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B n lepton and global X M uinHeeck Julian of B sym- ff Z B B ′ n etnnumber lepton and B /g − ↔ − ′ L rmTVu o10 to up TeV from ν g hlspewg1,610Hiebr,Germany Heidelberg, 69120 16, Philosophenweg rg, rohrnwlgtfre.W hwi particular in show We forces. light new other or L R urn ausfrtenurn aaeescnb found be can parameters neutrino the for values Current e.I n hng nt olc,udt,adimprove and update, collect, to the on on go then bounds and II Sec. cou- the constrains force. as severely new following, neutrinos the the right-handed light in to importance pling great of be will oe 3.Alocal A [3]. model rn ass and masses, trino lsl eae oesi e.I n nlycnld in 3. conclude Fig. finally and in IV presented V. other Sec. to Sec. in III, limits models these Sec. related of in closely applicability the TeV on 10 comment We to up masses ihvnsigMjrn hss[8]), parametrization phases standard Majorana the vanishing with (in charged- for relevant matrix current mixing Here, leptonic unitary breaking. the symmetry electroweak after i uaaculnst h ig doublet Higgs the to neutrinos couplings Yukawa left-handed active via the with particles Dirac form eutn ntenurn asmatrix mass neutrino the in resulting as unbroken an ehns ne h aeo etioeei vnfor even neutrinogenesis leptogenesis of elegant name conserved an the to under rise gives mechanism neutrinos of nature h erhfr∆( prominently for most search searches, the decade-long despite process cnrohsrrl 3 ]be osdrd e (fifth) [6], number new lepton considered; [5], number been or baryon 4] to coupled [3, for forces rarely searches has direct scenario to view of he ih-addnurnst h Mi re ocan- unbroken to For order in anomalies. 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M ′ 7 aeo orebe tde eoe u never but before, studied been course of have [7] 0–10 = 91 edleg emn and Germany Heidelberg, 69117 αβ B mtyb nrdcn he right- three introducing by mmetry n ec ee with never hence and , L = − U steol nml-reglobal anomaly-free only the is (1) L y B 10 13 αβ g U − [2]. ′ GeV. B B h St¨uckelberg the mass , V loo neetfrthe for interest of also eV, (1) |h V L − H R − eie ia neutrinos, Dirac besides ; L I MODEL II. U B i| ymtya natraiepoint alternative an as symmetry L sa nhscluiaymatrix. unitary unphysical an is − (1) hs et B Lett. Phys. etioesdul beta double neutrinoless 2 = ) L = B U opigstrength coupling − M < diag( L B eursteadto of addition the requires B Z − ′ m Dirac L < − 1 eas h Dirac the because , B m , 0GeV 10 B 739 L − − M m 2 ilto.This violation. L B etio.This neutrinos. ect m , 5–6 (2014) 256–262 , Z L arXiv:1408.6845 i )- ′ y hs states these , − , r h neu- the are αβ 3 ) L V U L g R violating † ′ α denotes Hν for R,β (1) Z , ′ 2 in Ref. [9]. Note that experimental upper bounds on the 0.8 neutrino masses of order eV imply very small values for had the Yukawa couplings y . 10−11 but pose no real inv | αβ| problem to the model. L 0.6 L

Dirac neutrinos aside, an unbroken U(1)B−L brings - ′ B with it only one more particle: the gauge Z , H 0.4 e+ e- coupled to the B L current jB−L jB jL via BR ′ ′ µ − ≡ − g ZµjB−L. All —including neutrinos—are de- 0.2 scribed by Dirac fermions after electroweak symmetry Μ + Μ - breaking, leading to vector-like Z′ couplings to these 0.0 mass eigenstates: 0.2 0.4 0.6 0.8 1.0 g′Z′ 1 uγµu + dγµd eγµe νγµν . (2) MGeV L ⊃ µ 3 − − fam ′ − − X    Figure 1: Z branching ratios to e+e (blue), µ+µ (blue, There are in particular no flavor changing neutral cur- dotdashed), neutrinos (green), and (red). ′ rents. The Z itself is allowed to have a mass MZ′ via the St¨uckelberg mechanism [10] without breaking the ′ (global) B L symmetry [3]. This mass is a free param- approach, Z shares the quantum numbers of the ω me- eter of the− model, not connected to a vacuum expecta- son, because it carries no isospin. It therefore has simi- lar decay modes [14], most importantly with highly sup- tion value; it is notably disconnected from, e.g., neutrino ′ 2 pressed Z ππ rate. Hadronic decays are still possible masses, and there exists no theoretically preferred value. → ′ 0 for M ′ > m , with the dominating channel Z π γ Lastly, the Z′ can kinetically mix with the hypercharge Z π → 1 Y ′µν below 600 MeV and partial width [14] boson through a coupling 2 sin χFµν F [12, 13], effec- ′ tively coupling the Z to the hypercharge current. We ′ 2 3 ′ 0 α αEM 3 mπ 2 2 ′ ′ will neglect this kinetic mixing angle χ in the following Γ(Z π γ)= 3 2 MZ 1 2 Fω(MZ ) . → 96π f − M ′ | | for simplicity, but comment on its effects in Sec. IV. π  Z  Let us make a couple of remarks to compare unbroken (5) B L with the better-studied case of spontaneously bro- 2 −1 Here Fω(s) (1 s/m iΓω/mω) denotes the Breit– ken− B L (“Majorana B L”), featuring three heavy ≃ − ω − − − Wigner form factor of the ω . For 600 MeV . and three light Majorana neutrinos. The decay width ′ MZ′ . 1 GeV, the dominant hadronic decay is Z into fermions is given by π+π−π0, with a rather complicated analytical expres-→ sion [14]. (Here, we also include the ρ width in the 2 2 ′ mf mf in order to cross the Z ρπ threshold.) ′ 1 ′ ′ Γ(Z ff)= 3 α MZ 1+2 2 1 4 2 Above GeV, s- channels open→ up and the Z′ → MZ′ ! s − MZ′ (3) becomes φ-like. From Fig. 1 we see that the hadronic 1, f = lepton, branching ratio for MZ′ < GeV is completely negligible × (1/3, f = quark, away from the ω resonance. This is in stark contrast to the hadronic decays of hidden [15], which feel with the B L fine-structure constant α′ g′2/4π. The the much broader ρ resonance. The reason for this is invisible width− of our Z′ is then governed≡ by the decay the isospin-violating coupling of hidden photons to elec- into the three light Dirac neutrinos ν = ν + ν : tric charge, allowing them to easily mix with both the ω L R (isospin 0) and the ρ (isospin 1) meson. ′ ′ ′ Regarding the baryon asymmetry of our Universe, let Γinv(Z )=3 Γ(Z νν)= α MZ′ , (4) × → us briefly sketch the B L conserving neutrinogenesis − which effectively counts the number of light neutrinos, in mechanism, following Ref. [2]: here it is crucial to note that the small Yukawa couplings y . 10−11 of our complete analogy to the invisible width of the Z, which | αβ| however only counts the number of light left-handed neu- Dirac neutrinos are insufficient to put the νR into thermal trinos. The invisible Z′ width of unbroken B L is hence equilibrium in the early Universe. The goal is then to − up to three times larger than in Majorana B L (de- create a lepton asymmetry ∆νR in the νR sector that is pending on the mass of the heavy neutrinos), due− to the formally canceled by an opposite asymmetry of the left- handed ∆νL = ∆νR , which can be achieved additional decay into νR. This is typically irrelevant for − Z′ collider searches, but will be very relevant for lighter with the introduction of e.g. two very heavy unstable Z′, and would of course be of great interest in case a Z′ scalar doublets. Since only the left-handed leptons are resonance is found. in thermal equilibrium with the SM plasma, a baryon ′ asymmetry ∆B will be generated by the sphalerons using For Z masses around ΛQCD 200 MeV, the hadronic Z′ decay is in principle more involved∼ than Eq. (3) be- ∆νL in typical leptogenesis manner. In this letter we cause of hadronization. In the vector-meson dominance will not be explicitly concerned with neutrinogenesis, so we ignore the additional scalars and constraints. Let us note though that the newly introduced gauge interactions via Z′ do not invalidate this mechanism, because they 2 ′ conserve the individual particle numbers and hence do Note though that small values for both g and MZ′ are techni- cally natural in the sense of ’t Hooft [11] in that all radiative not erase ∆νR or ∆νL ; rapid gauge interactions will of ′ corrections are again proportional to g and MZ′ , respectively. course put the νR in equilibrium and hence increase the 3 number of relativistic degrees of freedom, to be discussed A. Modified and fifth force searches in Sec. III. ′ Note that the gauge boson Z can in principle also be A distinct feature of the B L force, compared to stable enough to be . For this, we have to hidden photons in particular, is− the coupling to neutral make sure its lifetime τ surpasses that of our Universe, particles. As such, a light Z′ mediates a force between which requires astrophysical bodies that depends on their num- 3 1 1 1 ber and thus violates the weak equivalence principle. τ = > τUniverse , (6) Torsion-balance experiments [23, 24] set very strong lim- Γ ∝ α′M ′ ≃ 10−32 eV Z its on such long-range forces, which are most importantly the dominant decay mode for light Z′ being Z′ νν (un- also applicable for a completely massless Z′, resulting in ν ′ → less MZ′ < 2m , then Z 3γ is the only channel). lightest ′ −24 ′ −14 Using a misalignment mechanism→ in complete analogy to g < 10 at 95% C.L. for MZ < 10 eV . (7) the hidden photon case [16] (see also Ref. [17]) could then Distances λ 1/MZ′ from cm to 10 µm are constrained ′ make the Z a cold dark matter candidate. We will not by experiments≡ that test the gravitational inverse square discuss this scenario any further in this letter. law F 1/r2, which would be modified in the presence of a light∝ Z′ [24, 25]. At even smaller distances, it is the Casimir effect (or Van der Waals forces) that sets the III. LIMITS strongest limits on the modified 1/r2 law [26–28]. Neu- tron scattering limits are inferior to astrophysical con- 5 The main signature of unbroken B L, beyond Dirac straints in the mass range 1 eV < MZ′ < 10 eV, but −′ neutrinos, is the new gauge boson Z . As mentioned offer the best laboratory limits [29, 30]. above, both coupling strength and mass are free parame- ters, disconnected from other observables. One can there- ′ fore consider (g ,MZ′ ) in the range [0, 1] [0, 10 TeV] with B. Stellar evolution equal motivation. While still a lamppost× search, it is a mighty big one. For Z′ masses between 0.1eV and 0.1GeV, strong as- We will now collect and update bounds on the param- trophysical limits arise from the additional energy loss ′ eters g and MZ′ , ultimately resulting in Fig. 3. The provided by the Z′, either in the Sun, heavier stars, or first (and only) survey across all masses can be found red giants. These limits, dominant for 10 eV . MZ′ . in Ref. [7], which we refine and improve. Note that we 100 keV, can be readily translated from hidden- will set the kinetic mixing angle χ to zero in our anal- limits [31] via g′ ˆ= eχ, because the relevant coupling to ysis for simplicity, but comment on its effects later on (and ) has the same structure for elec- in Sec. IV. The mass range for B L below GeV was − tric charge and B L. Minor modifications arise due to recently covered in Ref. [18]; the mass range above MeV the B L coupling− to inside the star, which we in Ref. [19] (including kinetic mixing). However, none of neglect− here. the above have specifically considered unbroken B L, The stellar evolution is actually modified two-fold: a ′ − which has the Z coupled to Dirac neutrinos; this makes light Z′ can carry away energy, similar to hidden pho- especially Big Bang nucleosynthesis (BBN) bounds far tons, but the Z′ also enables the production of the ster- more important and deserves a discussion. ile νR; millicharged-particle limits [18, 32] then con- Many limits are obtained by translating well-known strain roughly g′ . 10−14 using red giants, which set hidden photon limits, see Ref. [20] for a recent review. the strongest limits in the range 0.1 eV . MZ′ . 10 eV. The main differences between these models are the invisi- White dwarf cooling via νR emission yields constraints ′ ble width of ZB−L, which reduces the (typically relevant) in the MeV–GeV region [33] of similar order as neutrino branching ratio into electrons and , and the addi- scattering experiments (next section). This white dwarf tional coupling to neutral particles (neutrons and neutri- limit is not shown in Fig. 3 in order to avoid cluttering. nos). Unfortunately the literature is inconsistent when From the supernova SN1987A we get very strong lim- it comes to the definition of the kinetic mixing angle of its on Z′ in the 100 keV–100 MeV range [34], again hidden photons, employing either mixing with the hyper- adopted from hidden photons. These limits do, how- charge or electromagnetic field strength tensor. The for- ever, neglect some important plasma effects and are ex- mer represents the proper gauge invariant structure and pected to change upon reanalysis. We show this limit in reduces to the latter in the low-energy limit. The two dashed lines in Fig. 3 to remind the reader of the sub- definitions differ by the cosine of the Weinberg angle— tleties involved. Once again, the fact that the Z′ cou- χEM = χY cos θW for small χ—and care has been taken ples to sterile νR yields an additional supernova cool- ′ below to ensure consistency in the resulting limits on g . ing bound; we adopt the limits from Ref. [35], based ′ Note that the limits on g and MZ′ are directly applica- on on-shell Z′ production in e+e− annihilations, using ble to any B L gauge boson coupled to Dirac neutrinos, BR(Z′ ν ν ) from Sec. II. For heavier Z′, off-shell − → R R even if the U(1)B−L is broken by n = 2 units (employed νR production becomes dominant, and one can derive 6 ′ in Refs. [21, 22]). They are also applicable to models bounds on MZ′ /g [36–38] of order TeV (not shown in where B L is broken by two units, but all six Majo- Fig. 3). rana neutrinos− are light, so that the invisible Z′ width does not change compared to the unbroken B L case discussed here. For Majorana B L with (some)− heavy − Majorana neutrinos, not all limits are applicable and we 3 The B − L contributions from electrons and protons cancel each will comment on that in due time. other in electrically neutral objects. 4

The stellar evolution bounds are of course not as re- E. BaBar fined as laboratory limits and should be understood as es- timates; there are no confidence levels attached to them. BaBar gives great limits on hidden photons in the ′ A dedicated analysis of the Z and ν interplay in stellar ′ R range 20MeV < MZ < 10 GeV [48] (rescaled to evolution is expected to improve on the given bounds, 95% C.L.) from the coupling to electrons in the process but lies beyond the scope of this letter. e+e− γZ′, Z′ e+e− (µ+µ−). Our larger invisi- ble width→ will reduce→ the bounds, similar to the beam dumps discussed above (by a factor of 2 below the C. Neutrino scattering threshold), so we translate

BR(A′ e+e−) Scattering of solar neutrinos on electrons in Borex- g′ ˆ= eχ HP → (11) ino [39] is sensitive to Z′ exchange, recently discussed BR(Z′ e+e−) s → in Refs. [18, 40, 41] for an MeV coupled to left-handed neutrinos and charged leptons. For sub- over the entire mass range. These are the strongest limits MeV masses, Gemma gives stronger limits [18], but still for masses 5GeV < MZ′ < 10 GeV. The fixed-target weaker than astrophysics. These limits surpass magnetic- experiments APEX [49] and MAMI [50] provide similar, moment limits, and make in particular a resolution of the slightly weaker, constraints, not shown in Fig. 3. muon’s g 2 anomaly impossible within gauged B L (or − − gauged U(1)L for that matter [42]). Neutrino–quark scat- ′ tering also yields strong limits of order MZ′ /g & TeV for F. Thermalization in the early Universe Z′ masses above 10GeV [7, 19]. Precise measurements have been performed by NuTeV [43], which may how- Big Bang nucleosynthesis (BBN) describes successfully ever suffer from systematic errors. We therefore take a our Universe at temperatures around MeV, and places ′ very conservative bound of MZ′ /g > 0.4 TeV, following strong bounds on the number of relativistic degrees of a recent reanalysis [44], about a factor 3–4 weaker than freedom. The latter are typically parameterized via Neff , previous limits [19, 45]. the effective number of neutrinos, predicted to be 3 by the SM. We take Neff < 4 as a conservative 95% C.L. limit from BBN [51], which in particular forbids the thermal- D. Beam dump ization of our three light right-handed neutrinos νR, and thus constrains the strength of the B L gauge interac- tions that would put them in equilibrium.− The interac- We take the 95% C.L. limits from hidden photons from tion rate of νR therefore has to be smaller than the Hub- Ref. [46], containing data from E774, E141, Orsay, KEK, 2 ble expansion rate H(T ) T /MPl around T 1 MeV. and E137. The lower (horizontal) limit on the coupling ∼ ∼ ′ Such reasoning has long since been used to constrain can be translated simply via g ˆ= eχ, because the number right-handed neutrino interactions [52, 53]. of events for small coupling is given by [47] To calculate the thermally averaged rate for ff ν ν induced by Z′ exchange we follow Ref. [54]: 2 R R gZ′ee ′ + − ↔ N Γ(Z e e ) , (8) 3 3 ∝ MZ′ → 2 d p d k Γ(ff νRνR) = 3 3 fν (p)fν (k) h ↔ i nνR (T ) (2π) (2π) independent of the invisible Z′ width. The width is on Z vMσ (s) , the other hand important for the upper limits (diagonal × νRνR→ff ′ χ 1/M ′ ), which correspond to a fast Z decay (with (12) ∼ Z Lorentz factor γ = E/M ′ ) inside the shielding of length Z k/T −1 L ; here, the number of events goes with with the Fermi–Dirac distribution fν (k) = (e + 1) , sh 3 2 the νR number density nνR = 3ζ(3)T /2π , and the

2 ′ Møller velocity vM. The interaction cross section σ gZ′ee ′ + − MZ N 2 BR(Z e e ) exp Lsh Γtotal , is to be evaluated at the center-of-mass energy s = ∝ M ′ → − E Z   2pk(1 cos θ), θ being the angle between the two col- (9) − liding νR particles. Unlike Ref. [54], we do not restrict ourselves to the limiting case MZ′ T , but rather use sensitive to Γtotal, and hence Γinv, due to the dominating ′ ≫ the full Z . For MZ′ T , on-shell production exponential factor. This shows that the upper limits have of Z′ becomes possible and the∼ interaction rate is reso- to be translated by comparing the total width. Below the ′ nantly enhanced, calculated most easily in the narrow- muon threshold, this simply corresponds to g ˆ= eχ/2, width approximation for the Z′ propagator because we have for 2m M ′ < 2m : e ≪ Z µ 1 π 2 ′ 2 ′ 2 2 2 2 δ(s MZ ) . (13) HP αχ B−L α − (s M ′ ) + M ′ Γ ′ → M ′ Γ ′ − Γ = M ′ , Γ = M ′ 3(ν)+1(e ) . Z Z Z Z Z total 3 Z total 3 Z −  (10) This step is equivalent to simply studying the inverse decay ff Z′, as has been done recently in Ref. [55] in The excluded area is hence smaller for unbroken B L a similar context.→ than maybe expected, due to the additional decay chan-− The thermally averaged interaction rate between right- nel into neutrinos. handed neutrinos and massless fermions f via s-channel 5

librium and go out before the Universe cools down to about T (νR) 150–200 MeV [56], they will contribute H ∼ ~T 2 to Neff in an entropy-suppressed fashion [52, 53], namely ∆Neff < 1, compatible with current data. For masses M ′ > 10 GeV, the ν decoupling before T (ν ) Z R R ∼ ~T 150 MeV then gives the well-known limits of the form ~1T ′ MZ′ /g > 6.7 TeV [54]. For 1 GeV < MZ′ < 10 GeV,

rate the limits on g′ strengthen considerably due to reso- nant Z′ production (see Fig. 3), down to g′ < 6 10−9 × at MZ′ = 1GeV. For lower masses, it becomes im- possible to decouple at T (νR), and we have to demand ~T5 that the (resonant) interaction rate is smaller than H(T ) for all temperatures 1 MeV

4 Assuming Maxwell–Boltzmann statistics for the νR allows for an 5 analytic evaluation of the narrow-width integral and replaces the Because of this, only the limits for MZ′ & 1 GeV are sensitive to 3 intermediate function in Eq. (15) by πx K1(x)/4ε. the precise bound we use for Neff . 6 from effective four-lepton operators with the final LEP 2 B. Kinetic mixing data [60], valid for M ′ 200GeV. Even stronger lim- Z ≫ its can likely be obtained from updated dedicated global Having ignored kinetic mixing up until now, let us com- fits to electroweak precision data [61, 62], but we take ment on it. Kinetic mixing is allowed in the Lagrangian Eq. (16) as the strongest limit on B L for Z′ masses 1 Y ′µν − via 2 sin χFµν F , unless some embedding into a Grand above 3 TeV. Unified Theory is assumed, which is incompatible with Not shown in Fig. 3 are the LEP constraints for masses our assumption of unbroken B L. Even if zero at some − close to the Z mass MZ′ MZ , which are stringent but scale, χ = 0 will be induced radiatively because we have very narrow [19]. ≃ particles6 charged under both U(1) groups [13], generat- ing a coupling of Z′ to hypercharge. This makes un- broken B L a three-parameter model (not counting neutrino masses− and mixing), described by g′, χ, and IV. COMMENTS MZ′ . Limits on these parameters have been discussed for MZ′ > 1 MeV in Ref. [19] (albeit without the cou- pling to light right-handed neutrinos). Let us therefore We have derived limits on our specific model of unbro- discuss the low-mass region. Note that our Z′ reduces to ken B L presented in Sec. II, but our results can be the familiar hidden photon in the limit g′ 0. − → ′ translated to other scenarios, too. Since an exhaustive For MZ′ MZ , kinetic mixing effectively induces Z list of related models is infeasible, we stick to the most mixing with≪ the photon, so the Z′ now couples to a lin- obvious ones. ear combination of B L and electric charge Q. If, for a certain Z′ mass, the B− L coupling dominates, we can apply the limits from this− letter; if the coupling to elec- tric charge dominates ( g′ e cos θ sin χ ), one can | | ≪ | W | A. Broken B − L simply take the limits from hidden photons [20]. The in- termediate regime, where the coupling strengths are of similar order, is more interesting; depending on the rela- The limits of Fig. 3 are directly applicable to models tive sign, the limits can either become stronger or cancel with spontaneously broken gauged U(1)B−L and Dirac each other out. In the most extreme case, the Z′ couples neutrinos, i.e. scenarios where B L is broken by n =2 ′ − 6 to the linear combination (B L) Q, which vanishes for units [21]. The scale MZ′ /g = n φ is then directly electrically charged leptons and− .− This is then a connected to the B L breaking vacuumh i expectation − non-chiral force acting exclusively on neutral fermions: value φ of a scalar φ. Even though (non-resonant) Dirac neutrons and neutrinos. Quite surprisingly, even this leptogenesish i works quite naturally at high B L breaking − restricted scenario is easily constrained: the fifth force scales in this scenario [22], this does not preclude sub-TeV (modified gravity) limits are all still applicable, as are Z′ bosons, because the coupling g′ can be small. Seeing ′ 9 the BBN bounds (νLνL νRνR being allowed); stellar as astrophysics is sensitive to scales MZ′ /g & 10 GeV evolution can still put constraints↔ on the coupling in the (Fig. 3), even those natural leptogenesis scales are probed intermediate mass range eV MZ , a Z coupled becomes invalid if all νR are heavy. The other limits to B L and hypercharge—corresponding precisely to the − do not change qualitatively, because they depend only case of U(1)B−L plus kinetic mixing—has been discussed slightly on the invisible Z′ width, and can be translated in Ref. [64]. straightforwardly. The high scales probed by astrophysics mentioned here are of course not only reminiscent of leptogenesis scales, V. CONCLUSION but also those of reheating and inflation, which can be linked to Z′ physics and dark matter (see for example Promoting the only ungauged anomaly-free symmetry Ref. [63]). A discussion goes unfortunately beyond the of the , U(1)B−L, to a local symmetry re- scope of this letter but certainly deserves attention. quires the introduction of three right-handed neutrinos, 7

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′ ′2 ′ Figure 3: Bounds on the U(1)B−L coupling constant g (fine-structure constant α = g /4π) and Z St¨uckelberg mass MZ′ (corresponding to a range λ = 1/MZ′ ) for vanishing kinetic-mixing angle χ = 0. The area on the upper left is excluded by limits from tests of the equivalence principle (EP, dark blue) [23, 24], gravitational inverse-square law (ISR, green) [24], Casimir effect (black) [26–28], neutron scattering (light green) [29, 30], energy loss via ν in red giants (RG/ν, red) [32], energy ′ ′ loss via Z in the Sun (Sun, yellow), horizontal branch stars (HB, orange), and red giants (RG, red) [31], SN1987A (Z ) (grey, dashed) [34], SN1987A (νR) (grey) [35], BBN (blue, Sec. III F), beam dump searches (BD, green) [46], neutrino– scattering (e–ν, purple) [18], BaBar (dark red) [48], neutrino–quark scattering (q–ν, brown) [44], ATLAS (LHC, black) [58], and LEP (orange) [59, 60]. Most (non-astrophysical) limits are at 95% C.L., see text for details. Also shown are diagonals at ′ 9 scales MZ′ /g = 100 GeV and 10 GeV for comparison (grey, dotdashed). which make neutrinos massive. Experimental searches giants), also applicable to models with broken B L. have yet to confirm the prevailing notion amongst the- We have shown in particular that successful Big Bang− orists that neutrinos are Majorana particles, and hence nucleosynthesis provides strong constraints in the mass B L broken by two units in nature. It is therefore range 10eV < M ′ < 10 GeV due to resonant enhance- − Z important to point out that there is currently no theo- ment of the thermalization rate ff νRνR, previously retical or phenomenological argument against an unbro- unexplored. ↔ ken U(1)B−L gauge symmetry, featuring Dirac neutri- nos and neutrinogenesis to explain the baryon asymme- try of the Universe. We can even give the new gauge ′ Acknowledgments boson Z a St¨uckelberg mass MZ′ , a new dimensionful parameter disconnected from other scales. In this let- ter we collected and updated constraints on the gauge The author gratefully thanks Joerg Jaeckel and coupling strength g′ in the entire testable mass range Joachim Kopp for discussions and comments on the 13 MZ′ = 0–10 eV, assuming for simplicity vanishing ki- manuscript, and Hye-Sung Lee and Sean Tulin for help- netic mixing with hypercharge. The excluded scales for ful correspondence. This work was supported by the ′ ′ a not-too-light Z range from MZ′ /g & 7 TeV (BBN and IMPRS-PTFS and by the Max Planck Society in the ′ 10 LEP) all the way up to M ′ /g & 3 10 GeV (red project MANITOP. Z ×

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