JHEP07(2020)235 Be 8 and 0 : Z E,Μ Springer Anomalies; July 2, 2020 July 31, 2020 May 22, 2020 : 2) : : E,Μ 2) − − G G , Accepted ( Received Published

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JHEP07(2020)235 Be 8 and 0 : Z e,µ Springer anomalies; July 2, 2020 July 31, 2020 May 22, 2020 : 2) : : e,µ 2) − − g g , Accepted ( Received Published . The model further includes L Be while being consistent with − 8 B b boson. The new ﬁelds and currents 0 Published for SISSA by https://doi.org/10.1007/JHEP07(2020)235 Z and the anomalous internal pair creation in [email protected] e , 2) and A.M. Teixeira − b g [email protected] , J. Orloﬀ Be nuclear transitions and . b 8 3 , which gives rise to a light L − 2005.00028 -breaking scalar can also successfully account for both ( B The Authors. L − c Beyond Standard Model, Neutrino Physics

B Be nuclear transitions, we explore a simple New Physics model, based on an Motivated by a simultaneous explanation of the apparent discrepancies in the , 8 J. Kriewald, [email protected] a Laboratoire de Physique de ClermontUniv. (UMR Clermont 6533), Auvergne, CNRS/IN2P3, 4 Av.E-mail: Blaise Pascal, F-63178 Cedex, Aubière France [email protected] Physik Department T70, Technische UniversitätMünchen, James-Franck-Straße 1, D-85748 Garching, Germany b a Open Access Article funded by SCOAP ArXiv ePrint: particularly predictive. Thereadily underlying incorporated idea into other of protophobic this U(1) minimal extensions of “prototype theKeywords: model” Standard can Model. be breaking of U(1) allow to explain thevarious anomalous experimental constraints. internal Interestingly, pair we find creationthe that new in the U(1) contributions of the the strong phenomenological constraints onthe the model’s combined parameter explanation space ultimately of render ( Abstract: light charged lepton anomalous magnetic dipolecreation moments, in and the anomalousextension internal pair of the Standardheavy Model gauge vector-like group fermion by fields, a as U(1) well as an extra scalar responsible for the low-scale C. Hati, Anomalies in towards a minimal combined explanation JHEP07(2020)235 23 10 20 17 11 1) transitions 19 M Be), with a significant 8 27 5 17 X h boson 13 0 : the model and 21 Z L 0 Be: impact for a combined explana- Be 7 8 − e,µ Z 8 – 1 – B 2) − g ]. 40 – 1 25 5 e,µ (spin-1) [ 0 1 2) Z ] on the measurement of the angular correlation of electron-positron internal − 41 g Several distinct probes have been used to look for the presence of the new mediators. tion of ( 6.1 Constraining the6.2 model’s parameters A combined explanation of ( 5.1 Experimental constraints5.2 on a light Majorana neutrinos: fitting the leptonic axial and vector couplings 2.1 Gauge sector 2.2 Lepton sector:2.3 masses and New mixings neutral current interactions: for relatively light new physics states.their results A [ few years ago,pair the creation Atomki (IPC) Collaboration for reported twoexcess nuclear being transitions observed of at Beryllium large atoms anglesunder for ( study one concerned of the them. decays The of magnetic the dipole excited ( isotriplet and isosinglet states, respectively Model (SM) gaugeand group, astroparticle has physics. beenfor a Numerous theoretically experimental longstanding well-motivated searches source light have(spin-1) mediators, of been or such interest, light dedicated as both to axions for look (spin-zero), particle dark photons Nuclear transitions are among the most interesting and promising laboratories to search 1 Introduction The possible existence of new interactions, in addition to those associated with the Standard 7 Concluding remarks A Electron-neutrino scattering data: analysis and fits 6 Addressing the anomalous IPC in 4 Explaining the anomalous IPC5 in Phenomenological constraints on neutral (vector and axial) couplings 3 New physics contributions to the anomalous magnetic moments Contents 1 Introduction 2 A light vector boson from a U(1) JHEP07(2020)235 ]). sig- , all 54 (1.1) ], due σ µν ˜ F 51 ) might ∗ µν decay. In 16(stat) . , ]. Be 0 0 8 aF ∗ 41 a g ] (although this , with 7.2 Be ◦ 8 and 53 , 0 64 MeV ; 15 MeV ∗ . . 52 =16.98 ]. Be pair correlation in the X 8 48 m = 17 = 18 − e 5(sys) MeV [ . + 0 e decays (only one set exhibiting Be. ∗ 8 Be in the context of simple U(1) ). The transitions are summarised Be 8 0 8 . The ]; moreover, it allowed constraining 01(16) MeV and its branching ratio ,E,E 6 . Be transition. Having a pseudoscalar Be 45 − 35(stat) 8 . ], resulting in another anomalous IPC – 0 + 8 10 = 0) = 0) 0 42 = 17 49 × Be isospin states ( ], relying on the exchange of a 16.7 MeV, , 8 (10 GeV) are excluded [ → ,T ,T X 48 / + + 55 1 + , m – 2 – 17 MeV and an interaction term =16.70 < 51 = 0 = 0 He [ = 6(1) X ≈ Be anomaly with an improved set up corroborated a 4 π π 8 eV, lie in a range similar to that suggested by the a γ m j j 5 ( ( Γ < g m 0 0 − / X Be Be 10 8 8 × GeV) 9 → → . 18 transition of Be was also revisited, and again no significant anomalies were 8 (10 = 3 + = 0) = 1) / , into the fundamental state ( X 0 ∗ -decay) to Γ with a mass of ,T ,T γ Be correspond to the spin-parity and isospin of the nuclear states, respec- → a + + 8 T − = 1 = 1 and π π 0 ]. A combined interpretation of the data of Be transition. and the latter can a priori be a scalar, pseudoscalar, vector, axial vector, or even ∗ j j ], the possibility of explaining the anomaly within nuclear physics was explored; however, the 8 ( ( 0 π 1 ∗ 47 50 ∗ Be j , 8 Be Be 46 8 8 If the anomalous IPC observations are to be interpreted as being mediated by a light Further anomalies in nuclear transitions have been observed, in particular concerning Recently, a re-investigation of the In ref. [ 1 20(syst) MeV, and Γ . can be partially circumventedA in potential the first presence explanation ofextensions of additional of the non-photonic the anomalous couplings SM IPC [ in was discussed inrequired [ form factors were found to be unrealistic for a nucleus like a spin-2 particle, providedscenario, it the hypothesis decays of into an intermediate electron-positron scalarconservation boson pairs. of has angular already For momentum been a dismissed in [ mediator parity the has conserving 1 been alsoaxion-like severely particle constrained (andcouplings disfavoured) in the by range experiments 1 — for an decay of a0 light particle: theanomalous corresponding mass and width, new state, normalised branching fractions than those of the preferred fitthe of 21.01 [ MeV 0 corresponding to the angularnificance. correlation of The electron-positron result pairs can at 115 also be potentially interpreted as the creation and subsequent of mixing betweensuggest the a two larger different preferred excited space mass suppression, for therefore theturn, potentially new it explaining mediator; can the further thisto null entail would the results significant lead new changes for particle to in mediating a the the large preferred anomalous phase quark IPC, (nucleon) corresponding couplings to significantly smaller the mass of the(normalised hypothetical to mediator the to 17.64 MeV transition of found [ an anomalous behaviour) in terms of a new light particle, in association with the possibility correlation of theanomalous 18.15 MeV result could transition; be potentially itintermediate interpreted light as was particle the creation subsequently with and mass pointed decay of out an unknown thatthe such earlier an results for the 18.15 MeV transition [ below, together with the associated energies: in which tively. A significant enhancement of the IPC was observed at large angles in the angular denoted JHEP07(2020)235 0 = π γA j (1.2) (1.3) (1.4) (1.5) → ] 0 ). As of 96 ]), a long [ π e 69 e, µ ] — an idea a 55 = ], light-by-light ]. Further ideas ` 80 56 – 2 ( / 72 ] ` 2) 71 . , ], typically leading to de- − ] has recently given rise to 12 . 70 g − ( 9 91 95 12 , , , − − 10 ≡ 90 94 11 10 ` 10 × ]). − experiment at FNAL [ a × ] it was pointed out that such hadronic × 68 10 µ – 36) 93 7) . . 2) × 57 Be. Interestingly this problem can be 2 − 8 7(0 . 88(0 g 73(28) . 2 . 0 ∼ 089(63) 180 , , (17 MeV), constrains its couplings to be strictly – 3 – ∼ − using Cs atoms [ SM µ O e a 592 ]; however, in [ 652 , , SM e α − 92 a 159 − , exp µ ] the possibility of a light pseudoscalar particle with a = 116 exp e 54 = 1 = a exp µ µ a = a exp e e a ∆ a deviation from the SM prediction, ∆ σ 5 . from the experimental result, σ ], and higher-order hadronic corrections [ 89 – 81 vector gauge boson. In [ + Extensions of the SM which include light new physics states coupled to the standard The favoured scenario of a new light vector boson is nevertheless heavily challenged by Recent results of a lattice study of leading order hadronic vacuum polarisation suggested that the was examined, while a potential explanation based on an axial vector particle (including = 1 2 − π Be); the generalisation towards a protophobic vector boson arising from a gauged U(1) discrepancy could bevacuum significantly polarisation contributions reduced could [ potentially leadin to other conflicts relevant with observables electroweak (hitherto fits, inducing in tensions good agreement with the SM). yet another discrepancy, this timeThe concerning experimental the measurement electron’s of anomalous the magnetic electron moment. anomalous magnetic moment currently exhibits a 2 scattering [ viations of about 3 Interestingly, a precise measurement of today (while expecting new datastanding from tension the persists new between ( the muon’s measured value [ and the SM prediction with improved hadronic vacuum polarisation [ charged leptons are aobservables, priori and expected even to havesevere) have other the implications anomalies, as potential for is to precision theof case tests address light of of those charged (solving, concerning leptons, leptonic the or anomalous usually at magnetic expressed moment least in rendering terms of less in Caesium (Cs) andneutrino-nucleus neutrino-electron scattering), scattering which (as force welltoo the as leptonic small non-observation couplings to of of accountcircumvented coherent the in for gauge the the boson presence anomalous towe of will IPC be additional pursue in vector-like and leptons explore as in noted our in work. [ view of negative searches for associateat production NA48/2 in which, rare for light meson a decays dark“protophobic”, (e.g., photon mass in stark8 contrast with the requirements toextension explain of the the anomalous SMis IPC (potentially also in with subject both to vector stringent and constraints from axial the couplings measurements to of the atomic SM parity fields) violation 0 an ab-initio computation forwere the put relevant forward form and factors) discussed was (see, carried for in instance, [ [ numerous experimental constraints: the dark photon hypothesis is strongly disfavoured in j JHEP07(2020)235 ] – ]. − for 98 g and 119 118 2 e 0 Z /m Be while 2 µ 8 m Be atoms, a 8 . In view of the µ opposite to that of a e a and ∆ associated with a low-scale e 0 a Z transitions. e further suggests the underlying presence − µ µ a , with the SM particle content extended by L – 4 – − B and ∆ e U(1) a × conversion), already in conflict with current data [ ]): in particular, certain scenarios have explored a chiral 2). We emphasise that even though we have considered a ]. It thus becomes particularly challenging to explain both Y e does not follow the quadratic na¨ıve behaviour − 117 e 97 − g – a couplings, arising from experimental searches, and which are µ U(1) 0 ∆ is also puzzling: not only is the sign of ∆ / 100 Z × µ by an extra scalar field. This “prototype model” offers a mini- a e,µ L and a ] experiments). L extension here — a minimal working “prototype model” — the general e − 3 L B 120 − SU(2) → B -neutrino couplings (the tightest bounds being due to the TEXONO [ breaking Higgs scalar can further saturate the discrepancies in both ( 0 × Z L − eγ, µ B (see for example [ → anomalies. In particular, a cancellation between the new contributions is crucial , but the ratio ∆ Our findings reveal that the interplay of the (one-loop) contributions of the In this work, we explore a simple New Physics model, based on an extended gauge µ ]. The observed pattern for ∆ to fermions are strongly constrained by experimental searches: the measurement of e,µ µ e,µ a a 0 combined explanation of the differentwhat anomalies renders concerns the the model electron ultimately ( particular predictive U(1) in idea can be straightforwardlyU(1) adopted extensions of and the incorporated SM. into other possible protophobic the U(1) 2) to reproduce theextensive observed limits pattern on of the further opposite constrained signs by of the ∆ requirements to explain the anomalous IPC in the atomic parity violation inin Caesium what proves concerns to be couplingsthe one non-observation to of of the coherent the electrons. neutrino-nucleusstraints most scattering stringent on impose constraints Likewise, equally neutrino-electron stringentand scattering con- CHARM-II and [ heavy vector-like fermion fields,breaking in of addition U(1) tomal the scenario light tobeing successfully consistent explain with the variousZ anomalous experimental internal bounds. pair However, creation the in couplings of the light in the loop (subject tocan be “generation-wise” mixing achieved, and between this SMfrom further and charged allows heavy lepton to vector-like flavour evade fields) violating the (cLFV) potentially very stringent constraints group SU(3) potentially lead to sizeable contributionscan for open the the leptonic magnetic door moments; tonon-trivial however, charged this couplings lepton to flavour violating bothfor interactions (new muons physics and fields electrons with Controlled can couplings of potentially electrons lead and to muons sizeable to rates beyond the Standard Model (BSM) fields within minimal SM extensions100 via a single newof particle New coupling Physics toattempts charged (NP) leptons have potentially [ been violating∆ recently the conducted universality to ofenhancement, simultaneously lepton due explain to flavours. the heavy vector-like tensions leptons Several in in the both one-loop dipole operator, which can joint behaviour of ∆ ∆ the magnetic dipole operatorinsertion (where of the the SM necessaryanomalies lepton) chirality [ simultaneously flip within appears a through minimal a flavour mass violation (MFV) hypothesis, or even Notice that in addition to being by themselves deviations from the SM expectations, the JHEP07(2020)235 , 4 i × . 2 k are ) Y (2.1) L L − B U(1) . 5 × (SU(2) is broken at L U(1) L L − × − can successfully ), with a charge B B 0 Y R SU(2) / ˜ Z N D f . × . While the first two ¯ f U(1) i U(1) h L − × f A (17 MeV). A successful X B L + gives rise to new gauge and , is given by Be atoms, including a discus- 0 ∼ O 8 U(1) L µν B 0 0 − SU(2) ˜ Z × F B ] further requires the presence of 2 m × µν . A summary of the key points and ˜ 55 F [ boson e,µ denotes the kinetic mixing parameter. k 0 2 L k 2) Z : the model − Gravity + L ; − B 0 − g ˜ µν B A 0 B ˜ F – 5 – µν , in which we detail the couplings of the exotic 0 , and ˜ boson F 2 i , which acquires a vacuum expectation value (VEV) 2 1 4 L ) X − Y − h and the U(1) is devoted to discussing how a light B 4 µν B ˜ F (U(1) correspond to the field strengths of the (kinetically mixed) µν thus, the model also includes three generations of isodoublet L ˜ F − µν 3 ) to ensure phenomenological viability in view of the constraints 0 4 1 B , section ˜ E F , and subsequently investigate how these impact the model’s param- 3 . and the U(1) , in particular the viable regimes allowing for a combined explanation ⊇ − 5 Be anomaly through a light U(1) 1 6 ˜ and 8 h and B gives rise to the triangular gauge anomalies — L A µν gauge kin. , L ˜ i F L − 3 ) B L 1, for each fermion generation. In the present model, the U(1) − − B . Such an extension with a locally gauged U(1) L Be anomaly, as well as the tensions in ( = 8 − B L The model is described in section In particular, constraints arising from neutrino-electron scattering experimentsWe and recall that atomic kinetic parity mixing always vio- appears at least at the one-loop level in models with fermions which (U(1) , responsible for a light vector boson, with a mass 3 4 − h X In the above, hypercharge boson lation require the addition of this exotic particleare content charged as under discussed both in U(1)s. more detail Here in we section parametrise these corrections through an effective coefficient, In the unbrokenbetween phase, the the hypercharge boson relevant part of the kinetic Lagrangian, including mixing and isosinglet vector-like leptons.tion properties under The the fieldsummarised extended in content gauge table of group SU(3) the model2.1 and the transforma- Gauge sector a low scale by av SM singlet scalar, explanation of the additional fields ( from various experiments; A automatically vanish fortribution the from SM additional field fields.achieve content, this relies the One on other of theB introduction the two of most require a SM conventional a singlet and (positive) neutral economical fermion con- ways ( to We consider a minimal gaugeU(1) extension of the SM gaugegauge-gravitational group, anomalies SU(3) in the theory,gauged which need U(1) to be cancelled. In particular, the constraints in section eter space in section of the further discussion are given in the Conclusions. 2 A light vector boson from a U(1) states to SM fields, anddescription their of impact the for new the contributions newgeneric to neutral way) charged current in lepton interactions. section anomalous After magneticexplain a moments the brief several (in reported a results onsion the of anomalous potentially IPC in relevant isospin-breaking effects. We revisit the available experimental JHEP07(2020)235 × (2.3) (2.4) (2.5) (2.6) (2.2) , ! , µν µν 0 0 µ , respectively; ˜ ˜ F L F L B − − B f 1 1 1 B L 1 3 1 3 1 3 0 1 1 2 ! Q − − − − k B f . 1 U(1) , 0 − µ Q and 0 µ L B k / ˜ f D f 2 k − B 1 . The gauge kinetic terms . (2.7) Y 3 ¯ 0 B − L f L 1 2 1 3 1 2 1 1 T i ε 2 k 1 6 2 3 1 2 0 0 − − −

1 − − − − − B − f B + 1 f U(1) g k − f X Q f q µν 1 0 Y Q L + 0 ˜ L − F q = g and = B 0 µν 0 0 µ ε f 0 2 2 1 1 1 2 1 2 1 1 = µν g i g Y F ˜ ˜ 0 F B i 0 SU(2) + µν + F µ 1 4 ˜ 1 4 µ c B − , B f , ε 0 − µ – 6 – = Y f 3 1 3 3 1 1 1 1 1 1 2 k 0 B Y µν L 0 2 k µν SU(3) i g 0 − − F ˜ i g B F 1 k − + µν g T + 1 µν F q ˜ µ F 1 4 ··· charge written as q 3 k = 2 − T + − L,R T L ) W + L ) µ = L f − + µ L − ∂ ,L 3 0 B B , d ε B µν , e = . R 0 R R X R L 0 L,R L SM L,R = L ˜ e µ d u gauge kin. h u N F ν i g T h L − E Field ˜ µ L D B µν ˜ + 0 B = ( = ( ˜ = F ` Q ··· 4 1 U(1) + L,R × − µ L Y ∂ µν ˜ F = U(1) µν µ . Field content of the model and transformation properties under the gauge group SU(3) ˜ × F D 1 4 L − in which we have introduced the following coupling strengths This transformation is obtainedangular by matrices a to Cholesky be absorbed decomposition,the into gauge allowing a covariant the redefinition derivative resulting of can the tri- then gauge be fields. written The as neutral part of by a linear transformation of the fields, the corresponding gauge couplingscan are be denoted cast by in matrix form as which can then be brought to the diagonal canonical form The relevant part of the gauge covariant derivative is given by with the hypercharge and Table 1 SU(2) JHEP07(2020)235 j R and e (2.8) (2.12) (2.13) (2.14) (2.15) (2.10) i L Z and will ¯ , E L 0 µ X − (the VEV Z . (2.11) h 0 B ) ε 5 1 2 X 0 )), one finds ij E L ε v # λ − 0 2.7 ., or ε B f 0 − , (2.9) . , c 2 θ . 0 w j R Q ) w ε i θ    L θ L µ µ X µ 3 0 − i L is the standard weak +H sin 2 h ¯ ` cos B B W sin h B R j w ε g 0 X 0 µ L θ    g ε + h , i L D 2 f ¯ ( ij L    ∓ 2 E 0 0 0 † λ θ θ g ) ! ε Q i 0 ' − − ( SM /e . − 0 X ˜ j h L sin R cos θ ie 2 + 1 -boson induced by h − 2 0 0 ij N h g 0 0 + θ 2 0 B k g B µ θ µ ε i c R − only appear at order + /v gauge coupling have been redefined as ¯ D + tan 2 A 0 2 sin N 0 2 = w j ). The resulting mass matrix, in which R 0 2 f cos 2 B θ L Z w g breaking), and X /v g L E 0 θ , w m h − 2.6 − 0 L i ) + ( θ L + 2 B p sin i B ie Q ¯ 2 B 0 B − L ij M 2 m cos g y g B + + 4 m cos 0 SM µ 1 2 − 2 SM ε h 0 – 7 – ' Z h h 0 + 4 ε 2 − θ ) 0 0 µ 2 ij , ε 2 0 f j θ

Z R h g D w Q 1 2 ( N w cos θ 2 − 0 sin † θ " i L w ε W ) j ¯ R ` w i θ 2 v θ ,M cos θ E defined in eq. ( 2 cos 0 = 2 w SM i SM L ε 0 θ µ v sin ˜ h ¯ h θ sin g sin E 2 h D = 0 − ij ν is the mass term for the − ij µ E g cos y ε f    4 D 2 X 3 M + + tan 2 = ( defined as v (corresponding to small kinetic mixing, cf. eq. ( responsible for U(1) = T 2 0 j R 0 − ( 0 L g ⊇ e θ ε X    j R coupling due to mixing with the − w breaking), the Lagrangian includes the following mass terms i L µ h Z,Z θ µ µ L ¯ 2 B ' ` 0 0 Z g L i L A Z ε 2 Z Z − mass gauge ¯ L SM cos B    i L M h ij L = 4 + ij ` M 0 y ,M ··· 2 B − − + m = = 0 µ ∂ A ' M lepton mass Corrections in the bosons, µ 5 L 0 D henceforth be neglected. 2.2 Lepton sector:Fermion masses masses (both and for mixings SMfollowing leptons and generic the terms additional in vector-like the leptons) Lagrangian arise from the The relevant terms of the gauge covariant derivative can now be expressed as in which the kinetic mixing parameter and the In the limit ofthe following small approximate expressionsZ for the mixing angle and the masses of the of the scalar singlet mixing angle. The mass eigenvalues of the neutral vector bosons are given by with the mixing angle in which the neutral bosons mixrelations amongst between themselves, physical can and be interaction diagonalised, gauge leading boson to states the following the physical fields. In thesubsequent broken U(1) phase (following electroweak symmetry breaking, and the with the covariant derivative Note that due to the above transformation, the mixing now appears in the couplings of JHEP07(2020)235 .      (2.17) (2.19) (2.18) v )) , (2.16) 2 L 3 E 3 X      M    M v 2 + X R 3 E R − 2 X R 2 E 2 E √ λ v e vv 3 L E E . The charged 4 v ) M 2 E M 4 y ( T L )) M λ    ) k 2 2 E − L c + √ 0 −    M M E . In what follows, L X 2 E 1 + E v v M λ M ,L √ 0 2 E eγ 2 E 0 3 E E + M λ , as well as the masses h 3 E √ M L ( M E 1 ,L 2 M → h 2 hM M X 2 c λ 2 ( L v v √ + L √ √ µ L 1 as the (small) expansion L M k hλ X λ M ]. Up to third order in the ν, N + and vv E 2 E )) X 2 L v √ 2 v M L ) 121 kM 0 √ λ v ( L E X M y , and ( )) λ + kλ , k 2 L ( 2 , kv T E L ,    R M ) X L,E 3 block (in generation space). The full h M v M + U R ( 3 E M )) 2 h E ` L × 2 3 L E ¯ M + ,E E M 3 L 2 M yv, hv ( L M M − R ( 3 X k 2 † 1 M + L L − v M 2 + √ 2 E 3 L ¯ 2 2 U E X L ,L L 3 L E √ λ v ), each coupling or mass term thus runs over M – 8 – R 4 M 2 L ( M L M = e kM 4 h λ 2 ¯ e ( E − + √ E ,( 2.15 − L λ L hM T diag ` X ( breaking, the mass matrices for the charged leptons = ) 1 M v M − L E 2 L M − L can be obtained by a perturbative expansion, justified    λ √ X − 2 L R ,E − R kM R B ( vv e − L M L E ], intergenerational couplings between the SM charged lep- L,R L 2 − λ    U ,L · L X 100 X ` e L vv E X vv ) X , we assume the couplings denote the three generations intrinsic to each lepton species. M v ) denote Yukawa-like interactions involving the SM leptons, heavy M 0 y v M y · 2 L L j 2 2 E 2 X L Z λ E h 2 X 2 √ E v λ 2 L √ M v M 3 2 L M E L M E 4 2 E M λ − L 4 ¯ ). In this study, we used λ λ M − E and E λ 4 3 − and L − − i 3 M E 3 X − E − L L,E 2 3 M L X v k 1 M ¯ 2 v L 1 M 3 L 2 M , M 3 E L √ λ E √ L λ 4 λ 4 ¯ e hλ , hλ 3, i.e. ( ( y           = ... = , to be diagonal, implying that the SM fields (neutrinos and charged leptons) of a = After electroweak and U(1) As emphasised in ref. [ L = 1 R ` mass U L,E U L and in view of relativelike size of leptons the ( SMparameters, lepton masses and and followed theperturbation the much series, heavier algorithm we ones thus of prescribed obtain the in vector- [ in which every entry shouldcharged be lepton understood mass as matrix a can 3 be (block-) diagonalised by a bi-unitary rotation where the rotation matrices given generation can only mix with vector-like fields ofand the neutrinos same can generation. beby cast the for following simplicity vectors: inlepton ( a mass “chiral matrix basis” is thus spanned, given for by each generation, tons and the vector-likeunacceptably fermions large should rates be forand very to cLFV further small, processes, circumvent in excessive asmediated flavour order by for changing the to neutral instance light current avoidM (FCNC) the interactions otherwise right-handed neutrinos and thethe vector-like beginning neutral of and this chargedcharged section, leptons; leptons and (i.e., as in mentioned 3 additionisosinglet in flavours), to vector-like the the leptons. three model SM includesi, j In generations three of eq. generations neutral ( of and isodoublet and in which JHEP07(2020)235 1, (2.26) (2.21) (2.23) (2.24) (2.25) X L v M M y , (2.20) , . (2.22) v L L ., ν ; contributions c y M             . L c 2 , c L as 0 − 0 2 X ν X M L v N B v L P v 2 + H ν L 2 X U y L √ λ M      2 j v 4 y 1 ν M √ L 2       λ − 2 α j X L 2 ) v √ 1 0 ν √ L M U λ ( 2 L 2 X L L M v 0 0 0 2 P , L 2 M 2 X L √ λ 1 µ ) L v 4 √ 0 0 2 1 γ M 2 X L , v      2 , v √ λ − √ c ν − 0 c 1 i α 0 . . ν ˜ 0 , U ) M ν 2 L y 1 N P † y 1 L L ν √ 2 M , U v U y 2      2 P M X ( 2 ν ν v · 2 v √ M X M √ i 2 ν y y U 0 0 0 0 ν T y y ¯ ν v ` y 3 block matrix. Following the same perturba- ν ν L 2 L ˜ 2 T X M P vy U y ν λ v v × y M L M =1 2 – 9 – U 12 vy 2 X · λ j       X = ' − v diag( √ 1 1 = − ν ν 9 − − =1 − ν 1 ˜ i X U diag ˆ m ν y C C 12 neutral lepton mass matrix is then given by 2 M L L = 2 ν M y y × ν 2 2 X e, µ, τ v X v U = c T c T 2 L α 0 0 X M , all other couplings are diagonal in generation space. Thus, ν v y M − ν L L − ) can be block-diagonalised via a single unitary rotation 2 L µ vy 0 0 X y λ v 2 2 √ X L T T W v 0 0 3 unity matrix. In turn, this allows defining the leptonic charged M − − 2 L 2 2.20 4 L L λ g × √ − c T c T − 1 N N =        T T = ν ν ), which can be itself diagonalised by a unitary matrix ], relying on the Majorana mass term of the singlet right-handed neutrinos, W ν ν L ˜ U 2, which is dynamically generated upon the breaking of U(1) denotes a 3 = = vy 128 – √ 1 ∝ / ν mass M 122 y L Concerning the neutral leptons, the symmetric (Majorana) mass matrix can be writ- X v in which current interactions as matrix ( The full diagonalisation of the 12 light (active) neutrino masses As already mentioned,neutrino we Yukawa couplings work underthe the entire flavour assumption structure that at with the origin the of exception leptonic of mixing is the encoded in the Dirac mass We notice that the lightnism (active) neutrino [ masses are generated∼ via a type-I seesaw mecha- from the vector-like neutrinosglected. arise only Up at to higher second orders order and in can the relevant therefore expansion be parameters, safely one ne- then finds for the tive approach, and in this casethe up mass to second matrix order of in eq. perturbations of ( with in which each entry again corresponds to a 3 ten as JHEP07(2020)235 0 ]) Z = 0, 133 (2.29) (2.27) – A q 3 block ε × 129 . (2.31) . For the up- and e b ,L ` . (2.30) , 0 a ` Z . (2.33) cb cb R L g U U , (2.32) − cb ν ac ac ) b ) U ,R ` cb ν † † 0 L R , a ` Z U ca j U U g ) . (2.28) ( ( f ∗ X ca ν L ) , which will be the key ingredients h U coupling formally vanishes, L L 0 ∗ 1 2 − ν A ij ( − − 0 B q ε Z U L B c B c = ( Z 5 Q − b γ L Q Q and ` L B c − a L L 0 − A + ` Q B c − − B Z V ij ε B B Q ε ε ε , g + Re – 10 – + + q µ L Im c c γ − b i L ,R ` ¯ ε Q B 0 f a − ε ε Q ε Q ` Z e B = g 3, corresponding to the lightest, mostly SM-like states ε 3 3 c , , , = X 2 2 2 + V qq , , are the effective couplings in units of 0 c , ε µ b Z X − X X refer to the mass eigenstates of different species: the lightest ,L ` =1 =1 0 J c c a ` Z = = = 1 V,A i ) the axial part of the g ε = = b b ν ν a a b i, j b V A ,L 1 2 ν ν ` ,R a, b, c ` u, d 0 0 a g g a ` Z ` Z = = g g (i.e. b ` f a V ` α j g ) 3) correspond to the isodoublet and isosinglet heavy vector-like leptons. ν , can be expressed as U 0 ( µ Z denotes a SM fermion (up- and down-type quarks, charged leptons, and neu- = 2 i α = 1) corresponds to the (mostly) SM charged lepton, while the two heavier ones J ) 0 µ f † L iZ U ( α a, b, c a, b, c P Similarly, the new couplings to the (Majorana) neutrinos are given by (i.e. This leads to the following vectorial and axial couplings In the above, the indices one ( On the other hand, andthe due lepton to sector the is different: mixingsleptons with the now the lead modified to vector-like left- mixings fermions, and between the right-handed different couplings situation species, for as for cast the below charged (for a given generation) down-type quarks ( while the vector part is given by boson, in which trinos) and the coefficients Having obtained all the relevantnow elements address to the characterise impact the of leptoncurrents, the and in additional gauge particular fields sectors, those and we to modified mediated address couplings by the to the anomalies the light new here neutral considered. The new neutral currents mediated by the lepton mass eigenstates, andPontecorvo-Maki-Nakagawa-Sakata (PMNS) 12 matrix neutral corresponds states). toof the The upper (not 3 necessarilyof unitary both [ charged and neutral lepton sectors). 2.3 New neutral current interactions: in which we have explicitly written the sums over flavour and mass eigenstates (9 charged JHEP07(2020)235 . or )). 1 0 Z (2.38) (2.37) 2.20 2), which ). , 2.33 = 1 , a, b L ) for the vector-like − ) and ( B ε L, E (with M , (2.41) 2.30 2 b , , R ν − L ), ( U , (2.35) , (2.36) − ) (2.39) L, E = 2 X L L B    λ v − ε − a RL 2 E 2.29 E α 2 to the charged leptons (SM- and ν B B C hv √ ) , (2.40) ε 2 E α , (2.34) ε M M X λ 2 X + = 1) one has h L RL 2 v 1 2 0 0 1 L α kv + √ C 2 X M LR 2 L α v − a, b − M 2 2 X − L α C 2 λ ( v 0 0 yv 2 X 2 √ 2 X L α 2 L α M v 1 2 − v LR λ 2 2 2 L α L α E α    M C λ 2 − X † L ( 2 L α 2 E α − M M v λ λ U – 11 – 2 1 2 1 E α ). Further notice that corrections to the tree-level 1 2 ` diag 2 E α = M = λ L † m + + + now explicitly denotes the SM lepton flavour. Note ) − P 2.19 ε ε ε g } = B 1 2 ε RL 2 S -boson appear only at higher orders in the perturbation X ), ( C g v √ ' ' − ' − ' − ' Z 2 α α α X α α = ( ` ` ` ν ,L e, µ, τ ` ,R 2.18 ,L v √ 0 0 0 α α α α α V ` A ` ` Z ` ν Z Z ε ε g LR g g ∈ { bosons, as well as the new heavy vector-like fermions, which can C − α X = h α ν α A ν and ε 0 Z as defined in eqs. ( can be dominant when compared to the SM ones. L,R X h U Finally, the scalar and pseudoscalar couplings of moments of the lightby charged the leptons, extra in thealso form propagate of in several the loop. one-loopNotice diagrams The that new mediated due contributions to areeven the schematically potentially illustrated in large figure couplings, the contributions induced by the couplings of the SMseries Higgs of and the mixing matrices, and are expected to3 be of little effect. New physics contributionsThe to field the content anomalous of magnetic the moments model gives rise to new contributions to the anomalous magnetic with and where of having imposed strictly diagonalstates. couplings The and “cross-species masses couplings” ( are defined in eqs.vector-like ( species) can be conveniently expressed as in which the subscript that flavour changing (tree-level) couplings are absent by construction, as a consequence corresponds to the Majorana mass eigenstatesFor with the purely lightest Majorana (mostly masses SM-like) (cf. eq. physical ( states ( Note that the vector part of the couplings vanishes for JHEP07(2020)235 . B B m /M ` and m ] 6 induced by = , (3.2) 134 λ e,µ , are given by , (3.3) , (3.1) dx 2) NS ` ) L,R # ! − a ) ` x ) i i g ρ ρ − − − ρ , (3.4) λ, x λ, 0 ( is the mass of the neutral ( 2 f ) ) is defined as follows F (1 + S G ,H 2 2 ` 2 B 2 ` m 2 S ) ρ λ X x λ, ρ ρ m ) m m h ( m 2 f x ) − ∗ F ∗ ) + ( − ` i A 2 ` i ρλ P 2 (1 , can be expressed as [ x g g π ρ π 2 2 4 4 ` i A x ` i P λ NV ` g g a 2 ) + ( − x λ (1 + R,L 2 ` 2 ) + λ ) + x i i ) (1 )] + is the vector (axial-vector) coupling − x ρ ) λ, ρ λ, ρ − A ( ( – 12 – − ( )(1 F x G V (1 g 2 ` 2 ` L,R 2 S 2 B 2(1 − ` dx ); m m m m − 1 (1 0 ∗ ∗ x 1.5 Z ` i ` i S 2 V 2 denoting the internal fermion mass and with )[ 1 2 g g π π 0 ), ( i x 4 4 f f ` i ` i S V g g − 1.3 M ) = " 9; however note that only fermions belonging to the same generation (but

,Z,W (1 0 i λ, ρ f x i Z ( ) are leptons; the photon can be attached to any of the charged fields. X 2 with X ,... 0 . As expected, such cancellations naturally rely on the interplay of 2 G 1 ` µ = , 0 = ) runs over all fermions which have non-vanishing couplings to the external leptons, a Z f, f /m 3.2 = 1 NV ` 1 2 i NS ` f a i a couplings. denotes the scalar (pseudoscalar) coupling and R,L M ∆ ∆ ` and ∆ ) ) = X P = e h ( a i S as defined in eqs. ( λ, ρ ρ g . Illustrative Feynman diagrams for the one-loop contributions to ( ( . Note that the loop functions of a vector or a scalar mediator have an overall ` and a F S 0 Generic new contributions due to a neutral scalar mediator (NS), ∆ Generic one-loop contributions generated by the exchange of a neutral vector boson Z The sum in eq. ( 6 the so that in general possibly of different speciesvanishing e.g., entry. SM leptons and isosinglet or isodoublet vector-like leptons) have a non- scalar positive sign, whereas theThis contributions allows of for axial a and partial pseudoscalaras cancellation between between mediators scalar vector are and and pseudoscalar negative. axial-vector ones, contributions,signs which as are of well crucial ∆ to explain the relative (opposite) with in which in which with ∆ is the mass of the exchanged vector boson. The function The internal states ( (NV) and a negatively charged internal fermion, ∆ Figure 1 the new states and couplings (with a possible mass insertion inside the loop or at an external leg). JHEP07(2020)235 (4.3) (4.4) (4.5) (4.1) (4.2) 7 , . 2 2 / ) 3 c 2 + b can be obtained production (and 0 transition depop- ( 0 0 Be atoms. M 8 Z Z − 0 , Z 2 ) a e m 15 MeV . , m 0 2 18 e ) Z . . c ) ) , m i 0 − − − q π m e Z µ 1 b ( + boson and the modified neutral γ m 0 24 i e ( 2 γ 0 q Z − 2 ) Γ / Γ Z 2 → V n 1 which are directly connected with an = ¯ a ε 0 λ 0 ≡ µ i Z + Z J ) 2 . The purely vector quark currents (see ) 0 ( | V p γ Be ) = V dd Z ε A ee 8 BR( ε ] it has been reported that a peak in the electron- ε | µ Be can be determined from a combination of i Be (cm). This gives rise to a lower bound on the Be p 48 8 = ( 8 8 5 + O eJ + 2 a, b, c – 13 – 3 − 3 2 | ( V ii | | → 0 → ε λ γ V uu 10 Z ∗ V ee ∗ ε k ε k | | | × u,d Be Be 2 = = X 3 8 ) i be sufficiently short lived for its decay to occur inside 8 . 1 V n 0 V n Γ( Γ( = ε ) = ε Z decay into a pair of electrons. At tree level, the latter is & − He, which could be explained by the creation and subsequent decay of + | 0 e (q) 4 0 + V ee µ V p Z Z and e ε ε J | In the isospin conserving limit, the normalised branching fraction Be. However, in the absence of any fit for normalised branching fractions we 8 V dd → = ( ε 0 state in ) ) 0 Z + − γ Be anomaly. A first bound on the couplings of the to electrons Z Γ( 8 V uu 0 ε Z Be + Be + 8 8 = 2 01 MeV 0 . → V p → ε ∗ ∗ )) are expressed as Be transitions. We begin by recalling that the relevant quark (nucleon) couplings 8 Be Be 8 8 2.27 The most important bounds clearly arise from the requirement that In what follows we discuss the bounds on the Firstly, let us consider one of the quantities which is extremely relevant for the IPC As already mentioned in the Introduction, in [ Γ( Γ( 7 positron pair angular correlationulating was the observed 21 in thea electromagnetically light forbidden particle in analogywill to not include this in our analysis. in which eq. ( is a particularly convenient observable becausein the the relevant ratio, nuclear giving matrix elements cancel the best fit value fordone the here normalised for branching both fractions the experimentally cases measured. ofIsospin This isospin conservation. is conservation and breaking. decay) complies with thefor (anomalous) the data on thenecessary to electron-positron explain angular the correlations anomalous IPC in explanation of the from the requirement thatthe the Atomki spectrometer, whose lengthcouplings is of the where is the defined Källénfunction as We proceed tocurrents discuss can how successfully the address the presence internal of pair creation a anomalyexcess light in — the widthgiven of by the 4 Explaining the anomalous IPC in JHEP07(2020)235 , ) 6 ], 6 − Be − ) + 8 4.4 2 10 144 (4.9) (4.6) (4.7) (4.8) 10 ) and q – (4.11) (4.10) ( 2 × 0 q × ,p Z ( 1 0 140 ,p F Z 1 . ) F ], which are k = 6(1) ( ) = = 6(1) p 2 γ 139 , u q γ Γ µ ( n Γ / , for a hypothetical 0 / 0 n p ) can be understood 0 Z e J Z M,p ε ν Z ) M q G 4.4 2 , V dd vs. ) it follows that the rel- i ε ∗ p µν ε σ Be . + 2 4.10 ) 8 . 2 µ | n q ) µ V uu 0 J ( ε , 0 − J ], it is convenient to parametrise ]). Notice that a large departure ,p | i e − Z 2 ∗ 55 + 2 44 µ + ( p F Be e − J 8 µ Be p h + ) denotes the spinor corresponding to 8 ) 10 | → = k µ n µ e J 0 ( 0 hadronic current as γ ε µ × p 1 ) J . From eq. ( Z ) | 0 u 2 = 0, since both the exited and the ground 2 + V dd . V dd Z q ε 1 i p ε Be ( ∗ ε 8 0 BR( + ,p ( h Z 1 – 14 – ,J Be e e 2 2 p + 2 F V uu 8 µ n | ε = = µ J 1 V uu ) | ≈ ε 0 J i i | + ∗ ∗ k V n ( ε µ = (2 = p p Be Be Be J 8 µ u i 8 8 n + h | | ε e = 0 we display the plane spanned by V p e J µ h ε Z µ EM µ = 0 2 | V ii J J J i | | ε and ) k ], and the normalised branching fraction Γ Be Be ( u,d V dd 8 8 p 48 ε h = X | h i In the above discussion it has been implicitly assumed that the + (q) 0 = µ Z V uu 0 J ]. The nucleon currents can be combined to obtain the isospin currents as | ε µ h Z ) 0 =17.01 MeV, and for the experimental best fit value Γ J 0 k 138 Z ( ], so that the proton matrix element can be written as = 2 – corresponds to a proton state and p m p i ε denoting protons and neutrons, one obtains ) Be are isospin singlets. Defining the 135 135 ≡ h 8 k ) leads to the following constraint from the protophobic limit is excluded by NA48/2 constraints [ ( µ )[ )[ p p 2 2 | | J p, n Be states are in fact isospin-mixed. In order to take the latter effects into account, q q p 4.4 8 =17.01 (16) MeV [ ( ( ε 0 0 0 mass of | ,p ,p Z Z Z 0 2 2 limiting case of a pure dark photon. Isospin breaking. states have a well-defined isospin; however,the as extensively noted in theisospin literature breaking in [ the electromagnetic transition operators arising from the neutron-proton Z (following the most recentof best fit values reporteddepicted in by [ the twoviable red protophobic vertical regime lines. still The currently allowed. region between The the horizontal latter dashed line corresponds denotes to the the eq. ( On the top left panel of figure in which evant nuclear matrix elements(in cancel the in isospin them conserving normalised limit). branching fraction of Therefore, eq. using ( the best fit values for the mass with In the isospin conservingstate limit, of Here a free proton.F The nuclear magnetic form factor is then given by as described below. Followingthe the matrix prescription element of for ref. nucleonsF [ in terms of the Dirac and Pauli form factors The cancellation of the nuclear matrix elements in the ratio of eq. ( JHEP07(2020)235 6 6 6 6 6 6 6 6 6 6 6 6 6 6 σ ------]). 10 10 10 10 10 10 10 10 10 10 10 10 10 10 × × × × × × × × × × × × × × 44 ) = 1. 4. 3. 2. 1. 2. 1. 6. 5. 5. 4. 3. 9. 8. 7. ]). − e 44 + e 0.015 0.015 → 0 ) for the isospin Z 17.5 MeV 4.3 17.01 MeV = ' = ' Z Z M M Isospin Violating Isospin Violating 0.010 0.010 =17.01 MeV, as well as 0 ' ' Z γ γ p p Z Z Γ Γ Γ Γ ϵ ϵ (see eq. ( m ] (for which we have taken a γ 0.005 0.005 Γ 55 / 0 Z 0.000 0.000 , following the uncertainties of [ 6 (following the fit values reported in [ −

6 , under the assumption BR(

0.015 0.010 0.005 0.000 0.015 0.010 0.005 0.000

10 γ −

n n ϵ ϵ Γ =17.5 MeV, for an experimental best ﬁt value × 0 10 / 0 Z 2 . Z × 0 m 6 6 6 6 6 6 6 6 6 6 6 6 6 6 ------– 15 – ∼ 10 10 10 10 10 10 10 10 10 10 10 10 10 10 × × × × × × × × × × × × × × γ 2. 1. 4. 3. 2. 1. 5. 4. 3. 6. 5. 9. 8. 7. Γ = 6(1) / 0 γ Z Γ / 0 Z 0.000 0.000 , in agreement with the values quoted in [ 6 0.005 0.005 − ' ' - - γ γ p p Z Z Γ Γ Γ Γ ϵ ϵ 10 × 2) . 0.010 0.010 - - 5(0 17.5 MeV . 17.01 MeV = ' = ' Z Z . On the left (right) panels, contour plots of the ratio Γ M M = 0 Isospin Conserved Isospin Conserved γ 0.015 0.015 - - Γ

0.000 0.015 0.010 0.005 0.005 0.000 0.010 0.015

n n ϵ ϵ Z Γ conservative estimate of the error in Γ interval for the experimental bestThe fit values region for between Γ theparameter two space, red as vertical lines allowedto by corresponds the NA48/2 to pure constraints, the dark while viablethe photon the protophobic experimental limit. horizontal region best dashed of On fitThe line the both value corresponds lower upper Γ panels panels illustrate we the have case taken in which Figure 2 conserving (violating) limit. The white dashed and solid lines correspond to the best fit and to the 1 JHEP07(2020)235 ]. ]. 144 144 , and 0 (4.13) (4.12) Z , 2 / 3 , 2 ]. In particu- ) can be defined =1 47 , we display the 2 , in turn leading 0 s 2 J,T 0 Z =17.01 MeV and for Ψ Be signal. In what Z 0 8 J and Z m 1 transition rate of the 15 MeV α . 1 ], m s M − 18 55 =0 − J,T ]. A comparison with the case 1 Ψ 44 2 [ plane, for J | ) can be obtained by computing the (corresponding to a heavier β 6 n V n ), as emphasised in [ 0 ε − J 0 ε = ∗ Z β 10 2 − s J ], and found to have potentially serious vs. Be V p × 8 Ψ p ε 55 Be excited state with an excitation energy and ε 8 J 95 ( . α – 16 – , = 6(1) =1 )+0 γ ), in the V n Γ J,T ε / 0 Ψ + Z 4.13 J in the isospin violating case. ), it is important to take into account the IPC null results Be. On the upper right panel of figure V p ∗ β 8 ]). This has led to changes in the relative strength of the ε n ε + Be 8 05 ( 146 , the physical states (with superscripts . =0 , 0 J | J,T 145 = Ψ ], this procedure may be used for the electromagnetic transitions of Be with its experimental value, using the matrix elements of ref. [ ) ]. To address this deficiency, an isospin breaking effect was intro- J ) 0 8 55 α γ Z 144 = 1 s J Be + Be + 8 Ψ 8 ] (following [ → → ∗ 55 ∗ ] Be Be Other than the best fit values for the mass of the mediator and normalised branch- For a doublet of spin 8 8 144 Γ( Γ( lar, in the presencekinematic of suppression, a thus finite implyingto isospin a a mixing, larger large the preferred phase latterfor mass space IPC the for suppression. null preferred the quark result Thisto (nucleon) would may significantly couplings call translate smaller to for into normalised the (further) a branching fractions significant when changes compared to the preferred fit allowed protophobic range of ing fraction for the18.15 MeV predominantly (here isosinglet denoted as for the predominantly isotriplet excited state ( and consequently, to newexplain bounds the on anomalous the IPC relevantisospin-violating in quark scenario (nucleon) of couplings eq. necessarythe ( to experimental best fitof value isospin Γ conservation (upper left plot) reveals a 15% modification with respect to the 17.64 MeV decay of This prescription leads to the corrected ratio of partial widths [ in the masses ofThe isospin multiplet isospin-breaking states, contributions e.g. havespurion, the been which nonzero incorporated regulates neutron-proton through the mass isospin-breaking thethrough difference. effects introduction within a of an “leakage” a isospin-invariant framework parameterparameter is (controlled subsequently by determined by non-perturbative matching effects). the resulting The “leakage” results are substantial, evenmatrix after element including [ theduced meson-exchange in currents the in hadronic thein form relevant ref. factor [ of theisoscalar electromagnetic and transition isovector operators transition themselves operators which appear as a result of isospin-breaking in which the relevantwidths mixing of the parameters isospin-pure statesAs using pointed the out Quantum in Monteisospin-mixed [ Carlo states (QMC) as approach well. [ However, the discrepancies with respect to the experimental implications for thefollows quark-level we summarise couplings the required most relevant to points, which explain will be the included inas the [ present study. mass difference was studied in detail in ref. [ JHEP07(2020)235 0 0 γ Z Z Γ (in / 0 0 (5.1) ∗ Z (4.14) (4.15) (4.16) Be 8 ], we obtain 55 ) = 1), [ − 6 e Be excited state). − + 8 e , the bottom panels 10 Be, there are exten- 8 2 → × Be. However, in view 0 8 5 . Z The non-observation of a ], as well as searches for dark = 0 . γ ) Γ 148 searches in beam dump exper- − , / , 3 ]. In figure 0 e , 0 − Z 16 + 3 44 Z e Γ − 147 − 10 8 10 boson 10 . → × 0 0 3 × × 6) − Z Z 1 . − 10 1 15) BR( × (3 < − 2 – 17 – 2 p . 1 ee A ε = (2 . | ≈ | | + V n V n V p ε 2 ε ε | | V ee + ε V p ε | production due to excessively feeble couplings; (ii) excessively 0 ]. Z 55 necessary to explain the anomalous IPC in 0 in electron beam dump experiments. Z 0 ]). Considering the benchmark value Z ) following the quoted uncertainties in [ 6 48 − decay, occurring even prior to the dump. Under assumption (i) (i.e. negligible 10 0 × Z 2 . To summarise, it is clearly important to further improve the estimation of nuclear 0 Since no public results are available to the best of our knowledge, we use the values quoted from a 8 ∼ rapid production), one finds the following bounds private communication in [ Searches for in experiments such as SLAC E141,photon Orsay bremsstrahlung and NA64 from [ electronfold and way: nuclei (i) scattering, absence can of be interpreted in a two- 5.1 Experimental constraintsThe on a most light relevantiments, constraints dark arise photon fromelectron scattering. bremsstrahlung negative and production, parity violation, and neutrino- If, and as discussedmust in satisfy the several previous requirements section,sive to the constraints explain new arising couplings the of from anomalouscouplings. fermions IPC various to In in experiments, the this both light section,independent regarding way, we and leptonic collect subsequently the and summarisingcase most hadronic the of important results light ones, Majorana of casting the neutrinos them new (which is in fit the carried a case for model- in the the model under consideration). 5 Phenomenological constraints on neutral (vector and axial) couplings This will allow determiningcouplings the of ranges the forof the the guesstimates bounds discussedranges on here, for the different in relevant couplings our quark (always numerical under (nucleon) the analysis assumption we BR( will adopt conservative ( illustrate the relevant parameterlimits (respectively space left for and right the plots). isospin conservingisospin and violation, isospin and violating performaddition to more the accurate currently fits available fits for for the the predominantly null isosinglet result of IPC in the following constraint in the isospin conserving limit, Leading to the above limits, we have used a conservative estimate for the error in Γ reported in [ JHEP07(2020)235 ) ], -, ' ee 4 0 149 (5.6) (5.4) (5.5) (5.2) (5.3) Z [ → m Be (see − 0 must be 8 e Z + ( V n 0 , e ε ) at KLOE- Z 71 + ee → ]. For . π 0 X → 151 . 0 → | Z + ( . 0 ) ) K 0 Z 0 − Z + from the above equation, . e 2 Z η | m ) Very important constraints + m ( e V p − ε e K → ) and | . ]. + 3 → + ee e ) 0 − φ − ) Z 150 e to electrons arise from the parity- 10 → → . N 6 + 0 0 7 0 × e − Be the neutron coupling Z Z − BR( Z 2 8 ( + 2 . 10 0 → 10 1 p , followed by the decay 0 Z × 4 BR( ( × γZ 20 MeV [ Z to quarks — already discussed in section − 4 Xγ 1 V dd 0 p . ε > 10 1 → Z A bound can also be obtained from the KLOE- . → BR( X 0 × | . – 18 – ) + π − 8 | m V p < couplings can be inferred from atomic parity vio- . e ], these yield required to explain the anomalous IPC in ε N A ee 6 0 2 | + | ε e Z A ee + V n & 156 = V ee ε ε | ε 2 ], to the following bound Z | | + + V dd (2 A ee 2 ε V p ε 139 | level [ V uu V ee ε + | ε ε + σ V uu 2 | A ee ε In addition to the (direct) requirements that an explanation of the 2 V ee | ε | 17 MeV [ πα ε 4 q ] ' ] experiment, as well as searches for 2 0 F 99 ]. At the 2 Z √ G 139 2 m | 155 – = | w 152 )), with the comparatively small allowed regime for Q ∆ ]. Currently, the most stringent constraint does arise from rare pion decays searches | 4.14 149 Be anomaly imposes on the couplings of the lation in Caesium, in particularCs atom from [ the measurement of the effective weak charge of the violating Møller scattering, measured17 MeV, at it yields the [ SLAC E158 experiment [ Further useful constraints on a light also often referred to as abounds “protophobic can scenario” also in be the obtained literature).and from other Further neutron-lead (subdominant) hadron scattering, decays, proton but fixed target we experiments willConstraints not arising take from them parity-violating experiments. into accounton in the the product present of study vector and axial couplings of the If one confronts the rangeeq. for ( it is clear thatsizeable. in order (This to enhancement explain of the neutron anomaly couplings in (or suppression of the proton ones) is at the NA48/2 [ 2 [ which lead, for Light meson decays. 8 important constraints on the latterinstance, arise this from is several the light case meson decay of experiments. searches For for Similar searches were also performedslightly heavier at candidates, BaBar, with although masses the latter were only sensitive to Searches for dark photon production. 2 experiment, which has searchedleading for to while (ii) (corresponding to fast decay) leads to JHEP07(2020)235 . 9 4 − 10 × obtained 8 from the exclude a . couplings. 6 | σ 0 V ee ε A ee ] was proposed Z ε | | ∼ 158 V ee ε | ]. For the anomalous and the anomalous IPC e 153 ], with the tightest bounds 2) 8 [ . 161 − 0 , which is particularly relevant – 9 g − ' 159 10 , , (5.7) 6 5 × − − 6 . Finally, neutrino-electron scattering pro- -couplings vanishes. The fits performed 10 10 2 0 Z × × . ]. (17 MeV) 8 4 | . . K 7 8 A ee 120 ε | – 19 – . . , | | | ] leads to the limits e µ 4%, which would lead to a bound on the axial coupling to ]. The bound which can be inferred from the result obtained . ν A ee ν e constructions the couplings are flavour-universal, the ε µ A ν | 149 0 A ν neutrino couplings [ 157 ε ε 0 | | Z Z , the effective weak charge of the Cs atom measurement ) dd ( V uu ε ] and KLOE-2 [ ]) do not play an important role in this scenario, due to vanishing neutrino 148 ] are thus not directly applicable to our study; consequently we have per- , we show the particular likelihood contours deviating 1 and 2 161 3 161 147 is an atomic form factor, with – – ]. While for some simple K 159 159 ]. As argued earlier, the axial coupling to electrons has to be negligibly small in 17 MeV, the most stringent bounds are in general due to the TEXONO experi- , 119 55 ' 159 In figure There are also measurements of the effective weak charge of other fermions, notably of the proton which 0 9 Z was performed by the Qweakby experiment Qweak [ is howevernew an measurement order of of the effective magnitudewith weak weaker charge an of than the anticipated the electron, relativeelectrons one the uncertainty comparable MOLLER from experiment to of the [ the 2 Caesium one measurement. from the For Caesium a measurement. leading to which we have assumedNote the that smallest allowed interference electron effects coupling refs. between [ the chargedvector and couplings. neutral currents (as Thereferred discussed to technical above in details can regarding be found the in calculation the appendix. and fitting procedure the SM prediction. Applyingfrom the NA64 constraints [ on the electron vector coupling ref. [ order to comply with constraintsthese from will atomic henceforth parity be violation set and, to for zero practical in purposes, ourbest analyses. fit point for the neutrino-electron scattering data, which is found to lie very close to In the present model, neutrinosing are Majorana flavour particles, conserving which pure impliesin vector that refs. the part [ correspond- of the formed new two-dimensional fits to simultaneouslyand constrain the muon axial neutrinos, couplings to and electron the vector coupling to electrons, following the prescription of extra fermion content inway our that model only leads the todata. couplings a to decoupling For electron of muon neutrinos the neutrinos,obtained can lepton from slightly be families the weaker constrained CHARM-II in but with experiment such nevertheless the [ a very TEXONO 5.2 relevant bounds can Majorana be neutrinos: fitting the leptonic axial and vector couplings Neutrino-electron scattering experiments. vides stringent constraints on the arising from the TEXONO andm CHARM-II experiments. In particular,ment for [ the mass range IPC favoured values of provides a very strong constraintfor on our scenario, asparticularly it challenging. renders a As we combined will explanationlarge subsequently region of discuss, of ( the the constraints parameter on space, leading to a “predictive” scenario for the in which JHEP07(2020)235 (5.8) (5.9) (5.10) (5.11) (5.12) (5.13) scattering data of 3 e − µ 10 ν ), are marked by dotted lines: CHARM-II TEXONO KLOE-2 Limit NA64 Limit , , , . , 5.7 3 9 6 5 , 3 − − − − 3 − 4 − − 10 10 10 10 10 10 10 × × × × µ × ν )). In turn, this will imply that very 2 6 8 4 × . . . . 15 1 2 2 7 8 | ` ν Be: impact for a combined expla- 2.14 ...... ` ⇓ ⇑ 8 A ν | | | | | | ε e | µ V n V p V ee A ee ν ν ε ε e ε ε µ | | | | A ν to matter: combining the required ranges of A ν – 20 – ε ε | allowed regions around the best fit point (respectively | . 0 5 . − 3 σ Z 3 10 (which we recall to correspond to a redefinition of − − ε 10 10 × × 2 e scattering data of TEXONO (red) and the ¯ ν 68 . e 0 e ν e,µ 6 − 2)

10 5 2 3 4

− − − −

| 10 10 10 10 −

ee ε V g -fit of the ¯ 2 ) = 1), χ − e + e . New → 0 To conclude this section, we list below a summary of the relevant constraints so nation of ( Z couplings to the specific structure of the present model. After taking the results of 0 As a first step,Z we apply the(negative) previously collider obtained searches model-independent forinfer constraints the an on exotic extremely the matter tightthe fields range effective into kinetic for account, mixing we parameter, will cf. eq. be ( able to 6 Addressing the anomalous IPC in couplings needed toother explain experiments, the we anomalous haveBR( established IPC the signal following with ranges the for the relevant couplings bounds (assuming from the TEXONO data mostly constrainsis the couplings responsible to for electron the neutrinos while constraints the on CHARM-II the data couplings to muon neutrinos. far inferred on the couplings of the Figure 3 CHARM-II (blue), displaying the 1-darker and and lighter 2- colours). The2 lower bound (dash-dotted of NA64 line) (dashed are line)obtained and also upper the upper shown, limits bound on with by the KLOE- the axial arrows coupling identifying to the neutrinos, viable cf. allowed eq. regions. ( The JHEP07(2020)235 ) ), = 2 L α n (6.5) (6.6) (6.7) (6.1) (6.2) (6.3) (6.4) ε 4.15 /M 2 X and v couplings. `` 2 L α V dd k ε λ for each lepton − + )) imply that for and (1 L V uu `` ε 5.7 M ∼ ). 14 GeV, since h 3 − ' . = 2 X X p (10 v ε v ε ) it follows that the bound , O L Be concern the physical mass , λ X 8 . However, the bounds of the . 3 5.6 h 3 and . 3 − − one requires − coupling to the SM-like neutrinos L X , 10 0 4 − v 10 6 10 . | − Z B × − L × ε × , ) and ( 10 01 − . 10 2 , . B 0 15) × 15) ε L 1 × | 8 2.31 − − . . e − .

B 6 | ε 17 MeV 2 X L 2 & L v = 2 8 (84) = = (2 − | – 21 – . ≈ 2 L 0 M | = (2 B ), and recalling that 7 0 V ee λ B L ε A νν ε Z | ε − . A νν m − + ε m B | 2.28 1 ε ε ≈ | |

A νν 0 ε | Z = = | | Be anomalous IPC, which are m p V n V 8 ε ε | | ] for neutrino-electron scattering (cf. eq. ( allows to suppress the couplings by a factor ), leads to 119 6.2 denoting SM flavours), hence implying 2.3 α as defined in eq. ( V qq ε ), with . On the other hand, from eqs. ( α , one obtains the following constraints on 2.34 , which should approximately be V dd 0 ε to neutrinos are identical to that of the neutron (up to a global sign), that is Z required to explain the 0 ). With In order to circumvent this problem, the effective In the absence of heavy vector-like leptons, there are no other sources of mixing in the Z + 2 A νν ε V uu 4.16 Thus, up togeneration a very good approximation, we can assume must be suppressed. Thesources additional of vector-like mixing leptons between open the distinctpling the species derived possibility in of of section neutral having leptons; the new (see effective eq. neutrino ( cou- the minimal allowed electron coupling for electron (muon) neutrinos.of As can be inferred, this is in clear conflict with the values which, in view ofTEXONO eq. experiment ( [ lepton section in additionthe to the PMNS. This would imply that the effective couplings of Furthermore, this implies an upper bound for the VEV of and the strength of its couplings( to nucleons (protons and neutrons),ε as given in eqs. ( anomalous magnetic moments, the latter requiring an interplay of the 6.1 Constraining the model’sThe primary parameters requirements to explainof the the anomalous IPC in little freedom is left to explain the experimental discrepancies in the light charged lepton JHEP07(2020)235 4. . 6 (6.10) ] limit ∼ 100 GeV coupling µ L 149 = 70 GeV 0 λ ∼ Z X h m , (6.8) 9 − 70) GeV. Therefore, as 10 − × . (6.9) 6 (50 ) suggest that the . 110 GeV, once the corrections 6 2 − 6.9 ∼ ∼ . 10

L × − 6 . B . ) and ( implies an upper bound on the vector- . In particular, the KLOE-2 [ ) can potentially lead to Landau poles at high- 2 ε ε π ] (under the assumption these dominantly E leading to 6.7 . (4 002 λ

. 2 with mass X O 2 10 L 0 162 v 2 X and a charged SM lepton. Taking into account 2 L X v 2 L M < – 22 – and h X λ 2 L | M h ε λ ), eqs. ( L | − λ 4 (for the ﬁrst generation, due to the very stringent . − 2 X 6 2 2.36 E v 2 X 100 GeV [ 2 , respectively. More importantly, the above discussion E ∼ v can also decay into two right handed neutrinos (modulo 2 E M L while for the second generation one only has ∼ λ 2 E e L M λ X λ λ h

11 ) and ( 2 1 ≈

and e E = 2.35 λ as is the case in our scenario. However, and given the similar decay E | λ X A ee ε h | now implies and ee ], one can roughly estimate that vector-like leptons with a mass pairs). This bound can be relaxed if other decay modes exist, for instance ε 0 = 90 GeV (which yields a physical mass Z (for the first lepton generation), E 163 , W ν E M ) for 55 λ = 5.3 should be sizeable L Further important constraints on the model’s parameters arise from the masses of the Together with eqs. ( In what follows, we will not explicitly includeCouplings the so flavour close indices, to as the it perturbativity would limit render of the notation too M 10 11 L,E as a benchmark value. Since cumbersome, but rather describe it in the text. energies, as a consequenceembedded into of an running ultra-violet effects. complete framework. To avoid this, the low-scale model here studied should be λ parity violation constraints), (We notice that smaller couplings,accommodated, still at complying with the all price imposedfermion of states.) constraints extending can In the still agreement particle be with content the to above include discussion, additional we further exotic choose smuon’s [ can decay into aa charged benchmark lepton choice and weto an fix the tree-level massdue of the to vector-like mixing leptons effects of are all generations taken into account). In turn, this implies that the couplings involving the and production mechanisms, adedicated more searches interesting for possibility sleptons (decaying is intocase to a neutralino of recast and vector-like the a leptons charged results decaying SMthe of into lepton) fact for LHC that the the vector-like lepton’s cross section is a few times larger than the selectron’s or vector-like leptons, which are boundedperturbativity from limit both of below and thelike above. couplings lepton On masses. the onevector-like lepton hand, On the masses the below decay other into hand, direct searches for vector-like leptons exclude to electrons is nowof almost eq. solely ( determined by Notice that this leadslike to lepton a tight couplings, correlationrenders between manifest the the isosinglet and necessitygenerations isodoublet of of vector- having heavy vector-like the leptons). additional field content (a minimum of two coupling which in turn implies from atomic parity violation in Caesium tightly constrains the isosinglet vector-like lepton JHEP07(2020)235 . L X Be X 8 (6.11) (6.12) We notice . 12 X , is introduced to X ε ] the authors have derived models. The new bounds L 164 ), thus avoiding the potential − B . 2.35 . ). In such a situation the longitudinal ), allows to derive the viability range W 0008 0008 . . 0 ) and ( 0 ) via anomalous couplings. , leading to an energy-enhanced emission of − 6.10 L − X . X 2.34 e,µ . should be close to its smallest allowed value ), leading to a signature strongly resembling ε ε 2) L M . – 23 – − has different couplings to fermions belonging to a given y . − B ε X g 002 0032 . . , if 0 0 X − − extension the heavy vector-like fermions do not contribute L , the discrepancies between SM prediction and experimental − couples to may also be broken at tree level, due to weak-isospin violation = 70 GeV, B 3 to electrons, see eq. ( X . 0 X X h Z + m extensions of the SM, such as U(1) ` ], such an enhancement can occur if the model’s content is such that a new set of `ν Be anomalies, and arise in general from an energy-enhanced emission X 8 164 — as can be seen from eqs. ( → ], which in turn implies the following range for L 55 W anomalies, let us notice that in the study of ref. [ vertices may break U(1) ¯ ν , or e,µ 002 [ . X Be, one must now consider whether it is still possible to account for the observed 2) W ` 8 + doublet and lacks the compensating coupling to the ]. In the present U(1) − e = 0 or . For the choice g L , Before finally addressing the feasibility of a combined explanation to the atomic eν L L ε As discussed in [ 164 ¯ ud − − , arising in U(1) 12 → B B W cancel potentially dangerous chiral anomalies.the Explicit SM Wess-Zumino gauge terms must symmetry, beMoreover, which the introduced in to SM turn reinforce current breaks that ( the U(1) SU(2) radiation from charged currentπ processes can be again enhanced, leading to very tight constraints from Introduction and in section observation have an opposite sign forvery electrons different and from muons, the expectation and na¨ıve exhibit (powers a of scaling the behaviour lepton mass ratio). heavy fermions with vector-like couplings to the SM gauge bosons, but chiral couplings to In view of thefrom stringent phenomenological constraints arguments on andIPC the from in parameter a space satisfactory oftensions explanation the in of the model, the electron imposed and anomalous muon both anomalous magnetic moments. As discussed both in the free set under the extendedbreaking gauge SU(2) group. Moreover, thereconstraints are inferred no for neutral vertices a explicitly possible energy-enhanced longitudinal emission of 6.2 A combined explanation of ( at addressing the (production) of the longitudinalthat component the prototype ( model hereof investigated [ departs in several pointsto from anomaly the cancellation, assumptions asneutrinos (and the independently, all SM the vector-like field fermions) constitute content a completely and anomaly- three generations of right handed and ( significantly stronger new constraints onX the parameter space ofcan new potentially (light) disfavour vector states, some well-motivated constructions, among which some aiming The combination of theon previous the constraint couplings with of thefor the one inferred from the KLOE-2 limit that of slepton pair production,ε current negative search resultsε then lead to constraints on a substantially large Majorana coupling JHEP07(2020)235 , , is `` 4 h e (for a = ` , as a ` | k ` h a ∆ | and , the (small) ` E h λ would imply taking and -induced contribution (which are consistent L can be slightly varied the regions complying λ X L,E 4 5 h X − M contributions. Leading to h 0 10 m Z × 8 and Be and respect all other imposed − 8 and can induce such an asymmetry, contribution, while a small positive = L,E ). Likewise, a similar effect occurs for (right). The ` 13 ε X M k µ h in the planes spanned by = ` and and ` and the large couplings between SM leptons ` 2) 3 0 h − Z − – 24 – 10 g × ), in most of the parameter space the new contribu- contributions to the electron and muon (left) and (for each generation) slightly modifies the slope of the with opposite signs; this requires nevertheless a large 0 . Moreover, a partial cancellation between the distinct L,E e = 2 e . In particular, notice that the negative electron ∆ Z L λ e,µ a 4 L = M (and in the remaining of our numerical analysis), we have 2) ) implies a strong relation between − ` 4 B − ε 6.9 g and , while an overall scaling to increase and ∆ and the E 5 µ a M gauge coupling and the kinetic mixing parameter have a very minor X h L remain essentially unconstrained − are considerably larger than what is suggested from experimental data. ` coupling for (green) and combined (blue) — is illustrated by the sharp kinks visible k B ` e,µ h X 2) h and arises from the cancellation of the scalar and the ` changes sign when the pseudoscalar dominates over the scalar contribution (for − level with the observation of ( h ` µ g a σ 2) contribution when the axial-vector contribution dominates over the vector one. The 0 − To conclude the discussion, and provide a final illustration of how constrained the This interplay of the different (new) contributions can be understood from figure Given the necessarily small mass of the Z g Being diagonal in generation space, we henceforth denote the couplings via a single index, i.e. (orange), : while for electrons eq. ( 13 0 `` A parameter space of this simple modelat becomes, the we 2 display in figure etc., for simplicity. mass-splitting between curves presented in figure (even) larger values forthat most the of model’s the parameterbenchmark space couplings values is in is severely only their constrained, viable allowed for so regions. a that comparatively any narrow (Notice departure band however from in the the parameter space.) constraints). We emphasise thatranges, as both a the consequence ofinfluence their on already the extremely contributions constrained in to the the allowed anomalous ranges).with magnetic respect lepton to Furthermore, moments the the proposed (when masses benchmark varied values, with only a minor impact on the results; a successfully induced by themuon flip ∆ of the signthe of numerical the results of figure taken as benchmark values with the criterion for explaining the anomalous IPC in the choices of the relevant Yukawa couplings the transition between positive (solidZ line) and negativein (dashed the line) logarithmic contributions plots — of from figure indeed leading to the desired ranges for the anomalouswhich magnetic illustrates moments. the function of the to ( diagrams can lead toaxial ∆ coupling to electrons,the which is couplings experimentally of excluded. thegeneration However, SM can an charged asymmetry overcome leptons in theg to problem, the generating vector-like statescouplings a belonging sizeable to “effective” the axial same coefficient and the heavier vector-like statestions ( to ( Firstly, recall that due(pseudo)scalar to contributions, a the cancellation opposite betweenoverall sign suppression the of latter of each contributions the ( allows loop for an functions for (axial) vector and JHEP07(2020)235 3 − ), ), ∼ 10 σ 002 and . e E 5 0 . X λ 6.1 σ comb. Z h , as a | interval = 0 ` 4 . = − a ` L 10 σ h e L ∆ × − | 9 λ B (orange) and ε 0 , 4 Z 4 − , which is varied − 10 E ` 10 × λ 8 µ × h 8 − is satisfied at 2 ; more recently, the elec- = 4 σ − µ ε 10 3 a . × ) region. Leading to this figure, 7 σ . For the muons, and although e (3 Be anomaly, and saturating the 8 is required by experimental data) σ 2) E − ) space succeeds in complying with 4 λ = 70 GeV, e g − k 10 X 7 8 9 6 × (right). Solid (dashed) lines correspond to 10 11 12 h − − − − 6 − − − −

10 10 10 10

and

10 10 10

µ m µ ∆ a e , L h = λ X couplings arising from atomic parity violation ` 0 /v lie within the 2 ) space for which ∆ L Z – 25 – ) µ µ k M ( 6 are forced into a strongly hierarchical pattern, at e , the comparatively larger freedom associated with − − a = 10 µ ` . A vertical opaque region corresponds to the ` × µ k k L 4 a h λ (left) and e 6 = − and 10 E = ` × ; the colour code denotes contributions from the λ 3 ` ` h a e h 6 − 10 × 2 = 90 GeV, . 8] (recall that for the electron anomalous magnetic moment, E coupling for , 4 ` M h = . 7 L 0 − X comb. Z h M 2) also started to display tensions between theory and observation (around 2 . Contributions to the anomalous magnetic moment of charged leptons, (green), as well as the combined one (blue). Horizontal (dotted) lines denote the 2 : only the narrow black band of the ( : ). The colour code reﬂects the size of the corresponding entry of − e = 10 5 X 6 g − ` 2) h 10 Finally, notice that the Notice that, as mentioned in the discussion at the beginning of the section (cf. k e, µ 10 11 12 13 14 (recall that no particular relation between remains strongly correlated with − − − − − regions of the electron and muon ∆ −

10 10 10 10 10

4). All remaining parameters are fixed to the same values used for the numerical analysis e = ∆ a µ E µ . σ g eral low-energy observables exhibitsignificance and tensions longevity. with The SMobservation discrepancy regarding predictions, between the the to anomalous SM variouslongstanding magnetic prediction anomaly, degrees moment and currently of experimental exhibiting of the atron tension muon ( around is 3 perhaps the most least in what concerns the first two generations. 7 Concluding remarks Despite the absence of a direct discovery of new resonances at high energy colliders, sev- all available constraints, whilecurrent both discrepancy addressing between the SM and IPC observationh on ( λ allows to identify a wider band in ( leading to figure the extremely stringent constraints on the and electron neutrino scatterings render( the model essentially predictive in what concerns and ` in the interval [1 6 positive (negative) values offrom ∆ 3 for which the combined contributions towe ∆ have selected a benchmarksection choice of parameters complying with all the constraints mentioned in Figure 4 function of the

JHEP07(2020)235

λ . ` 0 8 7 6 5 4 3 2 1 Z 2 − 10 3 )). In both panels − µ 10 ( e µ 4); for the right panel, . 4 2) − = 6 µ 10 k ` − ∼ g ( e E 5 − Be atoms, in particular an λ . 10 8 4 6 − ; moreover, this requires a nearly 10 e 2) level (i.e. 7 − − σ 10 1 2 3 4 (the dashed black line illustrates the value g (ﬁrst generation couplings of vector-like suggests the presence of a New Physics − − − −

10 10 10 10 e

σ µ h k e,µ 8, from dark violet to yellow). On the left panel, a at 2 at the 2 − e µ and – 26 – e 2) 2) 2 = 1 h − − − 10 E g g λ ( ` E parameter space: on the left (right) Be constrained the couplings of matter to the new . Other than the scalar field (whose VEV is responsible λ ` 8 4 L k − − 10 B vs. ` e h 2) e k − 6 g − ( 10 can only be achieved via a cancellation of the new (pseudo)scalar and e,µ 8 a − 10 . In this work, we have investigated the phenomenological implications of a BSM . Viable regions in 4). All other relevant parameters fixed as leading to figure . e,µ 10 6 − a The very stringent constraints arising from atomic parity violation lead to an extremely After having computed the modified couplings, and summarised the model’s contri- An interesting possibility is to interpret the atomic anomalies as being due to the 4 5 6 7 1 2 3 10 − − − − − − − ∼

10 10 10 10 10 10 10

e h µ E He transitions. tensions on ∆ (axial)vector contributions. strong correlation for the newleptons couplings to the SM Higgs) in order to comply with ( butions to the lightdressing charged the lepton anomalous anomalous magnetic IPCOnce moments, in all we remaining identified phenomenologicalspace, how constraints ad- one are is imposed led on to the an model’s extremely parameter tight scenario, in which saturating the (opposite-sign) construction in which the lightgroup vector via boson an arises additional fromfor U(1) a breaking minimal the extension new of U(1)),as the three of generations gauge heavy of vector-like Majorananew leptons right-handed matter are neutrinos, fields added as play an well to instrumentalmixing, the role and SM both in in field circumventing providing the content. additional very sources As stringent of discussed leptonic experimental here, constraints. the 4 presence of a lightnon-vanishing vector electroweak boson, couplings with to aon the mass ∆ standard close fields, to it 17 MeV. could also Should such have a an state impact have all the most intriguing sincethe instead light of lepton following masses, awhich scaling the na¨ıve proportional violates comparison to lepton of powers flavour ∆ lation of universality. was observed In for recentenhancement the years, of 18.15 an the MeV IPC anomalous nuclear at angular transition large corre- of angles, with a similar anomaly having been observed in the colour code denotes theonly value the of central black lineall complies the with coloured ( regionλ allows to satisfy ( Figure 5 JHEP07(2020)235 i P . The 5.1 ), with ]. Further 0 Z 165 ,, (A.1) (A.2) # # and correspond to the ], or studying the T T 2 2 ν ν i e e Z E E A m m 166 g , ]), which could allow a − − W G G and 167 + + ) . (A.3) i i G G V ee A g g − − ). The situation is slightly less 2 2 i L,E ee V ν ν λ g . T T E E µ )( the energy of the (anti)neutrino. The k i − − ν νν A 1 1 g E and − 2 − 2 + µ i h νν V couplings ( G G g ( – 27 – + + X i h 1 2 + 2 − P 0 − G G " " L e e π π − X W,Z,Z 4 4 m m ` = i ) = ) = = − − ¯ νe νe G ] decays (REDTOP experiment proposal [ → → η 159 − − transition) is sensitive to (neutral) muon axial couplings [ νe νe (¯ ( are defined as s 1 dσ dσ dT dT G → s is the recoil energy of the electron and T In general, the differential cross section for neutrino and antineutrino scattering can Future measurements of right-handed neutral couplings, or axial couplings, for the In the above, thedenoting the sum denominator runs of over thevector corresponding all and propagators; relevant axial vector couplings bosons of the (i.e. involved vector bosons to (anti)neutrinos and electrons. where coefficients be easily computed [ In this appendix weelectron-neutrino scattering, detail upon the which rely formalismdata the used used constraints arises presented for from in the section two different analysis experimental of set-ups, the CHARM-II available and TEXONO. data on JO and AMT acknowledgezon support 2020 within the research frameworkagreements and of No innovation the 690575 programme European and No under Union’s 674896. Hori- the Marie Sklodowska-Curie grant A Electron-neutrino scattering data: analysis and fits measurement of the axial couplings of muons. Acknowledgments CH acknowledges support from the DFG Emmy Noether Grant No. HA 8555/1-1. JK, parity-violation experiments carried in associationexperiments designed with to muonic test atoms. parity(ARC), As non-conservation (PNC) the an with example, measurement atomic in of radiativemuons capture the (2 forward-backward asymmetry ofpossibilities the include photon scattering radiated experiments, by muon such polarisation as in MUSE at PSI [ constraining for the muon anomalousdependence magnetic of moment, the albeit corresponding leading couplings, to a non-negligible second generation charged leptonsthough could this clearly further goes constrain beyond the the scope new of muon the couplings. present work, Al- one could possibly envisage non-perturbative regime for the JHEP07(2020)235 ] 119 contours σ 1 GeV, respec- . ) . (A.4) ); the maximum and 2 0 2 Z σ max = 19 m function , (A.12) ) is the neutrino flux, i ,T 2 + ¯ ν ¯ ν µ 2 ¯ , ν χ E T 1 ), (A.5) ( E dE e T e h φ # m , (A.11) dT (2 and 2 ν dσ dT = min( ∼ − , with the data. For neutrinos and 2 exp 2 ¯ , 0 1 T ], the event rate can be computed as T i, ¯ 1 exp T Z ¯ σ i, 7 GeV and T Z P . σ " 159 − ]). ) ∆ i To fit the data from the CHARM-II ex- , ¯ ν (for both σ is minimised, and its 1 The analysis of the TEXONO data [ = 23 E 2 w 168 2 ]), one can directly compare the differential ( 2 c , (A.7) i g φ χ 2 ) F µ i 2 ν 120 g 2 w – 28 – √ X G Z √ s E 2 8 h 1 4 = w T c = , (A.6) , (A.10) − e 4 − ρ , (A.8) ≈ − w final states are indistinguishable, a factor of 2 is present , (A.9) W A νν 2 (1 c g A w Z ν T ee V 2 g g c g e ε P 4 eε − − − 2 CHARM-II , and ¯ ) = = = 2 = = = = χ 2 2 ν 0 0 F Z Z Z g W Z Z ,T ee V νν ee A A ee V V G 2 νν 1 A g g g g g g T 8 √ ( R ≈ − W P are the bin edges for the electron’s recoil energy, 2 , 1 T runs over the different bins. The i the electron density of the target material and e in which ρ Data from theis TEXONO comparatively experiment. more involvedexperiment, than the that computation of ofanti-neutrino the flux. CHARM-II. binned Following the event approach Since rate of TEXONO ref. requires [ is knowledge a of the reactor reactor where around the minimum are computed. periment (extracted from tablecross-section, 2 averaged of over the ref. binnedantineutrinos, [ recoil the energy average energiestively. are Since no datadata correlation to from follow the a CHARM-II gaussian samples distribution, and is accordingly available, define we the assume all with all other remaining coefficientsfor Majorana vanishing. neutrinos In the order toin take the into (axial) account the neutrinoamplitudes fact coefficients involving that two (effectively neutrino allowing operators to [ doubleData the from contributions from the CHARM-II experiment. carry the following approximations For the case of the model under study, the vector and axial coefficients are given by Since the energy of the neutrinos is well below the masses of the relevant gauge bosons, we JHEP07(2020)235 is i ], in f which . 171 R 3 of the TEXONO e ρ ]. The resulting fit is curve we have taken the , (A.14) 119 0 ! Z ) ¯ ν , (A.16) ; for each of the latter, ], and the maximum values of i E 2 ( i ρ i 148 . (A.15) , f exp 1 . (A.13) i, − exp i ¯ ν i, R 147 X kg E R − 2 ) the neutrino spectrum. The remaining ¯

ν ¯ ν i ∆ 26 E + 2 E f,i R ]). In what concerns the neutrino spectra, 2 10 ( i E th i ρ M 8 MeV.) × f 170 ], leading to i W – 29 – i ], and is given in the form of numerical results for the − = X 77 . ] 119 P 2 172 = 2 max ' 169 T prediction. Leading to the 1 e π R 0 ρ , in particular in the results displayed in figure 4 Z ), which permits any use, distribution and reproduction in — the total thermal energy of the reactor, and 5.2 2 TEXONO function for the TEXONO experiment data, we again rely on ) = χ 8 MeV are parametrised by the exponential of a fifth degree ¯ th ν 2 E χ − W ( φ ]. Our result is as follows CC-BY 4.0 the fission energy and 119 can be defined as This article is distributed under the terms of the Creative Commons f,i E (Lower energies are not relevant for our study, since the TEXONO data max we display the experimental data obtained by TEXONO, together with the T 6 14 counts the different bins in the recoil energy. i In figure We have thus obtained the electron density of the detector material For completeness, we notice that the lower energy part of the spectrum, which is governed by slow 14 Attribution License ( any medium, provided the original author(s) and source are credited. neutron capture, hasapproximate been standard obtained fuel composition in of ref. pressurised [ water reactors. minimum couplings to electronsthe allowed by couplings NA64 to [ neutrinosused as in derived the from analysis the of section TEXONO data [ Open Access. where (fitted) SM curve as well as the Finally, and to define the the experimental data figure 16 of ref. [ consists of 10 equidistant bins between 3 experiment by fitting thein SM figure expectation 16 of of the ref. binned [ event rate to the SM curve given intrinsic parameters are corresponds to the distance between reactorexperimental and set-up detector (details can of the beand reactor found and depending general in on ref. thewhich [ spectra different between reactor 2 fuelpolynomial. constituents, we use the fit of ref. [ in which the sumsthe fission run rate, over the reactor fuel constituents The (anti)neutrino flux is given by [ recoil energy JHEP07(2020)235 4 , − as a 64 1 10 97 − (2008) 103 day 78 predictions, 1 0 − Z kg 1 (1999) 2439 − ]. (2007) 081101 82 26 Eur. Phys. J. C 8 , 10 Phys. Rev. Lett. × 98 , ]. Phys. Rev. D Phys. Rev. Lett. 77 , . SPIRE , New constraints on 2 A new constraint for the IN ' e Limits on new long range 7 ][ SPIRE ρ ]. Limit IN Search for an axionlike spin 0 SM, Z TEXONO data in units of MeV ][ R Phys. Rev. Lett. SPIRE , 6 IN Phys. Rev. Lett. ][ , MeV He co-magnetometer ]. 3 arXiv:0902.1056 T/ 5 ]. [ Limits on a nucleon-nucleon monopole-dipole – 30 – arXiv:0912.2175 [ SPIRE ], to which we superimpose our SM and ]. SPIRE IN IN 119 ][ 4 hep-ph/0610339 ][ [ )[ (2009) 423 SPIRE T (2010) 6 IN 680 ][ 91 3 Search for macroscopic CP-violating forces using a neutron EDM ]. (2007) 075006 005 000 025 020 015 010

......

arXiv:0809.4700 hep-ph/0606218

0 0 0 0 0 0

New constraints for CP-violating forces between nucleons in the range

−

day kg MeV / R −

− 75

[ [ JETP Lett. 1 1 1 SPIRE , Phys. Lett. B IN , [ ]. ]. arXiv:0808.2673 cm [ 1 . Data of the TEXONO experiment (neutrino rate SPIRE SPIRE IN IN coupling from spin relaxation of(2009) polarized 19 ultra-cold neutrons in traps spectrometer nuclear spin-dependent forces set with(2009) a 261801 K- cm – short-range forces coupling mass to[ intrinsic spin Preferred-Frame and CP-Violation Tests with092006 Polarized Electrons New CP-violation and preferred-frame tests(2006) with 021603 polarized electrons coupling of axion-like particlesPhys. to Rev. matter D via ultra-cold neutron gravitational experiments coupling using a paramagnetic salt[ with a dc SQUID V.K. Ignatovich and Y.N. Pokotilovski, A.P. Serebrov et al., G. Vasilakis, J.M. Brown, T.W. Kornack and M.V. Romalis, A.P. Serebrov, G.D. Hammond, C.C. Speake, C. Trenkel and A. Pulido Paton, B.R. Heckel, E.G. Adelberger, C.E. Cramer, T.S. Cook, S. Schlamminger and U. Schmidt, S. Baessler, V.V. Nesvizhevsky, K.V. Protasov and A. 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