PHYSICAL REVIEW D 101, 055014 (2020)

Dark CP violation and gauged lepton or baryon number for electroweak baryogenesis

Marcela Carena,1,2,3 Mariano Quirós ,4 and Yue Zhang 1,5 1Theoretical Physics Department, Fermilab, P.O. Box 500, Batavia, Illinois 60510, USA 2Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637, USA 3Kavli Institute for Cosmological Physics, University of Chicago, Chicago, Illinois 60637, USA 4Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology (BIST), Campus UAB, 08193 Bellaterra (Barcelona) Spain 5Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA

(Received 12 September 2019; accepted 19 February 2020; published 11 March 2020)

We explore the generation of the baryon asymmetry in an extension of the standard model where the 0 lepton number is promoted to a Uð1Þl gauge symmetry with an associated Z gauge boson. This is based on a novel electroweak baryogenesis mechanism first proposed by us in Ref. [M. Carena, M. Quiros, and Y. Zhang, Electroweak Baryogenesis From Dark CP violation, Phys. Rev. Lett. 122, 201802 (2019).]. Extra fermionic degrees of freedom, including a fermionic dark matter χ, are introduced in the dark sector for anomaly cancellation. The lepton number is spontaneously broken at high scale and the effective theory, containing the standard model, the Z0, the fermionic dark matter, and an additional complex scalar field S, violates CP in the dark sector. The complex scalar field couples to the Higgs portal and is essential in enabling a strong first order phase transition. Dark CP violation is diffused in front of the bubble walls and creates a chiral asymmetry for χ, which in turn creates a chemical potential for the standard model leptons. Weak sphalerons are then in charge of transforming the net lepton charge asymmetry into net baryon number. We explore the model phenomenology related to the leptophilic Z0, the dark matter candidate, the Higgs boson, and the additional scalar, as well as implications for electric dipole moments. We also discuss ð1Þ the case when baryon number U B is promoted to a gauge symmetry, and discuss electroweak baryogenesis and its corresponding phenomenology.

DOI: 10.1103/PhysRevD.101.055014

I. INTRODUCTION the discovery of the Higgs boson [9]. This gives a strong motivation to study EWBG in models with a dark sector The origin of the cosmic baryon asymmetry is a CP fascinating mystery for particle physics and cosmology. where SM gauge singlet particles source the required Electroweak baryogenesis (EWBG) [1–5] is an elegant violation. The main challenge of such realizations is possibility and predicts new physics beyond the standard finding an efficient mechanism to transfer the CP violation model (SM), near the electroweak scale, to trigger a from the dark sector to the visible sector in the early sufficiently strong first-order electroweak phase transition Universe, while still keeping contributions to EDMs (EWPT) and source enough CP violation. If the new sufficiently suppressed today. particles responsible for CP violation are charged under To this respect, an interesting scenario of dark sector CP the SM, they will also contribute to the electric dipole violation was presented in Ref. [10], where a Yukawa moments (EDM) at low energies [6]. Models that belong to interaction between a dark fermion and the SM fermion this class, including two Higgs-doublet models and super- doublets is responsible for communicating CP violation symmetric models, have been progressively receiving into the visible sector. Such a realization, however, leads to stronger and stronger constraints from the improved two-loop level contributions to EDMs. In turn, suppressing EDM measurements in recent years [7,8], especially after such contributions to the EDMs requires a finely tuned restoration of a global symmetry after the EWPT. In a recent short article [11], we presented the basic idea of a new EWBG mechanism in which the role of messenger Published by the American Physical Society under the terms of of the CP asymmetry can be played by a Z0 gauge boson the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to that couples to both the SM and the dark sector. The low- the author(s) and the published article’s title, journal citation, energy effective theory is a dark sector model containing a and DOI. Funded by SCOAP3. Dirac fermion χ (charged under the Z0) with a CP violating

2470-0010=2020=101(5)=055014(23) 055014-1 Published by the American Physical Society CARENA, QUIRÓS, and ZHANG PHYS. REV. D 101, 055014 (2020)

FIG. 2. Representative diagrams showing the loop generated electron EDM in two classes of models, where CP violation occurs through the interactions from electroweak charged par- ticles (left panel) or SM gauge singlets that couple to the Z0 (right μν panel). The gray blobs represent the loop generated hFμνF and FIG. 1. A schematic picture showing our model setup and the 0 0μν role played by each part in our proposed EWBG mechanism. hZμνZ effective vertices in the two cases, respectively. In the former case, the contribution to EDMs can occur at two-loop level via the Barr-Zee type diagrams. In the latter case, the contribution to EDMs must arise at more than two-loop level. coupling to a complex scalar field S. During a first-order phase transition, in the electroweak and the dark sectors l ¼ þ þ l ¼ þ involving both the Higgs field and the scalar S, a chiral- where Le Lμ Lτ and Lμ Lτ, which charge asymmetry in χ particles is first created. Through require the introduction of extra fermion fields the timelike component of the Z0 background (which is CP (anomalons) to cancel the gauge anomalies. Below odd, and also CPT odd), the χ asymmetry leads to a the spontaneous lepton number breaking scale, when chemical potential for all SM leptons. If the Z0 is suffi- part of the anomalon fields have been integrated out, ciently light, it mediates a long range force that extends into the low-energy effective theory is composed of the SM and a secluded dark sector. The two sectors are the region outside the bubble wall with unbroken electro- 0 weak symmetry. This chemical potential then biases the connected through the Z , which transfers the CP sphaleron processes and generates a net baryon asymmetry asymmetry to the observable sector, and the Higgs inside the bubbles. After the EWPT is completed, the Z0 portal interaction, responsible for inducing a first- background relaxes to 0 and the dark CP violation becomes order electroweak phase transition. It is worth noticing that, for our EWBG mechanism to secluded from the SM sector. A schematic plot of this setup 0 is shown in Fig. 1. work, the vector current that couples to the Z boson in the There are several distinct features of this model. effective theory must be anomalous with respect to the SM 0 ð2Þ (i) The Z gauge boson needs to be light, not much SU L gauge symmetry at the time of the EWPT. This is heavier than the electroweak scale, and not too achieved by (Boltzmann) decoupling the heavy anomalons weakly coupled to the SM leptons, for generating from the thermal plasma, such that only the SM fields are 1 sufficient baryon asymmetry. Therefore, the exist- kept populated at the critical temperature of the EWPT. ence of a light leptophilic Z0 serves as a smoking gun The effect of the anomalous current is to generate a non- of the proposed EWBG mechanism, and provides a vanishing chemical potential that triggers the electroweak well-motivated target for various experimental sphaleron processes to create a net baryon asymmetry. The searches, as we discuss below. above observation implies that our mechanism will not 0 (ii) Given that the CP violating interactions in the dark work, for example, if the Z is the gauge boson of the ð1Þ sector only involve SM gauge singlets, it follows U B−L symmetry (anomaly free in the presence of ð1Þ that, in the absence of any Yukawa couplings right-handed neutrinos), the hypercharge U Y, or linear involving both the SM and the dark sector particles, combinations thereof. The Uð1Þl lepton number sym- the two-loop Barr-Zee-type contributions to EDM metry we consider is anomaly free at high energy scales, [12,13] are forbidden. Indeed, we show that in this but it becomes anomalous after the spontaneous breaking framework, the leading contribution to EDMs must of the Uð1Þl gauge symmetry takes place and some of appear at least at the three-loop level, which is much the new fermions—otherwise responsible for anomaly less constrained by current EDM results. This point cancellation—are integrated out from the thermal plasma. is diagrammatically illustrated in Fig. 2. The effective theory below the mass of the heavy anomalons (iii) The particle χ in this model could serve as the dark

matter candidate, as we show in detail in this paper. 1 (iv) The simple model we have just discussed can be In other words, while heavy anomalons protect the gauge theory at zero temperature from gauge anomalies, through the embedded in an ultraviolet (UV) complete theory remaining Wess-Zumino terms [14], their abundance is Boltz- with gauged lepton number Uð1Þl, whose gauge mann suppressed at finite temperature so that they decouple from 0 boson is Zμ, for the two interesting benchmark cases, the thermal bath.

055014-2 DARK CP VIOLATION AND GAUGED LEPTON OR BARYON … PHYS. REV. D 101, 055014 (2020) is perfectly consistent, as gauge invariance is restored by the TABLE I. Fermion content (anomalons), and its quantum introduction of the Wess-Zumino terms [14].Thisisatthe numbers, in the anomaly-free model with gauged Uð1Þl sym- core of what makes our baryogenesis mechanism feasible. metry. q is a free (real) parameter. Similarly, our baryogenesis idea could also work for the ð3Þ ð2Þ ð1Þ ð1Þ ð1Þ Particle SU c SU L U Y U l gauged U B baryon number symmetry, which is also νi known to be anomalous with respect to the SM. R 1101 0 ¼ðν0 0 ÞT −1 2 q The content of this paper is organized as follows. In LL L;eL 12= 0 −1 q Sec. II we present our EWBG model, making explicit the eR 11 χ q structure of the extended dark fermion and scalar sectors R 110 L00 ¼ðν00 ;e00 ÞT 12−1=2 q þ N that interact with the SM particles through the Uð1Þ Z0 R R R g l e00 11−1 q þ N gauge boson and the Higgs portal. In Sec. III, we discuss L g χ 110q þ N the necessary steps for the first-order phase transition to L g occur, and the source of CP violation in the dark sector, as well as how the latter induces the actual mechanism of baryogenesis in the SM at the electroweak scale. In To pair up the other extra fermions and give them vectorlike Secs. IV and V we concentrate on the phenomenological masses (with respect to the SM gauge symmetries), a Φ aspects of our model and its possible signatures in current complex scalar is introduced carrying lepton number Ng. l ¼ The vacuum expectation value (VEV) of Φ, vΦ, sponta- and near future experiments, for the cases where 0 neously breaks the Uð1Þl, giving mass to the Z gauge Le þ Lμ þ Lτ and l ¼ Lμ þ Lτ, respectively. This includes the leptophilic Z0 searches, dark matter χ direct detection boson, as pffiffiffi searches, conditions for thermal freeze-out, bounds from 0 0 ¼ 2 ð Þ EDMs, and collider searches for dark scalar(s). We com- MZ Ngg vΦ; 2:1 ð1Þ ment on the case of gauged U B baryon number in Sec. VI. We reserve Sec. VII for our conclusions and and to the new fermions via the following Yukawa terms: provide some details of the calculation of the lepton ð ¯ 00 0 þ ¯00 0 þ χ¯ χ ÞΦ þ ð Þ asymmetry in Appendixes A and B. cLLRLL ceeLeR c χ L R H:c: 2:2

II. A MODEL WITH GAUGED Hereafter, for simplicity, we ignore the Yukawa couplings LEPTON NUMBER between lepton doublets and singlets with the Higgs boson (which would lead to subleading entries in the fermion As the starting point, we consider an extension of the SM mass matrix), as well as the potential Yukawa coupling with gauged lepton number symmetry Uð1Þl. Its gauge between the SM leptons and some of the new leptons (only 0 0 2 boson is called Z and its gauge coupling g . There are allowed for specific choices of q, for example, q ¼ 1), various choices to define the lepton number, l. The most which also helps to suppress new sources of lepton flavor l ¼ þ þ obvious choice is Le Lμ Lτ, where all three lepton violation [18]. 0 00 0 00 flavors are gauged universally. However, our baryogenesis Because LL;LR;eR;eL contain fermions charged under mechanism will also work if only a reduced number of the SM gauge group, which are constrained by the existing lepton flavors are gauged, e.g., l ¼ Lμ þ Lτ. In the LEP and LHC searches, we assume vΦ to be well above the following discussion, we keep the number of lepton flavors TeV scale and cL, ce to be of order 1, rendering these charged under the Uð1Þl as a free parameter, Ng, where particles sufficiently heavy. As a result, these particles Ng ¼ 3ð2Þ in the case l ¼ Le þ Lμ þ Lτðl ¼ Lμ þ LτÞ. could be integrated out at energy scales and temperatures of ð1Þ Because the Uð1Þl symmetry in the SM is anomalous order of the U l breaking scale. with respect to SUð2Þ × Uð1Þ , additional fermions For our baryogenesis mechanism to work, we assume L Y 0 0 (so-called anomalons) must be introduced for anomaly both the g and c χ parameters to be small, so that the Z χ χ cancellation. A minimal set of new fermion content [15–17] boson, as well as the L, R fermions, have masses around, is given in Table I, where q is an arbitrary real number. This or even below, the electroweak scale. In the forthcoming is the UV complete framework we consider. discussions, we also show that χ qualifies to be the dark The right-handed neutrinos νi ; ði ¼ 1; …;N Þ could pair matter candidate. R g 0 00 0 00 νi After integrating out the L ;L ;e ;e fermions, which up with the active neutrinos L in the SM, so that in this L R R L minimal setup the observed neutrino masses are Dirac.3 play a role in the anomaly cancellation mechanism, the Uð1Þl current involving only light degrees of freedom becomes anomalous at lower energy. As it is well known, 2Not to be confused with the SM hypercharge Uð1Þ gauge Y integrating out the anomalon fields leads to the introduction coupling, gY . 3The possibility of Majorana neutrinos is considered in of the Wess-Zumino (WZ) term [14], which is necessary for Sec. IV B. restoring the SM gauge invariance when calculating the

055014-3 CARENA, QUIRÓS, and ZHANG PHYS. REV. D 101, 055014 (2020) triangle diagrams in the effective theory.4 However, the which governs the lepton/baryon number violating spha- coefficient of the WZ term is not fixed but depends on the leron processes. This is the key ingredient of our baryo- convention, i.e., the momentum routings, and such con- genesis mechanism, which makes use of the Z0 field vention needs to be respected when calculating the triangle background, as we discuss in the following section. diagrams [19]. In particular, in the convention of “covariant Another way, which is the other key ingredient in our anomaly,” the coefficient of the WZ term vanishes [20]. baryogenesis mechanism, is the Higgs portal interaction Observe, however, that in the baryogenesis mechanism between S and H, discussed in this work all the relevant processes occur at 2 2 tree level, and therefore issues of gauge invariance and L ¼ −λSHjSj jHj ; ð2:6Þ appropriate loop momentum convention do not play a role, since they would only matter in one-loop processes that is responsible for triggering a sufficiently strong first- involving the Z0 (see, e.g., [21]). order electroweak phase transition. In addition to the above particle content, baryogenesis requires the presence of another complex (SM singlet) III. ELECTROWEAK BARYOGENESIS scalar S, which also carries lepton number N . We assume g MEDIATED BY THE Z0 BOSON that S is much lighter than Φ, and its VEV vS evolves, together with that of the Higgs field, during the electroweak In this section we consider how the different ingredients phase transition. In contrast, the VEV vΦ of Φ remains play their roles for successful electroweak baryogenesis. constant as the Universe evolves in the proximity of the We discuss successively the out of equilibrium condition in electroweak phase transition, since at these scales the field the phase transition, the new source of CP violation, and Φ is decoupled. In the presence of the S field, one can write the generation of the baryon asymmetry. down a Yukawa term that gives an additional mass to the χ fermion . It takes the form A. The phase transition(s) We consider a first-order electroweak phase transition χ¯ ðm0 þ λ SÞχ þ H:c:; ð2:3Þ L c R during which the Higgs VEV turns on, while the VEV of the S field varies at the same time. Such a scenario can be where the first term is given by m0 ¼ c χvΦ and λc is a (complex) Yukawa coupling. As a result, the mass of χ realized through the following steps in the history of our changes with the S field profile during the electroweak Universe. (1) At very high temperatures, all symmetries are phase transition, and, if the relative phase between m0 and λ S is physical, it serves as a source of CP violation in our restored. c (2) As the Universe cools down to the temperature baryogenesis mechanism. ∼ Φ hΦi¼ To summarize, our assumptions lead to a low-energy TΦ vΦ, the field acquires its VEV, vΦ, and the lepton number symmetry is broken. The effective theory below the Uð1Þl breaking scale (vΦ), which contains the SM fields plus the following new fields: nature of this phase transition is not relevant here, but the breaking of lepton number may possibly 0 proceed by a second-order phase transition. Zμ;S;χ ; χ : ð2:4Þ L R (3) As the Universe further cools down to a temperature T not far above the electroweak scale T , the S Among them, S and χ are SM gauge singlets and belong S EW L;R field first develops a VEV, hSi ≠ 0, when its mass to the dark sector. There are two possible portals for them to squared term (including the thermal corrections) interact with the SM sector. becomes negative, while the Higgs VEV remains One way is through the leptonic Z0 portal, 0, hHi¼0. The transition to this step could be a 0 0 μ μ ¯ μ simple crossover or just a second-order phase L ⊃ g Zμ½ðq þ N Þχ¯ γ χ þ q χ¯ γ χ þ L γ L g L L R R L L transition. ¯ μ þ lRγ lR; ð2:5Þ (4) At the critical temperature near the electroweak scale, Tc, a new minimum of the potential with where LL represents the SM left-handed lepton doublets hHi ≠ 0; hSi ≃ 0 emerges that turns into the true and lR represents the SM right-handed charged leptons. minimum (replacing the former one with hSi ≠ 0; 0 00 0 00 h i¼0 Here, after integrating out the heavy LL;LR;eR;eL fermion H ). This process must involve a first-order fields, the Z0 couples to an anomalous current with respect phase transition requiring the presence of a barrier ð2Þ to the SM gauge symmetries, in particular, the SU L, between both minima. The Universe tunnels from one vacuum to the other via bubble nucleation. 4A manifestation of the nondecoupling properties of fields that A schematic picture of the phase transitions in steps 3 acquire their masses only through a spontaneously breaking and 4 is depicted in Fig. 3 (left panel). It has been shown mechanism. [10,22,23] that the above evolutions could be realized

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values of the potential parameters, we just assume hereafter that they are such that they provide a strong enough first- order phase transition. Detailed model analyses can be found in Refs. [10,22].

B. The source of CP violation The scalar potential, and the χ-S Yukawa coupling terms introduced so far [see Eqs. (3.1) and (2.3)], do not FIG. 3. Schematic plot of the phase transitions. The left plot violate CP yet. This is because the scalar potential (3.1) shows the change of S and H VEVs during steps 3 and 4 is only a function of jSj and, as a result, we are allowed discussed in the text. The right plot shows their VEV profiles in to redefine the argument of S to remove the relative front of and behind the expanding bubble wall (shadowed region) phase between m0 and λ S in (2.3). Moreover, any during the electroweak phase transition in step 4. The bubble c overall phase of the χ mass term can be further removed interior is for z<0. by redefining the phases of χL and χR fields. Hence any CP violation effect in the Yukawa terms can be absorbed dynamically by the interplay among the terms in the scalar by field redefinitions, leaving no physical effect during potential describing the Higgs and the new scalar field S.At the phase transition. zero temperature, the scalar potential reads as5 In order to accommodate a physical CP violating effect, which is a necessary condition for baryogenesis, one option ð Þ¼λ ðj j2 − 2Þ2 þ λ ðj j2 − 2Þ2 þ λ j j2j j2 V H; S H H v S S vS SH S H : is to introduce terms in the potential depending on S, which hinders the redefinition of argðSÞ. The general form of these ð3:1Þ terms is

The conditions for H ¼ v, S ¼ 0 to be the global 2 2 2 δVðSÞ¼ρ S þ μ S þ λ3 jSj S þ H:c: ð3:3Þ minimum are S S S Naively, these terms violate the Uð1Þ gauged symmetry λ 4 λ 4 λ 2 2λ 2 ð Þ l Hv > SvS; SHv > SvS: 3:2 and are forbidden in the UV complete theory. However, in this model, one can write renormalizable, Uð1Þl invariant terms involving Φ and S,as At high temperatures, both H and S receive thermal cor- rections to their quadratic terms, a T2jHj2 and a T2jSj2, δ ðΦ Þ¼ðμ2 þ λ jΦj2ÞΦ þ λ0 Φ2 2 H S V ;S ΦS ΦS S ΦS S with a > 0. Thus, at very large T, the potential is H;S þ λ00 j j2Φ þ ð Þ minimized for hHi¼hSi¼0 (steps 1 and 2). Given that ΦS S S H:c: 3:4 the Higgs field couples to more degrees of freedom than S, Clearly, after Φ develops its VEV and the Uð1Þl symmetry it follows that aH >aS, and it is always possible to find an intermediate temperature where the Higgs quadratic term is is spontaneously broken, Eq. (3.4) can generate (3.3), ρ μ λ positive, while the S quadratic term is negative (step 3), leaving the coefficients S; S; 3S complex in general. In δ thus triggering a minimum with hSi ≠ 0; hHi¼0.Atlower this discussion, we neglect the backreaction of V on the Φ temperatures, however, the Higgs quadratic term also turns VEV of the field, which is a higher order effect in the negative. This implies that there should be a critical small vS=vΦ expansion. temperature where the two minima, (hSi ≠ 0; hHi¼0) In the following, for simplicity, we present in more detail μ and (hHi ≠ 0; hSi¼0), are degenerate, allowing for step the case where only S is nonzero. We could first use the 4 to occur. The Higgs portal interaction λ jSj2jHj2 in freedom of field redefinition to make the parameters m0 and SH μ2 λ Eq. (3.1) [or Eq. (2.6)], which is a cross quartic term, could S real and positive, but c in general remains as a complex δ ð Þ¼2μ2j j2 ½2 ð Þ then provide a tree-level temperature-dependent barrier that parameter. In this case, V S S S cos arg S is separates the two minima allowing for a first-order phase the only term in the potential for argðSÞ. It is always transition. As this phenomenon depends on the particular minimized for argðSÞ¼π=2, such that

2 2 5 δVðSÞ¼−2μ jSj : ð3:5Þ As the field Φ is integrated out at the electroweak scale, the S 2 2 2 2 presence of the Higgs portal terms λΦHjΦj jHj and λΦSjΦj jSj in (3.1) would amount to a simple redefinition of the mass terms The physical source of CP violation arises from the χ j j2 j j2 for H and S , thus not changing the general conclusion that mass term, M χ χ¯ L χR þ M χ χ¯ R χL, where follows. Of course in that case we would have to face a little ≫ hierarchy problem, arising from the fact that vΦ v; vS, which iθ M χ ¼ m0 þ λe jSj: ð3:6Þ can be mitigated, e.g., by assuming λΦH; λΦS ≪ 1.

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Here we make the phase of the second term explicit, with parametrizes its value after the completion of the phase θ ¼ argðλcÞþπ=2 and λ ≡ jλcj. During a first-order transition (z=Lω → −∞). The bubble wall width and electroweak phase transition, in the presence of a bubble velocity are denoted as Lω and vω, respectively. Here, wall, the magnitude of jSj is space-time dependent; hence we focus on the special case κ ¼ 1 where, after the phase having used the freedom to make m0 real, the phase of M χ transition (corresponding to z ≪ 0), the VEV of the S field is not removable. As is discussed in the following sub- completely turns off. This can be realized in the presence of χ μ2 2 section, this phase modifies the dispersion relations of L;R, the SS term in Eq. (3.3) as discussed above. We expect and their antiparticles, in a CP violating way [3], and the qualitative features of our results to hold when the other provides the key source of CP violation for baryogenesis. terms in δVðSÞ are turned on, so that κ ≠ 1. When minimizing the potential, we can combine The phase transition relevant quantities, including the Eq. (3.5) with (3.1) and repeat the discussions in wall width Lω, the wall velocity vω, the scalar field profile Sec. III A, which still hold with the replacement across the bubble wall, as well as the critical and nucleation 6 temperatures, Tc and Tn, μ2 2 → 2 þ S ð Þ We define the particle chiral asymmetries in the dark vS vS ; 3:7 λS sector as [3,5], at the nucleation temperature, provided conditions (3.2) hold after the shift (3.7).A 3 ξ ð Þ¼ ð − Þ special feature of considering only a nonzero μ in χ z 3 n χ n χc ; S L T L L Eq. (3.3) is that, after the electroweak phase transition, n χ 3 the VEV of S can relax to 0, and the mass of today is ξ χ ðzÞ¼ ðn χ − n χc Þ; ð3:10Þ R 3 R R uniquely determined by m0. Tn Alternatively, if the tadpole term ρSS is turned on in (3.3), one can still derive the physical CP violating phase where Tn is the temperature when bubbles emerge, n χL;R is similar to (3.6), but the VEV of S after the phase transition ξ ≡ the number density of chiral asymmetry, and χL;R Tn remains nonzero. The impact of a nonzero S VEV is only of μ χL;R defines the corresponding chemical potentials. The relevance for the contributions to EDMs, as is discussed in Yukawa interaction of χ with the S background violates Sec. IV D. So in many of our subsequent discussions we L;R CP but preserves a global symmetry Uð1Þ χ, whose current assume, unless explicitly mentioned, that ρS ¼ 0. μ μ μ is defined as J χ ¼ χ¯ Lγ χL þ χ¯ Rγ χR. As a result, although ξ ξ nonzero values for χL and χR can be generated by CP C. The varyogenesis mechanism ξ ð Þþξ ð Þ violation in the dark sector, the sum χL z χR z In this subsection, we discuss the microscopic particle vanishes. The space-time dependence in the absolute value physics processes for our baryogenesis mechanism to work. of the χ mass, jM χðzÞj, and its phase, argðM χÞ, near the All of them happen near the expanding bubble wall, during bubble wall play an important role by modifying the a first-order electroweak phase transition (step 4 of the early dispersion relations of χ particles and their antiparticles Universe history described in Sec. III A), when the L;R in a CP violating way. This affects the phase space Universe tunnels from the electroweak symmetric vacuum distribution of these particles. The resulting chiral asym- to the broken one via bubble nucleation. Such a phase metries evolve according to the diffusion equation transition involves the simultaneous changes in the SM Higgs field and the scalar field S. We first rewrite the χ − ξ00 − ξ0 þ Γ ðξ − ξ Þ¼ ð Þ mass term (3.6) with explicit spatial coordinate dependence D χL vω χL m χL χR SCPV; 3:11 (labeled by z) in the rest frame of the bubble wall, 6 ¼ 0 ¼ iθ Tc is defined as the temperature at which the H and H M χðzÞ¼m0 þ λe jSðzÞj; ð3:8Þ vðTcÞ minima are degenerate, whereas Tn is the temperature at which the phase transition occurs. They are, respectively, all where z is the distance from the bubble wall, as shown calculable as functions of the model parameters (see, e.g., in Fig. 3 (right panel). The z>0 ðz<0Þ region is the Ref. [24]). The main goal of this work, however, is to present electroweak symmetric (broken) phase located outside a new baryogenesis mechanism; hence we leave a detailed study of the strong first-order phase transition, and, in particular, the (inside) the bubble. Our discussion here is in the basis precise calculation of the value of T and the value of the Higgs ð λ θÞ n where m0; ; are all real parameters. We parametrize the field at Tn, vðTnÞ, for a future publication. The detailed analysis profile of jSðzÞj taking the form of the precise requirements on the model parameters for the phase transition is a straightforward task that however involves com- putationally intense calculations. In the present work, we assume jSðzÞj ¼ s0½1 þ κ tanhðz=LwÞ=2; ð3:9Þ that the model parameters are such that vðTnÞ=Tn ≳ 1, and we scan over a generous range of T values, as well as over other ð1 þ κÞ 2 j j n where s0 = is the value of S in the electro- relevant model parameters, including Lω and vω, as shown in weak symmetric phase (z=Lω → ∞), and s0ð1 − κÞ=2 Eq. (3.27).

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FIG. 4. Left panel: Chiral charge asymmetry in χL (opposite for χR) particles around the bubble wall, with parameters EQ 02 3 m0 ¼ s0 ¼ T ¼ 100 GeV, M 0 ¼ 1 GeV, λ ¼ 0.3, θ ¼ π=3, Lω ¼ 5=T , vω ¼ 0.1. Right panel: Δn ðzÞ=g T for the same values n Z n LL n of the parameters. For this plot we only show the result in the region z>0 because it corresponds to the range of integrals in Eq. (3.24) or (A4). where ð 0Þ means derivative with respect to z. The The solution to the above diffusion equation is formally 2 diffusion constant D is given by D ¼hv i=ð3ΓmÞ, with given by Γ ∼ λ2T =ð4πÞ, v is the particle velocity in the bubble Z m n ∞ wall rest frame, and hi is the thermal average over the ξ ð Þ¼ ð − Þ ð Þ ð Þ χL z dz0 G z z0 SCPV z0 ; 3:15 Fermi-Dirac distribution function fiðpÞ (i ¼ χL; χR) in the −∞ rest frame of the bubble wall, where the Green’s function GðzÞ satisfies the equation 1 ð Þ¼ ð Þ 00 0 fi p ð þ −μ Þ ; 3:12 −DG ðzÞ − vωG ðzÞþ2Γ GðzÞ¼δðzÞ: ð3:16Þ e E vωpz i =T þ 1 m The solution, continuous at the origin, is given by where μi is the chemical potential. The corresponding number density for χL, χR is defined as  −1 e−kþz;z≥ 0 Z ð Þ¼ D G z − ; 2 kþ − k− k−z 0 n ¼ d3pf ðpÞ: ð : Þ e ;z< i ð2πÞ3 i 3 13 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 8Γ ¼ vω 1 1 þ mD ð Þ k 2 : 3:17 The CP violating source term can be calculated using 2D vω Refs. [3,5] as   In the left panel of Fig. 4, we show the chiral asymmetry vω v χ ¼ z ½j ð Þj2ð ð ÞÞ000 distribution of L as a function of the z coordinate, for a SCPV 2 M χ z arg M χ z ΓmTn 2E given set of model and phase transition parameters.   λ½−2 þ ð2z Þ θ Unlike in the usual electroweak baryogenesis scenar- vω v m0s0 cosh sin ¼ z Lω ios, here the particle chiral charge asymmetry is gen- 2 3 4 z ; Γ T 2E Lωcosh ð Þ χ m n Lω erated in the dark sector through the particle, which is ð2Þ ð3:14Þ an SU L singlet and thus does not couple to the electroweak sphalerons. Moreover, for general values 2 2 2 of q, the gauge symmetry Uð1Þl forbids any renormaliz- where E ¼ p þjM χðzÞj . able operators through which the asymmetries in χ might Clearly, in Eq. (3.11), the source term S must be CPV be directly shared with the SM fermions that carry the nonzero in order to generate nonzero asymmetries in the ð2Þ 7 SU L charge. We here make the observation that, χ particle numbers, which are proportional to ξ χ , res- 0 L;R L;R thanks to the leptonic Z portal, which couples to both χ ð ð ÞÞ0 pectively. This requires a nonzero value of arg M χ z ; and the SM leptons, the CP violating effect in the dark i.e., the phase of the χ mass must not be a constant—it has sector can be transferred in a novel way to the observable to vary together with the S VEV along the z direction. sector. A quick glance at the form of the χ mass term in Eq. (3.8) shows that m0 has to be different from 0. We come back to 7As explained in the introduction, this aspect serves as a major this point near the end of this section when discussing the difference between our work and that in Ref. [10]. In our case, a numerical calculation of the baryon asymmetry and the new way of transferring the χ particle chiral charge asymmetry to scan over the parameter space. the visible sector is presented.

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χ χ 2 2 2 0 2 The main point here is that L and R carry different EQ NgTn g NgTn 0 8 Δn ðzÞ¼ μL ðzÞ¼ hZ0ðzÞi: ð3:21Þ Uð1Þl charges (q þ Ng and q, respectively). Conse- LL 3 L 3 quently, the above chiral asymmetries imply a net Uð1Þl charge density near the bubble wall as We show in the right panel of Fig. 4 the spatial distribu- tion of ΔnEQðzÞ for a given set of model and phase LL ρlðzÞ¼ðq þ NgÞ½n χ − n χc þq½n χ − n χc transition parameters. It is worth mentioning that the L L R R EQ 0 profiles Δn ðzÞ and hZ0ðzÞi depend on our assumption 1 3 LL ¼ N T ξ χ ðzÞ; ð3:18Þ 3 g n L of the bubble profile, Eq. (3.9). In the presence of the electroweak sphaleron processes, where use has been made of Eq. (3.10). The existence of which can change the lepton number, the actual SM lepton number asymmetry evolves toward its equilibrium value. this net Uð1Þl charge density yields a Coulomb back- 0 0 This evolution is governed by the following rate equation, ground of the Z potential, hZ0i. In the approximation of very large bubbles, this lepton number potential could be ∂Δn ðz; tÞ LL ¼ Γ ð − Þ½Δ EQð − Þ − Δ ð Þ calculated in cylindrical coordinates as sph z vωt n z vωt nL z; t ; ∂t LL L Z 0 ∞ ð3:22Þ 0 g hZ0ðzÞi ¼ dz1ρlðz1Þ exp ½−MZ0 jz − z1j; ð3:19Þ 2 0 MZ −∞ Γ where sph is the rate for the sphaleron process at the j ð Þj 0 nucleation temperature Tn. The second term on the right- where we neglect the impact of S z on the mass of Z , hand side of Eq. (3.22) represents the washout term, which which is mainly set by the value of vΦ at a much would drive the asymmetry to 0 if the sphaleron processes higher scale. 0 did not go out of equilibrium quickly enough. Assuming a The background of the vector field Z breaks the Lorentz strong first-order electroweak phase transition, where the symmetry and thus is a CPT violating effect, which is also condition vn=Tn ≳ 1 is fulfilled (vn is the Higgs VEVat the odd under the CP transformation. It retains certain simi- nucleation temperature T ), a good approximation for Γ larities to the spontaneous baryogenesis mechanism [25] n sph is that it is unsuppressed at any point z outside the bubble (also with gravitational baryogenesis [26]), where a time- wall, but becomes exponentially suppressed after the dependent (CPT violating) scalar field couples to the vector bubble wall has passed through taking this point to the current of a particle, and serves as its chemical potential.9 0 bubble interior, i.e., In our model, we use the timelike component of the Zμ gauge boson, whose CP and CPT violating background  Γ0 ∶ t < z=vω Γ ð − Þ¼ ð Þ is generated due to the microscopic interaction processes sph z vωt − : 3:23 Γ Msph=Tc ∶ between the dark sector particles and the bubble wall 0e t > z=vω described above. The Z0 background couples to the SM 0 Γ ≃ 120α5 ≃ 10−6 ¼ lepton current [see Eq. (2.5)]. As we see, given that this In Eq. (3.23), 0 wTn Tn [27], and Msph 4π ð2Þ vnB=g2 is the sphaleron mass in the broken phase, where current is anomalous with respect to the SM SU L gauge symmetry, it could bias the sphaleron process to work in B is a fudge factor [1] that depends on the Higgs mass, one direction. The Z0 background then yields a chemical and the weak coupling g2. In the SM, for the experimental 0 ≃ 1 96 potential for the SM leptons, value of the Higgs mass it turns out that B . .As discussed in detail in [28], the sphaleron rate in the 0 0 presence of an additional singlet depends on the parameters μ ðzÞ¼μl ðzÞ¼g hZ0ðzÞi: ð3:20Þ LL R in the VðS; HÞ potential, and could be calculated once this parameter dependence of the first-order phase transition is The thermal equilibrium asymmetry in SM lepton number worked out. would then be given by (considering left-handed lepton The solution to the rate equation takes the form [3] doublets) Z ∞ Γ0 Δ ¼ Δ EQð Þ −Γ0z=vω ð Þ nL dz nL z e : 3:24 8Note, their charges are not chosen by hand but, instead, L vω 0 L required by the anomaly cancellation conditions discussed in Sec. II and Table I. We refer the reader to Appendix A for more details on 9 0 Notice that the VEV of Z0 vanishes after the electroweak obtaining this result. At this point it is important to realize χ phase transition, as its value stems from the asymmetry in L;R that the final lepton number density, as given by Eq. (3.24), particles, which vanishes when argðM χÞ becomes a constant and is nonvanishing as a consequence of the fact that the the source of CP violation SCPV vanishes. Therefore at zero temperature our model does not contain any violation of Lorentz effective theory at the scale of electroweak baryogenesis symmetry. has an anomalous lepton number. Had we not integrated out

055014-8 DARK CP VIOLATION AND GAUGED LEPTON OR BARYON … PHYS. REV. D 101, 055014 (2020) any anomalon propagating in the UV theory, the final ΔnB ηB ¼ : ð3:25Þ lepton number density ΔnL would have been 0. This s statement is proven in detail in Appendix B. See also [11]. Because the sphaleron processes preserve B − L, equal The dark blue points in Fig. 5 show the working para- asymmetries are generated for baryon and lepton numbers, meter space where the observed baryon asymmetry [29] Δ ¼ Δ nB nLL . The entropy density of the Universe at the 2 3 η ≃ 0 9 10−10 ð Þ EW scale is s ≃ ð2π ÞgTc=45, where g ≃ gB þð7=8ÞgF ≃ B . × 3:26 Oð100Þ is the effective number of degrees of freedom at the EW phase transition. The final generated baryon-to-entropy can be generated. They are obtained by scanning over all ratio is then the model parameters in the following ranges,

−3 3 −2 MZ0 ;m0 ∈ ð10 ; 10 Þ GeV;s0;Tn ∈ ð100; 500Þ GeV; λ ∈ ð10 ; 1Þ; 0 −6 g ∈ ð10 ; 0.1Þ; θ ∈ ð−π=2; π=2Þ;Lw ∈ ð1=Tn; 10=TnÞ;vω ∈ ð0.05; 0.5Þ: ð3:27Þ

Here, the parameter m0 is the mass of the χ particle, is kinematically forbidden to decay into χ χ¯ . If created in assuming S has no VEV today. the laboratory, it decays into SM particles. This is a visible We display, in Fig. 5, the baryogenesis viable points in decay, and in the next section we confront these points with 0 0 the g versus MZ0 plane assuming Ng ¼ 3 (the case Ng ¼ 2 the existing, and near-future, Z searches. It is worth is independently exhibited in Sec. V), where the mass pointing out that the values of g0 of interest for successful parameters satisfy the relation m0 >MZ0 =2. The result baryogenesis are much smaller than 1; thus the back- shows that the smaller the Z0 mass, the smaller the value of reaction of Z0 particles on the bubble wall is negligible. 0 g in the allowed region. In particular, with MZ0 around On the other hand, we find that the resulting points with 0 0 −5 0 2 0 100 MeV, the gauge coupling g should be as small as 10 . m0 10 everywhere). This could be under- tional to ∼g =M 0 . In this case, m0 >MZ =2, the Z boson Z stood from the explicit expression for the source of CP violation for the baryogenesis mechanism SCPV. As dis- cussed in the paragraph below Eq. (3.12), the relevant CP violation source is proportional to the gradient of argðM χÞ along the z direction, where the VEV of S changes. Clearly, if the m0 term is very small, argðM χÞ 0 remains approximately θ, and ðargðM χÞÞ would be sup- pressed. To compensate for this suppression, larger values of g0 are needed. In this case, we find that the experimental constraints from invisibly decaying Z0 searches [30] are already strong enough to exclude almost the entire viable parameter space for baryogenesis. Therefore, we do not consider this case any further.

IV. PHENOMENOLOGY ¼ 3 FIG. 5. The parameter space of our model (assuming Ng ) In this section, we discuss the phenomenological con- that could generate the observed baryon asymmetry of the 0 sequences of the above-described baryogenesis mecha- Universe is covered by the blue points, in the g versus MZ0 nism. We show that generating the observed baryon parameter space. The colorful shaded regions have been excluded asymmetry in the model has a strong impact on the Z0 by the existing constraints from LEP, BABAR, electron g − 2, boson search, on the physics of χ as the dark matter beam dump, and neutrino-electron scattering experiments, as well as the measurement of flavor-changing K → π, B → K decay candidate, and on the electric dipole moments, as well as on rates. The yellow band is the favored region for explaining the possible LHC signals of the Higgs boson and the dark muon g − 2 anomaly. The black dashed line corresponds to the Higgs S. VEV vΦ equal to 1, 10 TeV. We consider in the parameter Throughout the discussions in this section, we assume 0 2 scanning the condition m0 >MZ = . the parameter Ng, the number of lepton flavors charged

055014-9 CARENA, QUIRÓS, and ZHANG PHYS. REV. D 101, 055014 (2020) under the Uð1Þl, to be equal to 3. We comment on the with these final state stringent limits can be set on the gauge differences in phenomenology if only two lepton flavors coupling g0. 0 are gauged, e.g., Lμ þ Lτ, in the upcoming Sec. V. The existing experimental constraints on a leptophilic Z are summarized in Fig. 5 for the Ng ¼ 3 model. These A. Searches for the leptophilic Z0 limits, altogether, set a lower bound on the Z0 mass of 0 around 10 GeV. Prospective Higgs factories [39] could First of all, let us recall that the presence of the Z boson 0 is the key for the success of our electroweak baryogenesis explore regions with larger Z masses. In addition, the gauge coupling g0 is indirectly con- mechanism. It needs to develop a CP (and CPT) violating ð1Þ background during the electroweak phase transition, which strained by requiring the anomalon fields for the U l CP symmetry to be sufficiently heavy. As discussed in Sec. II, permits transferring the violating effect from the dark ð1Þ sector to the SM leptons. In order to generate sufficient the gauged U l symmetry is broken by the VEV of a 02 2 scalar field Φ above the electroweak scale. The same VEV final baryon asymmetry, which is proportional to g =MZ0 , 0 0 defines the mass of the anomalon fields, as a function of the gauge boson Z cannot be too heavy and the coupling g their Yukawa coupling. To secure that the anomalon fields should not be too small, as shown in Fig. 5. 0 are already decoupled during the electroweak phase tran- At the same time, since the Z is the gauge boson for the sition, while avoiding the Yukawa couplings to be in the lepton number symmetry, it couples to the SM charged strongly coupled regime, values of vΦ above a few times leptons and neutrinos. Such a new vector particle has þ − the electroweak scale are required. In Fig. 5, we show been directly searched for at e e colliders, such as LEP indicative values of vΦ ¼ 1 and 10 TeV, respectively, where (both through resonances [31] and contact interactions we have used the Z0 mass given in Eq. (2.1). This shows [32]) and BABAR [33], as well as at electron beam dump that most of the experimentally allowed, EWBG-favored experiments [34], and neutrino experiments that are sensi- solutions are in the region of M 0 above 10 GeV. tive to neutrino-electron interactions (such as TEXONO) Z 0 Finally we comment that, in general, there is a kinetic [35]. The Z could also be exchanged at the loop level and mixing between the hypercharge gauge boson Bμ and the contribute to the anomalous magnetic moments of charged 0 new gauge boson Zμ,as leptons [36]. Many of these constraints are similar to, and could be translated from, the limits on dark photons 1 0 L ¼ − ðμÞ μν 0 ð Þ [35,37]. Because the Z now mainly couples to charged kin 2 c FY Fμν; 4:2 leptons and neutrinos, we reevaluate its branching ratios based on the following partial decay widths, where the coefficient cðμÞ receives renormalization at loop  sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi level. Its one-loop beta function takes the form [40] 02 2 2 g 2ml 4ml Γ 0 ¯ ¼ 0 1 þ 1 − 0 Z →ll MZ 2 2 ; ∂cðμÞ gYg 12π M 0 M 0 ¼ TrðYLÞ: ð4:3Þ Z Z ∂ log μ 12π2 g02 Γ 0 ¼ 3 0 ð Þ Z →νν¯ × 24π MZ ; 4:1 In the complete UV theory considered here, we have that TrðYLÞ¼−4ðq þ 3Þ. There is a special case, q ¼ −3, where l ¼ e, μ, τ. We neglect the Z0 decay into right- where the kinetic mixing parameter cðμÞ does not run at 10 handed neutrinos, assuming it is kinematically forbidden. energies above the Uð1Þl symmetry breaking scale, vΦ. 0 Because the Z boson in this model is hadrophobic, the For μ meson decays ( , J= , ) into Z only LL;LR;eL;eR,TrYL in the effective theory. This ¼ 0 apply through loop-level processes [35]. implies that even if we set cUV at high scale as the Moreover, because the Z0 couples to an anomalous boundary condition, it will be generated at low energies as SUð2Þ2 current with respect to L in the low-energy theory, 0 it makes important contributions to flavor-changing meson ð Þ ≃ þ gYg vΦ ð Þ 0 0 c MZ cUV 2 log ; 4:4 decays such as K → πZ and B → KZ through the Wess- 2π MZ Zumino term that occurs at two-loop level [38].11 For very 0 0 00 00 0 light Z , these decays are mainly into the longitudinal where we are assuming that the masses of LL;LR;eL;eR 0 component of the Z boson and the corresponding rates are are all of order vΦ, and compute the value of c at the MZ 1 2 0 mass scale where it is measured at the LEP experiment. enhanced by =MZ0. The Z boson then decays into charged 0 A nonzero kinetic mixing between Bμ and Zμ generates, lepton pairs or neutrinos. Following Ref. [38], we find that 0 after electroweak breaking, a mixing between Zμ and Zμ.

10 This could impact LEP observables including the Z boson The origin of right-handed neutrino masses is addressed in ρ Sec. IV B. mass (the parameter), the Z hadronic width, and the 11We thank Jeff Dror for pointing out to us the results in forward-backward asymmetries in leptonic Z decays. Ref. [38]. The analysis in [40] finds that cðMZÞ is constrained to

055014-10 DARK CP VIOLATION AND GAUGED LEPTON OR BARYON … PHYS. REV. D 101, 055014 (2020) be less than the percent level, with a much stronger in Fig. 5. See [43] for a recent calculation in a similar constraint in the region where Z and Z0 are nearly context. degenerate [41]. Compared to the EWBG favored region The viable alternative option for neutrino mass is to for g0 in Fig. 5, we find it easy to satisfy these constraints implement the seesaw mechanism by giving Majorana ν ν ∼500 provided cUV is small enough. masses to Ri . If all the Ri are heavier than MeV, To summarize, after taking into account all the above they decay before the big bang nucleosynthesis and have no 0 Δ constraints, the mass window of the Z for our baryogenesis effect in Neff [44]. However this option requires extend- mechanism to work is 10 GeV

≳ 10 0 ≫ ð Þ section is [46] vΦ TeV; for MZ TQCD: 4:6

ðσ Þ 0 0 0 ≪ vrel χ χ¯ →Z Z On the other hand, if MZ TQCD, the thermal averaged   2 3 2 M 0 = annihilation rate scales as T until the temperature falls Z 0 04 1 − 2 ν g m0 below the Z mass. In this case, requiring that R never ¼   i 2 2 2 64π M 0 reaches thermal equilibrium implies that MZ0 1 − Z 2m2    0  1=4 2 0 −5 MZ0 2 M 0 2 2 ≲ 10 0 ≪ ð Þ 18ð2q þ 3Þ þ Z ð2q − 9Þð2q þ 12q þ 9Þ g ; for MZ TQCD: 4:7 × 2 1 MeV m0 04 2 m0≫M 0 9g ð2q þ 3Þ Satisfying conditions (4.6) and (4.7) imposes strong con- ⟶Z ; ð : Þ × 32π 2 4 9 straints on the EWBG viable parameter space found MZ0

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This relation is shown by the red curve in Fig. 7 (left panel), for a particular value of q ¼ −3 (similar results hold for other values of q, as long as q is of order 1). Comparing with the blue and magenta dots, which are the phenom- enologically allowed points for successful baryogenesis (surviving the various constraints in Fig. 5), we find these values of g0 are too small to account for the correct dark matter relic density this way, unless the dark matter charge q value is unnaturally large. Hence, we need larger contributions to the dark matter annihilation cross section from additional channels. Next, we consider the s-channel Z0 exchange, as shown by Fig. 6 (upper-right diagram), where χ χ¯ annihilate into SM charged leptons and neutrinos. The corresponding cross section is (assuming the limit m0 ≫ MZ0 )

9g04ð2q þ 3Þ2 FIG. 6. Feynman diagrams for dark matter thermal freeze-out ðσ Þ þ − ¼ ð Þ vrel χ χ¯ →l l ;νν¯ 2 : 4:11 (first row) and direct detection (second row) in the model we 128πm0 consider. Time flows from left to right. Comparing this expression with Eq. (4.9), we find that ðσ Þ vrel χ χ¯ →lþl−;νν¯ is not sufficiently large, since it is para- χ χ¯ ðσ Þ ≫ 0 where vrel is the relative velocity between and particles metrically smaller than vrel χ χ¯ →Z0Z0 , for m0 MZ . The before the annihilation, and in the last step we take the limit 2 2 latter having an enhancement factor, m0=MZ0 , which arises ≫ 0 χ that m0 MZ . Requiring that obtains the observed relic from χ χ¯ mainly annihilating into the longitudinal compo- abundance [29] through this annihilation mechanism, nent of the Z0 boson. we get Finally, we consider the dark matter annihilation into the sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dark scalar S. Here we first derive the dark scalar spectrum and its couplings to the dark matter χ. The most general M 0 g0 ≃ Z : ð4:10Þ scalar potential of S is given by the sum of Eqs. (3.1) and 5.9 TeV × j2q þ 3j (3.3). We focus on the case where in (3.3) only the

FIG. 7. Confronting the electroweak baryogenesis favored parameter space (shown by the blue and magenta points) with dark matter observables, assuming the χ particle, which sources CP violation in baryogenesis, is also the dark matter candidate. All the blue and magenta points in the plots satisfy the constraints on the Z0 boson shown in Fig. 5. The magenta points are consistent with both the observed baryon asymmetry and the dark matter direct detection experiments, while the blue points fail to pass the latter constraint. In the left (right) panel, on the red curve (band), the χ particle could explain the correct relic density through the thermal freeze-out mechanism via the annihilation channel χ χ¯ → Z0Z0 (χ χ¯ → rr; aa; ra).

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μ2 2 þ quadratic term SS H:c: is present, and the VEV of S points obtained from the baryogenesis scan, and now relaxes to 0 when the dark matter freezes out (which shown in the λ versus m0 parameter space. This comparison typically occurs at temperatures below the electroweak makes it clear that there exists a viable region in the phase transition). In this case, CP can be violated in the parameter space where both successful electroweak baryo- dark sector as explained in Sec. III B. We can first redefine genesis and correct dark matter relic density are achievable. the phases of S and χL;R fields so that m0 and μS are real The favored region of dark matter mass is around a few parameters, but the λc coupling in Eq. (2.3) remains hundred GeV. iθλ complex in general. As before, we rewrite λc ¼ λe with λ and θλ being real parameters. In this basis, the complex 2. The direct detection scalar S is separatedpffiffiffi into its real and imaginary parts Direct detection of dark matter in this model could occur ¼ð þ Þ 2 0 0 S r ia = , where r and a are the physical mass through Z exchange. However, because the Z is the gauge eigenstates, with respective masses boson for lepton number, it does not directly couple to nucleons, implying that the dark matter-nucleon scattering 2 ¼ λ 2 − 2λ 2 þ 2μ2 Mr SHv SvS S; should occur through the loop of charged leptons that 2 ¼ λ 2 − 2λ 2 − 2μ2 ð Þ effectively act as a kinetic mixing between the Z0 and the Ma SHv SvS S: 4:12 photon, as shown in Fig. 6 (lower-left diagram). The 2 2 corresponding spin-independent cross section for this Conditions (3.2) and (3.7) guarantee that both Mr and Ma μ2 2 þ process is [47] are positive. Clearly, the presence of the SS H:c:   potential term breaks the degeneracy between r and a, 16α2α02ðq þ 3 2Þ2μ2 X 2 ≠ = p 2 Mr Ma. It is then straightforward to rewrite the Yukawa σ χ → χ ¼ fðq ;mlÞ ; p p 81πð 2 − 2 Þ2 interaction, Eq. (2.3), into those between r, a and the q MZ0 l¼e;μ;τ fermion χ, which takes the form ð4:16Þ

L ¼ λ iθλ χ¯ χ þ dark Yukawa e L RS H:c: 0 02 where α ¼ g =ð4πÞ, μ ¼ m0m =ðm0 þ m Þ is the r p p p ¼ pffiffiffi ðλ cos θλ χχ¯ þ λ sin θλ χ¯ iγ5 χÞ reduced mass of the dark matter and target nucleus system 2 (mp is the proton mass), and a þ pffiffiffi ð−λ sin θλ χχ¯ þ λ cos θλ χ¯ iγ5 χÞ: ð4:13Þ " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 4 2 ð 2 Þ¼ 5 2 þ 12 2 þ 6ð 2 þ 2 2 Þ 1 − ml f q ;ml 2 q ml q ml 2 With these interactions, we calculate the cross sections q q sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for χ χ¯ annihilating into rr, aa, and ra. The corresponding ! # 4 2 Λ2 Feynman diagrams are shown in Fig. 6 (upper-middle 1 − ml þ 3 2 × arccoth 2 q log 2 ; diagram). The sum of these annihilation cross sections is q ml

4 ð4:17Þ λ ð3 − cos4θλÞ ðσv Þ þðσv Þ þðσv Þ ≃ ; rel χ χ¯ →rr rel χ χ¯ →aa rel χ χ¯ →ra 256π 2 m0 where Λ is the cutoff scale corresponding to the renorm- 0 ð4:14Þ alization of the effective Z − γ kinetic mixing. We set Λ ¼ 1 TeV in our calculation, and assume q ∼ Oð1Þ. The where we assume that the final state particles r and a are typical square momentum transfer of the scattering is of much lighter than χ. Obtaining the correct relic density for order q2 ¼ −4μ2v2, where v ≃ 10−3 is the typical halo dark χ through this channel then requires λ to lie within the matter velocity.12 In Fig. 7, the points in magenta are window compatible with the present dark matter direct detection rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi constraints [48], and can generate the observed baryon m0 m0 asymmetry in the Universe. < λ < ð4:15Þ 1.4 TeV 1.0 TeV In addition, the dark matter direct detection could also be mediated by the scalar S (or equivalently the r, a mass for 0 < θλ < 2π. This relation is derived by assuming that eigenstates) and the Higgs boson exchange. If S has no the χ χ¯ → Z0Z0 and χ χ¯ → lþl−; νν¯ annihilation cross VEV today, the dark matter scattering is a loop-level sections discussed above are much smaller than the one process, as shown in Fig. 6 (lower-right diagram). In this in Eq. (4.14), and thus negligible when accounting for the case, the cross section arises from a loop suppressed Higgs total value of the thermal relic density. Region (4.15) is shown by the red band in Fig. 7 (right panel). Again, the 12In the case of the xenon nucleus target, we have μ ¼ ð þ Þ ∼ 130 blue/magenta dots are the phenomenologically viable m0mXe= m0 mXe , with mXe mp.

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0 ˜ 0μν FIG. 8. Two-loop generated hZμνZ vertex. portal interaction and is sufficiently small and can be neglected [49]. On the other hand, if S were to have a nonzero VEV, it would mix with the Higgs boson and the 0 ˜ 0μν FIG. 9. Two-loop generated electron EDM, from the hZμνZ dark matter scattering would occur at tree level. In such a 0 ˜ 0μν vertex (represented by the gray blob). In our model, the hZμνZ case, the direct detection constraints could become impor- is generated at two-loop level; see Fig. 8. The photon must be tant depending on the mass of S and the size of its mixing radiated from one of the internal propagators and that has to be an with the Higgs boson [50]. electron propagator because everybody else is electrically neutral.

D. Implications for electric dipole moments fields, there is no CP violation in the dark sector—all the We comment here on the implications of our baryo- parameters can be made real by field redefinitions—and genesis model for the electric dipole moment experiments. there is no contribution to any CP violating operators. It is generically expected that the CP violating interaction In the presence of dark sector CP violation, when the χ 0 ˜ 0μν between S and , required for successful baryogenesis, will contribution to the hZμνZ operator is nonzero, we could propagate at loop level to the standard model sector, giving use it to further generate the EDM for the electron, at the rise to EDMs. price of another two loops, as shown in Fig. 9. Unlike the The relevant interaction and mass terms for CP violation Barr-Zee-type diagrams for EDMs [12], here we must in the dark sector are given in Eqs. (2.3) and (3.3). We first attach both Z0 s to the electron line and the external photon consider the case where the VEVof S at zero temperature is to either of the internal electron propagators. μ2 2 þ 0 and only the SS H:c: term is present in Eq. (3.3).As By simple power counting, the resulting electron EDM is explained in Sec. IV C 1, the complex scalar S splits into its real and imaginary parts, yielding the physical mass eGFme 2 04 2 d ∼ ðλ λ g q Þ sinð2θλÞ eigenstates, r and a, respectively, and their interactions e ð16π2Þ4 SH with dark matter are given by Eq. (4.13).Ifθλ ≠ 0, the r and −30 2 04 2 ≲ 10 ðλ λ g q Þ sinð2θλÞe cm: ð4:19Þ a fields couple to both scalar (χχ¯ ) and pseudoscalar SH (χ¯ iγ5 χ) operators involving the χ fields. At the same time, This estimate is valid assuming that the r and a mass they also couple to the SM Higgs boson through the Higgs difference is around the electroweak scale. With the factor portal interaction, Eq. (2.6), 2 04 2 ðλSHλ g q Þ < 1, the resulting electron EDM is well below the current upper bound on d , which comes from the λ v e λ j j2j j2 ⊃ SH ð 2 þ 2Þ ð Þ ACME experiment [7]: d < 1.1 × 10−29e cm. As men- SH S H 2 h r a : 4:18 e tioned in the introduction, this is an appealing feature of our Then Eqs. (4.13) and (4.18) allow us to derive a CP model for electroweak baryogenesis that, unlike many 0 0 ˜ 0μν others, is safe from the EDM constraints, even if the CP violating Higgs-Z operator, of the form hZμνZ ,attwo- phase is of order 1. loop level, as shown in Fig. 8. Out of the two vertices where Finally, we comment on the case where the VEV of S at the dark scalars (r or a) are attached to the χ loop, one of zero temperature is nonzero. In this case, from the Higgs them needs to be the scalar coupling in Eq. (4.13) and the portal interaction, Eq. (2.6), there is a direct mixing other the pseudoscalar coupling, so that CP can be violated. 0 ˜ 0μν 0 ˜ 0μν between r and h fields. As a result, the hZμνZ vertex The resulting coefficient of the hZμνZ operator is 2 could be generated by replacing the scalar loo in Fig. 8 by proportional to λ sin θλ cos θλ. the r − h mixing, with only one r attached to the fermion It is worth noting that the nondegeneracy between r and loop via the pseudoscalar coupling, which becomes a one- a is the key for the coefficient of this operator to be loop diagram. The contribution to the electron EDM in this nonzero; otherwise the coupling structure in (4.13) would case reduces to three loops, lead to a complete cancellation between the two diagrams involving r and a, respectively. This cancellation could also −28 2 de ∼ 10 ðλSHvvS=Mr Þ sin θλe cm; ð4:20Þ be understood from a symmetry argument. Based on the δ 2 discussions in Sec. III B, if the V potential (containing where the factor ðλSHvvS=MrÞ is the mixing between r and μ2 2 SS term) vanishes, thus leading to degenerate r and a h. The Higgs boson rate measurements at the LHC requires

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0 FIG. 10. Feynman diagrams for the loop induced decay of r, a into two Z bosons (upperpffiffiffi left) and the production process gg → rr (or aa) via an off-shell Higgs boson (upper right). The cross section for the latter at s ¼ 13 TeV LHC is shown in the lower panel. this mixing must to be less than ≲20% [51]. This implies charged leptons sits on the Z0 resonance, so their invariant −29 that de ≲ 10 sin θλe cm, allowing the predicted EDM to masses all line up in the same energy bin corresponding to be closer to the current upper bound and giving a prospect the Z0 mass. Moreover, because r (and a) has both CP even for future electron EDM searches. and odd couplings with χ, the effective operators for its 0 0μν decay (after integrating out χ in the loop) are rZμνZ and 0 ˜ 0μν E. Possible LHC signals of the dark scalar(s) rZμνZ . The interference of the two decay amplitudes In this subsection, we comment on the possible collider allows us to probe CP violating observables in the final signals of the new scalar S in our model. Unlike the state charged lepton angular distributions, in analogy to electroweak phase transition discussion, where only the S using the golden channel of the Higgs decay to probe CP field background is relevant, here we consider the S violation [52]. excitations, being produced as particles. As mentioned in For the production of the new scalars r, a, we resort to Sec. IV D, the physical states from the S field are its real, r, the Higgs portal interaction, Eq. (2.6) or (4.18). If the S and imaginary, a, parts, which have different masses. field has no VEV today, there is a Z2 symmetry at this Their interactions with χ are given by Eq. (4.13); thus, vertex that requires that r or a must be pair produced. This if kinematically allowed, they could dominantly decay may occur at the LHC, or a prospective future hadron into χ χ¯ . However, as discussed in Sec. IV C, for the dark collider, through the gluon fusion process that creates an matter χ to freeze out effectively we need r and a to be off-shell Higgs boson, which later on splits into two r (or a) lighter than χ. In this case, they have to decay via a loop particles, as shown in Fig. 10 (upper right panel). The of χ into a pair of Z0 bosons, as shown in Fig. 10 (upper corresponding production cross section at the LHC is left panel).13 This could lead to a potentially interesting shown in the lower panel of Fig. 10. Quantitatively, 0 σ ∼ 10λ2 ∼0 1λ2 ≃ 150 signature because the Z boson, which is typically lighter gg→rr;aa SH fb ( . SH fb) for Mr;a GeV χ than (necessary for successful baryogenesis), has to (for Mr;a ≃ 300 GeV). After the decays of the r (or a) decay into SM charged leptons or neutrinos. Each decaying scalars, the final state could contain as many as four pairs of r or a could then produce as many as four charged leptons. charged leptons, which would provide a very striking There is important information about the model in these signal. A recent analysis [53] has shown that the multi- charged lepton decay products. First, each pair of the lepton final state data from the LHC [54] could already set useful limits on dark sector models. Comparing the 13A similar diagram makes in the standard model the “golden production cross section shown in Fig. 10 with the limits channel” decay h → γγ via a top quark loop. derived in [53], we find that the existing LHC data could

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differences and similarities for the EWBG predictions, as well as the phenomenological implications between this two flavor case and the previously studied three flavor case with l ¼ Le þ Lμ þ Lτ. The previous discussion on our recently proposed EWBG mechanism in Sec. III has assumed a generic value of Ng. The parametric dependence of the final baryon asymmetry to entropy ratio is given by

02 2 3 5 Δ g N T Lωα η ¼ nB ∝ g c W ð Þ B 2 ; 5:1 s MZ0 vω

from where one observes that, for a fixed value of MZ0 ,it 02 2 0 scales as g Ng; i.e., the favored values of g in the Ng ¼ 2 0 ∼1 5 ¼ 3 FIG. 11. Scanned points (blue) in the g −MZ0 plane, compatible case are . times larger than those in the Ng case. In with the observed baryon asymmetry of the Universe assuming Fig. 11, the blue points show the EWBG-favored region of 0 N ¼ 2. The colorful shaded regions have been excluded by the 0 g parameter space in the g versus MZ plane, obtained by existing constraints from the CCFR, Borexino experiments, and scanning over the model and phase transition parameters → πνν¯ → μμ the K and B K decay rate measurements, respec- given by Eq. (3.27). This figure is analogous to Fig. 5 for tively. The yellow band is the favored region for explaining the the Ng ¼ 2 case. muon g − 2 anomaly. The black dashed lines corresponding to vΦ Experimentally, the gauged Lμ þ Lτ model is interes- equal 1 and 10 TeV, two indicative values related to the anomalon 0 masses that need to be above the electroweak scale. ting because the Z does not couple to electrons at tree level. This helps to avoid most constraints discussed in Sec. IVA. There are, however, relevant constraints from neutrino trident production (CCFR) [56] and loop-induced already cover the region where the dark scalar (r or a)is solar-neutrino-electron scattering (Borexino) [35,57] that ∼200 λ ∼ Oð1Þ lighter GeV for SH . exclude the correspondingly labeled shaded regions in Finally, we comment on the case where S has a nonzero Fig. 11. In this model, the Borexino experiment stands VEV today. A nonzero VEV of S allows r-Higgs boson out to be the most important neutrino scattering experiment mixing implying that, in addition to the above pair pro- because the solar neutrino contains a νμ component. Like duction mode, r may be singly produced through mixing the L þ Lμ þ Lτ case, for small M 0 , this model is also via the gluon fusion channel. There are two possibilities to e Z strongly constrained by flavor-changing meson decays due consider: (a) the Higgs boson is produced off shell and to the anomalous Z0WW coupling [38]. The measurement subsequently mixes with r, that is, produced on shell as a 0 of K → πνν¯ and B → Kμμ decay rates has already new resonance and decays to a Z pair at tree level, leading excluded the cyan shaded region in Fig. 11. A prospective to four leptons in the final state. This is an interesting signature to be explored. The new r resonance can also high-energy electron-positron collider could probe the Z0 decay to SM final states, but this is further suppressed by an viable region of masses via the multimuon searches, BABAR additional r − h mixing factor. (b) The Higgs boson can be similar to the limit set by (not shown in Fig. 11 produced on shell and its decays can be modified through because it is superseded by CCFR) [58]. Lμ þ Lτ its mixing with r. Importantly, this has a direct impact on In view of neutrino cosmology, the gauged model has an attractive aspect where the scalars Φ and S precision measurements of the SM-like Higgs boson, by ð1Þ modifying the Higgs couplings to SM particles, allowing both carry U l charge 2. This allows them to directly give Majorana masses to the right-handed neutrinos, which for Higgs exotic decays, and affecting the di-Higgs pro- Δ is necessary for being consistent with the Neff bound in duction rate. In particular, the current bound [55] on Higgs 0 exotic decay h → 2Z0 → 4l is consistent with an order 1 cosmology and keeping the Z sufficiently light, as dis- 0 −2 cussed in Sec. IVB. However, with the minimal particle r − h mixing, for g ≲ 10 and vS ≲ 100 GeV. This region content given in Table I, the gauged Lμ þ Lτ model cannot of parameter space is just below the LEP bound shown in Fig. 5 and is an interesting benchmark for future collider generate realistic active neutrino masses and mixings. This searches. is mainly because the electron neutrino in this model is not charged under the Uð1Þl, which forbids it to mix with the μ and τ flavors unless a charge one scalar (named S0) under V. THE CASE OF GAUGED Lμ + Lτ Uð1Þl, with a nonvanishing VEV, is introduced. The In this section, we consider another incarnation of the relevant Yukawa interactions, and Majorana mass terms, gauged Uð1Þl model where only two lepton flavors are accounting for realistic neutrino masses and mixings take gauged, l ¼ Lμ þ Lτ, Ng ¼ 2. We comment on the the form

055014-16 DARK CP VIOLATION AND GAUGED LEPTON OR BARYON … PHYS. REV. D 101, 055014 (2020) X ee ¯ ˜ ν þ αβ ¯ ˜ ν þ ν¯c ν Yν LeH Re Yν LαH Rβ Mee Re Re α β¼μ τ X ; ; X þ 00 0ν¯c ν þ 0 Φν¯c ν þ ð Þ YeαS Re Rβ Yαβ Rα Rβ H:c:; 5:2 α¼μ;τ α;β¼μ;τ where we also have to introduce an electron flavored right- ν ð1Þ handed neutrino Re, which is a U l singlet and can have a bare Majorana mass Mee. The dark matter phenomenology in the gauged Lμ þ Lτ model is similar to that discussed in Sec. IV C, except that there could be an additional annihilation channel χ χ¯ → νRνR through an s-channel Φ or S exchange, if kinemat- ically allowed, as their Uð1Þl quantum numbers match for ¼ 2 Ng . These new annihilation channels introduce addi- ð1Þ FIG. 12. The parameter space of the gauged U B model that tional model dependence in the relic density calculations. could generate the observed baryon asymmetry of the Universe 0 Finally, the contribution to electron EDM in the gauged (blue points), in the g −MZ0 plane. The colorful shaded regions Lμ þ Lτ model is suppressed compared to the gauged Le þ have been excluded by the existing constraints from LHC 0 ϒ Lμ þ Lτ case, by the absence of Z -electron coupling. dijet searches (red), hadronic width of (magenta), and J=Psi (orange). The gray shaded region is the minimally excluded region by the LEP bound on electric charged anomalon VI. THE CASE OF GAUGED fields, assuming their Yukawa couplings with the VEV vΦ are BARYON NUMBER B pffiffiffiffiffi near the perturbative limit 4π. The black dashed lines corre- In this section we comment on an alternative Uð1Þ spond to vΦ equal to 1 and 10 TeV. extension of the standard model where the new electroweak baryogenesis mechanism proposed in this work could also constraints on the baryogenesis viable parameter space work. Here we consider gauging the baryon number, are shown in Fig. 12. The baryogenesis viable parameter ð1Þ U B, instead of the lepton number, under which the space in this model is the same as the blue points shown in 1 3 SM quarks carry charge = but leptons are neutral. An Fig. 5, except for a different set of experimental constraints interesting observation is that the same new fermion on the baryonic Z0 [59]. In particular, the LHC constraints ð1Þ 0 content as in Table I could also cancel all U B gauge 0 0 00 00 on the baryonic Z -quark coupling is much weaker than the χ χ 0 anomalies, where the LL, eR, R, LR, eL, L fields carry, LEP constraint on leptophilic Z [60]. This allows a wider ð1Þ under U B, the same charges assigned in Table I, 14 window for our EWBG mechanism to be successful. Ref. [16]. On the other hand, the right-handed neutrinos Because the Z0 in this case only couples to quarks, the νi ð1Þ R are now neutral under U B and they are just dark sector CP violation dominantly contributes to quark introduced for the purpose of giving mass to the neutrinos. EDMs, instead of the electron EDM, which are less An immediate consequence of this setup is that, without severely constrained. ð1Þ νi ’ participating in the new U B interactions, the R s will Like the gauged Uð1Þl model, here the dark fermion χ not be thermalized in the early Universe. Therefore, unlike could still be a thermal dark matter candidate. Its annihi- the Uð1Þl case, the Dirac neutrino mass scenario is lation channels are similar to those depicted in Fig. 6, Δ consistent with the cosmological constraints on Neff in except that the annihilation final states are quarks instead of ð1Þ the gauged U B model. leptons. On the other hand, direct detection constraints Z0 ð1Þ For electroweak baryogenesis, the baryonic 0 back- become much stronger because in the gauged U B model ground could still be generated from the χ-bubble-wall the Z0 directly couples to quarks and the dark-matter- interaction, which now serves as the baryon number nucleon scattering now occurs at tree level. For generic chemical potential for the SM quarks, instead of leptons values of q of order 1, the current direct detection limit on as in the Uð1Þl models. As a result, the Boltzmann equation the spin-independent dark-matter-nucleon scattering cross Eq. (3.22) will become directly 1 for the baryon asymmetry, section implies vΦ ≳ 20 TeV. This constraint is in tension Δ → Δ with the replacement nL nB, the thermal equilibrium with most of the EWBG favored points in Fig. 12.A Δ EQ q ¼ −3 2 asymmetry nB being identical to Eq. (3.21). It is worth possible way to alleviate this tension is to choose = noting that the baryon charge factor 1=3 for quarks is now in which case the dark-matter-Z0 coupling becomes an axial compensated by the number of colors. The existing current interaction and the corresponding dark-matter- nucleon scattering is suppressed by the incoming dark 14 ð1Þ matter velocity in the galactic halo. We keep the same notation as for U l, in spite of the fact 0 that these new states carry baryon number. Observe that they are Analogous to previous cases, because the Z couples ð2Þ2 all color singlets. to an anomalous current with respect to SU L in the

055014-17 CARENA, QUIRÓS, and ZHANG PHYS. REV. D 101, 055014 (2020) low-energy theory, it makes contributions to flavor- (1) CP is first violated in the dark sector, containing the 0 0 changing meson decays such as K → πZ and B → KZ χL;R fermions. Their mass term has an irreducible ð1Þ 0 as shown in Ref. [38]. However, in the U B model the Z phase that becomes time dependent only during the dominantly decays into quarks and antiquarks, while the first-order electroweak phase transition, involving decay into charged leptons could only occur through a Z0γ both the Higgs field and the dark scalar S,as kinetic mixing, and is subdominant if the kinetic mixing is described above. generated at loop level. As a result, the corresponding (2) This time-dependent CP violating mass generates flavor-changing constraints are much weaker and do not particle chiral asymmetries for χL;R in the dark appear in the range shown in Fig. 12. sector, which diffuse to the exterior of the bubble wall, where SM sphalerons are active. VII. CONCLUSION (3) By model construction, χL and χR carry different Uð1Þ charges. As a result, their chiral asymmetries One of the main challenges to electroweak baryogenesis l generate a net Uð1Þl charge density near the wall, models is that the required amount of CP violation can be 0 which yields a Coulomb background for the Z at odds with the improved limits on the electron and 0 gauge field. neutron electric dipole moments. In this work, we propose 0 (4) Given that the gauge field Z couples not only to the a model where electroweak baryogenesis is triggered by a 0 dark sector leptons but also to the SM leptons, it CP violating dark sector. During the electroweak phase generates a chemical potential for the SM leptons. transition, the CP violating effect is transferred from the (5) In the presence of sphaleron processes, which are dark to the visible sector at tree level via the background of active outside the bubble, the SM lepton number a Z0 gauge boson, whereas at zero temperature the trans- 0 asymmetry evolves towards its equilibrium value set mission of CPVeffects could be suppressed up to four-loop by the above chemical potential. level. This mechanism helps to alleviate the otherwise (6) As sphalerons preserve B − L, which originally was severe EDM constraints on the viable baryogenesis param- 0, they can change the generated SM lepton number eter space. into baryon number. Hence, a baryon number The Uð1Þ model we have considered is based on a l asymmetry is equally generated. gauged lepton number symmetry, where the anomaly (7) Inside the bubbles the sphaleron processes are sup- cancellation condition requires extending the SM sector pressed, and the baryon asymmetry generated at the with new fermions carrying lepton number. The lightest of phase transition is not washed out. This process sets these fermions plays the role of dark matter. After the the baryon asymmetry as an input for the initial spontaneous breaking of the gauged lepton number, once — condition in standard cosmology. all the new fermion fields (the anomalons) with the As for the phenomenology of the present model, the exception of the dark matter candidate—are integrated contributions to EDM are highly suppressed, below the out, the fermion content of the effective theory contains present experimental limits, and we do not expect to see all SM fermions, right handed neutrinos, and the dark 0 a positive signal in the next generation of experiments. matter. The force carrier of the new gauge interaction, Z , Instead, one of our main predictions, in particular, for the couples to the lepton number current involving all fermions 0 Uð1Þl model, is a leptophilic Z boson with mass below the in the effective theory, which is anomalous with respect to 0 ð2Þ TeV scale. The lighter the Z , the more weakly coupled it SU L, a key ingredient for the baryogenesis mechanism to work. should be, as shown in Fig. 5. It serves as a very well- To achieve a first-order electroweak phase transi- motivated target for a number of searches at near future tion we introduce a SM singlet S in the dark sector, and prospective experiments, such as BELLE II, NA64 μ which couples to the Higgs boson portal and may allow ( mode), and SHiP, as well as a possible Higgs factory. for a two-step phase transition in the early Universe. Accommodating a dark matter candidate within this Similar studies in the literature have shown that after an new EWBG mechanism provides an additional handle in probing this idea. Concerning the fermion candidate χ to initial transition from a trivial vacuum state ðvS; 0Þ at dark matter, we show that the annihilation cross sections very high temperatures, it is possible to trigger a strong 0 first-order transition to the electroweak vacuum ð0;vÞ, involving the new force carrier Z are too small. However thereby creating the out-of-equilibrium condition neces- the dark matter annihilation into the new scalar S comes sary for baryogenesis. A detailed analysis of the phase to the rescue, yielding the correct relic abundance via transition history and its relation to the proposed thermal freeze-out. Direct detection experiments also yield mechanism for electroweak baryogenesis will be pre- important information on the parameter space compatible sented elsewhere. with EWBG. The most relevant, straightforward contribu- The role of the dark sector CP violation in our baryo- tion comes from the Z0 exchange, which, given the genesis mechanism for the Uð1Þl model can be summa- leptophilic nature of this new gauge boson in the Uð1Þl rized in the following steps: model, implies that dark matter scattering occurs at loop

055014-18 DARK CP VIOLATION AND GAUGED LEPTON OR BARYON … PHYS. REV. D 101, 055014 (2020) level. Future direct dark matter searches, with an improve- APPENDIX A: EQUATION FOR THE ment of about 2 orders of magnitude over present bounds, LEPTON ASYMMETRY will provide an important test of the viable parameter space In this appendix we provide more details about solving in the Uð1Þ model of EWBG. l the sphaleron rate equation (3.22), to obtain the final Finally, there are novel collider signals from the new lepton/baryon asymmetry. We first rewrite Eq. (3.22) here, additional scalar S, which can be pair produced via an s-channel off-shell Higgs boson, or singly produced dΔn ðz; tÞ LL ¼ Γ ð − Þ½Δ EQð − Þ − Δ ð Þ through mixing with the Higgs boson. The former, pair- sph z vωt nL z vωt nL z; t ; dt L L production mode could lead to eight charged lepton final ð Þ states from the decays of the Z0 s. The latter, single- A1 production mode, instead, could yield four charged leptons. Γ 0 where the sphaleron rate sph was given in Eq. (3.23) for the For both cases, one could reconstruct the Z mass from the symmetric and broken phases. invariant mass of the charged lepton pairs. The new scalar S Several remarks are in order here. can also be virtually produced via mixing with the Higgs (i) We solve ΔnL for a generic point at a distance z from boson, altering the Higgs boson phenomenology. Current the moving bubble wall. We assume that the bubble bounds on the Higgs boson exotic decays still allow for a is formed at an initial time that we arbitrarily fix to large region of parameter space compatible with our t ¼ 0. The bubble wall passes through the point z at EWBG mechanism, and provide interesting opportunities ¼ 0 0 time t z=vω, and turns on the Higgs VEV at this for near-future searches in the Higgs decay to Z Z when point. We are interested in its final value, i.e., in kinematically allowed. principle, at t → ∞, after the bubble wall has passed Similar, corresponding, comments should apply to the through and bubble nucleation has taken place. ð1Þ U B model after replacing L by B and leptons by quarks. (ii) The electroweak sphaleron rate is strongly sup- χ ð1Þ However, for the DM candidate in the U B case, pressed in the broken phase for a strong first-order already present direct detection constraints make the phase transition, where the Higgs VEV at the scenario quite challenging. Nevertheless observe that it tunneling (or nucleation) temperature is vn ≳ Tn. is possible for χ to be only a fraction of the total dark matter − This behavior follows since e Msph=Tn ≪ 1, and in the Universe. In that case, the direct detection bounds, as hence Γ at the broken phase is negligible. At computed here for any of the models, would become less sph point z, instead, the sphaleron process is active, and stringent. Γ ð − Þ¼Γ ≠ 0 its rate is a constant, i.e., sph z vωt 0 , for the time window 0 ≤ t ≤ z=vω. ACKNOWLEDGMENTS (iii) As calculated, and shown in the left panel of Fig. 4, We thank Zackaria Chacko, James Cline, Bogdan the source term is peaked, and localized, around the Dobrescu, Bhaskar Dutta, Pavel Fileviez Perez, Paddy moving bubble wall. It is highly suppressed at large Fox, Ian Low, David Morrissey, and Tim Tait for useful instantaneous distance from the wall, i.e., for z discussions and correspondence. We are also grateful to greater than a few times the wall width Lω. Julian Heeck, Alexis Plasencia, and especially Jeff Dror, To solve Eq. (A1), we first get rid of the damping term on for very useful comments on the first version of this paper. the right-hand side with the redefinition This manuscript has been authored by Fermi Research Aðz; tÞ ≡ Δn ðz; tÞeΓ0t: ðA2Þ Alliance, LLC, under Contract No. DE-AC02-07CH11359 LL with the U.S. Department of Energy (DoE), Office of The differential equation for Aðz; tÞ is then Science, Office of High Energy Physics. The work of M. Q. dAðz; tÞ is partly supported by Spanish MINEICO under Grants Γ0t EQ ¼ e Γ0Δn ðz − vωtÞ; ðA3Þ No. CICYT-FEDER-FPA2014-55613-P and No. FPA2017- dt LL 88915-P, by the Government of Catalonia under Grant As explained in the second bullet above, this equation is No. 2017SGR1069, and by the Severo Ochoa Excellence only valid in the time window 0 ≤ t ≤ z=vω, as for larger Program of MINEICO under Grant No. SEV-2016-0588. values of t, Γ ≃ 0. The solution for Aðz; tÞ could be The work of Y. Z. is partly supported by the DoE under sph obtained by simply integrating the right-hand side over Contract No. DE-SC0007859. M. C. and Y. Z. thank the time, and then we could use Eq. (A2) to compute Aspen Center for Physics, which is supported by National Δn ðz; tÞ. For t ¼ z=vω,wehave Science Foundation Grant No. PHY-1607611, where part LL Z of this work was performed, and Colegio De Fisica Funda- z=vω 0 0 EQ 0 Γ0ðt −z=vωÞ Δn ðz; tÞj ¼ Γ0 dt Δn ðz − vωt Þe mental E Interdisciplinaria De Las Americas (COFI) for a LL t¼z=vω L 0 L travel support during the completion of this work. M. Q. Z Γ0 z thanks the Department of Physics, University of Notre ¼ Δ EQð Þ −Γ0y=vω ð Þ dy nL y e ; A4 Dame, where part of this work was done, for hospitality. vω 0 L

055014-19 CARENA, QUIRÓS, and ZHANG PHYS. REV. D 101, 055014 (2020) where in the second step, we have changed the integration chemical potential for the fields charged under it, and 0 0 variable from t to y ¼ z − vωt , the coordinate in the leads to the thermal equilibrium asymmetry in their number bubble wall center-of-mass frame. densities. Of particular interest to us are those for the ð2Þ Based on the above discussion, after the bubble wall SU L doublets, passes through the point z, the Higgs VEV turns on, and the 2 sphaleron process is highly suppressed. Consequently, the EQ 2 0 0 Δn ¼ Ng × 1 × Tcg hZ0i; Δ LL 3 quantity nLL is conserved in the broken electroweak phase. In other words, the created baryon/lepton asymme- 2 Δ EQ ¼ 1 q 2 0h 0 i → ∞ n 0 × × Tcg Z0 ; try freezes in, and we can derive that at t , LL 3 Z 2 Γ z EQ 2 0 0 0 EQ −Γ Δn 00 ¼ 1 × ðq þ NgÞ × Tcg hZ0i: ðB3Þ Δ ð Þ ≡ Δ ð ∞Þ ≃ Δ ð Þ 0y=vω LR 3 nL z nL z; dy nL y e : L L vω 0 L ðA5Þ In the context of EWBG, the electroweak sphaleron processes are responsible for changes in the lepton and 0 00 Therefore we define the asymmetry of the final lepton baryon numbers in the Universe. In the presence of LL;LR density in the Universe, integrating over all points z,as fields in the thermal bath, they also participate. The actual Z Z changes in the particle asymmetries are tied to each other, ∞ ∞ dΔnL ðzÞ Γ0 and satisfy the following relations, Δ ¼ L ¼ Δ EQð Þ −Γ0z=vω nL dz dz nL z e ; L 0 dz vω 0 L ∂ ∂ ∂ ∂ ðA6Þ ΔnB ¼ ΔnL ¼ 3 ΔnL0 ¼ −3 ΔnL00 ; ðB4Þ ∂t L ∂t L ∂t L ∂t R which is the result quoted in Eq. (3.24) in the main text. 0 where BL denotes the baryon number in left-handed SM Notice that we are integrating over all points z> , outside doublets. It is useful to define the “effective total lepton the bubble, as we are assuming that in the interior of the asymmetry” as bubble, z<0, Γsph ≃ 0.

Δn ðz; tÞ ≡ Δn ðz; tÞþΔn 0 ðz; tÞ − Δn 00 ðz; tÞ; L;eff LL LL LR APPENDIX B: THE CASE OF A 2 ðB5Þ NONANOMALOUS U(1)l ⊗ SU(2)L EFFECTIVE THEORY so that Eq. (B4) implies 0 Let us first consider the case where the masses of LL and L00 doublet fields are much smaller than the critical tem- ∂ 3 ∂ R Δn ðz; tÞ¼ Δn ðz; tÞ: ðB6Þ perature of the EWPT, and they are not integrated out. The ∂t BL 5 ∂t L;eff fermionic current J μ that Z0 couples to takes then the form Δ ð Þ The Boltzmann equation for nL;eff z; t satisfies XNg J μ ¼ ¯ γμ þ q ¯ 0 γμ 0 þðq þ Þ ¯ 00 γμ 00 ∂ LLi LLi LL LL Ng LR LR Δn ðz; tÞ¼Γ ðz − vωtÞ i¼1 ∂t L;eff sph þ; ðB1Þ ½Δ EQ ð − Þ − Δ ð Þ × nL;eff z vωt nL;eff z; t ; ¼ 1 ðB7Þ where LLi (i , 2, 3) are the SM lepton doublets, and the ð2Þ ellipsis represents the terms involving SU L singlet fields. The current J μ is nonanomalous with respect to Δ EQ ¼ Δ EQ þ Δ EQ − Δ EQ ð Þ nL;eff nL nL0 nL00 : B8 ð2Þ L L R the SM SU L, i.e.,

μ a b a ˜ b Equation (B3) then implies that a cancellation occurs ∂μJ ∝ trðlτ τ ÞW W Δ EQ ¼ 0 in Eq. (B8), leading to nL;eff . In this case, the ˜ Δ ð Þ ∝ ½Ng × 1 þ q − ðq þ NgÞtrðWWÞ¼0; ðB2Þ Boltzmann equation for nL;eff z; t has no source term, and assuming the Universe begins without any particle ˜ ð2Þ Δ where W (W) is the SU L field (dual field) strength, and asymmetries, no nL;eff is generated. In turn Eq. (B6) τa ð2Þ the Pauli matrices are SU L generators. implies that the baryon asymmetry cannot be generated. 0 Next, we assume the hZ0i background to be present One should note that such a conclusion is drawn by 0 00 during EWBG, still generated by the CP violating assuming the LL;LR fields to be relativistic degrees of χ-bubble-wall interaction, given by Eq. (3.19). Through freedom in the thermal bath during the EWPT. As pointed 0 the gauge interactions, the Z0 background serves as out in [11], the above cancellation is closely related to

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Eq. (B2), the conservation of the current J μ, with respect integrated out, the current that Z0 couples to in the low- ð2Þ to SU L. energy theory becomes On the other hand, if L0 ;L00 obtain a sufficiently large L R X3 Uð1Þ symmetry breaking mass through the Yukawa l Jμ ¼ L¯ γμL þ; ðB9Þ coupling to the Φ field as discussed in the main text, their Li Li i¼1 thermal number densities in Eq (B3) become Boltzmann ð2Þ suppressed. In this case, the above cancellation no longer which is anomalous with respect to SU L. In summary, occurs, and the proposed EWBG mechanism could be the created baryon asymmetry should be proportional to the 0 00 μ successful. In the limit when LL;LR are very heavy and nonconservation of the current J [11], as previously stated.

[1] V. A. Kuzmin, V. A. Rubakov, and M. E. Shaposhnikov, On [8] J. Baron et al. (ACME Collaboration), Order of magnitude the anomalous electroweak Baryon number non- smaller limit on the electric dipole moment of the electron, conservation in the Early Universe, Phys. Lett. 155B,36 Science 343, 269 (2014); W. C. Griffith, M. D. Swallows, (1985). T. H. Loftus, M. V. Romalis, B. R. Heckel, and E. N. [2] A. G. Cohen, D. B. Kaplan, and A. E. Nelson, Weak scale Fortson, Improved Limit on the Permanent Electric Dipole Baryogenesis, Phys. Lett. B 245, 561 (1990); G. R. Farrar Moment of Hg-199, Phys. Rev. Lett. 102, 101601 (2009); and M. E. Shaposhnikov, Baryon Asymmetry of the C. A. Baker et al., An Improved Experimental Limit on the Universe in the Minimal standard model, Phys. Rev. Lett. Electric Dipole Moment of the Neutron, Phys. Rev. Lett. 97, 70, 2833 (1993); Erratum, Phys. Rev. Lett. 71, 210 (1993); 131801 (2006). P. Huet and A. E. Nelson, CP violation and electroweak [9] J. Shu and Y. Zhang, Impact of a CP violating Higgs Sector: baryogenesis in extensions of the standard model, Phys. From LHC to Baryogenesis, Phys. Rev. Lett. 111, 091801 Lett. B 355, 229 (1995); Electroweak baryogenesis in (2013); S. Ipek, Perturbative analysis of the electron electric supersymmetric models, Phys. Rev. D 53, 4578 (1996); dipole moment and CP violation in two-Higgs-doublet A. Riotto, Towards a nonequilibrium quantum field theory models, Phys. Rev. D 89, 073012 (2014); M. Jung and approach to electroweak baryogenesis, Phys. Rev. D 53, A. Pich, Electric dipole moments in two-Higgs-Doublet 5834 (1996); M. Carena, M. Quiros, A. Riotto, I. Vilja, and models, J. High Energy Phys. 04 (2014) 076; T. Abe, J. C. E. M. Wagner, Electroweak baryogenesis and low-energy Hisano, T. Kitahara, and K. Tobioka, Gauge invariant Barr- supersymmetry, Nucl. Phys. B503, 387 (1997); M. Carena, Zee type contributions to fermionic EDMs in the two-Higgs J. M. Moreno, M. Quiros, M. Seco, and C. E. M. Wagner, doublet models, J. High Energy Phys. 01 (2014) 106; Supersymmetric CP violating currents and electroweak Erratum, J. High Energy Phys. 04 (2016) 161; S. Inoue, baryogenesis, Nucl. Phys. B599, 158 (2001). M. J. Ramsey-Musolf, and Y. Zhang, CP-violating phe- [3] J. M. Cline, M. Joyce, and K. Kainulainen, Supersymmetric nomenology of flavor conserving two Higgs doublet mod- electroweak baryogenesis, J. High Energy Phys. 07 (2000) els, Phys. Rev. D 89, 115023 (2014); K. Cheung, J. S. Lee, 018. E. Senaha, and P.-Y. Tseng, Confronting Higgcision with [4] M. Carena, M. Quiros, M. Seco, and C. E. M. Wagner, electric dipole moments, J. High Energy Phys. 06 (2014) Improved results in supersymmetric electroweak baryo- 149; L. Bian, T. Liu, and J. Shu, Cancellations Between genesis, Nucl. Phys. B650, 24 (2003); C. Lee, V. Cirigliano, Two-Loop Contributions to the Electron Electric Dipole and M. J. Ramsey-Musolf, Resonant relaxation in electro- Moment with a CP-Violating Higgs Sector, Phys. Rev. Lett. weak baryogenesis, Phys. Rev. D 71, 075010 (2005). 115, 021801 (2015); C.-Y. Chen, S. Dawson, and Y. Zhang, [5] J. M. Cline, Baryogenesis, in Les Houches Summer School Complementarity of LHC and EDMs for exploring Higgs —Session 86: Particle Physics and Cosmology: The Fabric CP violation, J. High Energy Phys. 06 (2015) 056; K. of Spacetime Les Houches, France (2006). Fuyuto, J. Hisano, and E. Senaha, Toward verification of [6] L. Fromme, S. J. Huber, and M. Seniuch, Baryogenesis in electroweak baryogenesis by electric dipole moments, Phys. the two-Higgs doublet model, J. High Energy Phys. 11 Lett. B 755, 491 (2016); M. Jiang, L. Bian, W. Huang, and (2006) 038; V. Cirigliano, S. Profumo, and M. J. Ramsey- J. Shu, Impact of a complex singlet: Electroweak baryo- Musolf, Baryogenesis, Electric dipole moments and dark genesis and dark matter, Phys. Rev. D 93, 065032 (2016);N. matter in the MSSM, J. High Energy Phys. 07 (2006) 002; Blinov, J. Kozaczuk, D. E. Morrissey, and C. Tamarit, Y. Li, S. Profumo, and M. Ramsey-Musolf, A comprehen- Electroweak Baryogenesis from exotic electroweak sym- sive analysis of electric dipole moment constraints on CP- metry breaking, Phys. Rev. D 92, 035012 (2015); C. Balazs, violating phases in the MSSM, J. High Energy Phys. 08 G. White, and J. Yue, Effective field theory, electric dipole (2010) 062. moments and electroweak baryogenesis, J. High Energy [7] V. Andreev et al. (ACME Collaboration), Improved limit on Phys. 03 (2017) 030; L. Bian and N. Chen, Cancellation the electric dipole moment of the electron, Nature (London) mechanism in the predictions of electric dipole moments, 562, 355 (2018). Phys. Rev. D 95, 115029 (2017); C.-Y. Chen, H.-L. Li, and

055014-21 CARENA, QUIRÓS, and ZHANG PHYS. REV. D 101, 055014 (2020)

M. Ramsey-Musolf, CP-violation in the two Higgs Doublet 86 (1991); E. W. Kolb and M. S. Turner, The early Universe, Model: From the LHC to EDMs, Phys. Rev. D 97, 015020 Front. Phys. 69, 1 (1990). (2018); C. Cesarotti, Q. Lu, Y. Nakai, A. Parikh, and M. [26] H. Davoudiasl, R. Kitano, G. D. Kribs, H. Murayama, and Reece, Interpreting the electron EDM constraint, J. High P. J. Steinhardt, Gravitational Baryogenesis, Phys. Rev. Lett. Energy Phys. 05 (2019) 059; D. Egana-Ugrinovic and S. 93, 201301 (2004); H. Davoudiasl, Gravitationally induced Thomas, Higgs Boson contributions to the electron electric dark matter asymmetry and dark nucleon decay, Phys. Rev. dipole moment, arXiv:1810.08631; M. J. Ramsey-Musolf, D 88, 095004 (2013). P. Winslow, and G. White, Color breaking Baryogenesis, [27] D. Bodeker, G. D. Moore, and K. Rummukainen, Chern- Phys. Rev. D 97, 123509 (2018). Simons number diffusion and hard thermal loops on the [10] J. M. Cline, K. Kainulainen, and D. Tucker-Smith, Electro- lattice, Phys. Rev. D 61, 056003 (2000). weak baryogenesis from a dark sector, Phys. Rev. D 95, [28] G. D. Moore, Measuring the broken phase sphaleron rate 115006 (2017). nonperturbatively, Phys. Rev. D 59, 014503 (1998);R. [11] M. Carena, M. Quiros, and Y. Zhang, Electroweak Baryo- Zhou, L. Bian, and H.-K. Guo, Probing the electroweak genesis From Dark CP violation, Phys. Rev. Lett. 122, Sphaleron with gravitational waves, arXiv:1910.00234. 201802 (2019). [29] N. Aghanim et al. (Planck Collaboration), Planck 2018 [12] S. M. Barr and A. Zee, Electric Dipole Moment of the results. VI. Cosmological parameters, arXiv:1807.06209. Electron and of the Neutron, Phys. Rev. Lett. 65, 21 (1990); [30] D. Banerjee et al. (NA64 Collaboration), Search for Erratum, Phys. Rev. Lett. 65, 2920 (1990). Invisible Decays of sub-GeV Dark Photons in Missing- [13] J. F. Gunion and R. Vega, The electron electric dipole Energy Events at the CERN SPS, Phys. Rev. Lett. 118, moment for a CP violating neutral Higgs sector, Phys. Lett. 011802 (2017); J. P. Lees et al. (BABAR Collaboration), B 251, 157 (1990). Search for Invisible Decays of a Dark Photon Produced in [14] J. Wess and B. Zumino, Consequences of anomalous ward eþe− Collisions at BABAR, Phys. Rev. Lett. 119, 131804 identities, Phys. Lett. 37B, 95 (1971). (2017). [15] P. F. Perez and M. B. Wise, Baryon and lepton number as [31] M. Tanabashi et al. (Particle Data Group), Review of local gauge symmetries, Phys. Rev. D 82, 011901 (2010); particle physics, Phys. Rev. D 98, 030001 (2018). Erratum, Phys. Rev. D 82, 079901 (2010). [32] J. Alcaraz et al. (DELPHI, OPAL, ALEPH, LEP [16] M. Duerr, P. F. Perez, and M. B. Wise, Gauge Theory for Electroweak Working Group and L3 Collaboration), A Baryon and Lepton Numbers with Leptoquarks, Phys. Rev. combination of preliminary electroweak measurements Lett. 110, 231801 (2013). and constraints on the standard model, arXiv:hep-ex/ [17] P. Schwaller, T. M. P. Tait, and R. Vega-Morales, Dark 0612034. matter and vectorlike leptons from gauged lepton number, [33] J. P. Lees et al. (BABAR Collaboration), Search for a Dark Phys. Rev. D 88, 035001 (2013). Photon in eþe− Collisions at BABAR, Phys. Rev. Lett. 113, [18] W. Altmannshofer, M. Carena, and A. Crivellin, Lμ − Lτ 201801 (2014). theory of Higgs flavor violation and ðg − 2Þμ, Phys. Rev. D [34] J. D. Bjorken, R. Essig, P. Schuster, and N. Toro, New fixed- 94, 095026 (2016). target experiments to search for dark gauge forces, Phys. [19] W. A. Bardeen and B. Zumino, Consistent and covariant Rev. D 80, 075018 (2009). anomalies in gauge and gravitational theories, Nucl. Phys. [35] M. Bauer, P. Foldenauer, and J. Jaeckel, Hunting all the B244, 421 (1984). hidden photons, J. High Energy Phys. 07 (2018) 094. [20] L. Rosenberg, Electromagnetic interactions of neutrinos, [36] H. Davoudiasl, H.-S. Lee, and W. J. Marciano, Muon Phys. Rev. 129, 2786 (1963). Anomaly and Dark Parity Violation, Phys. Rev. Lett. [21] P. J. Fox, I. Low, and Y. Zhang, Top-philic Z0 forces at the 109, 031802 (2012). LHC, J. High Energy Phys. 03 (2018) 074. [37] M. Battaglieri et al., US cosmic visions: New ideas in dark [22] J. R. Espinosa, T. Konstandin, and F. Riva, Strong matter 2017: Community report, arXiv:1707.04591. electroweak phase transitions in the standard model with [38] J. A. Dror, R. Lasenby, and M. Pospelov, New Constraints a singlet, Nucl. Phys. B854, 592 (2012); H. H. Patel, M. J. on Light Vectors Coupled to Anomalous Currents, Phys. Ramsey-Musolf, and M. B. Wise, Color breaking in the Rev. Lett. 119, 141803 (2017); Dark forces coupled to early Universe, Phys. Rev. D 88, 015003 (2013); C. Cheung nonconserved currents, Phys. Rev. D 96, 075036 (2017). and Y. Zhang, Electroweak cogenesis, J. High Energy Phys. [39] D. d’Enterria, Physics case of FCC-ee: Proceedings, Physics 09 (2013) 002; D. Curtin, P. Meade, and C.-T. Yu, Testing Prospects for Linear and other Future Colliders after the Electroweak Baryogenesis with future colliders, J. High Discovery of the Higgs (LFC15): Trento, Italy, Frascati Energy Phys. 11 (2014) 127. Phys. Ser. 61, 17 (2016); C.-S. S. Group, CEPC-SPPC [23] J. de Vries, M. Postma, J. van de Vis, and G. White, Preliminary Conceptual Design Report. 1. Physics and Electroweak Baryogenesis and the standard model Detector, Reports No. IHEP-CEPC-DR-2015-01, effective field theory, J. High Energy Phys. 01 (2018) No. IHEP-TH-2015-01, No. IHEP-EP-2015-01, 2015. 089. [40] C. D. Carone and H. Murayama, Realistic models with a [24] G. C. Dorsch, S. J. Huber, and T. Konstandin, Bubble wall light U(1) gauge boson coupled to baryon number, Phys. velocities in the standard model and beyond, J. Cosmol. Rev. D 52, 484 (1995). Astropart. Phys. 12 (2018) 034. [41] A. Hook, E. Izaguirre, and J. G. Wacker, Model independent [25] A. G. Cohen, D. B. Kaplan, and A. E. Nelson, Spontaneous bounds on kinetic mixing, Adv. High Energy Phys. 2011, baryogenesis at the weak phase transition, Phys. Lett. B 263, 859762 (2011); B. A. Dobrescu and C. Frugiuele, Hidden

055014-22 DARK CP VIOLATION AND GAUGED LEPTON OR BARYON … PHYS. REV. D 101, 055014 (2020)

GeV-Scale Interactions of Quarks, Phys. Rev. Lett. 113, Higgs boson production and decay rates and constraints on 061801 (2014). its couplings from a combined ATLASpffiffiffi and CMS analysis of [42] V. Barger, P. Langacker, and H.-S. Lee, Primordial nucleo- the LHC pp collision data at s ¼ 7 and 8 TeV, J. High synthesis constraints on Z0 properties, Phys. Rev. D 67, Energy Phys. 08 (2016) 045. 075009 (2003); D. K. Ghosh, G. Senjanovic, and Y. Zhang, [52] E. Accomando et al., Workshop on CP studies and Naturally light sterile neutrinos from theory of R-parity, nonStandard Higgs physics, arXiv:hep-ph/0608079;Y. Phys. Lett. B 698, 420 (2011); J. Hamann, S. Hannestad, Chen, R. Harnik, and R. Vega-Morales, Probing the Higgs G. G. Raffelt, and Y. Y. Y. Wong, Sterile neutrinos with eV Couplings to Photons in h->4l at the LHC, Phys. Rev. Lett. masses in cosmology: How disfavoured exactly? J. Cosmol. 113, 191801 (2014). Astropart. Phys. 09 (2011) 034. [53] E. Izaguirre and D. Stolarski, Searching for Higgs Decays to [43] K. N. Abazajian and J. Heeck, Observing Dirac neutrinos in as Many as 8 Leptons, Phys. Rev. Lett. 121, 221803 (2018). the cosmic microwave background, Phys. Rev. D 100, [54] A. M. Sirunyan et al. (CMS Collaboration), Search for 075027 (2019). electroweak production of charginos and neutralinos in [44] J. M. Berryman, A. de Gouva, K. J. Kelly, and Y. Zhang, pmultileptonffiffiffi final states in proton-proton collisions at Dark Matter and Neutrino mass from the smallest non- s ¼ 13 TeV, J. High Energy Phys. 03 (2018) 166. Abelian Chiral dark sector, Phys. Rev. D 96, 075010 (2017). [55] M. Aaboud et al. (ATLAS Collaboration), Search for [45] A. Atre, T. Han, S. Pascoli, and B. Zhang, The search for Higgs boson decays to beyond-the-Standard-Model light heavy Majorana neutrinos, J. High Energy Phys. 05 (2009) pbosonsffiffiffi in four-lepton events with the ATLAS detector at 030. s ¼ 13 TeV, J. High Energy Phys. 06 (2018) 166. [46] A. Krovi, I. Low, and Y. Zhang, Broadening dark matter [56] W. Altmannshofer, S. Gori, M. Pospelov, and I. Yavin, searches at the LHC: Mono-X versus Darkonium channels, Neutrino Trident Production: A Powerful Probe of New J. High Energy Phys. 10 (2018) 026. Physics with Neutrino Beams, Phys. Rev. Lett. 113, 091801 [47] P. J. Fox, R. Harnik, J. Kopp, and Y. Tsai, LEP shines light (2014). on dark matter, Phys. Rev. D 84, 014028 (2011). [57] T. Araki, S. Hoshino, T. Ota, J. Sato, and T. Shimomura, [48] E. Aprile et al. (XENON Collaboration), Dark Matter Detecting the Lμ − Lτ gauge boson at Belle II, Phys. Rev. D Search Results from a One Tonne × Year Exposure of 95, 055006 (2017). XENON1T, Phys. Rev. Lett. 121, 111302 (2018); X. Cui [58] J. P. Lees et al. (BABAR Collaboration), Search for a muonic et al. (PandaX-II Collaboration), Dark Matter Results From dark force at BABAR, Phys. Rev. D 94, 011102 (2016). 54-Ton-Day Exposure of PandaX-II Experiment, Phys. Rev. [59] H. An, X. Ji, and L.-T. Wang, Light dark matter and Z0 dark Lett. 119, 181302 (2017); D. S. Akerib et al. (LUX force at colliders, J. High Energy Phys. 07 (2012) 182; B. A. Collaboration), Results from a Search for Dark Matter in Dobrescu and F. Yu, Coupling-mass mapping of dijet peak the Complete LUX Exposure, Phys. Rev. Lett. 118, 021303 searches, Phys. Rev. D 88, 035021 (2013); Erratum, Phys. (2017). Rev. D 90, 079901 (2014); P. Fileviez Prez, E. Golias, R.-H. [49] T. A. Chowdhury, M. Nemevsek, G. Senjanovic, and Y. Li, and C. Murgui, Leptophobic dark matter and the Baryon Zhang, Dark matter as the trigger of strong electroweak number violation scale, Phys. Rev. D 99, 035009 (2019). phase transition, J. Cosmol. Astropart. Phys. 02 (2012) 029; [60] A. M. Sirunyan et al. (CMS Collaboration), Search for low J. M. Cline, K. Kainulainen, P. Scott, and C. Weniger, mass vector resonances decayingpffiffiffi into quark-antiquark pairs Update on scalar singlet dark matter, Phys. Rev. D 88, in proton-proton collisions at s ¼ 13 TeV, J. High Energy 055025 (2013); Erratum, Phys. Rev. D 92, 039906 (2015). Phys. 01 (2018) 097; ATLAS Collaboration, Search for light [50] M. B. Wise and Y. Zhang, Stable bound states of asym- dijet resonances with the ATLAS detector usingpffiffiffi a Trigger- metric dark matter, Phys. Rev. D 90, 055030 (2014); Level Analysis in LHC pp collisions at s ¼ 13 TeV, Erratum, Phys. Rev. D 91, 039907 (2015); Y. Zhang, CERN Report No. ATLAS-CONF-2016-030, 2016; A. M. Long-lived light mediator to dark matter and primordial Sirunyan et al. (CMS Collaboration), Searchpffiffiffi for dijet small scale spectrum, J. Cosmol. Astropart. Phys. 05 (2015) resonances in protonproton collisions at s ¼ 13 TeV 008; C. Kouvaris, I. M. Shoemaker, and K. Tuominen, Self- and constraints on dark matter and other models, Phys. interacting dark matter through the Higgs portal, Phys. Rev. Lett. B 769, 520 (2017); Erratum, Phys. Lett. B 772, 882 D 91, 043519 (2015). (2017); M. Aaboud et al. (ATLAS Collaboration), Search [51] M. Carena, Z. Liu, and M. Riembau, Probing the electro- −1 for new phenomena in dijetpffiffiffi events using 37 fb of pp weak phase transition via enhanced di-Higgs boson pro- collision data collected at s ¼ 13 TeV with the ATLAS duction, Phys. Rev. D 97, 095032 (2018); G. Aad et al. detector, Phys. Rev. D 96, 052004 (2017). (ATLAS and CMS Collaborations), Measurements of the

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