67. Vud , Vus the Cabibbo Angle, and CKM Unitarity

1

67. Vud , Vus the Cabibbo Angle, and CKM Unitarity
Revised March 2024 by E. Blucher (Chicago U.), G. D’Ambrosio (INFN, Napoli) and W.J. Marciano (BNL).
The Cabibbo-Kobayashi-Maskawa (CKM) [1,2] three-generation quark mixing matrix written interms of the Wolfenstein parameters (λ, A, ρ, η) [3] nicely illustrates the orthonormality constraintof unitarity, as well as the central role played by λ.








Vud Vus Vub
1 − λ2 /2λAλ3 (ρ − iη)

 

4
−λ
1 − λ2 /2
Aλ2
VCKM =  Vcd Vcs Vcb  = 
 + O(λ ) .
Vtd Vts Vtb
Aλ3 (1 − ρ − iη)−Aλ2
1(67.1)

That cornerstone parameter is a carryover from the two-generation Cabibbo angle, λ = sin(θCabibbo )= Vus . Its value is an important component in tests of CKM unitarity.
Up until 2003, the precise value of λ was controversial, with kaon decays (specifically K → πeν)branching fractions suggesting [4] λ ' 0.220, while indirect determinations via Vud obtained from
nuclear β-decays combined with unitarity preferred a somewhat larger λ ' 0.225 − 0.230. Thisdifference implied a 2 - 2.5 sigma deviation from the first row unitarity requirement|Vud |2 + |Vus |2 + |Vub |2 = 1,

(67.2)

a possible hint [5, 6] of new physics effects. Below, we describe the current status of Vud and Vus ,and their implication for the unitarity test in Eq. (67.2). (Since |Vub |2 ' 1.7 × 10−5 is negligibly
small, it is ignored in this discussion.) Eq. (67.2) is currently the most stringent test of unitarity in
the CKM matrix. As we shall see, it is again showing signs of a possible 2 to 3 sigma inconsistency.

67.1 Vud
Precise values of Vud have been obtained from superallowed nuclear, neutron, and pion beta
decays. Currently, the best determination of Vud comes from analysis of a set of 15 precisely
measured superallowed nuclear beta-decays [5, 6] (0+ → 0+ transitions). Measuring their halflives, t, and Q values gives the decay rate factors, f , which lead to a precise determination of Vudvia [7–12]. Based on several decades of dedicated studies, Hardy and Towner recently updated theaverage of the half-life, Ft, [6]|Vud |2 =K
2G2F (1 + ∆V
R )Ft

=

2912.95 ± 0.54
= 0.94815 ± 0.00060
Ft

for ∆VR = 2.454(19)%, (67.3)

where K/(~c)6 = 2π 3 ~ ln 2/(me c2 )5 = 8120.27648(26) × 10−10 GeV −4 s, ∆VR denotes the so-called
inner or universal electroweak radiative corrections (RC) to superallowed nuclear beta decays. A
dispersion relation (DR) calculational approach [13] to quantum loop corrections, specifically the
gamma-W box diagram, gives ∆VR = 0.02467(22). Actually the authors of Ref. [14] have combined
the different theoretical calculations leading to ∆VR = 0.02467(27) and
Vud = 0.97367(11)exp (13)∆V (27)NS (32)total (superallowed) .
R(67.4)As a result of its small uncertainty and more rigorous theoretical footing, we use this value
below. A somewhat different approach [15] found ∆VR = 0.02426(32). These recent values areroughly consistent. Both are larger than the 2018 PDG value of 0.02361(38). Implications andpossible nuclear physics modifications of those studies are still under scrutiny [16,17]. Note that an
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:21am

2

67. Vud , Vus the Cabibbo Angle, and CKM Unitarity

additional uncertainty (54)NS from nuclear structure (NS) [17] has been recently included [6] in thatmaster formula. This recent determination of Vud has shifted significantly down compared to the2018 PDG value of 0.97420(21). In addition, NS uncertainties are now the dominant contribution tothe overall uncertainty. Taken at face value, along with current Vus determinations (see subsection67.2), the reduced Vud would seem to violate the first row unitarity requirement, and thus suggest
the presence of deviations from unitarity or unaccounted uncertainties.
Measurements of the neutron lifetime, τn , and the ratio of axial-vector/vector couplings, gA ≡GA /GV , via neutron decay asymmetries combined with the inner radiative corrections can also be
used to determine Vud via the precise formula [14, 18]|Vud |2 =5024.7 s,
τn (1 + 3gA )(1 + ∆R )(67.5)

where ∆R = 0.03983(27) represents radiative corrections [14]. We present the Vud results using
either the the current published world averages [19] or the best measured values for λ [20] and forτn [21]. The average leads toτnave = 879.4(6) s (1.6 PDG scale factor)
avegA= 1.2756(13), (2.6 PDG scale factor),(67.6)

n,P DG|Vud| = 0.97441(28)τn (82)λ (13)∆R (88)TOT ,(67.7)

while the best values lead ton,best|Vud| = 0.97413(20)τn (35)λ (13)∆R (42)TOT .

(67.8)

Both central values are similar to the superallowed nuclear beta decay result reported above. Reconciliation with CKM unitarity suggests a shorter neutron lifetime near 878.5 s or a smaller gA . Wenote that the most precise recent neutron lifetime update reported [21], τn = 877.75(34) s, has an
uncertainty about half as big as the average given in Eq. 67.6. It corresponds to |Vud | = 0.9746(8),a value more in keeping with unitarity expectations; but too large an uncertainty from gA to bemeaningful. Future neutron studies [22] are expected to resolve any current inconsistencies andfurther reduce the uncertainties in gA and τn making them a potentially better way to determine
Vud without the nuclear physics uncertainties.
The PIBETA experiment at PSI measured the very small (O(10−8 )) branching ratio for π + →π o e+ νe with about ±0.6% precision. Its result gives [23]"BR(π + → e+ νe (γ))|Vud | = 0.9739(27)1.2325 × 10−4

#1

2

,

(67.9)

which is normalized using the very precisely measured BR(π + → e+ νe (γ)) = 1.2325(23) × 10−4 [7],rather than the theoretical branching ratio of 1.2350(2) × 10−4 , which if used, would increase|Vud | to 0.9749(27). Theoretical uncertainties in pion beta decay are very small [24], leaving open
more than an order of magnitude improvement of its experimental branching ratio before theory
uncertainties become a problem. Although challenging, improved measurements of pion beta decay
currently under discussion would allow this decay mode to compete with superallowed beta decays
and future neutron decay efforts for the most precise direct |Vud | determination.
31st May, 2024

3

67. Vud , Vus the Cabibbo Angle, and CKM Unitarity

67.2 Vus|Vus | may be directly obtained from kaon decays, hyperon decays, and tau decays. Early
determinations most often used K`3 decays:
ΓK`3 =

5
G2F MK``
SEW (1 + δK
+ δSU 2 )C 2 |Vus |2 f+2 (0)IK.
192π 3(67.10)Here, ` refers to either e or µ, GF is the Fermi constant, MK is the kaon mass, SEW is the short` is the mode-dependent long-distance radiative correction, f (0) is
distance radiative correction, δK+` is the phase-space
the calculated form factor at zero momentum transfer for the `ν system, and IK
integral, which depends on measured semileptonic form factors. For charged kaon decays, δSU 2
is the deviation from one of the ratio of f+ (0) for the charged to neutral kaon decay; it is zerofor the neutral kaon. C 2 is 1 (1/2) for neutral (charged) kaon decays. Most early determinationsof |Vus | were based solely on K → πeν decays; K → πµν decays were not used because of largeµ
uncertainties in IK. The experimental measurements are the semileptonic decay widths (basedon the semileptonic branching fractions and lifetime) and form factors (allowing calculation of the` , δphase space integrals). Theory is needed for SEW , δK
SU 2 , and f+ (0).
Many measurements during the last 20 years have resulted in a shift in |Vus |. Most importantly,
the K → πeν branching fractions are significantly different than much earlier PDG averages,
probably as a result of inadequate treatment of radiation in older experiments. This effect was firstobserved by BNL E865 [25] in the charged kaon system and then by KTeV [26,27] in the neutral kaon
system; subsequent measurements were made by KLOE [28–31], NA48 [32–34], and ISTRA+ [35].
Current averages (e.g., by the PDG [36] or Flavianet [37]) of the semileptonic branching fractions
are based only on recent, high-statistics experiments where the treatment of radiation is clear.
In addition to measurements of branching fractions, new measurements of lifetimes [38] and form
factors [39–43], have resulted in improved precision for all of the experimental inputs to |Vus |.
Precise measurements of form factors for Kµ3 decay make it possible to use both semileptonic
decay modes to extract Vus .
Following the analysis of Moulson [18,44] and the Flavianet group [37,45,46], along with recent
improvements in the QED radiative corrections [47], one finds [48], after including the isospin
violating effect, δSU 2 , the values of |Vus |f+ (0) in Table 67.1. The average of these measurements,
including correlation effects [18, 48–50] gives [14]f+ (0)|Vus | = 0.21656(35)FLAVIANET(67.11)

where the errors correspond to kaon experimental parameters and radiative corrections, respectively, leading, as we shall see, to |Vus | = 0.22330(35)exp (39)LAT (8)RC+IB (53)T OT .
Lattice QCD calculations of f+ (0) have been carried out for 2, 2+1, and 2+1+1 quark flavors
and range from about 0.96 to 0.97. Here, we illustrate recent FLAG (2020) updated averages [51]for 2+1 and 2+1+1 flavors:
f+ (0) = 0.9677(27) Nf = 2 + 1
f+ (0) = 0.9698(17) Nf = 2 + 1 + 1(67.12)Updating the result for Nf = 2 + 1 + 1 we obtain for Vus from Eq. (67.11) and Eq. (67.12),|Vus | = 0.2238(4)exp+RC (6)lattice (Nf = 2 + 1, K`3 decays)= 0.22330(4)exp+RC (4)lattice (Nf = 2 + 1 + 1, K`3 decays)
based on ref. [18] and ref. [48]31st May, 2024

(67.13)

4

67. Vud , Vus the Cabibbo Angle, and CKM Unitarity
Table 67.1: |Vus |f+ (0) from K`3, based on ref. [18, 48]
Decay Mode
K ± e3K ± µ3
KL e3
KL µ3
KS e3
KS µ3
Average (including correlation effects [18, 48])

|Vus |f+ (0)0.2169(6)0.2168(10)0.2162(5)0.2165(6)0.2169(8)0.2125(47)0.21656(35)A value of Vus can also be obtained from a comparison of the radiative inclusive decay rates for
K → µν(γ) and π → µν(γ) combined with a lattice gauge theory calculation of fK + /fπ+ via
Γ (K → µν(γ))|Vus |fK += 0.23871(20)|Vud |fπ+Γ (π → µν(γ))


1

2

(67.14)

with the small error coming from electroweak radiative corrections [52–54]; these corrections wereconfirmed by direct lattice calculation of the kaon and pion leptonic decay rates [53, 54].
A recent fit [14, 18] gives|Vus |fK += 0.27679(28)BR (20)corr .
|Vud |fπ+(67.15)

Employing the FLAG [51] lattice QCD averages for the isospin broken decay constants
fK += 1.1978(22) Nf = 2 + 1 + 1
fπ+(67.16)|Vus |= 0.23108(51) (Nf = 2 + 1 + 1, Kµ2 decays).
|Vud |(67.17)

leading to

Actually we can average the Cabibbo angle determinations from kaons: from K`3 decays in eqs.
67.11, 67.12 and from Kµ2 in eqs. 67.15, 67.16 with Nf = 2 + 1 + 1. Assuming 20% correlation
between (|Vus /Vud |fK /fπ ) and (fK /fπ ) we obtain
Vus = 0.22431(85)S = 2.5.

(67.18)The result shown in Eq. 67.18 is what we use when we consider the first row test of CKM unitarity.
The quoted uncertainty includes a scale factor of 2.5 to account for the discrepancies between thevalues of Vus obtained from K`3 and Kµ2 decays.
This is the current status of the Cabibbo Angle determination from the kaon sector, and theoretical and experimental progress is expected to clarify the situation soon.
It should be mentioned that hyperon decay fits suggest [55]
|Vus | = 0.2250(27) (Hyperon Decays)(67.19)modulo SU(3) breaking effects that could shift that value up or down. We note that a representative
effort [56] that incorporates SU(3) breaking found Vus = 0.226(5). Strangeness changing tau decays,
averaging both inclusive and exclusive measurements, give [57, 58]|Vus | = 0.2207(14) (Tau Decays) ,31st May, 2024

(67.20)5

67. Vud , Vus the Cabibbo Angle, and CKM Unitarity

which differs by about 2 sigma from the kaon determination discussed above, and would, if combined with Vud from super-allowed beta decays, lead to a 4 sigma deviation from unitarity. This
discrepancy results mainly from the inclusive tau decay results that rely on Finite Energy Sum
Rule techniques and assumptions, as well as experimental uncertainties. Recent investigation ofthat approach suggests a larger value for Vus , which is more in accord with other determinations [59].
Employing the values of Vud and Vus with an error scale factor of 2 from Eq. (67.4) and theaverage obtained after rescaling the errors from mEq. 67.18 leads to the unitarity consistency check|Vud |2 + |Vus |2 + |Vub |2 = 0.9983(6)(4),(67.21)

where the first error is the uncertainty from |Vud |2 and the second error is the uncertainty from theaverage |Vus |2 in Eq. 67.18 . One finds about an overall 2 sigma deviation from unitarity. (Thedeviation increases to 3 sigma if nuclear structure uncertainties are ignored.) That deviation could
be due a problem with |Vud | theory (RC or NS), the lattice determination of f+ (0) or new physics.

67.3 CKM Unitarity Constraints
The current 2 sigma experimental disagreement with unitarity, |Vud |2 +|Vus |2 +|Vub |2 = 0.9983(7),still provides strong confirmation of Standard Model radiative corrections (which range between
3-4% depending on the nucleus used) at a high significance level [60]. In addition, it implies constraints on “New Physics” effects at both the tree and quantum loop levels. Those effects could
be in the form of contributions to nuclear beta decays, K decays and/or muon decays, with thelast of these providing normalization via the muon lifetime [61], which is used to obtain the Fermi
constant, Gµ = 1.1663787(6) × 10−5 GeV−2 .
In the following examples, we illustrate the implications of CKM unitarity for (1) exotic muon
decays [62] (beyond ordinary muon decay µ+ → e+ νe ν̄µ ) and (2) new heavy quark mixing VuD [63].
Other examples in the literature [64, 65] include Zχ boson quantum loop effects, supersymmetry,
leptoquarks, compositeness etc.
References[1] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963).
[2] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).
[3] L. Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983).
[4] S. Eidelman et al. (Particle Data Group), Phys. Lett. B 592, 1-4, 1 (2004).
[5] I. Towner and J. Hardy, Rept. Prog. Phys. 73, 046301 (2010).
[6] J. C. Hardy and I. S. Towner, Phys. Rev. C 102, 4, 045501 (2020).
[7] W. J. Marciano and A. Sirlin, Phys. Rev. Lett. 71, 3629 (1993).
[8] A. Czarnecki, W. J. Marciano and A. Sirlin, Phys. Rev. D 70, 093006 (2004), [hep-ph/0406324].
[9] W. J. Marciano and A. Sirlin, Phys. Rev. Lett. 96, 032002 (2006), [hep-ph/0510099].
[10] I. Towner and J. Hardy, Phys. Rev. C 77, 025501 (2008), [arXiv:0710.3181].
[11] J. Hardy and I. Towner, Phys. Rev. C 79, 055502 (2009), [arXiv:0812.1202].
[12] J. Hardy and I. Towner, Phys. Rev. C 91, 2, 025501 (2015), [arXiv:1411.5987].
[13] C.-Y. Seng et al., Phys. Rev. Lett. 121, 24, 241804 (2018), [arXiv:1807.10197].
[14] V. Cirigliano et al., Phys. Lett. B 838, 137748 (2023), [arXiv:2208.11707].
[15] A. Czarnecki, W. J. Marciano and A. Sirlin, Phys. Rev. D 100, 7, 073008 (2019),[arXiv:1907.06737].
[16] C. Y. Seng, M. Gorchtein and M. J. Ramsey-Musolf, Phys. Rev. D 100, 1, 013001 (2019),[arXiv:1812.03352].
31st May, 2024

6

67. Vud , Vus the Cabibbo Angle, and CKM Unitarity[17] M. Gorchtein, Phys. Rev. Lett. 123, 4, 042503 (2019), [arXiv:1812.04229].
[18] M. Moulson, talk at CKM2023, https://indico.cern.ch/event/1184945/contributions/
5540813/attachments/2717802/4721036/CKM202023.pdf.
[19] R. L. Workman et al. (Particle Data Group), PTEP 2022, 083C01 (2022).
[20] B. Märkisch et al., Phys. Rev. Lett. 122, 24, 242501 (2019), [arXiv:1812.04666].
[21] F. M. Gonzalez et al. (UCNτ ) (2021), [arXiv:2106.10375].
[22] H. Abele, Prog. Part. Nucl. Phys. 60, 1 (2008).
[23] D. Pocanic et al., Phys. Rev. Lett. 93, 181803 (2004), [hep-ex/0312030]; A. Czarnecki, W. J.
Marciano and A. Sirlin, Phys. Rev. D 101, 9, 091301 (2020), [arXiv:1911.04685].
[24] X. Feng et al., Phys. Rev. Lett. 124, 19, 192002 (2020), [arXiv:2003.09798].
[25] A. Sher et al., Phys. Rev. Lett. 91, 261802 (2003), [hep-ex/0305042].
[26] T. Alexopoulos et al. (KTeV), Phys. Rev. Lett. 93, 181802 (2004), [hep-ex/0406001].
[27] T. Alexopoulos et al. (KTeV), Phys. Rev. D 70, 092006 (2004), [hep-ex/0406002].
[28] F. Ambrosino et al. (KLOE), Phys. Lett. B 632, 43 (2006), [hep-ex/0508027].
[29] F. Ambrosino et al. (KLOE), Phys. Lett. B 638, 140 (2006), [hep-ex/0603041].
[30] F. Ambrosino et al. (KLOE), Phys. Lett. B 636, 173 (2006), [hep-ex/0601026].
[31] B. Sciascia (KLOE), PoS HEP2005, 287 (2006), [hep-ex/0510028].
[32] A. Lai et al. (NA48), Phys. Lett. B 602, 41 (2004), [hep-ex/0410059].
[33] A. Lai et al. (NA48), Phys. Lett. B 645, 26 (2007), [hep-ex/0611052].
[34] J. Batley et al. (NA48/2), Eur. Phys. J. C 50, 329 (2007), [Erratum: Eur.Phys.J.C 52, 1021–
1023 (2007)], [hep-ex/0702015].
[35] V. Romanovsky et al. (2007), [arXiv:0704.2052].
[36] K. Olive et al. (Particle Data Group), Chin. Phys. C 38, 090001 (2014).
[37] M. Antonelli et al. (FlaviaNet Working Group on Kaon Decays), Eur. Phys. J. C 69, 399(2010), [arXiv:1005.2323]; For a detailed review, see; M. Antonelli et al., Phys. Rept. 494, 197(2010), [arXiv:0907.5386].
[38] F. Ambrosino et al. (KLOE), Phys. Lett. B 626, 15 (2005), [hep-ex/0507088].
[39] T. Alexopoulos et al. (KTeV), Phys. Rev. D 70, 092007 (2004), [hep-ex/0406003].
[40] E. Abouzaid et al. (KTeV), Phys. Rev. D 74, 097101 (2006), [hep-ex/0608058].
[41] F. Ambrosino et al. (KLOE), Phys. Lett. B 636, 166 (2006), [hep-ex/0601038].
[42] A. Lai et al. (NA48), Phys. Lett. B 604, 1 (2004), [hep-ex/0410065].
[43] O. Yushchenko et al., Phys. Lett. B 589, 111 (2004), [hep-ex/0404030].
[44] M. Moulson, PoS CKM2016, 033 (2017), [arXiv:1704.04104].
[45] E. Passemar, talk at CKM2018, https://zenodo.org/record/2565480.
[46] Y. Aoki et al. (Flavour Lattice Averaging Group (FLAG)), Eur. Phys. J. C 82, 10, 869 (2022),[arXiv:2111.09849].
[47] C.-Y. Seng et al. (2021), [arXiv:2103.04843].
[48] C.-Y. Seng et al. (2021), [arXiv:2107.14708].
[49] C.-Y. Seng et al., JHEP 07, 071 (2022), [arXiv:2203.05217].
[50] V. Cirigliano, talk at Cabibbo 60 - Convegno, Accademia Nazionale dei Lincei , https://www.
lincei.it/it/manifestazioni/cabibbo-60-convegno.
31st May, 2024

7

67. Vud , Vus the Cabibbo Angle, and CKM Unitarity[51] S. Aoki et al. (Flavour Lattice Averaging Group), Eur. Phys. J. C 80, 2, 113(2020), [arXiv:1902.08191]; A. Bazavov et al., Phys. Rev. Lett. 112, 11, 112001 (2014),[arXiv:1312.1228]; N. Carrasco et al., Phys. Rev. D 93, 11, 114512 (2016), [arXiv:1602.04113];
A. Bazavov et al., Phys. Rev. D 87, 073012 (2013), [arXiv:1212.4993]; P. A. Boyle et al.
(RBC/UKQCD), JHEP 06, 164 (2015), [arXiv:1504.01692]; A. Bazavov et al., Phys. Rev.
D 98, 7, 074512 (2018), [arXiv:1712.09262]; T. Blum et al. (RBC, UKQCD), Phys. Rev. D
93, 7, 074505 (2016), [arXiv:1411.7017]; R. Dowdall et al., Phys. Rev. D 88, 074504 (2013),[arXiv:1303.1670]; N. Carrasco et al., Phys. Rev. D 91, 5, 054507 (2015), [arXiv:1411.7908];
E. Follana et al. (HPQCD, UKQCD), Phys. Rev. Lett. 100, 062002 (2008), [arXiv:0706.1726];
A. Bazavov et al. (MILC), PoS LATTICE2010, 074 (2010), [arXiv:1012.0868]; S. Durr et al.,Phys. Rev. D 81, 054507 (2010), [arXiv:1001.4692]; S. Dürr et al., Phys. Rev. D 95, 5, 054513(2017), [arXiv:1601.05998]; V. Bornyakov et al. (QCDSF), Phys. Lett. B 767, 366 (2017),[arXiv:1612.04798].
[52] V. Cirigliano and H. Neufeld, Phys. Lett. B 700, 7 (2011), [arXiv:1102.0563]; W. J. Marciano,
Phys. Rev. Lett. 93, 231803 (2004), [hep-ph/0402299].
[53] D. Giusti et al., Phys. Rev. Lett. 120, 7, 072001 (2018), [arXiv:1711.06537].
[54] M. Di Carlo et al., Phys. Rev. D 100, 3, 034514 (2019), [arXiv:1904.08731].
[55] N. Cabibbo, E. C. Swallow and R. Winston, Phys. Rev. Lett. 92, 251803 (2004), [hepph/0307214].
[56] V. Mateu and A. Pich, JHEP 10, 041 (2005), [hep-ph/0509045].
[57] Y. S. Amhis et al. (HFLAV) (2019), [arXiv:1909.12524].
[58] Y. S. Amhis et al. (HFLAV), Phys. Rev. D 107, 5, 052008 (2023), [arXiv:2206.07501].
[59] R. J. Hudspith et al., Phys. Lett. B 781, 206 (2018), [arXiv:1702.01767]; P. Boyle et al. (RBC,
UKQCD), Phys. Rev. Lett. 121, 20, 202003 (2018), [arXiv:1803.07228].
[60] A. Sirlin, Rev. Mod. Phys. 50, 573 (1978), [Erratum: Rev.Mod.Phys. 50, 905 (1978)].
[61] D. Webber et al. (MuLan), Phys. Rev. Lett. 106, 041803 (2011), [arXiv:1010.0991].
[62] K. Babu and S. Pakvasa (2002), [hep-ph/0204236].
[63] W. Marciano and A. Sirlin, Phys. Rev. Lett. 56, 22 (1986); P. Langacker and D. London,
Phys. Rev. D 38, 886 (1988).
[64] W. Marciano and A. Sirlin, Phys. Rev. D 35, 1672 (1987).
[65] R. Barbieri et al., Phys. Lett. B 156, 348 (1985); K. Hagiwara, S. Matsumoto and Y. Yamada,
Phys. Rev. Lett. 75, 3605 (1995), [hep-ph/9507419]; A. Kurylov and M. Ramsey-Musolf, Phys.
Rev. Lett. 88, 071804 (2002), [hep-ph/0109222]; S. Bauman, J. Erler and M. Ramsey-Musolf,
Phys. Rev. D 87, 3, 035012 (2013), [arXiv:1204.0035].

31st May, 2024