Constraining Light Dark Photons from GW190517 and GW190426_152155
CConstraining Light Dark Photons from GW190517 and GW190426 152155
Diptimoy Ghosh ∗ and Divya Sachdeva † Department of Physics, Indian Institute of Science Education and Research Pune, India
Ultralight dark photons predicted in several Standard Model extensions can trigger the super-radiant instability around rotating black holes if their Compton wavelength is comparable to theBlackhole radius. Consequently, the angular momentum of the black hole is reduced to a valuewhich depends upon the mass and spin of the black hole as well as the mass of the dark photon.We use the mass and spin measurements of the primary black holes in two recently observed binaryblack hole systems: GW190517 and GW190426 152155 to constrain dark photon mass in the ranges1 . × − eV < m A (cid:48) < . × − eV and 1 . × − eV < m A (cid:48) < . × − eV respectively,assuming a timescale of a few million years from the time of formation of the binary black holesystem to the time of their merger. We also discuss an interesting X-ray binary system, MAXIJ1820 070, albeit with a relatively small value of the spin parameter. I. INTRODUCTION
The presence of dark matter (DM) in the universe andits dominance over luminous matter is well-established.In the absence of any DM candidate within the Stan-dard Model (SM), a variety of models have been proposedto explain this additional matter content by introducingnew degrees of freedom with masses ranging all the waydown to 10 − eV (lower bound comes from studies ofdwarf galaxy morphology [1]). One promising DM can-didate in the ultralight regime is the dark photon (DP),gauge boson of a U (1) gauge group which can acquireits mass through the Higgs or St¨uckelberg mechanism,proposed in various beyond the SM scenarios [2–4]. Byultralight DM, we mean particle of masses smaller than10 − eV; the mass range 10 − − − eV is often re-ferred to as fuzzy DM [5, 6] and the rest of the rangewith weak coupling to SM particle as weakly interactingslim particles [7–9]. While these states conform to certainobservational features of DM [10], they are also relevantin models explaining inflation and dark energy [11, 12].Being ultralight, they must be produced non-thermallyin the early Universe, for example via the misalignmentmechanism [8, 13], parametric resonance production ortachyonic instability of a (pseudo)scalar field [14–16], orfrom the decay of a cosmic strings [17], depending on howthey couple to the SM. The strongest constraints on ul-tra light gauge bosons with very weak coupling to the SMparticles come from equivalence principle tests, includingthose from the E¨ot-Wash group [18, 19] and Lunar LaserRanging experiments [20]. These are fifth-force exper-iments that do not assume dark photon as DM. Alongwith these, the parameter space corresponding to largercoupling with SM particles is constrained by a variety oflaboratory-based experiments, astrophysical and cosmo-logical observations (see Ref [21] for detailed review).Despite various efforts, the existence of DM is inferredonly through its gravitational effects on visible matter. In ∗ [email protected] † [email protected] this respect, the process of superradiance [22–32] aroundKerr black holes (BHs) provide a unique way of probingultralight bosons solely via their gravitational interac-tions. In this phenomenon, a gravitational bound state ofthe bosonic field develops which leads to a large extrac-tion of energy and angular momentum from a rotatingBH if the black hole radius is comparable to the boson’sCompton wavelength. Thus, the determination of spinand mass of BH can place a constraint on the existence ofthe ultralight bosons with a particular mass. There existseveral studies that exploit properties of stellar-mass as-trophysical BHs determined via their accretion observa-tions or Blackhole binary (BBH) system observed via X-ray or Gravitational wave signatures to unravel the exis-tence of ultralight vector fields. For example, Ref. [33, 34]exploit supermassive blackholes to disfavour the fuzzyDM mass region, Ref. [35–37] use observations of stellarmass BHs, X-ray binaries, and gravitational wave eventsto probe light scalar and vector bosons masses of differ-ent range. In the present work, we use the mass and spinmeasurements of the two recently observed BBH systemsviz. GW190517 [38] and GW190426 152155 [39], andof the X-ray binary MAXI J1820+070 [40] to constrainthe parameter space of DP complementing and extend-ing earlier studies. For the case of the first two BBHsystems, spin and mass are obtained from the data re-leased by LIGO-Virgo in their catalog GWTC-2 [41] andfor the third candidate, analysis is done by fitting thespectra obtained from Insight -HXMT [42–45]. II. SUPERRADIANCE OVERVIEW
We begin by recapitulating the key results related tothe process of superradiance for light vector fields thatare useful to our analysis. For an in-depth discussion ofthe subject, numerous resources [30–33, 35, 46–49] exist.Light non-relativistic dark photon of mass m A (cid:48) arounda Kerr blackhole of mass M BH with the angular velocity http://hxmten.ihep.ac.cn/ a r X i v : . [ a s t r o - ph . H E ] F e b f the BH horizon as Ω H may experience a superradiantinstability if the corresponding wave mode of frequency ω satisfies the following condition [24, 25, 30]: ωm < Ω H , (1)with m being the total angular momentum of boson alongthe BH’s rotating axis and Ω H being related to the di-mensionless spin parameter a ∗ ≡ J BH / ( G N M ) ∈ [0 , H = 12 G N M BH a ∗ √ − a ∗ , (2)where G N is Newton’s gravitational constant and J BH is the total angular momentum of the BH. For a signifi-cantly rotating blackhole, this condition implies that su-perradiance is effective only if the Compton wavelengthof the light vector is comparable to the radius of the blackhole which is given as, R BH = G N M BH (1 ± (cid:112) − a ∗ ) . (3)For growth of superradiant states, the boson need nothave any initial number density; quantum fluctuationsare sufficient to populate a boson cloud around the BH.The boson population will grow exponentially with time,extracting energy and angular momentum from the BH.The leading contribution to the growth rate comesfrom l = 0 and j = m = 1 mode [30, 35] and is given byΓ V = 4 a ∗ G N M m A (cid:48) . (4)For a BH starting with spin a ∗ , and the cloud extractingits angular momentum by an amount ∆ a ∗ = a ∗ , − a ∗ ,the final occupation number for the mode with azimuthalquantum number, m , is given by N m (cid:39) G N M ∆ a ∗ m . (5)Thus, if we require the superradiance rate to be fastenough, depleting the spin of a BH by ∆ a ∗ amountwithin the relevant timescale ( τ BH ), the following addi-tional condition should be satisfied for each mode:Γ V τ BH ≥ ln N m . (6)Although the l = 0 , j = m = 1 mode is the domi-nant one, higher- m modes would be relevant if growthof the ( m − th level stops within the lifetime of the BH.While for the dominant mode, analytic expression for thegrowth rate is available (eqn. 4), this has to be calculatednumerically for subdominant modes. Using Ref. [35], wecalculate the growth rates for different modes of weakly-coupled vector bosons of different masses and illustratetheir relevance as a function of the BH mass and spin infig 1.Using eqn. 6, an upper limit on the DP mass for anobservation of a BH mass and spin can be placed by M BH ( M ⊙ ) | a * | - eV2.5x10 - eV GW190517 j = j = j = MAXI J1820 + _ FIG. 1. The solid and the dashed curves satisfy eqn. 7 andeqn. 8 (with an equality sign) for j=1,2,3 level for two repre-sentative values of the vector field mass, m A (cid:48) ∼ − eV and m A (cid:48) ∼ . × − eV respectively. Each shaded region, thus,satisfies the superradiance condition. The black curve cor-responds to 1 σ contour for GW190517 taken from Ref. [38].The other two data points are the primary BH’s mass and spinmeasurements of the BBH system GW190426 152155 [39] andX-ray binary system MAXI J1820+070 [40] with 1 σ errors. ABH excludes the vector boson mass if it lies in the shaded re-gion, within experimental error. demanding that superradiance has not depleted the spinof the BH by ∆ a ∗ , m A (cid:48) < (cid:18) ln N m a ∗ G N M τ BH (cid:19) / . (7)A lower limit can be obtained from eqn. 1 by requiringthat superardiance is not effective enough to reduce thespin: m A (cid:48) > Ω H . (8)We will see in the next section how observation of BBHsin gravitational wave detectors and in X-rays can con-strain specific ranges of dark photon mass using eqn. 7and 8. III. RESULTS
In this section, we consider the observation of twoBBHs systems seen via gravitational wave signature andan X-ray binary as mentioned in the previous section toprobe a light DP mass. Taking a cue from Ref. [38, 50],we assume an inspiral timescale of 10 yr from the timethe binary black hole system is formed to the time theblack holes merge.For BHs with an astrophysical origin, Thorne has givena upper limit to the reduced spin a ∗ , ∼ .
998 [51]. Thislimit comes from the accretion of the surrounding gas2 .1 M ⊙ M ⊙ M ⊙ - - - - | a * | m A ' ( e V ) FIG. 2. Variation of bounds on the DP mass as a function ofspin of the primary BH of a BBH system GW190517 (green),GW190426 152155 (blue), and X-ray binary system MAXIJ1820+070 (red) corresponding to the inspiral timescale of10 yrs. The solid and dashed contours correspond to eqn. 8and eqn. 7 (with an equality sign) respectively. The regionbetween these curves is disfavoured. on a BH, and its balance with superradiance effects. Wetake this as the maximum spin a BH can have. The othermost important parameters in the present context are themass and the final spin of the primary BH, i.e the valuejust before it is merged. We use the results of the de-tailed analysis performed in Ref. [38] and Ref. [39] whichutilise the data provided on BBH systems GW190517and GW190426 152155 by GWTC-2 to obtain the spinof the primary BH as a function of its mass. These mea-surements are shown in fig. 1. We use the lower edge ofthe 1 σ contour of GW190517 and GW190426 152155 (asshown in fig. 1) to obtain conservative bounds.Following the discussion above and in the previoussection, a specific mass range of light vector boson isconstrained. In fig. 1, we show the effect of superradi-ance for representative values of vector boson mass. Thecurves satisfy eqn. 7 and eqn. 8 (with an equality sign)for j=1,2,3 level. As mentioned earlier, for a vector bo-son, the j = m = 1 , (cid:96) = 0 level dominates and thusthe bound can be obtained by considering just this level.For GW190517, for some part of the 1 σ contour (andfor larger dark photon mass, see fig. 1), higher- j modescan also contribute. We found numerically that includingthese increases the upper limit at most by a factor of 2.Below we quote the limits using the j = 1 level only forwhich analytic expressions are available.We find that the observations of two BBH systems dis-favours light bosons in the following ranges,1 . × − eV < m A (cid:48) < . × − eV , (9)1 . × − eV < m A (cid:48) < . × − eV . (10)In fig. 1, we also show the data point for the X-ray binaryMAXI J1820 070. Similar to the case of the other twoBBH candidates, using the lower edge (of data point)for spin measurement, we are able to constrain only asmall range of DP mass: 8 . × − eV < m A (cid:48) < . × − eV. With better spin measurements in thefuture, one can improve this bound. It is interesting tonote that even observations of relatively lower spin BHs,in principle, can provide useful limits, provided the spinmeasurement is precise enough.To better understand the results and implications ofsuperradiance, fig 2 shows the variation of the constraintsas a function of BH spin for different BH mass. Clearly,if the spin is higher, the constraints on the DP mass willbecome stronger. With increasing values of the BH mass,larger spin values would be required to superradiate effi-ciently. IV. CONCLUSION
In this short note, we have shown how the recent mea-surements of mass and spin of the primary BH from ob-servations of two new BBHs by LIGO/Virgo, and theobservation of an X-ray binary can probe the ultralightDP in different mass ranges due to the phenomenon ofsuperradiance. Using a combination of analytic and nu-merical results, we have found that the BBH observa-tions, GW190517 and GW190426 152155, disfavours themass ranges 1 . × − eV < m A (cid:48) < . × − eV and1 . × − eV < m A (cid:48) < . × − eV respectively.It should be noted that these bounds assume only grav-itational interactions. Thus, these results are valid onlyif the considered vector field is weakly coupled, i.e, pos-sible non-gravitational couplings with other particles, aswell as any non-trivial self-interactions should be negligi-ble compared to the gravitational interaction. For scalarbosons, it has been shown that self-interactions or cou-plings of bosons to other particles can affect the superra-diant instability, and significantly change the time scalerequired to extract a substantial amount of energy andangular momentum from the BH [52–59]. However, theeffect of such interactions for massive vectors has notbeen studied in detail.There are other caveats: analysis similar to ours isplagued with our lack of understanding of formation his-tory (thus lifetime and initial spin are largely uncertain)of a given BH binary; any given pair may not have su-perradiated enough if they merged too quickly or formedtoo close together. Moreover, the backreaction due to theDP cloud and perturbations due to the secondary BH onthe Kerr geometry can affect the final results. However,this effect is expected to be not significant. [31, 35, 49].As the sensitivity of the gravitational wave observato-ries improves, they will detect more or more such merg-ers, and thus, future observations are expected to coverlarger parameter space of DP mass and possibly find thefirst signature of such light bosons.3 CKNOWLEDGMENTS
The authors acknowledge support through the Ra-manujan Fellowship and the MATRICS grant of the De- partment of Science and Technology, Government of In-dia. [1] S.-R. Chen, H.-Y. Schive, and T. Chiueh, Jeans Analy-sis for Dwarf Spheroidal Galaxies in Wave Dark Mat-ter, Mon. Not. Roy. Astron. Soc. , 1338 (2017),arXiv:1606.09030 [astro-ph.GA].[2] M. Goodsell, J. Jaeckel, J. Redondo, and A. Ring-wald, Naturally Light Hidden Photons in LARGEVolume String Compactifications, JHEP , 027,arXiv:0909.0515 [hep-ph].[3] P. G. Camara, L. E. Ibanez, and F. Marchesano, RRphotons, JHEP , 110, arXiv:1106.0060 [hep-th].[4] K. Kaneta, H.-S. Lee, and S. Yun, Portal ConnectingDark Photons and Axions, Phys. Rev. Lett. , 101802(2017), arXiv:1611.01466 [hep-ph].[5] W. Hu, R. Barkana, and A. Gruzinov, Cold andfuzzy dark matter, Phys. Rev. Lett. , 1158 (2000),arXiv:astro-ph/0003365.[6] L. Hui, J. P. Ostriker, S. Tremaine, and E. Witten, Ul-tralight scalars as cosmological dark matter, Phys. Rev.D , 043541 (2017), arXiv:1610.08297 [astro-ph.CO].[7] P. Arias, D. Cadamuro, M. Goodsell, J. Jaeckel, J. Re-dondo, and A. Ringwald, WISPy Cold Dark Matter,JCAP , 013, arXiv:1201.5902 [hep-ph].[8] A. E. Nelson and J. Scholtz, Dark Light, Dark Matter andthe Misalignment Mechanism, Phys. Rev. D , 103501(2011), arXiv:1105.2812 [hep-ph].[9] K. Nakayama, Vector Coherent Oscillation Dark Matter,JCAP , 019, arXiv:1907.06243 [hep-ph].[10] D. J. E. Marsh and J. Silk, A Model For Halo FormationWith Axion Mixed Dark Matter, Mon. Not. Roy. Astron.Soc. , 2652 (2014), arXiv:1307.1705 [astro-ph.CO].[11] C. G. Boehmer and T. Harko, Dark energy as a massivevector field, Eur. Phys. J. C , 423 (2007), arXiv:gr-qc/0701029.[12] T. Koivisto and D. F. Mota, Vector Field Models of In-flation and Dark Energy, JCAP , 021, arXiv:0805.4229[astro-ph].[13] G. Alonso- ´Alvarez, T. Hugle, and J. Jaeckel, Mis-alignment \ & Co.: (Pseudo-)scalar and vector darkmatter with curvature couplings, JCAP , 014,arXiv:1905.09836 [hep-ph].[14] P. Agrawal, N. Kitajima, M. Reece, T. Sekiguchi,and F. Takahashi, Relic Abundance of Dark Pho-ton Dark Matter, Phys. Lett. B , 135136 (2020),arXiv:1810.07188 [hep-ph].[15] J. A. Dror, K. Harigaya, and V. Narayan, ParametricResonance Production of Ultralight Vector Dark Matter,Phys. Rev. D , 035036 (2019), arXiv:1810.07195 [hep-ph].[16] R. T. Co, A. Pierce, Z. Zhang, and Y. Zhao, Dark PhotonDark Matter Produced by Axion Oscillations, Phys. Rev.D , 075002 (2019), arXiv:1810.07196 [hep-ph].[17] A. J. Long and L.-T. Wang, Dark Photon Dark Matterfrom a Network of Cosmic Strings, Phys. Rev. D ,063529 (2019), arXiv:1901.03312 [hep-ph].[18] Y. Su, B. R. Heckel, E. G. Adelberger, J. H. Gundlach, M. Harris, G. L. Smith, and H. E. Swanson, New tests ofthe universality of free fall, Phys. Rev. D , 3614 (1994).[19] S. Schlamminger, K.-Y. Choi, T. A. Wagner, J. H. Gund-lach, and E. G. Adelberger, Test of the equivalence prin-ciple using a rotating torsion balance, Phys. Rev. Lett. , 041101 (2008).[20] J. G. Williams, S. G. Turyshev, and D. H. Boggs,Progress in lunar laser ranging tests of relativistic grav-ity, Phys. Rev. Lett. , 261101 (2004).[21] A. Filippi and M. De Napoli, Searching in the dark: thehunt for the dark photon, Rev. Phys. , 100042 (2020),arXiv:2006.04640 [hep-ph].[22] R. Penrose, Gravitational Collapse: the Role of GeneralRelativity, Nuovo Cimento Rivista Serie , 252 (1969).[23] S. Detweiler, Klein-gordon equation and rotating blackholes, Phys. Rev. D , 2323 (1980).[24] J. D. Bekenstein, Extraction of energy and charge froma black hole, Phys. Rev. D , 949 (1973).[25] J. D. Bekenstein and M. Schiffer, The many faces of su-perradiance, Phys. Rev. D , 064014 (1998).[26] C. W. Misner, Interpretation of gravitational-wave ob-servations, Phys. Rev. Lett. , 994 (1972).[27] W. H. Press and S. A. Teukolsky, Perturbations of a Ro-tating Black Hole. II. Dynamical Stability of the KerrMetric, Astrophys. J. , 649 (1973).[28] W. H. Press and S. A. Teukolsky, Floating Orbits, Su-perradiant Scattering and the Black-hole Bomb, Nature , 211 (1972).[29] T. J. Zouros and D. M. Eardley, Instabilities of massivescalar perturbations of a rotating black hole, Annals ofPhysics , 139 (1979).[30] A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper,and J. March-Russell, String Axiverse, Phys. Rev. D ,123530 (2010), arXiv:0905.4720 [hep-th].[31] R. Brito, V. Cardoso, and P. Pani, Black holes as particledetectors: evolution of superradiant instabilities, Classi-cal and Quantum Gravity , 134001 (2015).[32] R. Brito, V. Cardoso, and P. Pani, Superradiance: NewFrontiers in Black Hole Physics, Lect. Notes Phys. ,pp.1 (2015), arXiv:1501.06570 [gr-qc].[33] P. Pani, V. Cardoso, L. Gualtieri, E. Berti, andA. Ishibashi, Black hole bombs and photon mass bounds,Phys. Rev. Lett. , 131102 (2012), arXiv:1209.0465[gr-qc].[34] H. Davoudiasl and P. B. Denton, Ultralight BosonDark Matter and Event Horizon Telescope Observa-tions of M87*, Phys. Rev. Lett. , 021102 (2019),arXiv:1904.09242 [astro-ph.CO].[35] M. Baryakhtar, R. Lasenby, and M. Teo, Black Hole Su-perradiance Signatures of Ultralight Vectors, Phys. Rev.D , 035019 (2017), arXiv:1704.05081 [hep-ph].[36] V. Cardoso, ´O. J. Dias, G. S. Hartnett, M. Middle-ton, P. Pani, and J. E. Santos, Constraining the mass ofdark photons and axion-like particles through black-holesuperradiance, Journal of Cosmology and Astroparticle hysics (03), 043.[37] M. J. Stott, Ultralight Bosonic Field Mass Bounds fromAstrophysical Black Hole Spin, (2020), arXiv:2009.07206[hep-ph].[38] K. K. Y. Ng, S. Vitale, O. A. Hannuksela, and T. G. F.Li, Constraints on ultralight scalar bosons within blackhole spin measurements from LIGO-Virgo’s GWTC-2,(2020), arXiv:2011.06010 [gr-qc].[39] Y.-J. Li, M.-Z. Han, S.-P. Tang, Y.-Z. Wang, Y.-M. Hu,Q. Yuan, Y.-Z. Fan, and D.-M. Wei, GW190426 152155:a merger of neutron star-black hole or low mass binaryblack holes?, (2020), arXiv:2012.04978 [astro-ph.HE].[40] X. Zhao, L. Gou, Y. Dong, Y. Tuo, Z. Liao, Y. Li, N. Jia,Y. Feng, and J. F. Steiner, Estimating the black holespin for the X-ray binary MAXI J1820+070, (2020),arXiv:2012.05544 [astro-ph.HE].[41] R. Abbott et al. (LIGO Scientific, Virgo), GWTC-2:Compact Binary Coalescences Observed by LIGO andVirgo During the First Half of the Third Observing Run,(2020), arXiv:2010.14527 [gr-qc].[42] L. et al., The High Energy X-ray telescope (HE) onboardthe Insight-HXMT astronomy satellite, Science ChinaPhysics, Mechanics, and Astronomy , 249503 (2020),arXiv:1910.04955 [astro-ph.IM].[43] Z. et al., Overview to the Hard X-ray Modula-tion Telescope (Insight-HXMT) Satellite, Science ChinaPhysics, Mechanics, and Astronomy , 249502 (2020),arXiv:1910.09613 [astro-ph.IM].[44] C. et al., The Low Energy X-ray telescope (LE) onboardthe Insight-HXMT astronomy satellite, Science ChinaPhysics, Mechanics, and Astronomy , 249505 (2020),arXiv:1910.08319 [astro-ph.IM].[45] C. et al., The Medium Energy (ME) X-ray telescopeonboard the Insight-HXMT astronomy satellite, arXive-prints , arXiv:1910.04451 (2019), arXiv:1910.04451[astro-ph.IM].[46] H. Witek, V. Cardoso, A. Ishibashi, and U. Sperhake,Superradiant instabilities in astrophysical systems, Phys.Rev. D , 043513 (2013), arXiv:1212.0551 [gr-qc].[47] P. Pani, V. Cardoso, L. Gualtieri, E. Berti, andA. Ishibashi, Perturbations of slowly rotating black holes:massive vector fields in the Kerr metric, Phys. Rev. D , 104017 (2012), arXiv:1209.0773 [gr-qc].[48] W. E. East, Superradiant instability of massive vec-tor fields around spinning black holes in the relativisticregime, Phys. Rev. D , 024004 (2017).[49] W. E. East and F. Pretorius, Superradiant Instabil-ity and Backreaction of Massive Vector Fields aroundKerr Black Holes, Phys. Rev. Lett. , 041101 (2017),arXiv:1704.04791 [gr-qc].[50] K. K. Y. Ng, O. A. Hannuksela, S. Vitale, and T. G. F. Li,Searching for ultralight bosons within spin measurementsof a population of binary black hole mergers, (2019),arXiv:1908.02312 [gr-qc].[51] K. S. Thorne, Disk-Accretion onto a Black Hole. II. Evo-lution of the Hole, Astrophys. J. , 507 (1974).[52] A. Arvanitaki and S. Dubovsky, Exploring the string axi-verse with precision black hole physics, Phys. Rev. D ,044026 (2011).[53] H. Yoshino and H. Kodama, Bosenova collapse of axioncloud around a rotating black hole, Prog. Theor. Phys. , 153 (2012), arXiv:1203.5070 [gr-qc].[54] H. Yoshino and H. Kodama, The bosenova and axiverse,Class. Quant. Grav. , 214001 (2015), arXiv:1505.00714[gr-qc].[55] J. a. G. Rosa and T. W. Kephart, Stimulated Ax-ion Decay in Superradiant Clouds around PrimordialBlack Holes, Phys. Rev. Lett. , 231102 (2018),arXiv:1709.06581 [gr-qc].[56] T. Ikeda, R. Brito, and V. Cardoso, Blasts of Lightfrom Axions, Phys. Rev. Lett. , 081101 (2019),arXiv:1811.04950 [gr-qc].[57] M. Boskovic, R. Brito, V. Cardoso, T. Ikeda, andH. Witek, Axionic instabilities and new black hole solu-tions, Phys. Rev. D , 035006 (2019), arXiv:1811.04945[gr-qc].[58] H. Fukuda and K. Nakayama, Aspects of NonlinearEffect on Black Hole Superradiance, JHEP , 128,arXiv:1910.06308 [hep-ph].[59] A. Mathur, S. Rajendran, and E. H. Tanin, Clockworkmechanism to remove superradiance limits, Phys. Rev. D , 055015 (2020), arXiv:2004.12326 [hep-ph]., 055015 (2020), arXiv:2004.12326 [hep-ph].