U Boson Interpolating Between a Generalized Dark Photon Or Dark Z, an Axial Boson, and an Axionlike Particle

PHYSICAL REVIEW D 103, 035034 (2021)

U boson interpolating between a generalized dark photon or dark Z, an axial boson, and an axionlike particle

Pierre Fayet * Laboratoire de physique de l’École normale sup´erieure, 24 rue Lhomond, 75231 Paris cedex 05, France and Centre de physique th´eorique, École polytechnique, 91128 Palaiseau cedex, France

(Received 10 October 2020; accepted 16 December 2020; published 26 February 2021)

A light boson U from an extra Uð1Þ interpolates between a generalized dark photon coupled to Q, B, and Li (or B − L), plus possibly dark matter, a dark Z coupled to the Z current, and one ð1Þ axially coupled to quarks and leptons. We identify the corresponding U F symmetries, with 0 0 ¼ γ þ γ þ γ þ γ þ γ 0 þ γ F Y Y BB Li Li AFA F F dFd, Fd acting in a dark sector and F on possible semi- inert Brout-Englert-Higgs (BEH) doublets uncoupled to quarks and leptons. The U current is obtained ð1Þ from the U F and Z currents, with a mixing determined by the spin-0 BEH fields. The charge QU of chiral quarks and leptons is a combination of Q, B, Li, and T3L with the axial FA. It involves in general isovector and isoscalar axial terms, in the presence of two BEH doublets. A longitudinal U with axial couplings has enhanced interactions and behaves much as an axionlike particle. Its axial −7 couplings gA, usually restricted to ≲ 2 × 10 mUðMeVÞ, lead to effective pseudoscalar ones ¼ 2 ¼ 21=4 1=2 ¼ θ gP gA × mq;l=mU GF mq;l A. A is proportional to an invisibility parameter r cos A induced by a singlet v.e.v., possibly large and allowing the U to be very weakly interacting. This allows for a very small gauge coupling, expressed with two doublets and a singlet as 00 −6 g =4 ≃ 2 × 10 mUðMeVÞ r= sin 2β. We discuss phenomenological implications for meson decays, neutrino interactions, atomic-physics parity violation, naturally suppressed π0 → γU decays, etc. The ð4Þ U boson fits within the grand unification framework, in symbiosis with a SU es electrostrong symmetry broken at the grand unification scale, with QU depending on Q, B − L, FA, and T3A through three parameters γY , γA, and η.

DOI: 10.1103/PhysRevD.103.035034

I. A NEW LIGHT GAUGE BOSON as to be very weakly interacting [1]. One can search for it in eþe− annihilations, K, ψ, and ϒ decays, beam dump Could there be a neutral light gauge boson in nature, experiments, neutrino scatterings, parity-violation effects, other than the photon? We have long discussed the possible etc. It may also serve as a mediator for the annihilation of existence of such a new boson called the U boson, with dark matter particles, providing sufficient annihilations to weak or very weak axial and/or vector couplings to quarks allow for dark matter to be light [3,4], circumventing the and leptons [1]. The vector part in the U current is generally Hut-Lee-Weinberg mass bound of a few GeVs [5,6].Itmay obtained, after mixing with the Z current, as a combination contribute to anomalous magnetic moments of charged of the conserved baryonic, leptonic and electromagnetic leptons, providing potential interpretations for possible currents [2]. An axial part may be present when the deviations from their standard model (SM) expected values electroweak breaking is generated by two Brout-Englert- [7]. The search for such light bosons, very weakly coupled Higgs (BEH) doublets, at least, with different gauge to standard model particles and providing a possible bridge quantum numbers. The new gauge boson then behaves to a new dark sector, has developed into a subject of intense very much, in the low mass limit, as an axionlike particle, experimental interest [8,9]. which may be turned into mostly an electroweak singlet so The many different aspects of the physics of a new light gauge boson make it worthwhile to discuss and relate, *ENS, Université PSL, CNRS, Sorbonne Université, Université within a single unified framework, the cases of a dark de Paris, F-75005 Paris, France photon [10] or generalized dark photon vectorially coupled to a combination of electromagnetic, baryonic, and leptonic Published by the American Physical Society under the terms of currents, and of spin-1 bosons with axial couplings, often the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to closely mimicking spin-0 axionlike particles (ALPs). the author(s) and the published article’s title, journal citation, The properties of the new boson, denoted by U, will be and DOI. Funded by SCOAP3. determined in a general and systematic way, under minimal

2470-0010=2021=103(3)=035034(28) 035034-1 Published by the American Physical Society PIERRE FAYET PHYS. REV. D 103, 035034 (2021) hypothesis, from the gauge symmetry principle, depending 00 ξ FA − 2β ð − 2θ Þ ð Þ on the number and properties of the Brout-Englert-Higgs g cos 2 cos T3L sin Q ; 4 fields responsible for symmetry breaking; and, also, on a possible grand unification of strong, electromagnetic, and up to additional vector couplings associated with Q, B, and weak interactions, that would further restrict its couplings. Li. tan β ¼ v2=v1 denotes the ratio of the two doublet No reference to kinetic mixing is needed nor even useful, v.e.v.’s. This provides the axial couplings the weak hypercharge Y being already (independently of dark matter) a natural contributor to the extra-Uð1Þ sym- g00 cos ξ metry generator F, next to other operators like baryonic and ¼ ð1 2βÞ ð Þ gA 4 cos ; 5 leptonic numbers B and Li, an operator FA acting axially on quarks and leptons, Fd acting in a dark sector, and possibly F0 acting on semi-inert doublets uncoupled to quarks and reexpressed as leptons (if present). μ ð1Þ 1 2 m r cot β ðfor u; c; tÞ; The extra U gauge field C slightly mixes with the 21=4 = U ð Þ μ GF × 6 standard model one Z0 (and not with the photon field, in |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}2 r tan β ðfor d; s; b; e; μ; τÞ; this description), into −6 2 × 10 mUðMeVÞ μ μ μ U ¼ cos ξ C þ sin ξ Z0; ð1Þ thanks to the proportionality relation between g00 cos ξ and ¼ θ ≤1 θ ¼j j 2β with cos ξ ≃ 1 in most of the situations considered here. We mU. r cos A , defined from tan A Fσ w=v sin ,is ð1Þ shall show how, by coupling to a combination of the extra- an invisibility parameter, small if the extra U pisffiffiffi broken Uð1Þ and Z currents, the spin-1 U boson includes the at a high scale by a large singlet v.e.v. hσi¼w= 2, further special cases of a dark photon, coupled to the electromag- contributing to mU. We can then reconstruct from (3) the netic current through effective pseudoscalar couplings of quarks and leptons as Y β ð Þ 00 ξ þ − 2θ ¼ 00 ξ 2θ ð Þ −6 r cot for u; c; t ; g cos T3L sin Q g cos cos Q; 2 g ≃ 4 × 10 m ðMeVÞ × 2 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} P q;l β ð ; μ τÞ ϵe r tan for d; s; b e; ; : ð7Þ or of a generalized dark photon coupled to Q, B, and Li currents; or of a dark Z coupled to the Z current; or of a A small r is associated with a large singlet v.e.v., light boson with axial couplings; and how, more generally, significantly increasing the U boson mass in a two-doublet it interpolates between all these situations. model, according to We shall find the corresponding expressions of quark and lepton couplings, obtained as a general linear combination 00 2β hσi 00 2β of Q, B, and Li with the weak hypercharge T3L and the g v sin g v sin mU ¼ ⟶ ; ð8Þ axial FA. The seven chiral quark and lepton couplings, for 2 2r one generation, satisfy two independent relations. This pffiffiffi ¼ −1=2 provides in particular a universality relation gAd gAe with v ¼ðGF 2Þ ≃ 246 GeV, disregarding here between axial couplings of down quarks and charged cos ξ ≃ 1 for simplicity [11]. Conversely it leads, for any leptons, as well as a constraint on the neutrino couplings, given mU, to a significantly smaller value of the extra-Uð1Þ determined in this framework by the other quark and lepton gauge coupling g00, expressed proportionally to the invis- couplings. ibility parameter r according to the formula We shall also discuss how axial couplings, originating from T3L and FA when present (providing isovector and g00 r isoscalar contributions, respectively), are at the source of ≃ 2 × 10−6 m ðMeVÞ : ð9Þ 4 U sin 2β enhanced interactions for a longitudinal U boson having a small mass m . These may be described by effective U ¼ 1 β ¼ 1 pseudoscalar couplings obtained from the axial ones by It involves, for r and tan , the benchmark 2 ≃ 2 10−6 ð Þ value mU= v × mU MeVpffiffiffi, corresponding to 2 002 8 2 ¼ð 2 þ 02Þ 8 2 ¼ 2 mq;l g = mU g g = mZ GF= , relevant in the gP ¼ gA ; ð3Þ mU discussion of nonstandard neutrino interactions, as compared with standard ones [1]. This illustrates also which are similar to those of an invisible axion, or axionlike how the U interactions may be arbitrarily small for small particle [1]. The U couplings to quarks and leptons are r, for any given mU, in a 2-doublet þ singlet model, with obtained in a two-doublet þ one-singlet model [11] from in particular

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g002 G m2 r2 nearly “invisible”. It is expressed in terms of its doublet and ≃ pFffiffiffi U : ð10Þ 8ðm2 − q2Þ 2 m2 − q2 sin22β singlet components as U U pffiffiffi ¼ θ þ θ ð 2 σÞ ð Þ After discussing many of the phenomenological effects a cos A A sin A Im ; 14 ψ ϒ π0 of such a light U boson, including , , , K, and B its couplings to quarks and leptons being proportional to decays, neutrino interactions, atomic-physics parity viola- the invisibility parameter tion, new long-range forces, etc., we shall express its couplings in the framework of grand unification. They r ¼ cos θA < 1; ð15Þ ð4Þ are constrained by a SU es electrostrong symmetry relating the photon with the eight gluons and commuting as in (7). Its production rates are proportional to ð1Þ r2 ¼ cos2 θ , with, in particular, when the axial generator with U U, valid at the grand unification scale. All quark A and lepton couplings can then be expressed in terms of FA participates in the gauging so that a longitudinal U three arbitrary parameters. The vector couplings, that boson gets produced much like an invisible axion, would be proportional to the SUð4Þ -invariant es B ½ψðϒÞ → γ þ U Q − 2ðB − LÞð11Þ ≃ r2 B ½ψðϒÞ → γ þ standard axion; etc: ð16Þ at the grand unification scale (so as to vanish for up quarks, These rates are sufficiently small if r ¼ cos θA is small belonging with their antiquarks to electrostrong sextets), enough, i.e., if the pseudoscalar a, turned into the longi- appear at lower energies as a general linear combination of tudinal degree of freedom of the light U boson, is mostly an Q and B − L [including in particular the more specific case electroweak singlet. of a protophobic coupling to Q − ðB − LÞ]. B. The U current, combination of the II. THE GENERAL PICTURE FOR A LIGHT extra-Uð1Þ and Z currents SPIN-1 U BOSON In this approach, initiated forty years ago, the SUð3Þ × ð2Þ ð1Þ A. An axionlike behavior of a light U boson SU × U Y gauge group of the standard model (SM) is ð1Þ The longitudinal polarization state of a light U boson extended to include an extra U F factor. Its gauge field Cμ, taken to be very weakly coupled, mixes in general with with axial couplings undergoes enhanced interactions, by a μ μ the Z∘ field of the standard model, according to factor ≈ k =mU which may be large for small mU. In fact 00 the U does not decouple even if its gauge coupling g gets μ μ μ U ¼ cos ξ C þ sin ξ Z∘ ; ð17Þ very small, its mass mU getting very small as well, its longitudinal polarization state interacting with amplitudes with a small mixing angle ξ given by [2] ∝ 00 μ g k =mU having finite limits. It behaves then very much 00 X 2 00 as the corresponding Goldstone boson, a spin-0 axionlike ξ ≃ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig ϵ ð Þ vi ¼ g Γ ð Þ tan 2 02 iF hi 2 : 18 particle a, but it has no γγ decay mode. Its effective g þ g i v gZ pseudoscalar couplings to quarks and leptons gP are ð1Þ The mixing is determined by the extra U F obtained from its axial ones g through the correspondence 00 A coupling g , and by a sum on the Brout-Englert- 2 Higgs doublets h , denoted by Γ. The h ’s are doublets ¯ μ mf ¯ i i g Uμ fγ γ5f → g a fγ5f; ð12Þ ð Þ¼ϵ ¼1 ’ A A m withpffiffiffi weak hypercharges Y hi i and v.e.v. s U 2 vi= P, contributingp toffiffiffi the mixing with the Z, with so that [1] 2 ¼ 2 ¼ 1 ð 2Þ ≃ ð246 22 Þ2 v i vi = GF . GeV . Extra singlets, when present, can contribute significantly to the 2mql gPql ¼ gAql : ð13Þ U mass but not to its mixing with the Z, in the small mass mU limit. The U current is thus obtained in a general way as the ð1Þ The resulting enhancement of U interactions for small following linear combination of the extra-U F and Z currents: mU could lead to too large effects. One way to avoid them is to diminish these effective pseudoscalar couplings by μ 00 introducing, next to the two doublets h1 and h2 considered J ≃ g cos ξ U in this case, a spin-0 singlet σ, now often known as a “dark X 2 1 μ þ ϵ ð Þ vi ð μ − 2θ μ Þ Higgs” field; it acquires a sufficiently large v.e.v. as × 2 J iF hi 2 J3 sin Jem : F v L compared to the electroweak scale, so that the extra |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}i Uð1Þ symmetry gets broken “at a large scale” [1]. This Γ turns the equivalent axionlike pseudoscalar from an active ð Þ field A into a mostly inert one a, very weakly interacting or 19

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ξ ð1Þ In the situations usually considered here the mixing angle transforming under U F. The resulting current for quarks, is small so that cos ξ ≃ 1. leptons, and dark matter is then obtained from (17)–(19) as Two classes of models are particularly interesting and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi representative. With F taken as the weak hypercharge Y J μ ¼ ξ 2 þ 02 ð μ − 2θ μ ÞþJ μ U sin g g J3 sin Jem (plus a possible dark matter contribution), a single doublet L Ud as in the standard model (or two as in the supersymmetric X 2 ≃ 00 ξ ϵ ð Þ vi ð μ − 2θ μ ÞþJ μ ¼ ¼ 1 c g cos iF hi 2 J3L sin Jem Ud: standard model, with F Y for h2 and h1) so that v Γ ¼ 1, and an extra singlet σ to generate the U mass, the i 1 μ μ ð24Þ 2 JY and J3L terms in Eq. (19), both parity violating, μ recombine into J . The mixing of the Y current with em Only semi-inert doublets contribute to Γ when the other the Z current reconstructs exactly the electromagnetic doublets coupled to quarks and leptons have F ¼ 0, a semi- current, with inert doublet h0 taken with F ¼ 2 leading to Γ ¼ 2v02=v2 ¼ 2 2β0 J μ ≃ 00 ξ 2θ μ þ J μ ð Þ sin in (24). Such a U boson simply coupled to the Z U g cos cos Jem Ud: 20 weak neutral current may be referred to as a dark Z boson [14–16]. Again, other contributions proportional to Q, B, “ ” The U boson appears as a dark photon with a coupling Li, and FA currents may be present as in (21), (23). 00 ξ 2θ ¼ χ μ μ g cos cos e tan to the electromagnetic current. For Γ ¼ 0, C and Z0 do not mix, the U current for When F includes also, in addition to Y, terms proportional quarks, leptons, and dark matter being simply given by to baryon and lepton numbers, the U appears as a 00 generalized dark photon coupled to a linear combination J μ ¼ g ðγ μ þ γ μ þ γ μ þ γ μ ÞþJ μ ð Þ U AJA YJY BJB L JL Ud: 25 of Q, B, and Li currents [2,12], 2 i i μ 00 2 μ 1 μ μ μ J ≃ g cosξ cos θJem þ ðγBJ þγL J Þ þJ : ð21Þ U 2 B i Li Ud C. The U charge as a general combination of Q, B, Li, T3L, and FA In contrast F may be taken as an axial symmetry We shall discuss more precisely in Sec. VI how the extra- −1 2 þ1 2 ð1Þ generator FA (equal to = and = for left-handed U F symmetry to be gauged may be generated by an and right-handed quark and lepton fields, respectively), arbitrary linear combination of the axial generator FA with plus a possible dark matter contribution. This requires two Y, B, and Li, plus a dark matter contribution, and a possible 0 BEH doublets at least, taken as h1 and h2 with tan β ¼ γF0 F term acting on semi-inert BEH doublets uncoupled to v2=v1 as in supersymmetric extensions of the standard quarks and leptons. This extra-Uð1Þ generator is then model [13]. These two doublets, taken with Y ¼ ∓1 and given by FA ¼ 1, may then be rotated independently thanks to 0 ð1Þ ð1Þ F ¼ γ F þ γ Y þ γ B þ γ L þ γ F þ γ 0 F : ð26Þ U A and U Y. We thus have, as seen from (19) with A A Y B Li i d d F 2 2 2 Γ ¼ðv2 − v1Þ=v ¼ − cos 2β [11], ð1Þ This includes specific combinations for which U F does μ 1 μ μ μ μ not act on left-handed quarks and leptons but only on right- J ≃g00 ξ J − 2βðJ − 2θJ Þ þJ ; ð Þ U cos 2 A cos 3L sin em Ud 22 handed ones, such as Y B − L or, including B and L contributions as in (21), ¼ − ð Þ i T3R 2 2 ; 27 μ 00 1 μ μ 2 μ J ≃ ξ − 2β ð − θ Þ or ð3B þ LÞ=2 þ FA, equal to þ1 for right-handed quark U g cos J cos J3 sin Jem 2 A L and lepton fields. ¼ ∓1 00 1 μ μ μ For two active doublets h1 and h2 with Y and þ g cos ξ ðγBJ þ γL J ÞþJ : ð23Þ 2 B i Li Ud FA ¼ 1, plus a singlet σ, the mixing angle ξ in (18) is given by Doublet BEH fields uncoupled to quarks and leptons but 00 with nonvanishing v.e.v.’s, referred to as “semi-inert”,may pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig μ tan ξ ≃ ðγY − γA cos 2βÞ: ð28Þ also participate in the mixing, again with C very weakly g2 þ g02 coupled to quarks and leptons through a linear combination of the Y, B, Li, and FA currents. As a special case it is even Remarkably, it allows us to unify within a single formula possible that the Uð1Þ gauge field Cμ does not couple to the situations described in (20)–(23), with vector and/or F μ quarks and leptons at all, while still mixing with Z∘ as in axial couplings of the U boson, by expressing the U current (17), (18) thanks to the v.e.v.’s of semi-inert doublets in (19) as the sum of three contributions:

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γ ð1 μ þ μ − 2θ μ Þ γ 2θ μ (1) a vector current linear combination of Q, B, and Li Recombining Y 2JY J3L sin Jem into Ycos Jem, currents, as for a generalized dark photon in (21) J μ we find the general expression of U as [2,12]; μ (2) a combination of a universal axial current JA 1 μ − 2θ μ μ 00 2 μ μ μ with the standard weak neutral current J3L sin Jem J ≃ g cos ξ γY cos θ Jem þ ðγBJ þ γiJ Þ U 2 B Li as in (22), as for an axial boson mixed with the Z [11]; γ μ μ μ þ g00 cos ξ A J þ η ðJ − sin2θ J Þ ð35Þ (3) and a dark matter contribution [3,4]. 2 A 3L em This reads: 00 1 μ μ þ g cos ξ ðγ 0 J 0 þ γ J Þ: 2 F F d d μ 00 2 μ 1 μ μ J ≃ g cos ξ γY cos θ Jem þ ðγBJ þ γiJ Þ U 2 B Li It involves a general linear combination of electromagnetic 1 00 μ μ 2 μ and Z currents with baryonic and leptonic currents, and an þ g cos ξγ J − cos 2β ðJ3 − sin θ JemÞ A 2 A L axial current, with an additional dark matter contribution. 00 g μ The mixing with the Z current is determined by the þ cos ξγ J : ð29Þ γ η – γ ¼ η ¼ 0 2 d d parameters Y and in (30) (33). For A we recover a U vectorially coupled to Q, B, and Li.ForγA ≠ 0 ˜ In a general way we can decompose F as γYY þ F, we recover the U current as an axial one mixed with the Z so that current and combined with Q, B, and Li currents. For γA ¼ 0, γY and η ≠ 0 we get a mixing between a dark X 2 v photon and a dark Z, also coupled to baryonic and leptonic Γ ¼ γ þ ϵ F˜ ðh Þ i ¼ γ þ η; ð30Þ Y i i 2 Y currents. i v This corresponds to a U charge, associated with the 00 00 with coupling g cos ξ (in most cases close to g ), obtained from the Z − U mixing in (17) as X 2 0 vi η ¼ ϵ ½γ F þ γ 0 F ðh Þ ; ð31Þ i A A F i 2 1 gZ 2 i v Q ¼ F þ tan ξ ðT3 − sin θ QÞ: ð36Þ U 2 g00 L leading to It reads in the small mU limit 00 ξ ≃ g ðγ þ ηÞ ð Þ tan Y ; 32 1 0 gZ Q ≃ ½γ F þ γ Y þ γ B þ γ L þ γ 0 F þ γ F U 2 A A Y B Li i F d d 2 as in (28). With the usual active doublets h1, h2 (Y ¼ ∓1, þðγY þ ηÞðT3L − sin θ QÞ; ð37Þ 0 FA ¼ 1), a semi-inert one h (taken for future convenience with F0 ¼ 2Y ¼ 2), and a singlet σ not contributing to η,we reexpressed as have 1 2 − 2 02 Q ≃ γ cos2θ Q þ ðγ F þ γ B þ γ L Þ v2 v1 v U Y 2 A A B Li i η ¼ γ þ 2γ 0 ð Þ A v2 F v2 38 2 1 0 2 0 2 0 þ η ðT3L − sin θ QÞþ ðγF0 F þ γdFdÞ; ¼ −γA cos 2β cos β þ 2γF0 sin β : ð33Þ 2

This reduces to η ¼ − cos 2β for h1, h2, σ (choosing J μ 02 2 2 0 in agreement with expression (35) of U. γA ¼ 1), and η ¼ 2v =v ¼ 2 sin β for a SM-like doublet 0 γ 0 ¼ 1 hsm and a semi-inert one h (choosing F ), with a possible singlet. D. Special cases of a U unmixed with the Z The U current is then expressed as When Γ ¼ γY þ η ¼ 0 so that ξ ¼ 0, there is no mixing μ ð1Þ μ μ between the standard model Z0 and the extra-U C J ≃ g00 cos ξ μ ≡ μ U gauge fields. Z Z0 remains coupled to quarks and 1 leptons as in the SM, and Uμ ≡ Cμ is coupled to the μ μ μ μ μ μ μ × ðγAJ þ γYJ þ γBJ þ γiJ þ γF0 J 0 þ γdJ Þ ð1Þ 2 A Y B Li F d extra-U current JF as in (25). This is the case, in particular, for an axial gauge boson unmixed with the Z μ μ þðγ þ ηÞð − 2θ Þ ð Þ (with γ ¼ 0, γ ¼ 1, and η ¼ − cos 2β ¼ 0 for v1 ¼ v2), Y J3L sin Jem : 34 Y A which may also be coupled to B and Li [1].

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This may also occur with a semi-inert doublet h0, for and 2 0 γA ¼ 0 and Γ ¼ γY þ 2γF0 sin β ¼ 0. As a very special case we may take γ ¼ −γ 0 ¼ 1 with SM-like and semi- γ γ Y F ð Þ ¼ γ 2θ þ B þ Li 0 ¼1 QU V Y cos Q B Li inert doublets hsm and h having opposite F and 2 2 0 μ μ equal v.e.v.’s (tan β ¼ 1) so that C and Z0 do not mix. In 1 1 þ η − 2θ − ð − Þ ð Þ all such situations with no mixing we get from (34) the U 2 sin Q 4 B L ; 43 current γA η ðQ Þ ¼ F − T3 : 00 U A 2 A 2 A J μ ¼ g μ U 2 JF 00 With T3 ¼ T3L þ T3R, T3A ¼ −T3L þ T3R, and T3L ¼ g μ μ μ μ μ μ ð − Þ 2 ¼ ðγAJ þ γYJ þ γBJ þ γiJ þ γF0 J 0 þ γdJ Þ: T3 T3A = , one can also write, if one prefers, 2 A Y B Li F d ðT3 Þ ¼ −T3 =2, and ðT3 Þ ¼T3=2¼Q=2−ðB−LÞ=4. ð Þ L A A L V 39 Let us consider as a check η ¼ −γY so that there is no mixing with the Z. Then It corresponds to 8 γ γ B Li < ðQUÞ ¼ γY½Q − ðT3LÞ þ B þ Li; F 1 0 V V 2 2 ¼ ¼ ðγ þ γ þ γ þ γ þ γ 0 þ γ Þ ð Þ QU 2 2 AFA YY BB iLi F F dFd ; 44 : γA ðQ Þ ¼ F − γ ðT3 Þ : ð40Þ U A 2 A Y L A acting on quarks and leptons as a linear combination of Y, Summing the two we get for the quark and lepton couplings, with Q − T3 ¼ Y=2, B, and Li (including possibly T3R) with the axial FA. This L ¼ 2 unmixed expression of QU F= is also easily recovered γ γ γ γ 1 from the general expression (38), under the no-mixing ¼ A þ Y þ B þ Li ¼ ð Þ QU 2 FA 2 Y 2 B 2 Li 2 F; 45 constraint η ¼ −γY.

which provides back QU ¼ F=2 as in (38), (40), in the E. Vector and axial parts in the U charge absence of mixing with the Z. Returning to the general situation of the U current in (34) the couplings of chiral quark and lepton fields are F. More on the vector couplings expressed as g00 cos ξ Q , with Q given in (38) as a linear U U The vector couplings involve a linear combination of the combination of Q, B, L , T3 , and F (the colorless i L A conserved B; L , and electromagnetic currents. The pure neutral symmetry generators in the standard model and i dark photon case corresponds to the specific direction its extensions) [17]: ð1; 0; 0; 0; 0; 0; 0Þ in the seven-dimensional parameter γ γ space ¼ðγ 2θ − η 2θÞ þ B þ Li QU Y cos sin Q 2 B 2 Li γ ðγ ; γ ; γ ; γ ; ηÞ: ð46Þ þ A þ η ð Þ Y B Li A 2 FA T3L: 41 As a result of the mixing with Z we recognize in the The couplings are expressed in terms of five independent vector contribution to the U charge the part corresponding parameters if we assume lepton universality, γY, γB and γL, to the weak charge [11,18], defined as 00 γA, and η, all of them multiplied by g cos ξ. This allows for 2 many interesting situations, with in particular the possibil- ðQ Þ ¼ðT3 Þ − sin θ Q Z V L V ity of reduced couplings to neutrinos, and to protons. In 1 1 ¼ − 2θ − ð − Þ ð Þ addition to vector couplings involving Q, B, and Li,we 2 sin Q 4 B L : 47 also get in general axial couplings, involving isoscalar as well as isovector contributions. When conventionally renormalized by a factor 4 into Let us separate the parts corresponding to vector and ¼ þ þð − Þ 2 axial couplings, with Q T3L T3R B L = so that ¼ 4ð Þ ¼ð2 − 4 2θÞ − ð − Þ ð Þ QWeak QZ V sin Q B L ; 48 þ − − 2θ ¼ T3L T3R − 2θ − T3R T3L T3L sin Q 2 sin Q 2 it reduces to the usual definition of the weak charge for a ð Þ¼ð1 − 4 2θÞ − nucleus, QWeak Z; N sin Z N [19]. 1 1 T3 2 ¼ − sin2θ Q − ðB − LÞ − A ; ð42Þ For sin θ having the grand unification value 3=8 [20] the 2 4 2 vector couplings of the U are fixed by

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1 1 with the Z. All axial couplings are given by the two Qgut ¼ Q − ðB − LÞ: ð49Þ ZV 8 4 universal expressions

They vanish for up quarks, as do the Z vector couplings to 8 > g00 cos ξ up quarks. We shall come back to this in Sec. IX,in <> ¼ ¼ ðγ − ηÞ gAþ gA uct 4 A ; connection with a SUð4Þ electrostrong symmetry relating ð53Þ es > 00 ξ the photon with the eight gluons. :> ¼ ¼ ¼ g cos ðγ þ ηÞ 2 gA− gAeμτ gA dsb 4 A : With sin θ close to 1=4 however, the QZ contribution to the vector couplings in (47), close to be proportional to B − L − Q (rather than to B − L − Q=2 for sin2 θ ¼ 3=8), They satisfy the general universality relations is significantly smaller for the proton than for the neutron “ ” g ¼ g ¼ g ; [11], and thus naturally protophobic , instead of being Au Ac At ð Þ “u-phobic” at the GUT scale. 54 gAe ¼ gAμ ¼ gAτ ¼ gAd ¼ gAs ¼ gAb:

G. P, C, and CP properties of the U boson These are in fact a consequence of the hypothesis that up There are different representatives within the general quarks on one hand, down quarks, and charged leptons on class of models considered, associated with different the other hand, all acquire their masses from the same BEH choices for the gauged extra-Uð1Þ quantum number F in doublet, as in the standard model, or from h2 and h1 as in (26), the doublet and singlet v.e.v.’s, etc. The resulting U supersymmetric theories. However, should we use more current in (19)–(25), (29) or (35) may be purely vectorial, doublets and allow for different axial generators acting purely axial, or in general appears as a combination of separately on leptons of the three families, and on quarks, ð1Þ vector and axial parts, with to participate in the U F gauging (cf. Sec. VI B), then the universality relations (54) and subsequent constraints on −− ’ vectorially coupled U∶ CP ðas the photonÞ; the axial gA s would have to be abandoned. ð50Þ ∶ þþ For a purely isoscalar axial coupling or a purely isovector axially coupled U CP : one, all axial couplings have the same magnitude If both couplings are present the U interactions are C- and jg j¼jg j¼jg μτj; ð55Þ P-violating, the U still having CP ¼þ. Its interactions A uct A dsb Ae generally preserve CP except for very small effects, but the two active doublets h1 and h2 (Y ¼ ∓1, F ¼ 1), including a possible very small CP-violating monopole- A with the singlet σ, lead to a combination of isoscalar dipole spin-dependent interaction [21]. 00 and isovector couplings. Normalizing g so that γA ¼ 1, Let us now concentrate on the axial couplings. 2 2 2 η ¼ðv2 − v1Þ=v ¼ − cos 2β, we recover [11] III. AXIAL COUPLINGS AND EFFECTIVE 1 ðJ μ Þ ¼ 00 ξ μ − 2β ð μ Þ ð Þ PSEUDOSCALAR ONES U ax g cos 2 JA cos J3L ax ; 56 A. Universal expressions of axial couplings providing The rearrangements leading to expressions (29), (35) of the current may also be understood by considering the axial 00 00 2 μ g cos ξ g cos ξ cos β part in the quark and lepton contribution to J , g ¼ ð1 cos 2βÞ¼ × . ð57Þ U A 4 2 2β sin μ γ μ μ ðJ Þ ¼ 00 ξ A þ η ð Þ ; ð Þ U ax g cos 2 JA J3L ax 51 The upper sign þ is relative to up quarks, and the lower one − to down quarks and charged leptons. – which is independent of γ , and obviously of γ and γ .It All these expressions (51) (57) for the axial couplings Y B Li ð4Þ leads to the axial couplings g ¼ðg − g Þ=2, are also compatible with grand unification and the SU es A R L electrostrong symmetry, as we shall see in Sec. IX. 00 00 g cos ξ g ¼ g cos ξ ðQ Þ ¼ ½ γ ∓ η ; ð52Þ 00 A U A 4 A B. g as proportional to mU and to the invisibility ↑↑ parameter r c isoscalar isovector Let us consider two BEH doublets h1 and h2 (or hsm and h0) with different gauge quantum numbers, breaking γ γ γ ðQ Þ ð1Þ independently of Y, B, and Li . It corresponds to U A in spontaneously the U U symmetry and making the U (43), with the γA and ∓ η terms providing isoscalar and massive, its current having an axial part. Its mass, initially isovector contributions, the latter induced by the mixing given in the small mU limit by

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00 00 g cos ξ v g cos ξ 1 2 m ≃ 2β ð Þ ¼ ∓ 2β0 ≃ 21=4 = U ð∓ β0Þ ð Þ mU 2 sin ; 58 gA 2 sin GF 2 r tan : 65 where tan β ¼ v2=v1, gets increasedpffiffiffi by the effect of an In general axial couplings may be parametrized 1 ¼ 21=4 1=2 ≃ extra singlet v.e.v. hσi¼w= 2. It can then be expressed proportionally to the benchmark value =v GF −6 −1 as [11] 4 × 10 MeV , according to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 00 ξ 00 ξ 2β g cos 2 2 2 2 g cos vsin 1=4 1=2 mU −6 m ¼ v sin 2β þ Fσw ¼ ; ð59Þ g ¼ 2 G A ≃ 2 × 10 m ðMeVÞA: ð66Þ U 2 2 r A F 2 U with the invisibility parameter A, proportional to the invisibility parameter r ¼ cos θA,is given for the two active doublets h1, h2 transforming under v sin 2β ¼ θ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ Uð1Þ ,by r cos A 2 2 2 2 ; 60 A v sin 2β þ Fσw Aþ ¼ r cot β;A− ¼ r tan β; ð67Þ as we shall rediscuss in Secs. VII and VIII. c In such a situation involving axial couplings we can then and for a SM-like doublet h (or h2, h with the same 00 sm 1 view, conversely, the gauge coupling g as proportional to quantum numbers) with a semi-inert one h0,by mU and r, with ¼ ∓ β0 ð Þ 00 A r tan : 68 g cos ξ m r 1 2 r ¼ U ¼ 21=4G = m : ð61Þ 2 v sin 2β F U sin 2β C. From axial couplings g to pseudoscalar ones g It also reads A P A longitudinal U behaves, in the small mass limit, much 00 as an axionlike particle with effective pseudoscalar cou- g r cos ξ ≃ 2 × 10−6 m ðMeVÞ ; ð62Þ plings to quarks and leptons obtained from (13), (53), (66) as 4 U sin 2β 00 2mql g cos ξ gPql ¼ gAql ¼ mql ðγA ∓ ηÞ which reduces to the simple benchmark value mU 2mU 1 2 00 ¼ 21=4 = ð Þ g GF mql A: 69 ≃ 2 × 10−6 m ðMeVÞ r ð63Þ |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} 4 U −6 4 × 10 mqlðMeVÞ for tan β ¼ 1, disregarding for simplicity cos ξ ≃ 1 [1]. This is of interest, in particular, when discussing nonstandard We get from gA in (57), (64) the effective pseudoscalar 00 couplings of a longitudinal U to quarks and leptons, neutrino interactions with gðνLÞ and gðeÞ ≈ g =4, as done in Secs. VA, VB. As a result of (62) the axial couplings expressed in the small mU limit as (57) read, in the small mU limit, 1 2 2m 21=4G = m r cot β; g ¼ g ql ¼ F u ð70Þ β Pql Aql 1 4 1=2 1 2 m r cot ; mU 2 = β ¼ 21=4 = U ð Þ GF mde r tan ; gA GF × 64 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}2 r tan β: as for a quasi-invisible axion; or similarly, with a semi-inert 2 × 10−6m ðMeVÞ U doublet h0,

We can also consider a SM-like doublet hsm (possibly 1=4 1=2 0 c gPql ¼ ∓2 G mql r tan β ; ð71Þ replaced by h2 and h1, taken with the same quantum F numbers so that γ ¼ 0), and a semi-inert one h0, choosing A which vanish in the limit v0 → 0, for which the semi-inert for convenience, without restriction, the normalization 0 00 0 0 doublet h becomes inert. of g so that h has F ¼ F ¼ 2Y ¼ 2 with γF0 ¼ 1. F is here taken as in Eq. (26) with γA ¼ 0. Then η ¼ 2 02 2 ¼ 2 2 β0 0 ¼ γ D. Recovering the pseudoscalar couplings v =v sin . With hsm and h having F Y c of a longitudinal U and γY þ 2 differing by the same jΔFj¼2 as earlier for h1 and h2, mU is still given by the same Eq. (59) as in [11] with These effective pseudoscalar couplings gP of a longi- β β0 γ γ γ replaced by , independently of Y, B, and Li . The axial tudinal U boson, proportional to the invisibility parameter couplings in (53), now isovector, read r ¼ cos θA, may also be obtained directly from the couplings

035034-8 U BOSON INTERPOLATING BETWEEN A GENERALIZED … PHYS. REV. D 103, 035034 (2021) of the equivalent pseudoscalar a, as found from its expres- ΓðU → ff¯Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sion (14). Indeed in the casep offfiffiffi (70) the Goldstone field 2 2 2 ¼ 2 ð− β 0 þ β 0Þ m 4m 2m 4m eliminated by the Z is zg Im cos h1 sin h2 , ¼ U 1 − f 2 1 þ f þ 2 1 − f 2 g 2 g 2 and the pseudoscalar a reads 12π m Vf m Af m sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU U U 2 pffiffiffi m 4m ¼ 2 ½ θ ð β 0 þ β 0Þþ θ σ ð Þ ¼ U 1 − f a Im cos A sin h1 cos h2 sin A : 72 24π 2 mU 2 2 0 0 λ ¼ m 6m Withp theffiffiffi Yukawa couplings of hp1 ffiffiffiand h2 expressedpffiffiffi as de × ðg2 þ g2 Þ 1 − f þ g g f : ð76Þ 2 ð βÞ λ ¼ 2 ð βÞ 2 0 Lf Rf m2 Lf Rf m2 pmdeffiffiffi = v cos and u mu = v sin , Im h1 and U U 2 Im h0 have pseudoscalar couplings to down quarks and 2 The resulting lifetime varies considerably depending charged leptons, and up quarks, m =ðv cos βÞ and de principally on m and g00, especially with g00 proportional m =ðv sin βÞ, respectively. We thus recover directly the U u to m as given by Eq. (62) in a 2-doubletþ1-singlet model, couplings (70) as U leading to 8 m 1 2 < g ¼ u cos θ cot β ¼ 21=4G = m r cot β; r2 Pu v A F u Γ ∝ g002m ∝ G m3 : ð77Þ ð73Þ U U F U 22β : m 1 2 sin g ¼ de cos θ tan β ¼ 21=4G = m r tan β: Pde v A F de This leads in particular, sufficiently above threshold and depending on the axial and/or vector couplings considered, They are the same as for a standard axion [22,23], multiplied and accessible channels, to typical estimates like by the invisibility parameter r, as for a quasi-invisible axion [1,24,25] (see also Sec. VII B). −9 −8 0 ð10 to 10 Þ s Similarly in the case of a semi-inert doublet h leading τ ≈ ð78Þ m ðMeVÞ3r2 to (71)pffiffiffiffi, the Goldstone field eliminated by the Z is U ¼ 2 ð β0 0 þ β0 00Þ zg Im cos hsm sin h , so that 2 3 2 [reexpressed as τ ≈ 10 y=ðmUðeVÞ r Þ for a very light U pffiffiffi decaying into neutrinos]. The resulting decay length, ¼ 2 ½ θ ð− β0 0 þ β0 00Þþ θ σ ð Þ a Im cos A sin hsm cos h sin A : 74 22β ¼ β γ τ ∝ pU sin ð Þ 0 l U Uc 4 2 ; 79 The Yukawa couplingspffiffiffi of the SM-like doublet hsm, GFmU r with v.e.v. v cos β0= 2, are enhanced by 1= cos β0 as 0 0 is of the order of [1] compared to the SM. hsm (respectively hsm, with opposite imaginary part) plays the role of h0 (respectively h0) 2 1 p ðMeVÞ in giving masses to up quarks (respectively down quarks l ≈ .7 m× U ; ð80Þ 0 m ðMeVÞ4r2 and charged leptons). Multiplying by − cos θA sin β from U (74) and including the extra − sign for down quarks and þ − νν¯ charged leptons we recover the pseudoscalar couplings of a for an axially coupled boson decaying into e e or ≃ 6 ¼ in (71) as pairs, e.g., l m for a 20 GeV boson of mass mU 7 MeV with r ¼ 1. The decay length l plays an essential role in the derivation mql 0 1 4 1=2 0 g ¼ ∓ cos θ tan β ¼ ∓ 2 = G m r tan β : of experimental constraints in terms of m and g00 (or Pql v A F ql U r= sin 2β), in particular from beam dump experiments. ð Þ 75 They require l to be sufficiently short, or sufficiently large, for the U boson, if sufficiently produced, to remain unde- E. U lifetime and beam dump experiments tected. This originally allowed to exclude, from the results of early beam dump experiments at Brookhaven [26,27],an Let us give the partial widths for U decays into quarks axially coupled U boson in the 1 to 7 MeV mass range, in the ¼ and leptons, from their vector and axial couplings gV simplest case r ¼ tan β ¼ 1 [1]. Beam dump experiments 00 ξ ð Þ ¼ 00 ξ ð Þ g cos QU V and gA g cos QU A obtained from (43) can now constrain the existence of such particles in the 1–10 [see also later (81)–(84)]. From the decay widths – 2 2 or up to 1 100 MeV mass range, depending on the couplings Γ ¯ ¼ð 12πÞð þ Þ into massless particles ff mU= gVf gAf , and considered and the strength of their interactions [8,9]. 3 3 including the phase space factors ð3β − β Þ=2 and β for We now present in Secs. IV and V the main conse- vector and axial couplings, respectively, we have with quences of this analysis for U boson phenomenology. β ¼ ¼ð1−4 2 2 Þ1=2 ð3 − β2Þ 2 ¼ 1 þ 2 2 2 f vf=c mf=mU , f = mf=mU, We leave the more technical aspects on the choice of the

035034-9 PIERRE FAYET PHYS. REV. D 103, 035034 (2021) extra-Uð1Þ symmetry to be gauged, and on the generation IV. EFFECTS OF AXIAL COUPLINGS: ψ, ϒ, K, B of the U mass and mixing angle ξ, for the subsequent DECAYS, PARITY VIOLATION, … Secs. VI–VIII. We finally discuss in Sec. IX how the U can A. Quark and lepton couplings find a natural place within the grand unification (and supersymmetry) frameworks, and resulting implications Let us write the couplings of chiral quark and lepton for its couplings. fields, as obtained from (41),

8 h γ γ γ η i < ¼ 00 ξ ðγ 2θ − η 2θÞ þ B þ Li − A gL g cos Y cos sin Q 2 B 2 Li 4 2 ; h i ð Þ : γ γ γ 81 ¼ 00 ξ ðγ 2θ − η 2θÞ þ B þ Li þ A gR g cos Y cos sin Q 2 B 2 Li 4 ; valid in the small mU limit. The vector and axial couplings gV ¼ðgL þ gRÞ=2 and gA ¼ðgR − gLÞ=2, defined according to 1 − γ 1 þ γ γμ 5 þ γμ 5 ¼ γμð þ γ Þ ð Þ gL 2 gR 2 gV gA 5 ; 82 are 8 h γ γ 1 1 i < 00 2 B Li 2 gV ≃ g cos ξ γY cos θ Q þ B þ Li þ η − sin θ Q − ðB − LÞ ; 2 2 2 4 ð Þ : γ ∓ η 83 ≃ 00 ξ A gA g cos 4 ; 8 > ðψ → γ Þ 2 as also seen from (43). The vector couplings may also be > B U GpFffiffiffimc 2 −4 2 1 1 <> þ − ≃ Cψ Aþ ≃ 8 × 10 Cψ Aþ; reexpressed using Q − 2 ðB − LÞ¼T3 ¼2 ,as Bðψ → μ μ Þ 2πα ð87Þ > ðϒ → γ Þ 2 γ > B U GFmb 2 −3 2 00 2 γ L : ≃ pffiffiffi CϒA− ≃ 8 × 10 CϒA−; ≃ ξ γ θ þ B þ i ðϒ → μþμ−Þ gV g cos Y cos Q 2 B 2 Li B 2πα 1 þ η − 2θ ð Þ so that 4 sin Q : 84 −5 2 Bðψ → γUÞ ≃ 5 × 10 Cψ Aþ; ð Þ −4 2 88 The seven couplings of chiral quark and lepton fields Bðϒ → γUÞ ≃ 2 × 10 CϒA−: within a single generation are expressed in (41) as a linear Cψ and Cϒ, expected to be larger than 1=2, take into combination of the five charge operators Q, B, L, T3L, and FA. They verify two independent relations: account QCD radiative and relativistic corrections. We obtained long ago ¼ gAe gAd; pffiffiffiffiffiffiffiffi ð85Þ j j ≲ 1 5 10−6 ð Þ ðν Þ¼ ð Þþ ð Þ − ð Þ gA uct . × mU MeV = Binv; g L g uL gV e gV d : pffiffiffiffiffiffiffiffi ð89Þ j ¼ j ≲ 8 10−6 ð Þ gAeμτ gA dsb : × mU MeV = Binv: The second one is related to the fact that the vector part in the U current is a combination of Q, B, and L currents. For The improved experimental limits for a light U [28–32], purely vector couplings to Q, B, and L it reduces to gðν Þ¼g ðuÞþg ðeÞ − g ðdÞ. It extends for three gen- Bðψ → γ þ invisible UÞ < 7 × 10−7; L V V V ð Þ erations into −6 90 Bðϒ → γ þ invisible UÞ ≲ 10 ; g μτ ¼ g ; now imply Ae A dsb ð Þ 86 pffiffiffiffiffiffiffiffi gðνiLÞ − gVðliÞ¼gðuL;cL;tLÞ − gVðd; s; bÞ: jAþj < .17= B ; pffiffiffiffiffiffiffiffiinv ð91Þ j j ≲ 1 A− . = Binv; B. Limits on gA’s and gP’s from ψ and ϒ decays corresponding, as seen from (64), (66),to Upper limits on axial couplings gA and associated ψ ϒ pffiffiffiffiffiffiffiffi pseudoscalar ones gP from and decays are rather j j ≲ 3 4 10−7 ð Þ gAc . × mU MeV = Binv; stringent. With the correspondence r cot β → Aþ, pffiffiffiffiffiffiffiffi ð92Þ j j ≲ 2 10−7 ð Þ r tan β → A−, we get the branching ratios [1,7], gAb × mU MeV = Binv:

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They may be combined into Binv denotes the branching ratio for the U to be undetected pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi in the experiment considered, due to a long enough lifetime j j ≲ ð 13 16Þ ð Þ AþA− . to . = Binv; 93 or to invisible decayp modesffiffiffiffiffiffiffiffi into νν¯ or light dark matter 2 particles. For small Binv complementary results can be for m in the 0 to 1 GeV=c mass range. When Aþ ¼ þ − þ − U obtained from visible U decays into e e or μ μ , r cot β and A− ¼ r tan β this reads pffiffiffiffiffiffiffiffi depending on mU. ≲ ð 13 16Þ ð Þ Furthermore, if the axial couplings are purely isoscalar or r . to . = Binv; 94 isovector so that jgAej¼jgAtj, the even stronger constraint independently of tan β. A longitudinal U boson is then that may be derived on the latter [see later Eqs. (122), (123) required to behave mostly as a spin-0 electroweak singlet, in Sec. IV F] can also be applied to the electron, then as anticipated long ago. leading to When the axial part in the U current originates totally, μ either from the universal axial current J , isoscalar, or from jg j ≲ 2 × 10−9m ðMeVÞ; A Ae U ð98Þ the mixing with the Z, then jgA uctj¼jgAeμτj¼jgA dsbj as in −9 jgPej ≲ 2 × 10 : (55). The Yukawa couplings of a in (13) are all proportional to quark and lepton masses up to a possible sign, with D. An extra Uð1Þ broken at the ≈ 108 GeV scale? mql 1 4 1=2 j j¼ ≃ 2 = What about even smaller values of gAe and gPe? A very gPql A GF mql A v light U boson with energies in the keV range and very ≃ 4 10−6 ð Þ ð Þ × mql MeV A: 95 small effective pseudoscalar coupling to electrons gPe ≈ 3 10−12 ψ ϒ × might possibly be responsible for the excess and decays then cease to give complementary electronic recoil events recently observed in XENON1T, if indications on the couplings to c and b quarks. It is thus ψ attributed to a solar axionlike particle rather than to some important that decay results [29] be expressed independ- other effect [34]. Such a small g , and associated axial ently of ϒ results [32], to be applicable in a model- Pe coupling independent way (not just with next-to-minimal-supersymmetric-standard-model-like situations in mind). The m −15 ϒ g ≃ U g ≈ 3 × 10 m ðkeVÞ; ð99Þ present limit from decays then implies Ae 2m Pe U pffiffiffiffiffiffiffiffi e j j¼j j ≲ 1 ð Þ Aþ A− . = Binv: 96 would then correspond, using the expression of the This applies in particular to semi-inert doublet models effective pseudoscalar coupling to the electron as with isovector axial couplings, for which A ¼ ∓ r tan β0 as ≃ 2 10−6 ð β β0Þ ð Þ in (75). Then a small A does not necessarily imply a small r gPe × r × tan or tan ; 100 as tan β0 may also be small. One might even have r ¼ 1 i.e., 0 from Eqs. (64), (66), (69) or (75),to no extra singlet,p withffiffiffi a small β , i.e., a smallpffiffiffi semi-inert 0 2 2 ≃174 doublet v.e.v. v = (as compared with v= GeV). ¼ ð β β0Þ ≈ 1 5 10−6 ð Þ This is not surprising as for v0 → 0 the semi-inert doublet A− r tan or tan . × ; 101 model tends towards an inert doublet one with a in (74) ≃ 2 10−6 ≈ 3 10−12 becoming decoupled from quarks and leptons, even with no so that gPe × A− × (leaving aside pos- extra singlet. Still such a possibility (not strongly motivated) sible astrophysical constraints). j j gets disfavored in view of the stronger limits on jAþj¼jA−j Such a small value of A− may be associated with a very small value of the invisibility parameter r [1] originating from K and B decays [see later Eq. (121) in Sec. IV F], then 6 requiring v0 to be much smaller than v. from a large singlet v.e.v. typically ≈ 10 times the electroweakpffiffiffi scale. More precisely this large singlet v.e.v. C. Axionlike couplings to e’s must be very small w= 2, obtained from (60) as Let us move to the discussion of effective pseudoscalar v sin 2β v 2sin2β axionlike couplings of the U boson to electrons. The jFσjw ≃ ≃ r A− universality relations (54) imply that gAe and gAμ must ϒ ≃ 1 6 108 ð2 2βÞ ð Þ obey the same bounds as obtained for gAb from decayspffiffiffiffiffiffiffiffi . × GeV × sin ; 102 þ j j ≲ 1 [33], or for gAs from K decays, thus, with A− . = Binv as deduced from [32], should then be of the order of ≈ 108 GeV. [In the absence ¼ 1 pffiffiffiffiffiffiffiffi of a singlet, i.e., with r one may also consider a very jg j ≲ 2 × 10−7m ðMeVÞ= B ; small semi-inert v.e.v. v0 ≈ 1.5 × 10−6v ≃ 370 keV as seen Ae pUffiffiffiffiffiffiffiffi inv ð97Þ jg j ≲ 2 × 10−7= B : from (101), but this would require a large amount of fine Pe inv tuning.]

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E. Parity-violation effects in atomic physics corresponding to A positive aspect of a very strong constraint on g ,asin Ae jg g j (98), is that it facilitates considerably satisfying another Ae Vq ≲ 10−3 ð Þ 2 GF; 104 constraint from parity-violation effects in atomic physics. mU This one requires [18], for the change in the weak charge for m larger than a few MeV=c2. QW of the cesium nucleus to be less than 1 [35], U qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi More precisely the axial coupling of the electron gAe and the vector coupling g of an average quark in the nucleus jg g j ≲ 10−7m ðMeVÞ; ð103Þ Vq Ae Vq U are obtained from (43), (53) as

8 γ þ η > ¼ 00 ξ A < gAe g cos 4 ; ð Þ > γ 1 1 105 :> 3ð þ Þ ¼ 00 ξ γ 2θ þ B ð þ Þþη − 2θ − Z N gVq g cos Y cos Z 2 Z N 4 sin Z 4 N ; allowing to express ΔQ , in present notations, as [18], ΔQ W W ¼ r2tan2β0Kðm Þ: ð111Þ Q U pffiffiffi W 3ð þ Þ 2 2 gAe × Z N gVq ≲1% ΔQ ¼ Kðm Þ: ð106Þ This effect is as experimentally required if the W 2 U ≲ 1 GF mU invisibility factor r is smallpffiffiffi enough, typically . ; or if the semi-inert v.e.v. v0= 2 is small enough (typically ≲ 17 ¼ 1 β0 KðmUÞ takes into account the effect of the U mass, varying GeV for r ) so that tan is small, or also if ð Þ ≲ 025 ≲ 1 between about .5 for mU of a few MeV, approaching 1 for K mU is small, with e.g., K . for mU . MeV, mU larger than about 50 MeV. atomic physics experiments loosing some of their sensi- Let us check this general formula on the very specific tivity for a too light U boson. However, the factor A ¼ 0 case of a dark Z, the comparison with the Z exchange r tan β gets in any case more strongly constrained from B amplitude being immediate in this case. Ignoring for and K decay experiments than from atomic-physics parity simplicity cos ξ ≃ 1, we have in this approximation, with violation, in the case of a dark Z. γ ¼ γ ¼ γ ¼ γ ¼ 0, η ¼ 2sin2β0, More generally returning to Eqs. (103)–(106) we get in a Y B Li A 00 2 0 0 similar way, for gAe ≃ðg =2Þsin β ≃½mU=ð2vÞ×ðrtanβ Þ 00 00 and g ∝ g00=2 ≃ ðm =vÞ × r= sin 2β0, g g 2 Vq U gAe ¼ η; 3ðZþNÞgVq ¼ ½ð1−4sin θÞZ−Nη; ð107Þ 4 4 ΔQ 1 W ∝ r2 Kðm Þ: ð Þ 2β0 U 112 so that QW cos Similarly with the active doublets h1, h2 leading to m2 g002 ≃ ð 00 2Þ 2 β ≃ Δ ¼ Z η2½ð1 − 4 2θÞ − ð Þ ð Þ gAe in (57), (64), we get, with gAe g = sin QW 2 2 sin Z N K mU : 108 ½ ð2 Þ ð βÞ gZ mU mU= v × r tan , Δ 1 ð Þ QW 2 It is proportional to QW Z; N , and may be reexpressed ∝ r Kðm Þ: ð113Þ 00 Q cos2β U with tan ξ ¼ðg =gZÞη from (32),as W

The exact expressions depend on γY, γB as well as on γA and m2 η Δ ¼ 2ξ Z ð Þ ð Þ ð Þ , in particular through gVq in (105). QW tan 2 QW Z; N K mU ; 109 mU F. Other limits from meson decays as immediately found for a dark Z of mass m rather than U A light U boson may be produced in meson decays. m , coupled to the Z current with relative strength tan ξ Z A preliminary estimate of the decay rate for Kþ → πþU [14,15]. With was proposed in [1] before the top quark was known or 00 00 even imagined to be so heavy, based on ξ mZ ¼ g mZ η ¼ g v η ¼ r η ¼ β0 ð Þ tan 0 r tan 110 ( mU gZ mU 2mU sin 2β −6 gAs ¼ 2 × 10 mUðMeVÞ A−; 2 ð114Þ from (18), (30) and (61), (65), the relative effect of a dark Z ms −4 gPs ¼ gAsl ≃ 6 × 10 A−; on QW is given by mU

035034-12 U BOSON INTERPOLATING BETWEEN A GENERALIZED … PHYS. REV. D 103, 035034 (2021) and For a purely isoscalar or purely isovector axial coupling this leads to a very small effective pseudoscalar coupling to the 1 g2 ð þ → πþ Þ ∝ Ps ≃ 10−8 2 ð Þ electron, B K U π 4π A−; 115 m leading to consider an order of magnitude estimate in jg j¼ e jg j ≲ 2 × 10−9; ð123Þ −8 2 −5 2 Pe Pt the ≈ 10 A− (up to possibly ≈ 10 A−) range. This led mt tentatively, from a limit still allowing for the much smaller axionlike electron BðKþ → πþ þ invisibleUÞ < ð.73 to ≈ 1Þ× 10−10 ð116Þ coupling ≃ 3 × 10−12 tentatively discussed in Sec. IV D. for m < 100 MeV [36] (see also [37]), in the absence of U V. NEUTRINO INTERACTIONS, π0 → γU DECAYS, cancellation effects involving large couplings to the top, to U-INDUCED FORCES, … jA−j ≲ .1 for an invisible U, i.e., [38] pffiffiffiffiffiffiffiffi A. Nonstandard neutrino interactions, j j ≲ 2 10−7 ð Þ ð Þ 00 gAs × mU MeV = Binv: 117 with a small g ∝ mU The exchanges of a new U boson would lead to This is also what is now found for jgAbj from ϒ decays in (92). nonstandard neutral-current neutrino interactions, a prime The top quark, however, is very heavy, leading to potential effect discussed forty years ago [1], that would not exist for a pure dark photon, uncoupled to neutrinos. 2 mt mt They must be somewhat smaller than the standard ones, gPt ¼ gAt ¼ Aþ ≃ .7 Aþ; ð118Þ mU v otherwise they would have been detected already. The 2 −2 2 comparison between nonstandard and standard neutrino resulting in ð1=πÞ g =4π ≃ 1.2 × 10 Aþ. For processes Pt 002 ð8 2 Þ involving the coupling of a longitudinal U to a t quark the interactionspffiffiffi involves at first comparing g = mU with 2 ¼ 2 ð8 2 Þ branching ratios may be expressed as GF= gZ= mZ . It leads to a benchmark value ð þ → πþ Þ 1 2 00 B K U gPt −2 2 g −3=4 1=2 −6 ∝ ≃ 1.2 × 10 Aþ: ð119Þ ¼ 2 G m ≃ 2 × 10 m ðMeVÞ; ð124Þ BðB → KUÞ π 4π 4 F U U These ratios have been estimated in [14,15] in the special such that γ ¼ γ ¼ γ ¼ γ ¼ 0 case of a dark Z, for which Y B Li A , η ¼ 2 2 β0 ξ ≃ β0 ¼ g002 g2 G sin , tan mZ=mU r tan Aþ as in (110), ¼ Z ¼ pFffiffiffi ; ð125Þ δ 8 2 8 2 then denoted in this specific case by . The results may mU mZ 2 be transposed to general situations dealing with the 00 production a light U from the axial top quark current, with g naturally proportional to mU. Furthermore in the allowing us to express the rates (119) in terms of Aþ,evenif presence of two doublets and a singlet with an invisibility this one is in general no longer to be identified with δ ¼ parameter r ≤ 1, we have as in [11] 00 tan ξ mZ=mU [due to γY in tan ξ ¼ðg =gZÞðγY þ ηÞ in ξ ≠ 0 g00 m r r (32)]; and indeed tan for a dark photon, which has cos ξ ¼ U ≃ 2 × 10−6 m ðMeVÞ ð126Þ no significant axial couplings. 4 2v sin 2β U sin 2β We can then write þ þ −4 2 (cf. also Sec. VII). It serves in the evaluations of the U BðK → π UÞ ≈ 4 × 10 Aþ; 00 ξ ð Þ couplings g cos QU,asin(64) for the axial couplings gA. 2 120 BðB → KUÞ ≈ :1 Aþ: The couplings to neutrinos should then generally be small, especially for small values of mU. Indeed from νee jAþj This leads to stronger bounds on , such as, from kaon scattering experiments at low energies at LAMPF and decays and for mU ≲ 100 MeV, pffiffiffiffiffiffiffiffi LSND [40,41] we got, some time ago, a limit on the j j ≲ 10−3 ð Þ product of couplings gðν ÞgðeÞ ≃ g002Q ðν ÞQ ðeÞ [3,7], Aþ = Binv; 121 L U L U pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ≲ 10−4 −6 and even down to = Binv, depending on mU [39]. jgðνLÞgðeÞj ≲ 3 × 10 mUðMeVÞ Considering also charged decay modes it is reasonable to −3 for m ≳ a few MeVs: ð127Þ retain the estimate jAþj ≲ 10 , corresponding to U 8 −5 −9 This limit, of about 3 × 10 for m ≃ 10 MeV, applies < jg j ≲ 2 × 10 m ðMeVÞ; U At U in particular for a coupling to B − L, which should then 2m ð122Þ −6 : t −3 jg − < 3 10 m ð Þ or jgPtj¼ jgAtj ≲ .7 × 10 : also verify B L × U MeV . It remains rel- mU evant in present discussions of neutrino-electron scattering

035034-13 PIERRE FAYET PHYS. REV. D 103, 035034 (2021) experiments (including from XENON1T [34]), which (should it be due to new physics…), depending also lead to slightly more constraining limits ≲ ð1 to 3Þ× on the values of QUðνLÞ and QUðeÞ [44–46]. −6 10 mUðMeVÞ for mU ≳ a few MeVs (depending on the (2) The U-induced amplitudes in (128) may be reduced experiment considered), down to ≲ ð4 × 10−7 to 3 × 10−6Þ thanks to a small value of the invisibility parameter r g00 for mU ≲ 1 MeV [42–46]. associated with a large singlet v.e.v., with expressed proportionally to mU and r as in (61), 00 B. Three mechanisms for small nonstandard thereby further decreasing the value of g for any neutrino interactions given mU as shown by (126). (3) These amplitudes may also be reduced owing to More precisely, with two BEH doublets U-exchange small or even vanishing values for the U charges of amplitudes between two particles 1 and 2 are found [using neutrinos. This was also initiated in [1], noting that expression (61) of g00 cos ξ] to be proportional to the U charge of neutrinos in (41) would vanish for a þ 2 γ ¼ 1 002 2ξ 2 2 coupling to FA L= corresponding to A , g cos GF mU r Q 1Q 2 ¼ pffiffiffi Q 1Q 2; γ ¼ 1=2, without mixing with Z (γ ¼ η ¼ 0). 8ð 2 − 2Þ U U 2 2 − 2 22β U U L Y mU q mU q sin We can also consider couplings to B, Q, and ↑↑↑ 1 L − 2T3L, or equivalently T3R ¼Q−T3L −2ðB−LÞ, ð1Þð2Þð3Þ which all vanish for νL. In general QU is expressed in (41) as a five-parameter (assuming lepton univer- ð128Þ sality) linear combination of Q, B, L, T3L, and FA, pffiffiffi so that we have for neutrinos to be compared with SM amplitudes, ∝ G = 2. This F expression plays an essential role to ensure that these γ γ η ðν Þ ≃ 00 ξ L − A þ ð Þ nonstandard neutrino interactions are sufficiently small. g L g cos 2 4 2 : 130 Beyond the choice of a small extra-Uð1Þ gauge coupling, 00 with g naturally proportional to mU as seen above,pffiffiffi having Thus this expression small as compared with GF= 2 may be realized using one or the other of three mechanisms gðν Þ¼0 ⇔ Q ¼ linear combination of (possibly combined) acting on the factors (1), (2), or (3) L U ð131Þ þ 2 in (128) [1]: pffiffiffiffiffiffiffiffi Q;B;T3R; and L FA: 2 (1) A sufficiently small mU compared to jq j in the experiment considered, which motivated very early Some of these operators may be replaced by other 2 j 2j a light or even very light U boson, so that mU= q is combinations like FA þ T3L or L − 2T3L, ….Ifwe small. For the scattering of low-energy neutrinos on exclude FA a vanishing coupling to neutrinos simply electrons at rest, this means,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for example, favoring corresponds to having QU as a linear combination of 2 values of mU smaller than meErecoil. Q, B, and T3R. In view of the recent interest for interpreting the An early example of such a situation is obtained for an possible excess of events with electronic recoil axial current combined with the Z current as in (22) [11], energies of a few keVs observed by XENON1T with γA ¼ 1, η ¼ − cos 2β, so that ≈ 2 5 [34], this would mean, with Erecoil . keV, favoring values of m ≲ 50 keV. Such a light U would 1 1 U gðν Þ¼g00 cos ξ − − cos 2β ; ð132Þ also make U effects generally too small to be L 4 2 detected in experiments performed at a higher q2, 2 ð− 2Þ 2β ¼ −1 2 β ≃ π 3 η ¼ 1 2 thanks to the small mU= q factor in (128). vanishing for cos = .For = , = , For mU ≈ 50 keV the central value of the gauge 00 ≈ 4 10−7 1 coupling g in (125), (126) is × , so that ¼ ð þ − 2θ Þ “ ” ð Þ QU 2 FA T3L sin Q is neutrinophobic: 133 00 −7 mU ≲ 50 keV with g ≈ 4 × 10 ⇒ g002 G The possibility of a small or even vanishing QU is some- ≈ ð.5 to 1Þ pFffiffiffi ð129Þ thing to keep in mind when discussing nonstandard 8ðm2 − q2Þ 2 U neutrino interactions and interpreting possible excesses, ≈ 2 5 especially if it turns out that there are no such excesses. for Erecoil . keV, allowing for a comparison with standard solar neutrino interaction amplitudes. Not π0 → γ very surprisingly when dealing with corrections to C. U decays and possible protophobia weak-interaction cross sections, this is what may be The singlet positronium state (parapositronium), with needed for a possible interpretation of the excess C ¼þ and decaying into γγ, can also decay into γU

035034-14 U BOSON INTERPOLATING BETWEEN A GENERALIZED … PHYS. REV. D 103, 035034 (2021) through the vector coupling of the electron but not the axial 1 Q ≃ ðF þT3 −Q=4Þ neutrino-and protophobic; ð140Þ one, with a branching ratio U 2 A L 2 in the sense indicated above. Still our purpose is not to 1 gVe Bð S0 Ps → γUÞ ≃ 2 ; ð134Þ U e emphasize a uncoupled or little coupled to neutrinos or protons to avoid experimental constraints; nor to overtry 0 interpreting possible excesses as signs of a new boson, but in the small m limit [1]. In a similar way the π can decay U rather to identify regions, in a multidimensional parameter into γU through the vector couplings of the U to u and d space ðg00;m ; γ ; γ ; γ ; γ ; η; …Þ, for which various quarks, with a branching ratio [47] U Y B Li A experiments may be sensitive, or not, to a U boson, 2g þ g 2 g ðpÞ 2 depending on its properties. Bðπ0 → γUÞ ≃ 2 Vu Vd ¼ 2 V : ð135Þ e e D. A possible U boson near 17 MeV ?? Axial couplings do not contribute, the axial quark current The example of a ≈ 10 MeV boson mediating the being a C ¼þoperator, as indicated in (50). annihilations of light dark matter particles ≈ 4 MeV was The π0 → γU decay rate gets suppressed for small given in [3], as an illustration of how the Hut-Lee-Weinberg ð Þ ≃ 00ð Þ ð Þ gV p g QU V p . This may be obtained with a small lower mass bound of a few GeVs [5,6] could be circum- value of the gauge coupling g00, naturally proportional to vented thanks to a new interaction, effectively stronger than mU as we saw. This may also be obtained with a suppressed weak interactions but only at lower energies. Long before value of the U charge of the proton, as naturally obtained 2015, the Atomki group has been looking in 8Be transitions long ago [11] as a result of Z − U mixing effects, with a U for such a light U boson decaying into eþe− [51]. Possible ð Þ ¼ η ð Þ charge QU V QZ V in (43), (47), (48), where signs that might be interpreted as a neutral isoscalar axial- vector U boson with Jπ ¼ 1þ already showed up in 2012 at ð Þ ¼ð Þ − 2θ ≈ 3σ 5σ QZ V T3L V sin Q [52], then at more than in 2015, with a mass of 2 1 1 16.7 MeV=c [53]. Complementary indications appeared ¼ − 2θ − ð − Þ ð Þ 4 2 sin Q 4 B L : 136 recently in He transitions [54]. A light U with vector couplings [2,12], taken as π0 →γ Then protophobic to satisfy U decay constraints [49,50], has been advocated as a possible interpretation of these ð Þ ð Þ events [55,56], also subject to the necessity of simulta- gV p ¼ QWeak p ≃ 4 2θ − 1 ≃−0 07 ð Þ ð Þ ð Þ sin . ; 137 neously avoiding too large effects in neutrino scatterings, gV n QWeak n and other requirements and constraints [57], as from NA64 [58]. Other interpretations involving axial couplings have using the recent values of p and n weak charges [48]. The “ ” been discussed, as e.g., in [59]. The possibility of a U U boson is then naturally protophobic . Such an inhibition boson with both axial and vector couplings, a naturally π0 → γ of U decays gets increasingly favored by the recent protophobic behavior originating from the mixing with the 48 2 limits from the NA = experiment [49,50], in particular, Z [11], and even reduced couplings to neutrinos as in (132), requiring (133), (140), may also be noted. Still more experimental information is desirable before going too far in discussing jg ðpÞj ¼ g00 cos ξ jðQ Þ ðpÞj ¼ jϵ ej V pUffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV Vp interpretations for these results, which have not been ≲ 3 × 10−4= BðU → eþe−Þ: ð138Þ confirmed independently yet.

Equation (137) for gVðpÞ is valid when the vector part in E. Spin-dependent forces, EP violations, the U current mostly originates from the mixing with and a possible link with inflation the Z as in (22) or (24) (including the very specific case of a Other possible manifestations of a spin-1 U boson ð Þ ¼ð1 − 2θÞ − 1 ð − Þ pure dark Z), with QZ V 2 sin Q 4 B L in include a CP-conserving dipole-dipole interaction [11] (48), (136) approaching the protophobic combination of the same type as for an axion [60], thanks to ½Q − ðB − LÞ=4 for sin2 θ close to 1=4. Eq. (13), and a CP-violating monopole-dipole interaction We note that the U charge in (133), corresponding to leading to a mass-spin coupling [21,61], both originating from an axial part in the U couplings. 00 μ g cos ξ μ μ μ A very to extremely light (or even massless) U boson J ¼ ðJ þ J − 2θ J ÞðÞ U 2 A 3L sin em 139 with extremely small couplings to a combination of B, L, and Q could lead to a new long-range force. By adding is, for sin2 θ close to 1=4, its effects to those of gravitation, it could be detected

035034-15 PIERRE FAYET PHYS. REV. D 103, 035034 (2021) through apparent violations of the Equivalence Principle electromagnetic current as in (19)–(21), as a linear combi- [2,11,62,63]. This constrains its gauge coupling to be nation (21) of the B, Li, and electromagnetic currents with a extremely small, g00 ≲ 10−24 [64]. possible dark matter current [2,12]. The extreme smallness of this gauge coupling may be There are additional possibilities with two or more BEH related, within supersymmetric theories, with a very large doublets. We often concentrate on the remarkable situation 00 00 extra-Uð1ÞpAbelianffiffiffiffiffi ξ D term [13,65]pffiffiffiffiffi, where the scale for which two doublets, taken as parameter ξ00 is proportional to 1= g00. It corresponds to h0 hþ a huge vacuum energy density that may be at the origin of ¼ 1 ¼ 2 ð Þ h1 − ;h2 0 ; 143 the very fast inflation of the early Universe, in connection h1 h2 with an extremely weak new long-range force, as already pointed out in [66]. The strength of the new force may give masses to charged leptons and bottom quarks, and up then be related with the inflation scale Λ through a quarks, respectively, through the usual trilinear Yukawa relation like couplings as in supersymmetric extensions of the standard model, allowing for these theories to be made supersym- m 2 metric if desired [13]. These couplings, proportional to g00 ≈ sparticle ; ð141Þ Λ h1eRlL, h1dRqL, and h2uRqL (þh:c:), originate within supersymmetry from the trilinear superpotential leading to g00 of the order of ∼10−24 for sparticle masses in 15 16 W ¼ λ ¯ þ λ ¯ þ λ ¯ ð Þ the ∼1 to 10 TeV range with Λ ∼ 10 to 10 GeV [64]. eH1EL eH1DQ uH2UQ: 144 We shall now discuss in Sec. VI the general expression of the F symmetry generator which may be gauged, in Their gauge invariance requires connection with the Brout-Englert-Higgs sector responsible 8 Fðh1ÞþFðl Þ − Fðe Þ¼0; for gauge symmetry breaking. In Sec. VII we return to the < L R Fðh1ÞþFðq Þ − Fðd Þ¼0; ð145Þ simple case of F as an axial generator FA as initially : L R ð Þþ ð Þ − ð Þ¼0 motivated by the 2-BEH structure of supersymmetric F h2 F qL F uR : theories, recalling how a longitudinal U boson behaves ð1Þ much as an axionlike particle. This led us to make it very This allows for the gauging of an axial U A symmetry, weakly interacting as in (14) thanks to a “dark Higgs” with singlet σ, with the extra Uð1Þ broken at a higher scale. We FAðh1Þ¼FAðh2Þ¼1; discuss in Sec. VIII the mass of the U boson and its mixing ð146Þ ð Þ¼− 1 ð Þ¼1 with the Z occurring for v2 ≠ v1, leading to the U current FA qL;lL 2 ;FA qR;eR 2 ; (23). We then extend these results to an extra-Uð1Þ generated by an arbitrary linear combination of the axial as in supersymmetric extensions of the standard model known as the USSM, with the μH2H1 mass term replaced generator FA with Y, B, and Li, plus a dark matter 0 λH2H1S S contribution, and a possible γ 0 F term acting on semi- by a trilinear coupling with a singlet superfield F ¼ −2 inert BEH doublets uncoupled to quarks and leptons. We having FA [13,67]. ð1Þ finally deal with grand unification in Sec. IX. This gauging of an extra-U may be done as well for Fðh1Þ ≠ Fðh2Þ, i.e., with F extended to a linear combination of F and Y, γ F þ γ Y, possibly combined with B VI. WHICH EXTRA-Uð1Þ SYMMETRY? A A A Y and Li. Indeed choosing in a supersymmetric extension of 0 ð1Þ A. A combination of FA; Y; B; Li; F , and Fdark the standard model with an extra-U gauge group [13] ð Þ¼ Which Uð1Þ symmetry generators may be gauged is equal values of F for the two BEH doublets, F h1 ð Þ¼1 restricted by the invariance of the Yukawa couplings F h2 , was only considered as a simplifying ð1Þ μ responsible for quark and lepton masses, and depends assumption, so that the extra-U gauge field C be on the number and characteristics of Brout-Englert-Higgs axially coupled to quarks and leptons. ; ; doublets, coupled or uncoupled to quarks and leptons. With In fact, the seven values of F for (h1, h2 qL, lL uR, dR, a single doublet, responsible for their masses as in the SM, eR), for a single generation, related by the three Eqs. (145), any extra-Uð1Þ generator F must act on quarks and leptons depend on four parameters and may be expressed as ¼ γ þ γ þ γ þ γ as a combination of the weak hypercharge Y with baryon F AFA BB LL YY. Equations (145), extended and lepton numbers B and L , to the three families and taking into account CKM mixing i angles between quarks, imply that the gauged quantum ¼ γ þ γ þ γ þ γ ð Þ numbers of chiral quark and lepton fields, and of the two F BB Li Li YY dFd: 142 doublets h1 and h2, may be expressed as The U current is then obtained in the small mass limit, F ¼ γ F þ γ B þ γ L þ γ Y; ð147Þ after mixing with the Z current reconstructing the A A B Li i Y

035034-16 U BOSON INTERPOLATING BETWEEN A GENERALIZED … PHYS. REV. D 103, 035034 (2021) with One might go even further by generating quark and lepton masses through unconventional dimension ≥ 5 multilinear Fðh1ÞþFðh2Þ Fðh2Þ − Fðh1Þ γ ¼ ; γ ¼ : ð148Þ Yukawa couplings involving additional singlets charged A 2 Y 2 ð1Þ under U F, so that Eqs. (145) would no longer have to be ð ; Þ This includes in particular couplings to satisfied. The F values for h1;h2; qL;lL uR;dR;eR , and, subsequently, the resulting values for the gA’s and gV ’sof Y B − L 3B þ L quarks and leptons would then become unrelated parame- T3 ¼ − ; and=or þ F ; ð149Þ R 2 2 2 A ters. There is also the possibility of flavor-changing neutral which act only on right-handed quark and lepton fields. current couplings of the U boson, however a potential source Even in these cases the resulting U current, obtained as a of difficulties. Although we shall restrict this study to the F ð1Þ generator in (150) for simplicity, it can be extended to more combination of the extra-U F and Z currents, in general involves left-handed quarks and leptons as well through the general situations. This is usually done, however, at the price ðγ þ ηÞ μ of motivation, elegance, and predictivity, with full general- term proportional to Y JZ in Eq. (34), unless this one is absent as in (39), (40). ity implying a total loss of predictivity. We can still add extra terms to expression (147) of F, provided they vanish for quarks, leptons and the BEH C. An extra singlet contributing to mU doublets responsible for their masses. This leads to We also introduce a neutral singlet σ, intended to provide 0 a small mass for the U boson if the electroweak breaking is F ¼ γ F þ γ B þ γ L þ γ Y þ γ 0 F þ γ F : ð150Þ A A B Li i Y F d d induced by a single BEH doublet (or two or more but with Fd acts only on particles and fields in a dark or hidden the same gauge quantum numbers) [2,12]; or to provide an sector. F0 may act on semi-inert BEH doublets h0, extra contribution to its mass as in (59) if the axial generator uncoupled to SM quarks and leptons but contributing to FA contributes to F, to make the longitudinal degree of the mixing with the Z as in Eqs. (18), (30)–(33). freedom of a light U with axial couplings sufficiently We may even consider identifying Fd ¼ γRR with the weakly interacting. This gauging involving FA, possibly in generator of continuous R-symmetry transformations—the combination with Y, originates from the supersymmetric R progenitor of R-parity Rp ¼ð−1Þ in the supersymmetric standard model with a singlet superfield S coupled through λ ¼ 1 standard model. Indeed F, including its possible R con- a H2H1S superpotential, with FA for H1 and H2 and ¼ −2 tribution, is generally intended to be spontaneously broken FA for the singlet S [13,67]. One may also consider S so that the U acquires a mass, then also allowing for the without coupling it to H1 and H2. Independently of ð1Þ R-carrying gluinos and gravitino to acquire U R breaking supersymmetry, one may consider a dimension-3 coupling 2 masses, after spontaneous supersymmetry breaking. R does ∝ h2h1σ with FAσ ¼ −2, or a quartic coupling ∝ h2h1σ not act on SM particles but on their superpartners, including with FAσ ¼ −1, or more generally consider Fσ as an neutralinos and gravitinos which are potential dark matter arbitrary parameter. candidates, then coupled to the U boson which may act as We express the doublet and singlet v.e.v.’sas the mediator of their annihilations. ( pffiffiffi pffiffiffi 0 0 hh1i¼v1= 2; hh2i¼v2= 2; B. Further possibilities for an extra Uð1Þ pffiffiffi ð152Þ F hσi¼w= 2; One can generalize further expression (150) of F, still using conventional dimension-4 trilinear Yukawa cou- with plings, by considering that up quarks, down quarks, and the three charged leptons may acquire their masses from up v1 ¼ v cos β;v2 ¼ v sin β; ð153Þ to five different spin-0 BEH doublets (still allowing to avoid unwanted flavor-changing neutral current effects). −1 2 β¼1 ¼ ¼ 2−1=4 = ≃ 246 where tan =x v2=v1,andv GF GeV. This would allow one to replace the single axial generator ð2Þ ð1Þ ð1Þ F acting universally on quarks and leptons as in (146) by The SU × U Y × extra-U F covariant derivative is A given by up to four different ones FAq and FAli acting separately on quarks, and on leptons of the three families, leading to 0 00 ¼ ∂ − þ g þ g ð Þ ¼ γ þ γ þ γ þ γ þ γ iDμ i μ g T:Wμ 2 YBμ 2 FCμ : 154 F AFAq Ali FAli BB Li Li YY 0 þ γ 0 þ γ ð Þ F F dFd: 151 μν No “kinetic-mixing” term proportional to BμνC needs to be The resulting axial couplings are modified accordingly, considered, as it may be eliminated by returning to an those of down quarks and charged leptons being no longer orthogonal basis, with Cμ redefined as orthogonal to Bμ. systematically equal as in (53), (57), (66). The Cμ coupling involves in general a contribution from the

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μ g00v g00 g00 weak hypercharge current JY, already taken into account in pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mU ¼ ¼ mW ¼ mZ; ð157Þ expressions (142) or (150) of F. 2 g g2 þ g02

D. About anomalies so that There are additional constraints from the requirement of anomaly cancellation. They allow for the gauging of Y, G g2 g2 þ g02 g002 pFffiffiffi ¼ ¼ ¼ : ð Þ B − L, and L − L , in the presence of right-handed 8 2 8 2 8 2 158 i j 2 mW mZ mU neutrino fields νR. We may also consider axial generators such as FA in (146), (150), assuming anomalies to be The neutral current effects of the U, for example in neutrino canceled, e.g., by mirror fermions, or exotic fermions scattering experiments, are then generally too large if the U similar to those in 27 representations of Eð6Þ (decomposing is heavy. If it is light, however, neutral current amplitudes, ¯ ¯ into 16 þ 10 þ 1 ¼ 10 þ 5 þ 1 þ 5 þ 5 þ 1), or using proportional to other mechanisms. In particular one may consider the ð6Þ 002 2 002 same neutral currents as in E , associated with T3L;Y, g G m g ½ − 5 ð − Þ − ¼ pFffiffiffi U ≃− for m2 ≪ jq2j; ð159Þ Y 2 B L , and FA, and those associated with Li Lj. 8ðm2 − q2Þ 2m2 − q2 8q2 U This leads to U U 00 0 can be sufficiently small if g is small enough, with the U F ¼ γ F þ γ − ðB − LÞþγ Y þ γ 0 F þ γ F 2 A A B L Y F d d light as compared to jq j in the experiment considered, þ δeðLe − LτÞþδμðLμ − LτÞ: ð155Þ providing an early motivation for making a new neutral gauge boson light (cf. Sec. III E). 00 It correspondsP to the earlier expression (150) of F with The gauge coupling g and axial couplings to quarks and γ ¼ −3γ the relation i Li B. Still we intend to perform the charged leptons may then be expressed proportionally to present study in a way as general as possible, irrespectively mU, reading in this first simple situation as of the constraints from anomaly cancellation, which may 00 g g m m 1 2 m involve presently unknown fermion fields; or, otherwise, ¼ ¼ U ¼ U ¼ 21=4 = U gA GF one may also choose to exclude the axial generator FA from 4 4 mW 2v 2 the gauging by taking γA ¼ 0. −6 ≃ 2 × 10 mUðMeVÞ. ð160Þ VII. STARTING WITH AN AXIAL U This benchmark value appears in many discussions of a A. ν scatterings and beam dump experiments new light gauge boson as a mediator of nonstandard We now return for a short time to the simple gauging ofp theffiffiffi neutrino interactions, or produced in meson decays or beam dump experiments, or contributing to atomic-physics axial generator F ¼ FA in (146), choosing v1 ¼ v2 ¼ v= 2 so that the extra Uð1Þ gauge field Cμ does not mix with the parity violation, etc. Its axial couplings are then expressed μ μ μ as in (66) as a fraction of this benchmark value, according to standard model one Z∘ ¼ cos θ W3 − sin θ B . This results μ μ in an axially coupled Uμ ≡ Cμ,withJ ¼ðg00=2ÞJ as from U A ≃ 2 10−6 ð Þ ð Þ (19). The gauging of this extra Uð1Þ, allowing to rotate gA × mU MeV A: 161 independently h1 and h2 in supersymmetric theories, was originally done in connection with spontaneous supersym- Such a U boson axially coupled with full strength metry breaking, triggered by the extra-Uð1Þ ξ00 D00 term [13]. as in (160), i.e., with A ¼ 1, was constrained long This already provided a possible motivation for a very small ago to be heavier than 7 MeV to be sufficiently short- coupling g00 and a large supersymmetry-breaking scale from lived in beam dump experiments [1], with a decay rate 00 00 Γ ∝ 002 ∝ 3 ¼ βγ τ ∝ a very large ξ parameter inversely proportional to g , g mU mU, and a decay length l c 002 00−2 −2 ∝ −4 associated with a huge vacuum energy density ∝ 1=g . g mU mU (cf. Sec. III E). It had also to be lighter The left-handed and right-handed projectors ð1 − γ5Þ=2 than 300 MeV, to sufficiently benefit from propagator ð1 þ γ Þ 2 ¼ −1 2 þ1 2 2 and 5 = are combined with FA = and = effects in neutrino-electron scattering experiments for mU for the chiral quark and lepton fields, respectively, as in smaller than the relevant jq2j under consideration at the 00 (146). The U current [including g =2 from the covariant time. This left, forty years ago, a window of opportunity derivative in (154)] is expressed for quarks and leptons as between 7 and 300 MeV for an axially coupled boson, in the absence of an extra singlet allowing for jAj to be 00 00 X J μ ¼ g μ ¼ g −ν¯ γμν þ ¯γμγ ð Þ significantly smaller than 1. This early example illustrates U 2 JA 4 L L f 5f : 156 ude the complementary roles of neutrino scatterings at lower jq2j, and beam dump experiments, in the search for new Without extra singlet v.e.v. the U mass is, for v1 ¼ v2, light gauge bosons.

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B. The need for an invisible axionlike behavior C. “Invisibility” and high scale symmetry or The longitudinal degree of freedom of an ultrarelativistic supersymmetry breaking pffiffiffi U boson with axial couplings would behave much as an This mechanism relying on a large singlet v.e.v. w= 2 axionlike particle, with effective pseudoscalar couplings to making the U boson almost “invisible” also led us 00 “ ” quarks and leptons given by Eq. (13). With gA ¼ g =4 as in independently [1] to the invisible axion mechanism (160) they would read, for equal doublet v.e.v.’s, [24,25] for an anomalous global Uð1Þ. We had already used this property of a very light gauge 2m ¼ ql ¼ 21=4 1=2 ≃ 4 10−6 ð Þ boson behaving as a Goldstone boson (as in the standard gPql gA GF mql × mql MeV ; 0 → 0 → 0 mU model in the limit g, g , mW, mZ with v fixed), and interacting with amplitudes inversely proportional to ð162Þ the symmetry breaking scale. It provided a laboratory to establish this equivalence property for supersymmetric as reconstructing the pseudoscalar couplings of a standard well as for gauge theories, with, furthermore, the gauge axion A, for v2 ¼ v1 [22,23]. particle, either fermionic or bosonic, interacting quasi- We thus considered very early a large singlet v.e.v. pffiffiffi “invisibly” if the supersymmetry-breaking or gauge- hσi¼ 2 w= , increasing mU to symmetry breaking scale is large [68]. 3 2 00 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 00 Avery light spin- = gravitino then behaves very much as g 2 2 2 g mW the corresponding (“eaten-away”) spin-1=2 goldstino. It can mU ¼ v þ Fσw ¼ ; ð163Þ 2 g r also be made almost “invisible”, i.e., with very reduced interactions ∝ 1=d or 1=F, thanks to a very large super- defining the invisibility parameter [1] symmetry-breaking scale parameter (d or F), i.e., a very large auxiliary-field v.e.v. within an extra-Uð1Þ, dark, or v ¼ θ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ≤ 1 ð Þ hidden sector. This also provides an early decoupling r cos A 2 2 2 ; 164 v þ Fσw of a very weakly interacting gravitino. For an extra-Uð1Þ this is associated with a very small gauge coupling g00 (and the equivalent pseudoscalar a in (14) being mostly an effective couplings of the equivalent goldstino) [13,64,66]. ¼ θ θ ¼ electroweak singlet for small r cos A, with tan A Supersymmetry, spontaneously broken “at a high scale” in a jFσjw=v. 00 hidden sector, gets softly broken in the visible one. The axial coupling gA ¼ g =4 in (156) gets expressed proportionately to mU and r, with VIII. U MASS AND MIXING ANGLE WITH Z 00 ¼ g ¼ g mU ≃ 2 10−6 ð Þ ð Þ A. U as an axial boson mixed with the Z gA 4 4 r × rmU MeV : 165 mW We now discuss more precisely mixing effects between μ μ the extra-Uð1Þ gauge field C and the Z∘ of the standard The effective pseudoscalar couplings (13) of a longi- model, when the two doublets have different v.e.v.’s so that U tudinal , the U current picks up an extra term proportional to μ μ μ J ¼ J − 2θ J ðv2 − v2Þ=v2 ¼ 2m Z 3L sin em, proportionally to 2 1 ¼ ql ¼ 21=4 1=2 − 2β gPql gA GF rmql cos , as expressed in Eqs. (19), (23), (29). We first mU consider the gauging of the axial generator FA, with h1 and ≃ 4 10−6 ð Þ ð Þ × rmql MeV ; 166 h2 having Y ¼∓ 1, and F ¼ FA ¼ 1, and a singlet σ (now often referred to as a “dark Higgs” field). When h1 and h2 μ μ get proportional to r ¼ cos θA, in agreement with (14).For acquire different v.e.v.’s C mixes with the Z∘ of the 2 agivenmU, the amplitudes involving a U, proportional to standard model through the mass terms gV , gA,orgP, and thus to r, are sufficiently small if r is small enough, i.e., jFσjw large enough compared to the ð μ þ 00 μÞ2 2 ð− μ þ 00 μÞ2 2 electroweak scale. The U decay rate gets reduced, leading L ¼ gZZ∘ g C v1 þ gZZ∘ g C v2 to a lifetime and decay length increased by a factor 1=r2, m 8 8 002 2 2 00−2 −1 −3 −2 g Fσw μ τ ∝ g m ∝ m r ; þ Cμ C ; ð168Þ U U ð167Þ 8 ¼ βγ τ ∝ 00−2 −2 ∝ −4 −2 l c g mU mU r ; 2 ¼ 2 þ 02 μ allowing for a sufficiently weakly coupled U boson of any with gZ g g . This includes a pCffiffiffi mass term 00 mass to satisfy direct experimental constraints, including mC ¼ g jFσj w=2, induced by hσi¼w= 2, and leads to from beam dump experiments. the mass2 matrix [11]

035034-19 PIERRE FAYET PHYS. REV. D 103, 035034 (2021) 1 g2 ðv2 þ v2Þ −g g00ðv2 − v2Þ or M2 ¼ Z 1 2 Z 2 1 4 − 00ð 2 − 2Þ 002ð 2 þ 2 þ 2 2Þ 00 gZg v2 v1 g v1 v2 Fσw g cos ξ v sin 2β ! m ¼ : ð178Þ g2 g g00 cos2β U 2 r v2 Z Z ¼ 2 2 Fσw : We have defined, as in (164) for β ¼ π=4, the invisibility 4 g g00 cos2β g002 cos22β þ sin22β þ Z v2 parameter ð Þ 169 v sin 2β r ¼ cos θ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð179Þ A 2 2 2β þ 2 2 Its mass eigenstates are v sin Fσw θ ¼ðj j Þ ð 2βÞ 00 μ μ μ with tan A Fσ w = v sin . g can be expressed Z ¼ cos ξ Z∘ − sin ξ C ; ð Þ proportionally to mU and r as in (61), with μ μ μ 170 U ¼ sin ξ Z∘ þ cos ξ C ; 00 ξ with g cos −6 r ≃ 2 × 10 m ðMeVÞ ; ð180Þ 4 U sin 2β μ μ μ Z∘ ¼ cos θ W − sin θ B ; 3 ð Þ μ μ μ 171 A ¼ sin θ W3 þ cos θ B ; where r is small when jFσjw is large compared to v sin 2β. 00 4 ≃ 2 10−6 as in the standard model, so that g = × mU (MeV) occurs as the natural benchmark value for the extra-Uð1Þ gauge coupling, as seen for 0 1 0 10 1 μ μ nonstandard neutrino interactions in Secs. VA and VB. Z cξcθ −cξsθ −sξ W3 B μ C B CB μ C Altogether we have in this case @ A A ¼ @ sθ cθ 0 A@ B A: ð172Þ 00 μ μ 8 sξcθ −sξsθ cξ g U C > tan ξ ≃−pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos 2β; > 2 þ 02 > g g The Z − U mixing angle ξ is given by > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 þ 02 þ 002 22β > g g g cos v mZ∘ < m ≃ ≃ ; 2 00ð 2 − 2Þ Z 2 cos ξ gZg v2 v1 ð181Þ tan 2ξ ¼ : ð173Þ > 00 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 2 − 002Þð 2 þ 2Þ − 002 2 2 > g cos ξ gZ g v1 v2 g Fσw > ≃ 22β 2 þ 2 2 > mU sin v Fσw ; > 2 > In the small mU limit and considering the contribution from :> v sin 2β r ¼ θ ≃ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : the last two terms in the CC matrix element in (169) as a cos A 2 2 2 2 v sin 2β þ Fσw perturbation, ξ is approximately given by [11] Equations (61), (180) determining the gauge coupling g00 g00 v2 − v2 g00 ¼ θ ξ ≃ 2 1 ¼ ð− 2βÞ ð Þ in terms of mU, the invisibility parameter r cos A and tan 2 2 cos : 174 ’ β þ gZ v2 þ v1 gZ the ratio of spin-0 v.e.v. s tan , in a two-doublet one- singlet model, remain valid under more general circum- This is a special case of the general formula (18) for any stances, through essentially the same calculation. set of BEH doublets, referred to as φi with Y ¼þ1,or equivalently h , with Y ¼ ϵ ¼1 [2], ð Þ i i B. General case: U as an extra U 1 F gauge boson mixed with Z 00 X 2 00 X 2 ξ ≃ g ðφ Þ vi ¼ g ϵ ð Þ vi ð Þ ð1Þ tan F i 2 iF hi 2 : 175 The extra-U generator F may involve as in (150),in gZ v gZ v i i addition to the axial FA, terms proportional to Y, B, and Li, 0 a (light) dark matter contribution γdFd, and a γF0 F term Taking the last two terms in the CC matrix element of acting on semi-inert doublets uncoupled to quarks and 2 M as a perturbation, we identify leptons. There is no need to consider a kinetic-mixing term μν μ ∝ BμνC , immediately eliminated by redefining C as ð 2 þ 02Þ 2 ð 2 þ 02 þ 002 22βÞ 2 μ 2 ≃ g g v ¼ g g g cos v ð Þ orthogonal to B . mZ 4 2ξ 4 ; 176 cos For h1 and h2 taken with Y ¼ ∓1 and F ¼ Fi, and a singlet σ with F ¼ Fσ, the mixing angle ξ is given in the then get, from these last two terms, small g00 limit by (18),

002 2 00 2 g cos ξ 2 2 2 2 g 2 2 ≃ ð 2β þ ÞðÞ tan ξ ≃ ðF2sin β − F1cos βÞ: ð182Þ mU v sin Fσw 177 4 gZ

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σ The quantum numbers of h1, h2, and are g00 tan ξ ≃ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðγ − γ cos 2βÞ; 2 þ 02 Y A g g ¼ γ − γ F1 A Y; g2 þ g02 Fσ ¼ γ F σ þ γ F σ: ð183Þ 2ξ ≃ ¼ γ þ γ A A d d cos 2 02 002 2 ; F2 A Y; g þ g þ g ðγ − γ cos 2βÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiY A g2 þ g02 þ g002ðγ − γ cos 2βÞ2v m ≃ Y A ≃ Z∘ σ ð1Þ mZ ; This allows for to transform under U A and/or under a 2 cos ξ ð1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dark U d symmetry generated by Fd. (The calculation, g00 cos ξ ¼ γ þ γ ¼ γ ∓ γ c ≃ γ2 22β 2 þ 2 2 formulated with F AFA YY Y A for h1 and mU 2 Asin v Fσw ; h2, applies as well for a SM-like doublet hsm and a semi- 00 0 0 ξ 2β ¼ γ þ γ 0 γ γ þ 2 g cos v sin inert one h , with F YY F F taken as Y and Y , ≃ jγAj ; respectively.) 2 r jγ j 2β The mass terms (168) read A v sin r ¼ cos θ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : A γ2 2 22β þ 2 2 Av sin Fσw 1 2 L ¼ ½ μ − 00ðγ − γ 2βÞ μ2 v ð189Þ m 2 gZZ∘ g Y A cos C 4 For γ ≠ 0 and normalizing g00 to γ ¼ 1, this reduces 1 002 A A g 2 2 2 2 2 μ for γ ¼ 0 to (181), the U being coupled as in (156) þ ðγ sin 2β v þ Fσw Þ CμC ; ð184Þ Y 2 4 A to an axial current, plus a contribution induced by the mixing with Z, for tan β ≠ 1.Forβ ¼ π=4 there is no mixing ¼ 2 ¼ and the mixing angle ξ is given in the small mass limit by andpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi we recover mZ gZv= as in the SM and mU 00 2 2 2 00 (182), which reads g v þ Fσw =2 ¼ g v=ð2rÞ as in (163). In the other limit sin 2β → 0, e.g., through a small v1 with β ≃ π=2 and 2β ≃−1 00 cos ,p weffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi approach a one-doublet-v.e.v. situation g ≃ 2 þ 02 þ 002 2 ≃ 00 ξ j j 2 tan ξ ≃ ðγY − γA cos 2βÞ: ð185Þ with mZ g g g v= , mU g cos Fσ w= , gZ and vector couplings of the U to massive up quarks. 00 For γA ≠ 0 and normalizing g to γA ¼ 1, we have again 2 as in (180), independently of γ , γ , and γ , The first mass term in (184) leads to Y B Li 00 ξ g cos ≃ 2 10−6 ð Þ r ð Þ 4 × mU MeV 2β ; 190 2 þ 02 sin 2 ≃ g g 2 m 2 v η ¼ − 2β Z 4cos ξ with cos . 2 þ 02 þ 002ðγ − γ 2βÞ2 When FA does not participate in the gauging so that g g g Y A cos 2 γ ¼ 0 00 γ ¼ 1 ≃ v : ð186Þ A , and normalizing g to Y , the U would remain 4 massless without the singlet σ. Then we have 8 g00 μ μ μ 2 > tan ξ ≃ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ∝ CμC ¼ð ξ U − ξ Z Þ > 2 02 The second one, cos sin and > g þ g considered as a perturbation, leads to <> pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g2 þ g02 þ g002 v ð Þ > m ≃ ; 191 > Z 2 002 2 > g cos ξ > 00 2 2 2 2 2 2 : g cos ξ jFσj w m ≃ ðγ sin 2β v þ Fσw Þ: ð187Þ ≃ U 4 A mU 2 ; the 3 × 3 mixing matrix (172) reducing to [2,12] It is independent of γY, and reduces to (177) for γA ¼ 1.We 8 0 00 > gW3 − g B − g C define as in (14), (179) r ¼ cos θ , given by > Z ≃ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; A > 2 þ 02 þ 002 > g g g <> 0 g W3 þ gB jFσj w A ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð Þ θ ¼ ð Þ 2 02 192 tan A : 188 > g þ g jγ j sin 2β v > A > 00 0 2 02 > g ðgW3 − g BÞþðg þ g ÞC :> U ≃ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; g2 þ g02 g2 þ g02 þ g002 We summarize these results on the U mass, mixing angle ξ, and invisibility parameter r by with the U vectorially coupled to quarks and leptons.

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C. Exact determinations of mU and ξ leads to The mass2 matrix corresponding to (184), which Π ½ 2 þ 02 þ 002ðγ þ ηÞ2 2 includes the element 2 ≃ Σ − ≃ g g g Y v mZ Σ 4 002 2 2 g 2 2 2 2 2 2 ¼ ½ðγ −γ Þ þðγ þγ Þ þ mZ∘ mCC 4 A Y v1 A Y v2 Fσw ≃ ; ð201Þ cos2ξ 002 2 2 2 g v Fσw ¼ ðγ −γ cos2βÞ2 þγ2 sin22βþ ; ð193Þ 4 Y A A v2 and at next order to reads Π g002v2 m2 ≃ ≃ 2ξ H U Σ − Π cos 4 2 2 − 00ðγ þ ηÞ Σ 2 v gZ gZg Y M ¼ ; ð194Þ 002 2 4 00 002 2 g cos ξ 2 2 2 2 2 −gZg ðγY þ ηÞ g ½ðγY þ ηÞ þ H ≃ ðγ 2β þ Þ ð Þ 4 Asin v Fσw : 202 2 2 γ þ η ¼ F2 β − F1 β ¼ γ − γ 2β where Y sin cos Y A cos , with 2 2 2 It gives back mZ and mU in (186), (187), in which mU could 2 2 be directly obtained as ¼ γ2 2 2β þ Fσw ð Þ H A sin 2 : 195 v 2 2 ðm Þ g002v2 m2 ≃ cos2ξ m2 − Z∘C ≃ cos2ξ H: ð203Þ m2 and m2 are obtained from their product and sum, U CC 2 4 Z U mZ∘Z∘ ðg2 þ g02Þv2 g002v2 Π ¼ H; There is no need to consider further corrections if the U is 4 4 Π2 Σ4 ≃ 4 4 light, = mU=mZ being very small. 2 02 002 2 2 c ½g þ g þ g ðγY þ ηÞ þ HÞ v With γ ¼ðF2 þ F1Þ=2 ¼½Fðh2Þ − Fðh Þ=2, this Σ ¼ ; ð196Þ A 1 4 expression of mU reads, in any two-doublet þ singlet c situation where the two doublets (called h and h2,orh and so that 1 h0) have the same Y ¼ 1, rffiffiffiffiffiffiffiffiffiffiffiffiffiffi Σ 4Π sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 ;m2 ¼ 1 1 − ; ð197Þ 00 ξ ð 0Þ − ð Þ 2 Z U 2 Σ2 ≃ g cos F h F h 2 22β þ 2 2 ð Þ mU 2 2 v sin Fσw : 204 also valid for a heavy U. The mixing angle ξ is given by This also illustrates how two doublets with the same F and 00 2gZg ðγY þ ηÞ Y act as a single one, breaking only one neutral gauge 2ξ ¼ ; ð Þ 2 tan 2 − 002 ½ðγ þ ηÞ2 þ 198 symmetry generator, thus providing no contribution to m . gZ g Y H U 0 reducing to (173) for γY ¼ 0; γA ¼ 1, and η ¼ − cos 2β, D. Application to a semi-inert doublet h η ¼ −γ 2β and to (185) for small mU, with A cos . In particular, let us consider a SM-like doublet h with ξ pffiffiffi sm These formulas giving the exact values of mU and in a β0 2 ¼ γ 0 v.e.v. v cos = andpffiffiffi F Y,andasemi-inertoneh general two-doublet + one-singlet situation may also be 0 0 with v.e.v. v sin β = 2, taken with F ¼ 2; γF0 ¼ 1,and applied to not-so-light or even heavy U bosons, that could ¼ γ þ 2 ð 0Þ − ð Þ¼2 þ − F Y . With the same difference F h F hsm be detected through their decays U → μ μ , as in LHCb c as for Fðh2Þ − Fðh1Þ¼2γA ¼ 2 before, mU is still given, for example [69,70]. γ γ γ 2 2 independently of Y, B,and Li ,byEqs.(177), (187), (204), m and m can be developed in the small mU limit, Z U qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi according to 00 ξ ≃ g cos 2 22β0 þ 2 2 8 mU 2 v sin Fσw > Π Π2 <> m2 ¼ Σ − − þ; 00 ξ 2β0 Z Σ Σ3 ¼ g cos v sin ð Þ ð199Þ 2 ; 205 > Π Π2 Π3 r : m2 ¼ þ þ 2 þ: U Σ Σ3 Σ5 leading as before to

A first approximation, g00 cosξ m r r ≃ U ≃ 2 ×10−6 m ðMeVÞ ; ð206Þ 4 2v sin2β0 U sin2β0 Π g002v2 m2 ≃ ≃ H; ð200Þ U Σ 4 just changing β into β0.

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2 All this remains true if we further replace h by h2 and 2 0 2 w sm H ¼ sin 2β þ Fσ : ð213Þ c γ ¼ 0 ¼ 1 ¼ γ 2 h1 (taken for Apffiffiffi with thepffiffiffi same Y and F Y), v with v.e.v.’s v2= 2 and v1= 2 such that This provides ξ givenpffiffiffiffi in the small mass limit by (208), 8 m ≃ ðg00 cos ξ v=2Þ H by (205), and m ≃ g v=ð2 cos ξÞ ¼ β ¼ β β0 U Z Z < v1 u cos v cos cos ; as in (186). ¼ β ¼ β β0 ð Þ : v2 u sin v sin cos ; 207 v0 ¼ v sin β0: IX. THE U CHARGE, GRAND UNIFICATION, AND AN ELECTROSTRONG SYMMETRY η ¼ 2 02 2 ¼ 2 2 β0 ξ In both cases v =v sin , the mixing angle We recall that, as an effect of our assumptions on the being given by BEH sector, the couplings of the U boson considered here, 00 flavor-diagonal, do not contribute directly to proton decay g 2 0 or neutron-antineutron oscillation amplitudes. tan ξ ≃ ðγY þ 2sin β Þ; ð208Þ gZ Further restrictions on the U couplings may be obtained in the framework of grand unification [20], if we demand ≃ ξ and the mass mU by (205), with mZ mZ∘ = cos as in the extra-Uð1Þ generator F to commute with SUð5Þ within (189), (201). a SUð5Þ × Uð1Þ gauge group, so that γ ¼ 0 0 σ F In these situations for which A (with hsm, h and , c 0 or h1, h2, h , and σ), the axial part in the U current, 5 0 F ¼ γ F þ γ Y − ðB − LÞ þ γ 0 F þ γ F : ð Þ isovector, originates from the mixing with the Z, inde- A A Y 2 F d d 214 γ γ γ pendently of Y, B, and Li . The U charge of quarks and leptons is a linear combination of Q, B, Li, and T3L, the The vanishing of the vector coupling of the U to up quarks, γ ¼ 0 2 θ ¼ 3 8 ð4Þ current being given by Eq. (35) with A , i.e., for sin = , is an effect of a SU es electrostrong ð5Þ ð1Þ symmetry group within SU × U F [71], relating the ð ¯ Þ μ 00 2 μ 1 μ μ photon with the eight gluons. uL; uL transforms as an J ≃ g cos ξ γY cos θ Jem þ ðγBJ þ γiJ Þ U 2 B Li electrostrong sextet so that the up quark contributions to the U current should then be axial (or vanish identically in the þ η ð μ − 2θ μ Þ ð Þ small m limit), as for the Z current itself [12]. Indeed for J3L sin Jem : 209 U γA ≠ 0 the antiquintuplet and quintuplet v.e.v.’s hh1i and hh2i break spontaneously [71]: 2 γY cos θ and η provide, respectively, effective measures of the pure dark photon and dark Z contributions to the U hh1i; hh2i current. ð5Þ ð1Þ ⟶ ð4Þ ð Þ SU × U F SU electrostrong; 215 To illustrate explicitly the calculation of mU, adding the 2 c M∘ matrix obtained from h alone, or h and h2, with sm 1 ð ∓1=3 Þ Y ¼ 1 and F ¼ γ , giving masses to the quartet Y ;W and singlet Z and Y U bosons, with the U remaining light for small g00, and the 4=3 ð4Þ 2 2 − 00γ X massless at this stage. SU es, commuting with 2 u gZ gZg Y M∘ ¼ ; ð210Þ Uð1Þ × Uð1Þ , is further broken to SUð3Þ × Uð1Þ 4 − 00γ 002γ2 Z U QCD QED gZg Y g Y by an adjoint 24 or an equivalent mechanism, as within ¼ 1 ¼ 2 0 N or N supersymmetric (or higher-dimensional) that would lead to a massless U, the similar one from h grand unified theories. (with Y ¼ 1 and F ¼ γY þ 2), With a single quintuplet hsm acquiring a v.e.v. (or two or even four as in supersymmetric theories but with the same 02 2 − 00ðγ þ 2Þ 02 v gZ gZg Y gauge quantum numbers) one has with γ ¼ 0 the sym- M ¼ ; ð211Þ A 4 00 002 2 −gZg ðγY þ 2Þ g ðγY þ 2Þ metry breaking pattern hh i and the singlet contribution, we get as in (194) ð5Þ ð1Þ ⟶sm ð4Þ ð1Þ ð Þ SU × U F SU es × U U: 216 2 2 − 00ðγ þ2 2β0Þ U 2 v gZ gZg Y sin The can then stay massless, or acquire a mass, possibly M ¼ ; very small if g00 is very small, from the v.e.v. of the dark 4 −g g00ðγ þ2sin2β0Þ g002½ðγ þ2sin2β0Þ2 þH Z Y Y singlet σ, if present. ð212Þ In all such cases the extra-Uð1Þ and Z currents, and resulting U current obtained from their combination, are with invariant, above the grand unification scale, under the

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ð4Þ SU es electrostrong symmetry group relating the photon This one becomes manifest for the vector couplings for with the eight gluons. It is spontaneously broken into sin2 θ having the grand unification value 3=8 [20]. We then ð3Þ ð1Þ SU × U QED at the grand unification scale, with the have X4=3 and Y1=3 gauge bosons acquiring large masses, Q − 2ðB −LÞ related in the simplest case by [66,71] ð Þ ¼ð5γ þ ηÞ ¼ð5γ þ ηÞ gut ð Þ QU V Y 8 Y QZV: 220 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4Þ m ¼ m2 þ m2 : ð217Þ It is proportional to the SU es invariant combination Y W X ð Þ ¼ appearing in (49), and as the weak charge QZ V 4 QWeak= in (47), (48), (136) [12], This construction also provides a natural solution for the 8 triplet-doublet splitting problem in grand unified theories, > 0 6 ¼ð ¯Þ <> for u; u L; with the doublet mass parameters automatically adjusting Q − 2ðB − LÞ gut ¼ ¼ − 1 for 4 ¼ðd; eþÞ ; to zero so as to allow for the electroweak breaking. This QZV 8 > 8 LþR :> 1 mechanism can be transposed to (or originates from) þ 4 for 1 ¼ νL½þνR: supersymmetric grand unified theories in higher dimen- ð221Þ sions, for which mX ≈ πℏ=Lc [72]. Independently of grand unification, and of the possible 00 It vanishes, as required for a vectorial charge, for the values of g and mU, supersymmetry can provide the ð ¯Þ u; u L sextet (the vector coupling at the grand unification associations [66,71] scale then appearing as “u-phobic”). ð Þ At the same time QU A in (219) involves FA and T3A, ↔2 ð 4Þ ↔1 ð 5Þ 0 ð Þ ð4Þ U or uinos or spin- BEH bosons; 218 both invariant under the SU es electrostrong symmetry group, with 8 all with mass mU as long as supersymmetry is unbroken; > 6 ¼ðu; u¯Þ ; this is also in agreement with gauge-BEH unification, > L 1 < 4 ¼ð þÞ allowing to view BEH bosons as extra spin-0 states for d; e L; FA ¼ − for ð222Þ massive spin-1 gauge bosons. We then have to take into 2 > 4¯ ¼ðd;¯ e−Þ ; :> L account additional superpotential and supersymmetry- 1 ¼ ν ½þν¯ 00 L L ; breaking terms, allowing for very small g and mU even with a very large extra-Uð1Þ ξ00D00 term. (There is also the and further possibility, especially for very small g00 in N ¼ 2 8 00 > − 1 6 ¼ð ¯Þ supersymmetric GUTs, of viewing ξ as a dynamical > 2 for u; u L; > quantity, very large in the first moments of the < þ 1 4 ¼ð þÞ 2 for d; e L; Universe.) Some of these uinos (and associated spin-0 T3 ¼ T3 − T3 ¼ ð223Þ A R L > þ 1 4¯ ¼ð¯ −Þ bosons) may remain lighter than the U through a see-saw > 2 for d; e L; :> 1 mechanism, in the presence of susy-breaking gaugino or − 2 for 1 ¼ νL½þν¯L: Higgsino mass terms. ð Þ ð Þ QU V and QU A in (43), now restricted by the grand This shows how a new U boson, originating from an extra ð1Þ ð5Þ unification and electrostrong symmetries, depend only on U F symmetry commuting with SU and mixed with ð1Þ the three parameters γY, γA, and η, instead of up to seven the Z, is associated with a U U symmetry commuting γ ¼ −γ ¼ 5 γ with a SUð4Þ electrostrong symmetry group associating before. They read, with Li B 2 Y, es the photon with the eight gluons [12,71].ForγY ¼ γB ¼ γ ¼ 0 we still have a situation compatible with grand ∶ Li GUT γ η 8 5 unification with QU depending only on A and as in [11] > 2 η > ðQUÞ ¼ γY cos θ Q − ðB − LÞ (or just on as for a dark Z), its vectorial part becoming > V 4 2 <> almost protophobic for sin θ close to 1=4. 1 1 ð219Þ þ η − 2θ − ð − Þ > 2 sin Q 4 B L ; > X. CONCLUSIONS :> γ η ð Þ ¼ A − The standard model of particle physics is a great QU A FA T3A; 2 2 construction for understanding the world of particles and interactions, but it leaves many questions unanswered; and ð Þ ¼ − 2 with T3L A T3A= . The axial couplings, already given cannot be taken as a complete theory, among them, the in (53) independently of sin2 θ, and of grand unification, are nature of dark matter, possibly made of new particles not the same for down quarks and charged leptons, as also present in the SM, and of dark energy. The latter is related ð4Þ required by the SU es electrostrong symmetry. with the cosmological constant and vacuum energy density,

035034-24 U BOSON INTERPOLATING BETWEEN A GENERALIZED … PHYS. REV. D 103, 035034 (2021) extremely large in the very first moments of the Universe These extra Uð1Þ’s can provide a bridge to a new dark but extremely small now. Furthermore, if we know four sector, allowing for dark matter particles to annihilate and types of interactions, other ones may well exist. opening the possibility for them to be light. The associated Essentially all extensions of the standard model motivated light neutral gauge boson, called U forty years ago, may by these questions involve new particles and fields, and new manifest under different aspects, depending on its mass, the symmetries. The main approach has been, for several vector and/or axial character of its couplings, the strength decades, to turn to higher energies to search for new heavy of its interactions, its lifetime, and decay modes, etc. particles. The discovery of the BEH boson at 125 GeV was a We have presented a unified framework relating the triumph for the standard model, but superpartners in its different aspects of such a new gauge boson. Its possible supersymmetric extensions have not been found yet, and couplings appear in the visible sector as linear combina- ≈ ℏ compactification scales =Lc may well be far away; so if tions of Q, B, and Li, with the weak isospin T3L, and an “ ” there must be new physics , where is it? axial generator FA. This relies on a minimal set of Acomplementaryapproach,whichhasgrownintoasubject hypotheses, illustrating the crucial role of the BEH sector of intense interest, involves moderate or even low energies, in determining the properties of the new boson. Its current ð1Þ searching for new interactions and their mediators. This may is a linear combination of the extra U F and Z currents, involve interactions significantly weaker than weak inter- determined by the Z − U mixing angle ξ. This one, small if actions, down possibly to the strength of gravity and even less. the U is light, is given in thisP limit by a sum on BEH Using symmetries has good chances to remain a valid ξ ≃ ð 00 Þ ϵ ð Þ 2 2 ¼ð 00 Þ doublets, with tan g =gZ i iF hi vi =v g =gZ guiding principle for discussing new particles and inter- ðγY þ ηÞ, so that (disregarding for simplicity cos ξ ≃ 1) actions, all four known ones being associated with gauge or space-time symmetries. Supersymmetry provides an exten- 1 J μ ≃ 00 μ þðγ þ ηÞð μ − 2θ μ Þ sion of space-time to fermionic coordinates, with a new U g 2 JF Y J3L sin Jem Uð1Þ R R symmetry acting chirally on them. It leads to γ γ γ ¼ð−1ÞR 00 μ μ B μ Li μ A μ parity, Rp , at the origin of the stability of the ≃ g γYJem þ ηJ þ J þ J þ J Z 2 B 2 Li 2 A lightest supersymmetric particle and of its possible role as a μ dark matter candidate. þ J þ: ð224Þ ð1Þ dark Just as U R acts chirally on gauginos and Higgsinos, ð1Þ one can consider a U A symmetry acting chirally on The U boson may manifest as a generalized dark photon matter fields, as also suggested by the 2-BEH doublet also coupled to B and Li, a dark Z boson, an axial boson, structure of supersymmetric theories. Both chiral sym- and a (quasi-“invisible”) axionlike particle. It connects metries play an important role in discussions of sponta- these different aspects and interpolates between them, the neous supersymmetry breaking, possibly through the ξD dark photon case corresponding for example to the specific term of an extra Uð1Þ, and they might get related within direction (1, 0, 0,0,0; 0, 0) in a 7-dimensional space ð1Þ ð4Þ extended supersymmetry, enlarging U R up to SU . parametrized by One may also consider extra space dimensions, which naturally combines with supersymmetry. The possibility of γ γ ðγ γ γ Þ; γ η ð Þ Y; B; L ; Lμ ; Lτ A and : 225 performing translations or moving along compact dimen- e sions in the higher-d space-time may result in additional The U may also couple to dark matter (in particular if the R Uð1Þ’s, as associated with (central) charges in the extended current is involved in the gauging), possibly more strongly supersymmetry algebra, and gauged by graviphotons that than to ordinary particles. Invisible decay modes into light may be described by extra components of the gravitational dark matter may be then favored. ð1Þ’ field. Many such U s tend to be generated from (super) In the visible sector we have the correspondences: string theories, and it is natural to expect their remnants to act in 4 dimensions. Wemay also extend directly theSMgroupbeyond SUð3Þ× γY → dark photon; ð2Þ ð1Þ ð5Þ SU × U Y,orSU for a grand unified theory. 9 ð1Þ η → ð “ ”Þ The minimal approach involves an extra-U F symmetry, dark Z = quasi- invisible 00 focusing here on small values of its gauge coupling g .Its → isoscalar ; axionlike behavior possible generators naturally appear as linear combinations of γA → 5 axial boson of longitudinalU Y, B,andL,orB − L, with an axial FA. Y − 2 ðB − LÞ and FA ð5Þ have a special status as they both commute with the SU γ ¼ − γ → B−L gauge boson; ð1Þ B L grand unification group. The resulting U U generator ð4Þ etc: commutes with a SU es electrostrong symmetry group, spontaneously broken to SUð3Þ × Uð1Þ at the grand QED ð Þ unification scale, possibly through extra dimensions. 226

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2m The U charge is expressed for vector couplings as a ¼ ql ¼ 21=4 1=2 − gP gA GF A combination of Q with B and Li,orB L in a grand m U unified theory, r cot β; ≃ 4 × 10−6 m ðMeVÞ × ð230Þ γ γ ql r tan β: ð Þ ¼ γ 2θ þ B þ Li QU V Y cos Q 2 B 2 Li 1 2 1 This reconstructs the same pseudoscalar couplings as for a þ η − sin θ Q − ðB − LÞ : ð227Þ 2 4 quasi-invisible axion, with r ¼ cos θA as the invisibility parameter. With a semi-inert doublet h0, one has, simi- ≃ ∓ 4 10−6 ð Þ β0 When the η term from the mixing with Z dominates, the U larly, gP × mql MeV r tan . ð Þ ψ ϒ has a nearly protophobic behavior, with QU V almost , , K, and B decays lead to strong limits on axial 0 −7 proportional to B − L − Q, which may favor a small π → couplings, typically ≲ 2 × 10 mUðMeVÞ or even down to γ −9 U decay rate. In the framework of grand unification, all ≲ 2 × 10 mUðMeVÞ. Such a strong limit on gAe may vector and axial couplings of the U may be expressed as in facilitate satisfying the atomic-physics parity-violation con- (219) in terms of three parameters only, γY, γA, and η, straint. The U couplings to neutrinos may also be very small, 00 instead of the seven ones in (225). thanks to a small g naturally proportional to mU and r;orto Axial couplings a small QUðνLÞ,ifQU is close to a combination of Q, B, T3R, and L þ 2FA. The neutrino and electron couplings, 00 proportional to m rQ , are constrained from ν − e scatter- ≃ 00ð Þ ≃ g ðγ ∓ ηÞðÞ U pUffiffiffiffiffiffiffiffiffiffi gA g QU A 4 A 228 ≲ ð1 3Þ 10−6 ð Þ ing experiments to gνL ge to × mU MeV for mU ≳ 1 MeV, depending on mU and on the experiment may be isoscalar, or isovector from the contribution of the Z considered, much in line with expression (229) of g00=4 ≃ ð 00 4Þð1 −6 current, or a mixing of both, as with gA g = as 2 × 10 mUðMeVÞ × r= sin 2β. cos 2βÞ in a two-doublet þ one-singlet model. The extra- The extra-Uð1Þ coupling g00 may vary between sizeable ð1Þ 00 −24 U gauge coupling g , then proportional to mU and to the values for larger m ’s ∼ a few ten GeVs, down to ≲ 10 ¼ θ U invisibility parameter r cos A, is given by for an extremely light or even massless U boson. It could then be detectable through apparent violations of the 00 g mU r −6 r Equivalence principle. The extreme weakness of such a ≃ ≃ 2 × 10 m ðMeVÞ: ð229Þ 4 2v sin 2β sin 2β U new long range force might be related, within supersymmetry, with a very large energy scale ≳ 1016 GeV, asso- It is naturally very small if the U is very light, or if r is small ciated with a huge vacuum energy density that may be at from a large singlet v.e.v., with the extra-Uð1Þ symmetry the origin of the inflation of the early Universe. broken at a large scale. The possible existence of a new interaction, that may fit Axial couplings of the U, when present, are respon- within the grand unification framework and have connec- sible for enhanced interactions of a longitudinal U, inter- tions with the fundamental structure of space-time, is both a acting in the small mU limit with effective pseudoscalar fascinating theoretical subject and a very rich field for couplings experimental investigations.

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