we study the parallel phase and the coincident phase of D-brane systems with the compactification of one closed modulus. D-brane systems with two phases are described by different 4-folds in terms of Type-Ⅱ/F-theory duality, and the phase transitions are related by the blow-up from a 4-fold with singularities to a 4-fold without. In terms of gauge theory, the phase transition corresponds to the enhancement of gauge group U(1)×U(1)→ U(2) connecting the Coulomb branch and the Higgs branch. For the sextic and octic with two D-branes, using mirror symmetry and Type-Ⅱ/F theory duality, A-model superpotentials are obtained from the B-model side for the two phases, and the U(1) Ooguri-Vafa invariants for the parallel phase and U(2) Ooguri-Vafa invariants for the coincident phase are extracted from the A-model superpotential. The difference between the invariants of the two phases is evidence of the phase transition between the Coulomb branch and the Higgs branch.
In this paper we study the parallel phase and the coincident phase of D-brane systems with the compactification of one closed modulus. D-brane systems with two phases are described by different 4-folds in terms of Type-Ⅱ/F-theory duality, and the phase transitions are related by the blow-up from a 4-fold with singularities to a 4-fold without. In terms of gauge theory, the phase transition corresponds to the enhancement of gauge group U(1)×U(1)→ U(2) connecting the Coulomb branch and the Higgs branch. For the sextic and octic with two D-branes, using mirror symmetry and Type-Ⅱ/F theory duality, A-model superpotentials are obtained from the B-model side for the two phases, and the U(1) Ooguri-Vafa invariants for the parallel phase and U(2) Ooguri-Vafa invariants for the coincident phase are extracted from the A-model superpotential. The difference between the invariants of the two phases is evidence of the phase transition between the Coulomb branch and the Higgs branch.
2018, 42(9): 093105. doi: 10.1088/1674-1137/42/9/093105
Abstract:
In this paper we calculate the total and fiducial cross sections as well as differential distributions for the Higgs-strahlung or VH process pp → VH → lνl/l- l+ + H, (V=W or Z, l=e, μ) including QCD and electro-weak corrections up to next-to-leading order before and after reweighting photon PDFs of NNPDF2.3qed, NNPDF3.0qed, MRST2004qed, CT14QEDinc, and LUXqed at the LHC with 13 TeV and Higgs boson mass MH=125 GeV. The predictions from the various photon PDFs before and after reweighting against each other are in good agreement. The photon PDF uncertainties of the photon-induced cross sections decrease significantly with the reweighting PDFs.
In this paper we calculate the total and fiducial cross sections as well as differential distributions for the Higgs-strahlung or VH process pp → VH → lνl/l- l+ + H, (V=W or Z, l=e, μ) including QCD and electro-weak corrections up to next-to-leading order before and after reweighting photon PDFs of NNPDF2.3qed, NNPDF3.0qed, MRST2004qed, CT14QEDinc, and LUXqed at the LHC with 13 TeV and Higgs boson mass MH=125 GeV. The predictions from the various photon PDFs before and after reweighting against each other are in good agreement. The photon PDF uncertainties of the photon-induced cross sections decrease significantly with the reweighting PDFs.
2018, 42(9): 093106. doi: 10.1088/1674-1137/42/9/093106
Abstract:
In this work, we make the first study of electroweak baryogenesis (EWBG) based on the LHC data in the CP-violating next-to-minimal supersymmetric model (NMSSM) where a strongly first order electroweak phase transition (EWPT) is obtained in the general complex Higgs potential. With representative benchmark points which pass the current LEP and LHC constraints, we demonstrate the structure of EWPT for those points and how a strongly first order EWPT is obtained in the complex NMSSM where the resulting gravitational wave production properties are found to be within the reaches of future space-based interferometers like BBO and Ultimate-DECIGO. We further calculate the generated baryon asymmetries where the CP violating sources are (1):higgsino-singlino dominated, (2):higgsino-gaugino dominated or (3):from both sources. It is shown that all three representing scenarios could evade the strong constraints set by various electric dipole moments (EDM) searches where cancellations among the EDM contributions occur at the tree level (higgsino-singlino dominated) or loop level (higgsino-gaugino dominated). The 125 GeV SM like Higgs can be either the second lightest neutral Higgs H2 or the third lightest neutral Higgs H3. Finally, we comment on the future direct and indirect probe of CPV in the Higgs sector from the collider and EDM experiments.
In this work, we make the first study of electroweak baryogenesis (EWBG) based on the LHC data in the CP-violating next-to-minimal supersymmetric model (NMSSM) where a strongly first order electroweak phase transition (EWPT) is obtained in the general complex Higgs potential. With representative benchmark points which pass the current LEP and LHC constraints, we demonstrate the structure of EWPT for those points and how a strongly first order EWPT is obtained in the complex NMSSM where the resulting gravitational wave production properties are found to be within the reaches of future space-based interferometers like BBO and Ultimate-DECIGO. We further calculate the generated baryon asymmetries where the CP violating sources are (1):higgsino-singlino dominated, (2):higgsino-gaugino dominated or (3):from both sources. It is shown that all three representing scenarios could evade the strong constraints set by various electric dipole moments (EDM) searches where cancellations among the EDM contributions occur at the tree level (higgsino-singlino dominated) or loop level (higgsino-gaugino dominated). The 125 GeV SM like Higgs can be either the second lightest neutral Higgs H2 or the third lightest neutral Higgs H3. Finally, we comment on the future direct and indirect probe of CPV in the Higgs sector from the collider and EDM experiments.
2018, 42(9): 093107. doi: 10.1088/1674-1137/42/9/093107
Abstract:
In this study, Higgs and Z boson associated production with subsequent decay is attempted in the framework of alternative left-right model, which is motivated by superstring-inspired E6 model at CEPC and future linear colliders. We systematically analyze each decay channel of Higgs with theoretical constraints and latest experimental methods. Due to the mixing of scalars in the Higgs sector, charged Higgs bosons can play an essential role in the phenomenological analysis of this process. Even though the predictions of this model for the signal strengths of this process are close to the standard model expectations, it can be distinct under high luminosity.
In this study, Higgs and Z boson associated production with subsequent decay is attempted in the framework of alternative left-right model, which is motivated by superstring-inspired E6 model at CEPC and future linear colliders. We systematically analyze each decay channel of Higgs with theoretical constraints and latest experimental methods. Due to the mixing of scalars in the Higgs sector, charged Higgs bosons can play an essential role in the phenomenological analysis of this process. Even though the predictions of this model for the signal strengths of this process are close to the standard model expectations, it can be distinct under high luminosity.
2018, 42(9): 094001. doi: 10.1088/1674-1137/42/9/094001
Abstract:
We propose to deploy limits that arise from different tests of the Pauli Exclusion Principle:i) to provide theories of quantum gravity with experimental guidance; ii) to distinguish, among the plethora of possible models, the ones that are already ruled out by current data; iii) to direct future attempts to be in accordance with experimental constraints. We first review experimental bounds on nuclear processes forbidden by the Pauli Exclusion Principle, which have been derived by several experimental collaborations making use of various detector materials. Distinct features of the experimental devices entail sensitivities on the constraints hitherto achieved that may differ from one another by several orders of magnitude. We show that with choices of these limits, well-known examples of flat noncommutative space-time instantiations of quantum gravity can be heavily constrained, and eventually ruled out. We devote particular attention to the analysis of the κ-Minkowski and θ-Minkowski noncommutative spacetimes. These are deeply connected to some scenarios in string theory, loop quantum gravity, and noncommutative geometry. We emphasize that the severe constraints on these quantum spacetimes, although they cannot rule out theories of top-down quantum gravity to which they are connected in various ways, provide a powerful limitation for those models. Focus on this will be necessary in the future.
We propose to deploy limits that arise from different tests of the Pauli Exclusion Principle:i) to provide theories of quantum gravity with experimental guidance; ii) to distinguish, among the plethora of possible models, the ones that are already ruled out by current data; iii) to direct future attempts to be in accordance with experimental constraints. We first review experimental bounds on nuclear processes forbidden by the Pauli Exclusion Principle, which have been derived by several experimental collaborations making use of various detector materials. Distinct features of the experimental devices entail sensitivities on the constraints hitherto achieved that may differ from one another by several orders of magnitude. We show that with choices of these limits, well-known examples of flat noncommutative space-time instantiations of quantum gravity can be heavily constrained, and eventually ruled out. We devote particular attention to the analysis of the κ-Minkowski and θ-Minkowski noncommutative spacetimes. These are deeply connected to some scenarios in string theory, loop quantum gravity, and noncommutative geometry. We emphasize that the severe constraints on these quantum spacetimes, although they cannot rule out theories of top-down quantum gravity to which they are connected in various ways, provide a powerful limitation for those models. Focus on this will be necessary in the future.
2018, 42(9): 094101. doi: 10.1088/1674-1137/42/9/094101
Abstract:
The shell correction effects on the α decay properties of heavy and superheavy nuclei have been studied in a macroscopic-microscopic manner. The macroscopic part is constructed from the generalized liquid drop model (GLDM), whereas the microscopic part, namely, the shell correction energy, brings about certain effects on the potential barriers and half-lives under a WKB approximation, which is emphasized in this work. The results show that the shell effects play a significant role in the estimation of the α decay half-lives within the actinide region. Predictions of the α decay half-lives are then generated for superheavy nuclei, which will provide useful information for future experiments.
The shell correction effects on the α decay properties of heavy and superheavy nuclei have been studied in a macroscopic-microscopic manner. The macroscopic part is constructed from the generalized liquid drop model (GLDM), whereas the microscopic part, namely, the shell correction energy, brings about certain effects on the potential barriers and half-lives under a WKB approximation, which is emphasized in this work. The results show that the shell effects play a significant role in the estimation of the α decay half-lives within the actinide region. Predictions of the α decay half-lives are then generated for superheavy nuclei, which will provide useful information for future experiments.
2018, 42(9): 094102. doi: 10.1088/1674-1137/42/9/094102
Abstract:
By incorporating hidden scale symmetry and hidden local symmetry in the nuclear effective field theory, combined with the double soft-pion theorem, we predict that the Gamow-Teller operator coming from the space component of the axial current should remain unaffected by the QCD vacuum change caused by the baryonic density, whereas the first forbidden beta transition operator coming from the time component should be strongly enhanced. While the latter has been confirmed for some time, the former was given support by a powerful recent ab initio quantum Monte Carlo calculation for light nuclei, which also confirmed the old "chiral filter hypothesis." Formulated in terms of the Fermi-liquid fixed point structure of strong-coupled nuclear interactions, we offer an extremely simple resolution to the long-standing puzzle of the "quenched gA," gAeff≈ 1[1], found in nuclear Gamow-Teller beta transitions, giant Gamow-Teller resonances, and double beta decays.
By incorporating hidden scale symmetry and hidden local symmetry in the nuclear effective field theory, combined with the double soft-pion theorem, we predict that the Gamow-Teller operator coming from the space component of the axial current should remain unaffected by the QCD vacuum change caused by the baryonic density, whereas the first forbidden beta transition operator coming from the time component should be strongly enhanced. While the latter has been confirmed for some time, the former was given support by a powerful recent ab initio quantum Monte Carlo calculation for light nuclei, which also confirmed the old "chiral filter hypothesis." Formulated in terms of the Fermi-liquid fixed point structure of strong-coupled nuclear interactions, we offer an extremely simple resolution to the long-standing puzzle of the "quenched gA," gAeff≈ 1[1], found in nuclear Gamow-Teller beta transitions, giant Gamow-Teller resonances, and double beta decays.
2018, 42(9): 095101. doi: 10.1088/1674-1137/42/9/095101
Abstract:
We constrain three cosmological models-the concordance cold dark matter plus cosmological constant (∧CDM) model, the power-law (PL) model, and the Rh=ct model-using the available local probes, which include the JLA compilation of type-Ia supernovae (SNe Ia), the direct measurement of the Hubble constant (H(z)), and the baryon acoustic oscillations (BAO). For the ∧CDM model, we consider two different cases, i.e. zero and non-zero spatial curvature. We find that by using the JLA alone, the ∧CDM and PL models are indistinguishable, but the Rh=ct model is strongly disfavored. If we combine JLA+H(z), the ∧CDM model is strongly favored over the other two models. The combination of all three datasets supports ∧CDM as the best model. We also use the low-redshift (z<0.2) data to constrain the deceleration parameter using the cosmography method, and find that only the ∧CDM model is consistent with cosmography. However, there is no strong evidence to distinguish between flat and non-flat ∧CDM models by using the local data alone.
We constrain three cosmological models-the concordance cold dark matter plus cosmological constant (∧CDM) model, the power-law (PL) model, and the Rh=ct model-using the available local probes, which include the JLA compilation of type-Ia supernovae (SNe Ia), the direct measurement of the Hubble constant (H(z)), and the baryon acoustic oscillations (BAO). For the ∧CDM model, we consider two different cases, i.e. zero and non-zero spatial curvature. We find that by using the JLA alone, the ∧CDM and PL models are indistinguishable, but the Rh=ct model is strongly disfavored. If we combine JLA+H(z), the ∧CDM model is strongly favored over the other two models. The combination of all three datasets supports ∧CDM as the best model. We also use the low-redshift (z<0.2) data to constrain the deceleration parameter using the cosmography method, and find that only the ∧CDM model is consistent with cosmography. However, there is no strong evidence to distinguish between flat and non-flat ∧CDM models by using the local data alone.
2018, 42(9): 095102. doi: 10.1088/1674-1137/42/9/095102
Abstract:
In this paper, we present a gauge inflation model based on the orbifold M4×S1/Z2 with non-Abelian SU(2) gauge symmetry, which is probably the simplest model in this category. As the inflaton potential is fully radiatively generated exclusively by gauge self-interactions, the model is predictive; thus, it is protected by gauge symmetry itself, without the introduction of any additional matter fields or arbitrary interactions. We show that the model fully agrees with the recent cosmological observations within the controlled perturbative regime of gauge interactions, g4 ≲ 1/(2πRMP), with the compactification radius (10 ≲ RMP ≲ 100):the expected magnitude of the curvature perturbation power spectrum and the value of the corresponding spectral index are in perfect agreement with the recent observations. The model also predicts a large fraction of the gravitational waves, negligible non-Gaussianity, and a sufficiently high reheating temperature.
In this paper, we present a gauge inflation model based on the orbifold M4×S1/Z2 with non-Abelian SU(2) gauge symmetry, which is probably the simplest model in this category. As the inflaton potential is fully radiatively generated exclusively by gauge self-interactions, the model is predictive; thus, it is protected by gauge symmetry itself, without the introduction of any additional matter fields or arbitrary interactions. We show that the model fully agrees with the recent cosmological observations within the controlled perturbative regime of gauge interactions, g4 ≲ 1/(2πRMP), with the compactification radius (10 ≲ RMP ≲ 100):the expected magnitude of the curvature perturbation power spectrum and the value of the corresponding spectral index are in perfect agreement with the recent observations. The model also predicts a large fraction of the gravitational waves, negligible non-Gaussianity, and a sufficiently high reheating temperature.
2018, 42(9): 095103. doi: 10.1088/1674-1137/42/9/095103
Abstract:
We investigate the constraints on total neutrino mass in the scenario of vacuum energy interacting with cold dark matter. We focus on two typical interaction forms, i.e., Q=βHρc and Q=βHρ∧. To avoid the occurrence of large-scale instability in interacting dark energy cosmology, we adopt the parameterized post-Friedmann approach to calculate the perturbation evolution of dark energy. We employ observational data, including the Planck cosmic microwave background temperature and polarization data, baryon acoustic oscillation data, a JLA sample of type Ia supernovae observation, direct measurement of the Hubble constant, and redshift space distortion data. We find that, compared with those in the ∧CDM model, much looser constraints on ∑mν are obtained in the Q=βHρc model, whereas slightly tighter constraints are obtained in the Q=βHρ∧ model. Consideration of the possible mass hierarchies of neutrinos reveals that the smallest upper limit of ∑mν appears in the degenerate hierarchy case. By comparing the values of χmin2, we find that the normal hierarchy case is favored over the inverted one. In particular, we find that the difference △χmin2 ≡ χIH; min2-χNH; min2 > 2 in the Q=βHρc model. In addition, we find that β=0 is consistent with the current observations in the Q=βHρc model, and β < 0 is favored at more than the 1σ level in the Q=βHρ∧ model.
We investigate the constraints on total neutrino mass in the scenario of vacuum energy interacting with cold dark matter. We focus on two typical interaction forms, i.e., Q=βHρc and Q=βHρ∧. To avoid the occurrence of large-scale instability in interacting dark energy cosmology, we adopt the parameterized post-Friedmann approach to calculate the perturbation evolution of dark energy. We employ observational data, including the Planck cosmic microwave background temperature and polarization data, baryon acoustic oscillation data, a JLA sample of type Ia supernovae observation, direct measurement of the Hubble constant, and redshift space distortion data. We find that, compared with those in the ∧CDM model, much looser constraints on ∑mν are obtained in the Q=βHρc model, whereas slightly tighter constraints are obtained in the Q=βHρ∧ model. Consideration of the possible mass hierarchies of neutrinos reveals that the smallest upper limit of ∑mν appears in the degenerate hierarchy case. By comparing the values of χmin2, we find that the normal hierarchy case is favored over the inverted one. In particular, we find that the difference △χmin2 ≡ χIH; min2-χNH; min2 > 2 in the Q=βHρc model. In addition, we find that β=0 is consistent with the current observations in the Q=βHρc model, and β < 0 is favored at more than the 1σ level in the Q=βHρ∧ model.
2018, 42(9): 095104. doi: 10.1088/1674-1137/42/9/095104
Abstract:
The possible variation of the electromagnetic fine structure constant, αe, at cosmological scales has aroused great interest in recent years. Strongly lensed gravitational waves (GWs) and their electromagnetic counterparts could be used to test this variation. Under the assumption that the speed of a photon can be modified, whereas the speed of a GW is the same as predicted by general relativity, and they both propagate in a flat Friedman-Robertson-Walker universe, we investigated the difference in time delays of the images and derived the upper bound of the variation of αe. For a typical lensing system in the standard cosmological models, we obtained Bcosθ ≤ 1.85×10-5, where B is the dipolar amplitude and θ is the angle between observation and the preferred direction. Our result is consistent with the most up-to-date observations on αe. In addition, the observations of strongly lensed GWs and their electromagnetic counterparts could be used to test which types of alternative theories of gravity can account for the variation of αe.
The possible variation of the electromagnetic fine structure constant, αe, at cosmological scales has aroused great interest in recent years. Strongly lensed gravitational waves (GWs) and their electromagnetic counterparts could be used to test this variation. Under the assumption that the speed of a photon can be modified, whereas the speed of a GW is the same as predicted by general relativity, and they both propagate in a flat Friedman-Robertson-Walker universe, we investigated the difference in time delays of the images and derived the upper bound of the variation of αe. For a typical lensing system in the standard cosmological models, we obtained Bcosθ ≤ 1.85×10-5, where B is the dipolar amplitude and θ is the angle between observation and the preferred direction. Our result is consistent with the most up-to-date observations on αe. In addition, the observations of strongly lensed GWs and their electromagnetic counterparts could be used to test which types of alternative theories of gravity can account for the variation of αe.
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