Viable Twin Cosmology from Neutrino Mixing
VViable Twin Cosmology from Neutrino Mixing
Csaba Cs´aki, ∗ Eric Kuflik, † and Salvator Lombardo ‡ Laboratory for Elementary Particle Physics, Cornell University, Ithaca, NY 14850, USA
Twin Higgs models solve the little hierarchy problem without introducing new colored particles,however they are often in tension with measurements of the radiation density at late times. Herewe explore viable cosmological histories for Twin Higgs models. In particular, we show that mixingbetween the SM and twin neutrinos can thermalize the two sectors below the twin QCD phasetransition, significantly reducing the twin sector’s contribution to the radiation density. The requisitetwin neutrino masses of O (1 −
20) GeV and mixing angle with SM neutrinos of 10 − − − canbe probed in a variety of current and planned experiments. We further find that these parameterscan be naturally accessed in a warped UV completion, where the neutrino sector can also generatethe Z -breaking Higgs mass term needed to produce the hierarchy between the symmetry breakingscales f and v . INTRODUCTION
Twin Higgs (TH) models provide an elegant solutionto the hierarchy problem without introducing new statesthat are charged under the SM gauge symmetries [1]. In-stead, a mirror sector with its own SU(3) × SU(2) × U(1)gauge symmetry is assumed. The Z symmetry relatingthe SM and mirror sectors protects the Higgs mass fromlarge radiative corrections, with the twin partners can-celling the SM quadratic divergences at one loop. Othervariations of neutral naturalness include [2–8]. While thisidea is very efficient at hiding new physics from the LHCand future colliders, it often leads to tension with cosmo-logical observations due to the appearance of new lightrelativistic degrees of freedom (DOF), namely the twinphoton and twin neutrinos.The standard assumption of TH models is that onlythe Higgs portal connects the SM and the mirror sec-tors. This maintains thermal equilibrium between thetwo sectors down to temperatures of a few GeV, belowwhich the twin sector decouples [9]. At this point thetwin and SM sectors have similar energy densities, andthe twin photon and neutrinos contribute significantly tothe radiation density at late times. In particular, theMirror Twin Higgs (MTH) model—the scenario wherethe mirror sector is a full copy of the SM—predicts anexceedingly large contribution to the correction of the to-tal radiation density (usually expressed in terms of ∆ N eff ,as measured from Big Bang Nucleosynthesis (BBN) andthe Cosmic Microwave Background (CMB)).Recently, several solutions have been proposed for thecosmological problems of the Twin Higgs, including theFraternal Twin Higgs (FTH) [10], hard Z -breaking inthe Yukawa couplings [11], and SM reheating from a lightright-handed neutrino [12]. Further cosmological aspectsof Twin Higgs models, including dark matter, have beenstudied in [11–19].In this paper we propose that the neutrino portal canalso naturally be used to connect the twin and the SMsectors. Mixing between the SM and twin neutrinos ap-pears in many simple implementations of the twin neu- trino sector. We show that such mixing can lower thedecoupling temperature between the two sectors, poten-tially delaying decoupling past the scale of the twin QCDphase transition. When the decoupling of the two sectorshappens between the twin and ordinary QCD phase tran-sition scales, the contribution of the twin sector to ∆ N eff is strongly reduced, since at the time of equilibrium thereare fewer relativistic DOFs in the twin sector comparedto the SM. We explore in detail the dependence of ∆ N eff on the decoupling temperature, as well as the relationbetween the decoupling temperature and the twin neu-trino masses and mixings with the SM neutrinos. We findthat reasonably sized mixing between the two sectors oforder sin θ ∼ − − − , and twin neutrino masses of O (10) GeV, can result in a viable cosmological scenariofor TH models.We also show that a composite Twin Higgs (CTH) UVcompletion [20–24] can naturally incorporate twin neu-trino masses and mixings with the SM neutrinos of thedesired magnitude. By using this CTH framework wedemonstrate that the neutrino sector can also automat-ically generate the Z -breaking Higgs mass term neededto produce the hierarchy between the symmetry breakingscales f and v for the same parameters that result in aviable cosmology.This paper is organized as follows: We first investi-gate the dependence N eff on the decoupling temperatureof the twin sector. Then we calculate the decouplingtemperature as a function of the twin neutrino massesand their mixings with the SM neutrinos, followed byan overview of the possible new experimental signals ofthe various TH scenarios. Next we present realistic massand mixing patterns in the neutrino sector, followed byan implementation of this sector in the holographic CTHsetup. We close the paper by a discussion of the Z -breaking effects in the Higgs potential. Various appen-dices contain the details of the RS construction, the re-sulting warped mass spectrum, the effect of Majoranamasses on the spectral functions, and finally the detailsof the full Coleman-Weinberg calculation for the neutrinosector. a r X i v : . [ h e p - ph ] M a r DARK RADIATION IN TWIN HIGGS MODELS
We begin by describing the contributions to the radia-tion density of the universe at late times in various typesof Twin Higgs models and compare those to the experi-mental bounds. Later we will show how to use mixing inthe neutrino sector to obtain viable scenarios.The total radiation density of the universe is typicallyparameterized in terms of the effective number of neu-trino species, N eff , defined as ρ r ≡ ρ γ (cid:32) (cid:18) (cid:19) / N eff (cid:33) , (1)where ρ γ is the observed radiation density and N SMeff =3 .
046 is the value predicted in the SM from standardneutrino decoupling.For a twin sector identical to the SM, the total en-ergy density of the universe doubles, leading to ∆ N eff ≡ N eff − N SMeff (cid:39) .
4, which is very strongly excluded by thePlanck result of N eff = 3 . ± .
23 [25]. Of course, wealready know that the twin sector cannot be identical tothe SM sector, since at the very least the Higgs vev ratiosobey f /v >
1. The simplest solution to avoid the N eff constraint would be to raise the mass of all the light twinparticles, which would remove the twin contributions to N eff . This however is not possible: the twin electron (orthe twin tau for the case of the FTH) would not be able toannihilate away and would overclose the universe. There-fore at least one of the twin states must remain light toallow the annihilation of the twin electrons.Since there are necessarily contributions to ∆ N eff , weneed to refine the prediction by taking into account thetemperature difference between the two sectors: the valueof N eff will be determined by g (cid:48) (cid:63) , N eff = N SMeff + 47 (11 / / g (cid:48) (cid:63) (2)where g (cid:48) (cid:63) = (cid:88) i s i g i (cid:18) T (cid:48) i T (cid:19) (3)is the number of effective degrees of freedom in the twinsector weighted by the relative temperatures of each com-ponent T (cid:48) i compared to the SM, s i = 1(7 /
8) for a boson(fermion), and the sum runs over the relativistic twinDOFs at late times. The ratio of the temperature ofthe dark sector to the SM temperature can be calculatedassuming separate entropy conservation in both sectorsafter decoupling [26], T (cid:48) T = (cid:18) g (cid:63)s ( T ) g (cid:48) (cid:63)s ( T ) g (cid:48) (cid:63)s ( T d ) g (cid:63)s ( T d ) (cid:19) / for T < T d (4)where T d is the temperature at decoupling. Next we willconsider various possible options for lowering ∆ N eff indifferent realizations of Twin Higgs models. FIG. 1: Contribution to ∆ N eff with heavy twin neutrinos( m ˜ ν = 10 GeV) and massless twin photon, when varyingΛ (cid:48) / Λ. Each band corresponds to varying f/v from 5 (top ofband) to 20 (bottom of band). The smallest contribution isobtained when the two sectors decouple between their QCDphase transitions. The dashed bounded regions correspondto one and two generation Twin Higgs models. The Planckconstraint of N eff = 3 . ± .
23 [25] is also shown.
The simplest approach to lowering ∆ N eff in TwinHiggs models is to raise the mass of the twin neutri-nos above a few GeV. This will remove the twin neu-trino’s contributions to g (cid:48) (cid:63) ( T d ) and to g (cid:48) (cid:63) ( T ), resultingin a smaller ∆ N eff , where the additional energy densityat late times arises entirely from the twin photon. Thiscan be naturally achieved by lowering the twin seesawscale, but comes at a price of an additional source for Z breaking. In a later section, we show that this couldpotentially also be the source of the Z breaking in theHiggs potential generating f /v ∼ a few.The resulting predictions for ∆ N eff depend signifi-cantly on the value of the decoupling temperature, T d .This a consequence of the large change in the number ofdegrees of freedom during the twin (and ordinary) QCDphase transition (PT). If decoupling happens between thetwo PTs, then ∆ N eff will be strongly reduced.The twinQCD PT strongly reduces the degrees of freedom withinthe twin sector, while dumping its entropy into both sec-tors as they are still in equilibrium. Then, if by the timethe SM QCD PT occurs, the two sectors have thermallydecoupled, the entropy of the SM QCD PT will be trans-ferred only to the SM bath, raising the SM temperaturerelative to the twin temperature. This suppression willbe reflected in the final value of ∆ N eff measured at latetimes.In Fig. 1 we show the predictions for ∆ N eff for theMTH model with a massless twin photon, and with themasses of the three twin neutrinos raised to m ˜ ν = 10GeV, as a function of the decoupling temperature T d .Here f /v has been varied from 5 to 20, and the ra-tio of twin to SM QCD phase transition temperatures,Λ (cid:48) / Λ, between 2 and 10 (where Λ (cid:48) / Λ = 5 is obtainedin MTH with an O (10%) splitting of the SU(3) gaugecouplings [15]). The minimum contribution to ∆ N eff de-pends significantly on f /v for any decoupling temper-ature, as it determines whether or not the light twinstates—namely the twin pions, muons and electrons—arerelativistic at the time of decoupling. There is no strongdependence of the minimum contribution to ∆ N eff onΛ (cid:48) / Λ; however, for Λ (cid:48) / Λ ∼
1, the value of T d required toavoid the constraint has to lie in a narrow range betweenthe two QCD PT scales.We learn that without introducing an additional sourceof Z breaking in the twin Yukawa couplings, MTH is intension with the Planck constraint on ∆ N eff . In orderto satisfy the 2 σ bounds of ∆ N eff = 0 . f /v (cid:38)
20 isrequired, which would imply reintroducing tuning intothe Higgs potential. Additionally, we show N eff for TwinHiggs models with one or two generations. The one gen-eration model can be thought of as a FTH model withgauged hypercharge. Stage 3/4 CMB experiments shouldbe able to highly constrain Twin Higgs models with a sin-gle light state, either a twin photon or neutrino, indepen-dent of T d , f /v , and Λ (cid:48) / Λ, since such scenarios contributea minimum ∆ N eff (cid:38) THERMAL DECOUPLING FROM NEUTRINOMIXING
In this section, we explore the consequences of mix-ing between SM and twin neutrinos on the cosmology ofTwin Higgs models. We show that for sufficiently largemixing angles, the neutrino-twin neutrino scattering pro-cesses in Fig. 2 may be the last to efficiently transfer en-ergy between the sectors, thereby lowering the decouplingtemperature and potentially reducing the contribution to N eff .Twin neutrino mixing induces interactions, mediatedby SM EW gauge bosons, between the twin neutrinosand the SM leptons: L int( ν, ˜ ν ) = g √ (cid:96)γ µ P L ( c θ ν + s θ ˜ ν ) W + µ + h . c . + g c w ( c θ ¯ ν + s θ ¯˜ ν ) γ µ P L ( c θ ν + s θ ˜ ν ) Z µ , (5)where c θ = cos θ , s θ = sin θ , and θ is the neutrino mixingangle. Throughout this section we will assume that oneonly twin neutrino mixes with a SM neutrino, while theresults are easily generalized to more complicated mixing.Energy transfer between the two sectors is most efficientwhen scattering between sectors involves relativistic, andtherefore abundant, particles. Thus for this discussion wewill only need to consider the pions, light charged leptons,and neutrinos in each sector.In order to estimate the decoupling temperature, wefollow [27] and calculate the fractional energy transfer ν ˜ ν Z νν ν ˜ ν Z νν FIG. 2: Diagrams (in the interaction basis) responsible forthermalizing the SM and twin sector. Mass insertions cor-respond to an insertion of a mixing between a SM and twinneutrino. rateΓ E ( T ) = n i n j (cid:104) σv ∆ E (cid:105) ij → k(cid:96) ρ n i m i Γ i → jk ρ decays and inverse decays(6)where n i ( n (cid:48) i ) and ρ i ( ρ (cid:48) i ) are the equilibrium number den-sity and energy density distributions of the particles in-volved, and ρ is the average density of all the particlesinvolved. The thermally averaged energy transfer rate is n i n j (cid:104) σv ∆ E i (cid:105) ij → k(cid:96) ≡ (cid:90) d Π i d Π j d Π k d Π (cid:96) (2 π ) δ ( p ) × f i f j (1 ± f k )(1 ± f (cid:96) ) | M | ∆ E i , (7)where d Π i ≡ g i d p i (2 π ) E i is the Lorentz invariant phase-space volume. The matrix elements are to be averagedover initial and final degrees of freedom. To find the de-coupling temperature, the thermally averaged rates needto be calculated. This involves numerically integratingthe high dimensional phase-space integrals in Eq. (7).When performing the integrals numerically, we follow thetechniques given in Appendix A of [27].Thermal decoupling occurs when the energy transferrate is no longer fast compared to the expansion,Γ E ( T d ) (cid:39) H ( T d ) , (8)at which point the energy transfer process begins losingto the expansion of the universe and freezes out. Wehave explicitly solved the full Boltzmann equations andverified that Eq. (8) well-approximates the correct decou-pling temperature.The two most significant energy-transfer processes formixing angles s θ (cid:46) − are the semi-annihilations ˜ νν ↔ νν and decays and inverse decays ˜ ν ↔ νf ¯ f where f isa light SM fermion. One may expect that twin pion de-cays and scattering may also be significant, as they arealso only suppressed by one mixing insertion. However,these occur via off-shell twin weak bosons and are sup-pressed by ( v/f ) relative to the twin neutrino processesmediated by SM EW bosons. - - - - - FIG. 3: Contours of the cosmologically preferred region 0 . < T d < purple region shows the 95% C.L. lim-its from DELPHI on sterile neutrinos produced from Z -decaysat LEP [28]. We also show projected reaches from displacedsearches (see next section) as dashed curves. Projections forSHiP [29] ( green ), DUNE [30] ( yellow ), and FCC-ee [31, 32]( red ) are taken from [33]. The LHC reach at √ s = 13 TeVwith 300 fb − , with searches for lepton jets ( blue ) and trilep-tons ( brown ) are also shown [34]. The exclusion region andprojection curves are only valid if the twin-neutrino cannotdecay into light twin particles, for instance, in the FTH (1generation) scenario. The energy transfer rate from decays and inverse de-cays is Γ ˜ ν ↔ νf ¯ f = α s θ m ν πc w s w m Z . (9)A fairly good analytic solution for T d in Eq. (8) can beobtained if decoupling happens when some of the par-ticles involved are non-relativistic, and the phase-spaceintegrals can be performed analytically. In the limit that T (cid:28) m ˜ ν , we find for the semi-annihilation process, (cid:104) σvE ˜ ν (cid:105) ˜ νν → νν = πα s θ m ν c w s w m Z . (10)Using Eqs. (9) and (10) in Eqs. (7) and (8), we find thatthe process ˜ νν ↔ νν thermally decouples at m ˜ ν T d (cid:39) . (cid:16) s θ − (cid:17) (cid:16) m ˜ ν
10 GeV (cid:17) (cid:16) g (cid:63)s (cid:17) (cid:18) T d . (cid:19) (11)and ˜ ν ↔ νf ¯ f decouples at m ˜ ν T d (cid:39) . (cid:16) s θ − (cid:17) (cid:16) m ˜ ν
10 GeV (cid:17) (cid:16) g (cid:63)s (cid:17) (cid:18) T d . (cid:19) . (12)Thus for most of the parameter space, decays and in-verse decays decouple later than the semi-annihilation. Ideally, thermal decoupling happens before the QCDphase transition, but below the scales of µ (cid:48) , π (cid:48) and Λ (cid:48) ,around 0 . (cid:46) T d (cid:46) m ˜ ν and sin θ , usingthe full numerical phase space integrals. As expected,decays and inverse decays provide the last scattering,except for small twin neutrino masses m ˜ ν (cid:46) N eff . Optimal values of decou-pling occur for m ˜ ν ∼
10 GeV and sin θ ∼ − − − . SIGNATURES AND CONSTRAINTS
The first signal of the Twin Higgs sector may comefrom a measurement of N eff at late times. As evidentfrom Fig. 1 the full MTH model is already in tension withdata from Planck, while the two-generation model will beprobed soon by Stage 3 CMB experiments. The singlegeneration FTH model, with a massless twin photon ortwin neutrino, may not be probed until future Stage 4CMB experiments.In addition to the CMB measurements of N eff , weshould look for other opportunities to discover the TwinHiggs. The most well-explored direction is to exploit theHiggs portal connecting the SM and twin sectors, leadingto displaced Higgs decays for the FTH [10, 35–40]. Wecan also look for the exotic states present at the scale ofthe composite and supersymmetric UV completions [41–44].As we have argued that a sizable mixing between thetwin and standard neutrino sectors could be present, weshould also be able to probe the twin sector via the neu-trino portal. The twin neutrinos can be produced in de-cays of the Z -boson and heavy mesons at high and lowenergy machines. However, if there are twin particleslighter than the twin neutrinos, the neutrinos will prefer-entially decay into twin particles which will ultimately beinvisible to detection, leading to missing energy signals.If no twin particles are kinematically accessible to thetwin neutrino, e.g. as in a one generation twin model,the twin neutrino will decay back into the visible sector,providing another possible window for detection of TH.The decay width corresponding to the range of massesand mixing angles which minimize N eff for a one-generation twin model with massless twin photon cor-respond to macroscopic lifetimes for the twin neutrinos, τ ˜ ν = 0 .
38 mm (cid:18) − s θ (cid:19) (cid:18)
10 GeV m ˜ ν (cid:19) . (13)This allows the twin neutrinos to be probed at theLHC via displaced vertex searches [34] and the proposedMATHUSLA detector [45]; and at fixed target and beamdump experiments such as DUNE [30] and SHiP [29].Future high energy e + e − machines may also probe thisparameter space [31, 32]. The projected reach of theseexperiments are depicted in Fig. 3 alongside the cosmo-logically preferred region. NEUTRINO MASSES AND MIXING
The low energy mass-matrix involving the SM neutri-nos ν R , ν L and the twin neutrinos ˜ ν R , ˜ ν L , is in general anarbitrary 4x4 matrix (where for simplicity we suppressflavor in this and later sections). The form of this massmatrix can be quite complicated in general. In whatfollows, we draw motivation from the Randall-Sundrum(RS) setup that will be described below and consider twosimple benchmark scenarios for neutrino masses and mix-ings. We find that only the second scenario can lead tolarge enough mixing for the neutrino-interactions to ther-malize the two sectors (sin θ > − ), as we will be shownbelow. Two Seesaws : The simplest case to consider is thesetup where the only source of Z breaking and leptonnumber violation are the different seesaw scales. In theRS setup this corresponds to the case when the onlysource of Z breaking and lepton number violation arethe right-handed neutrino masses localized on the UVbrane. The effective neutrino mass and mixing terms arethen L = 12 M ν R ν R + 12 ˜ M ˜ ν R ˜ ν R + M D ν R ˜ ν R + m D ν R ν L + ˜ m D ˜ ν R ˜ ν L + h.c. (14)Assuming M D (cid:28) M, ˜ M , the light SM neutrinos and twinneutrinos have a typical see-saw mass set by the scales m ν (cid:39) m D M , m ˜ ν (cid:39) ˜ m D ˜ M , (15)respectively. Mixing between the twin and SM neutrinosis induced via the UV brane Dirac mass M D leading toa mixing angle sin θ (cid:39) M D (cid:112) M ˜ M (cid:114) m ν m ˜ ν , (16) In the minimal FTH, if the twin neutrino is heavy, it mustdecay—via its mixing with the SM neutrino—before BBN, alsomotivating macroscopic lifetimes. where m ν and m ˜ ν are the light SM and twin neutrinomasses. This mixing will be bounded by sin θ (cid:28) − due to the small neutrino mass ratio, and will generallynot be large enough to thermalize the two sectors whenthe twin neutrino becomes non-relativistic. SM seesaw, Dirac twin neutrinos:
Another inter-esting limit is the case when only the SM neutrinos aresee-sawed, while the twin neutrinos are pseudo-Dirac. Inthe 5D setup this corresponds to the situation when thesinglet twin neutrino is strongly IR-localized and thuscannot feel the UV-localized Majorana masses, while L − ˜ L is broken only on the UV brane. The effectiveneutrino masses and mixings are L = 12 M ν R ν R + m D ν R ν L + ˜ m D ˜ ν R ˜ ν L + m ˜ ν L ν L + h.c. (17)Integrating out ν R in this limit, L eff = − m D M ν L + ( mν L + ˜ m D ˜ ν R )˜ ν L + h.c. (18)and ˜ ν L and the linear combination mν L + ˜ m D ˜ ν R thenform a pseudo Dirac pair with mass ∼ ˜ m D for ˜ m D (cid:29) m ,and the SM neutrino acquires a Majorana mass m D / M .The mixing angle in this case issin θ (cid:39) m m D . (19)Compared the previous two see-saw scenario, this mixingis not limited by the neutrino mass ratio and can be quitelarge. TWIN NEUTRINOS IN A WARPED UVCOMPLETION
We have shown in the previous sections that the neu-trino sector of Twin Higgs models can have a significanteffect on its cosmology. Here we explore the most well-known UV completion of Twin Higgs models based on awarped extra dimension known as the “holographic com-posite Twin Higgs” (CTH) [20]. This setup can addressboth the structure of the twin neutrino masses and mix-ings, as well as the effect of the Z breaking necessarilypresent in the neutrino sector on the Higgs potential. The setup of the holographic CTH
The holographic composite Twin Higgs model is basedon a 5D RS [46, 47] setup with an AdS backgroundmetric ds = ( R/z ) ( dx − dz ), where R is the AdScurvature, the UV brane is at z = R and the IR braneat z = R (cid:48) . The SO(8) global symmetry required for theTwin Higgs mechanism is incorporated as a bulk gaugesymmetry (along with QCD and twin QCD, which play SMSM Z ~ SO (8) IRUV SO (8) ! SO (7) SO (8) ! SO (7) SO (8) ! ⌫ R , ⌫ L , ˜ ⌫ L ˜ ⌫ R [ SU (2) ⇥ U (1)] FIG. 4: An illustration of the structure of the neutrino sectorin the warped UV completion. no role in the lepton sector). On the UV brane, boundaryconditions for the gauge fields break the bulk gauge sym-metry down to the gauge symmetries of the Twin Higgs:SO(8) → SU(2) L × SU(2) mL × U(1) Y × U(1) mY . This break-ing pattern ensures the correct light gauge boson spec-trum . On the IR brane, SO(8) is broken down to SO(7).The gauge generators broken on both branes correspondto broken global symmetries and result in seven Gold-stone bosons arising from the A component of the cor-responding gauge fields. Six of the seven Goldstones areeaten by SM and twin gauge W and Z bosons, and oneremains as the physical pseudo-Nambu-Goldstone bosonHiggs.We additionally gauge ( B − L ) − ( ˜ B − ˜ L ) in the bulk.This extra gauge symmetry ensures that L − ˜ L leptonnumber is a good symmetry in the bulk and on the IRbrane (the CFT interpretation of this statement is thatthe CFT itself preserves this combination of lepton andtwin lepton number). To make sure this does not resultin new light degrees of freedom, the symmetry is brokenon the UV brane. The UV brane is the only source of L − ˜ L violation; thus all Majorana masses arise from UV-localized operators. An illustration of the structure of thewarped UV completion can be seen in Fig. 4.The two relevant mass scales of the 5D model are R (cid:48) and f . The UV-brane location, R (cid:48) , sets the KK scale( M KK ≈ /R (cid:48) ), while f is the global symmetry breakingscale for SO(8) → SO(7): f = 2 g ∗ R (cid:48) . (20) Although we do not do so, one could eliminate the twin photonfrom the spectrum by breaking the mirror hypercharge U(1) mY on the UV brane. Here g ∗ is the the dimensionless 5D SO(8) gauge cou-pling ( g ∗ = g R − / ) which parametrizes the interactionstrength of the KK modes and sets the ratio M KK /f . The neutrino sector of the holographic CTH
The SM and twin SU(2) L doublet leptonsare embedded in two separate vectors ofSO(8) ⊃ SO(4) × SO(4) m ⊃ SU(2) L × SU(2) R × SU(2) mL × SU(2) mR as Ψ = 1 √ e L + . . .ie L + . . .ν L + . . .iν L + . . . ... , Ψ m = 1 √ ...˜ e L + . . .i ˜ e L + . . . ˜ ν L + . . .i ˜ ν L + . . . (21)and the right handed neutrinos are introduced as SO(8)singlet fermions Ψ = ν R , Ψ m = ˜ ν R . (22)Here and throughout we neglect flavor indices. One canunderstand the discussion below as pertaining to one twinneutrino generation, while generalizing to multiple gen-erations is straightforward.The symmetry breaking patterns on the branes, alongwith the L − ˜ L symmetry, will determine which low-energy mass terms exist for the would-be zero modes.Since UV-localized mass terms are the only source of L − ˜ L breaking, the effective zero-mode low-energy La-grangian, after integrating out the KK modes, is L = 12 M ν R ν R + 12 ˜ M ˜ ν R ˜ ν R + M D ν R ˜ ν R + m D ν R ν L + ˜ m D ˜ ν R ˜ ν L + m ˜ ν L ν L + h.c. (23)Depending on the size of the mass-terms in Eq. (23),this matches onto either Eq. (14) or Eq. (17). In the limitwhere mixing terms between the SM and twin sectors arenegligible ( M D , m → M , ˜ M ) will be large, and we arrive atthe two seesaws of Eq. (15). We note, however, that thissituation is tuned: M D is naturally the same size as M This g ∗ should not be confused with the number of relativisticdegrees of freedom g (cid:63) . The strong coupling limit g ∗ ∼ π reducesthe tension with flavor constraints in the quark sector [21] andalso creates separation between M KK and f . At the same timethis limit does not increase the Higgs potential tuning becausethe dominant quantum corrections to the Higgs mass are cut offby the twin top at a scale ∼ f . and ˜ M . If instead, the right-handed twin neutrino isvery IR-localized, with all other neutrino fields kept UV-localized, ˜ M and M D are exponentially suppressed by R/R (cid:48) (see Eq. (30)), and we match onto Eq. (17). Z -breaking effects on the Higgs potential Finally, we explore the consequences of breaking theTwin Higgs Z symmetry in the neutrino sector by alarge difference between the SM and twin seesaw massscales. The Z breaking in the neutrino sector reintro-duces quadratic sensitivity to the cutoff scale into theHiggs mass term from one-loop diagrams involving thewould-be zero mode neutrinos and KK modes. Withoutthe warped UV completion, one might naively expect thecontributions from the Z breaking in the neutrino sec-tor to be large—the right-handed neutrino mass providesa large scale which could potentially feed into the Higgspotential. However, in the RS UV completion, the Z breaking reintroduces quadratic sensitivity to M KK butnot to the seesaw scales themselves.If the Z symmetry is badly broken in the low-lyingKK modes, one naively expects the contribution to theHiggs potential to be δm h ∼ y g ∗ π M KK , (24)where y is an order one factor arising from the overlap ofthe KK modes and Higgs. However, this overestimatesthe corrections to the Higgs mass since it does not takeinto account the collective effect of all the KK modes—atlarge momenta, where the KK modes become important,the exponential suppression becomes important. There-fore, we must calculate the Higgs potential in the 5Dtheory, taking into account all of the KK modes.We present the detailed calculation the of 1-loop con-tributions to the Higgs potential in the Appendix. Sincethe full formulae for the corrections to the potential interms of the RS parameters are lengthy and not very il-luminating, we present here the approximate expressioninstead: We will be interested in the case where the twinright-handed neutrino is strongly IR localized and left-handed neutrino is UV localized, as this is the case thatcan generate large mixing between the SM and twin neu-trinos. In the limit that c m (cid:29) − .
5, the contributionto the Higgs mass is dominated by the twin sector and The m term can be forbidden naturally by gauging B − L and˜ B − ˜ L separately in the bulk. given approximately by δm h (cid:39) g ∗ π M KK (2 c m − (cid:18) RR (cid:48) (cid:19) c m − (cid:34) (cid:18) e (cid:19) × I − c m ( x ) (cid:32) I + c m ( x ) + I + c m ( x ) I − c m ( x ) | m ν | I − c m ( x ) (cid:33) (cid:35) − (25)where x = M KK R (cid:48) and the numerical value of the termin [ . . . ] − is O (0 . e − p/M KK . The analogous formula with c m ( c m ) → c ( c ) also holds for the subdominant SM con-tribution. Compared to the naive estimate, there is awarping down by a factor of (2 c m − / R/R (cid:48) ) c m − .This is the same suppression factor that appears in thesize of the mixing angle, sin θ in Eq. (31), and twin-neutrino mass, ˜ m D in Eq. (30) when the twin neutrino isstrongly IR localized. The numerical value of the contri-bution to the Higgs mass for a twin neutrino with mass m ˜ ν = 10 GeV and mixing sin θ = 10 − is small for mostparameters. However Eq. (25) is an approximate expres-sion and breaks down for c m very close to 1/2, i.e., wherethe twin neutrino becomes heavy. There are points in pa-rameter space where a single twin neutrino can have siz-able mixing, and the KK modes contribute significantlyto the Z -breaking Higgs mass term.In addition, the different generations of twin neutri-nos are not necessarily expected to be degenerate, and aheavy twin neutrino can give a large contribution to δm h if m ˜ ν ∼
200 GeV. For the case of a heavy twin-neutrino,the contribution to δm h can be well approximated by thewould-be zero mode alone. Calculating the standard one-loop contribution to the Higgs potential from the Yukawacoupling m ˜ ν f h ˜ ν L ˜ ν R , we find δm h (cid:39) π (cid:18) m ˜ ν f (cid:19) M KK (26) (cid:39) (260 GeV) (cid:16) m ˜ ν
200 GeV (cid:17) (cid:18) M KK
10 TeV (cid:19) (cid:18) f /v (cid:19) . Acknowledgments —
We thank Kaustubh Agashe,Marco Farina, Michael Geller, Yonit Hochberg, SungwooHong, Jessie Shelton, Ofri Telem and Yuhsin Tsai foruseful discussions. C.C., E.K. and S.L. are supported inpart by the NSF grant PHY-1316222. EK is supported bya Hans Bethe Postdoctoral Fellowship at Cornell. Thiswork was initiated at the Aspen Center for Physics, whichis supported by NSF grant PHY-1066293.
APPENDIX
RS setup
Here we present more details of the embeddings of theleptons into the warped model. All fermions are intro-duced as 5D bulk fields, corresponding to 4-componentDirac spinors, i.e.
Ψ = ( χ, ¯ ψ ), where both χ and ψ areleft-handed 2-component Weyl spinors. The fields explic-itly depicted in Eq. (21) will contain zero modes in the χ components of the bulk fields (SM and twin left-handedleptons). Their χ components are assumed to have (+ , +)boundary conditions at the UV and IR branes to ensurethe existence of a zero mode in χ , while the fields sup-pressed in Eq. (21) are assumed to have ( − , +) boundaryconditions to avoid extra light fermions in the spectrum.Fields containing right-handed zero modes ( ν R , ˜ ν R , e R ,˜ e R ) are assumed to have ( − , − ) boundary conditions forthe χ -component, in order to allow for a zero mode inthe ψ component of the corresponding bulk field.The vector of SO(8) contains two SU(2) L doublets, q ±± , where the first and second subscript denotes thefield’s T L and T R quantum numbers, respectively, andtwo SU(2) mL doublets p ±± ,Ψ = √ ( q ++ + q −− , iq ++ − iq −− , q + − + q − + , iq + − − iq − + ,p ++ + p −− , ip ++ − ip −− , p + − + p − + , ip + − − ip − + ) . We identify hypercharge as Y = T R and twin hyper-charge as Y m = T mR , so q + − has the quantum numbersof ν L and q −− has the quantum numbers of e L . For thetwins states, we should identify p + − with ˜ ν L and p −− with ˜ e L . In addition to the fields defined earlier, right-handed electrons are embedded into the antisymmetric representation of SO(8), Ψ , Ψ m .Each 5D field has a bulk Dirac mass cR ¯ΨΨ, where thedimensionless parameters c , c m , c , c m , c , c m con-trol the localization properties of the zero modes. Weconsider parameter space where all lepton fields are UVlocalized (corresponding to elementary leptons in the 4Dlanguage), except the singlet twin neutrino which we al-low to be either UV localized ( c m < − ) or IR localized( c m > − ), the latter case corresponding to compositeright-handed twin neutrinos.The symmetry breaking patterns allow mass terms onthe UV and IR branes, which will determine the massesand mixings among the light neutrinos. On the UVbrane, the most general renormalizable Lagrangian al-lowed by the gauge symmetries and the boundary condi-tions includes the mass terms L UV = 12 M ν R ψ ν R ψ ν R + 12 M ˜ ν R ψ ˜ ν R ψ ˜ ν R + M (cid:48) ψ ν R ψ ˜ ν R + h.c. (27)where the brane mass parameters M ν R , M ˜ ν R and M (cid:48) aredimensionless and are generically 3 × Z symmetry relating the two sectors is broken onthe UV brane. In particular, Z is broken on the UVbrane if M ν R (cid:54) = M ˜ ν R . The Majorana mass terms M ν R and M ˜ ν R for the singlet neutrinos lead to the warpedseesaw mechanism [48–50]. The CFT interpretation ofthe warped seesaw mechanism (see [50]) is that while theCFT itself is lepton-number preserving, the elementarysector breaks lepton number at a high scale.Using the interpretation that the CFT is Z preserving,we require that the IR brane localized mass terms are Z invariant. On the IR brane SO(8) is broken to SO(7),under which Ψ decomposes as Ψ + Ψ . We can thenwrite the following SO(7) invariant IR brane localizedmass terms L IR = − (cid:18) RR (cid:48) (cid:19) [ m ν ( χ ψ ν R + χ m ψ ˜ ν R ) + m (cid:48) χ χ m + m e ( χ χ + χ m χ m )] + h.c. (28)which will be responsible for the Dirac masses of the lep-tons. The effect of the χ ψ ν R operator is to modify theboundary conditions on the IR brane such that the right-handed singlet neutrino zero mode in ψ ν R is partially ro-tated into χ . Once A acquires a vev (correspondingto EWSB), the two would-be zero modes in χ acquire aDirac mass from Ψ (cid:104) A (cid:105) Ψ . Similarly χ χ m rotates thezero mode in χ ˜ ν L (living in χ m ) into χ and leads to aDirac mass between χ ν L and χ ˜ ν L after EWSB. Mass spectrum
In this Appendix we describe the mass terms for thewould-be zero mode leptons to lowest order in
R/R (cid:48) and v/f . We first observe that all KK states can be integratedout, leaving only the effective Lagrangian for the would-be zero modes L eff = 12 M ψ (0) ν R ψ (0) ν R + 12 ˜ M ψ (0)˜ ν R ψ (0)˜ ν R + M D ψ (0) ν R ψ (0)˜ ν R (29)+ m D χ (0) ν L ψ (0) ν R + ˜ m D χ (0)˜ ν L ψ (0)˜ ν R + mχ (0) ν L χ (0)˜ ν L + h.c. which matches onto Eq. (23). Majorana mass terms χ (0) ν L χ (0) ν L and χ (0)˜ ν L χ (0)˜ ν L do not appear even after EWSBsince the Higgs sector preserves L − ˜ L (a result of thefact that B − L − ˜ B + ˜ L is a gauge symmetry in the bulkand IR brane).Expressing the (dimensional) mass terms for thewould-be zero mode neutrinos in terms of the RS pa-rameters, we find m D (cid:39) g ∗ vm ν a c , − c (cid:0) RR (cid:48) (cid:1) c − c − ˜ m D (cid:39) g ∗ fm ν × (cid:40) a c m , − c m (cid:0) RR (cid:48) (cid:1) c m − c m − c m < − ia c m , − c m (cid:0) RR (cid:48) (cid:1) c m − c m > − m (cid:39) g ∗ vm (cid:48) a − c , − c m (cid:0) RR (cid:48) (cid:1) c + c m − M (cid:39) − (1 + 2 c ) M νR R ˜ M (cid:39) M ˜ νR R × (cid:40) − (1 + 2 c m ) c m < − (1 + 2 c m ) (cid:0) RR (cid:48) (cid:1) c m c m > − M D (cid:39) M (cid:48) R × (cid:40) √ a c ,c m c m < − √ ia c ,c m (cid:0) RR (cid:48) (cid:1) + c m c m > − (30)where a x,y = (cid:112) (1 + 2 x )(1 + 2 y ) /
2. Note that keepingterms higher order in v/f results in the replacement g ∗ v → g ∗ f sin( vf ) and g ∗ f → g ∗ f cos( vf ), as expectedfrom a pseudo Goldstone (Twin) Higgs. In the absence ofmixing, the neutrino mass is given by m D / M . These re-sults agree with the exact result obtained from the lowestzero of the spectral functions, which takes into accountthe mixing of the KK modes and zero modes within eachSO(8) multiplet. In terms of the RS parameters, themixing angle in Eq. (19) issin θ (cid:39) (2 c m − (cid:18) RR (cid:48) (cid:19) c m − c m (cid:18) vf (cid:19) (cid:18) m (cid:48) m ν (cid:19) . (31)The electrons (and muons/taus), acquire the followingmass terms after EWSB, m e (cid:39) g ∗ vm e (cid:40) a c , − c (cid:0) RR (cid:48) (cid:1) c − c − c < − ia c , − c (cid:0) RR (cid:48) (cid:1) c − c > − (32)and similarly for m ˜ e . Coleman-Weinberg potential for Majorana KKspectrum
The calculation presented in the main text of the Z -breaking effects in the Higgs potential from the neutrinosector is utilizing the full expression of the Coleman-Weinberg potential. However due to the appearance ofthe Majorana masses for the right handed neutrinos thestandard techniques (which assume Dirac fermions) forevaluating the CW potential for KK theories have to beaugmented. Here we explain how to deal with a Ma-jorana KK spectrum in general, and present the actualcalculation in the next Appendix.The general expression of the Coleman-Weinberg po-tential for KK theories takes the form [51] V = ( − F N (cid:88) n (cid:90) d p (2 π ) log( p + m n ) (33) where N is the number of DOFs at each level of the KKtower (3 for a gauge boson, 4 for a Dirac fermion) and m n ( v ) are the KK masses which depend on the Higgsvev. The sum is usually turned into a contour integral inthe the complex m -plane, resulting in an integral over thespectral function ρ ( m ), whose zeros encode the KK spec-trum. The spectral function is determined by solving theequations of motion (EOM) and applying the boundaryconditions to obtain a quantization condition.The function ρ ( m ) has simple poles along the real axiscorresponding to the KK spectrum, so the the sum overthe KK masses can be performed via a contour integral inthe complex m -plane using zeta function regularizationtechniques [52, 53]. For Dirac KK modes, ρ ( m ) ≡ ρ ( m )(as consequence of the exact degeneracy of the left andright 2-component spinors which make up each Dirac KKstate). Thus for the Dirac case one can perform the KKsum by summing over the positive masses in Eq. (33) andsetting N = 4 to account for the DOFs associated withboth chiralities. This is equivalent to a contour integralwhich only encloses the the positive zeros of the spectralfunction in the complex m plane. However, in the pres-ence of Majorana mass terms on the UV brane, the KKspectrum becomes pseudo Dirac, and the zeros of ρ ( m ) at m (cid:39) ± m do not pair up exactly (and as a consequence ρ ( m ) is a function of m , not m as in the Dirac case).Therefore, one must also include the negative mass solu-tions in the sum in Eq. (33), or equivalently the zeros of ρ ( m ) along the Re( m ) < h -independent contribution tothe cosmological constant and performing two contourintegrals, one to enclose the poles on the Re( m ) > m ) < m axis ( m = ip ) V = ( − F N (4 π ) (cid:90) ∞ dp p Re log[ ρ ( ip )] (34)where ρ ( ip ) is generally complex since ρ is a function of m rather than m and N = 2 for Majorana KK modes. Coleman-Weinberg potential for the warped twinseesaw
We first present the spectral functions ρ ν ( m ) ( ρ mν ( m )),which encode the exact masses of the SM (twin) neutrinoKK spectrum. In the presence of ν -˜ ν mixing, there isonly one spectral function incorporating both the SMand twin neutrino KK towers. However for the purposeof calculating the CW potential we can set the mixing tozero ( M (cid:48) → m (cid:48) →
0) since the effects from the mixingare expected to be small. The two spectral functions areparameterized by two form factors:0 ρ ν = 1 + f ν sin (cid:18) hf (cid:19) (35) ρ mν = 1 + f mν cos (cid:18) hf (cid:19) . (36)The sin ( h/f ) terms are generated by the SM neutrinosector, while the pieces with cos ( h/f ) are generated bytwin neutrinos.First, we look for separable solutions to the bulk EOM,assuming the ansatz χ = (cid:88) n g n ( z ) χ n ( x ) and ¯ ψ = (cid:88) n f n ( z ) ¯ ψ n ( x ) . (37)In the presence of (cid:104) A (cid:105) , the bulk EOMs are coupled. Wecan, however, perform a field redefinition (which resem-bles a gauge transformation within the bulk) to remove (cid:104) A (cid:105) from the EOMs [54, 55]. The transformation thatdoes this is the Wilson line Ω( z ) = e − ig (cid:82) dz (cid:104) A (cid:105) . Wedefine hatted fields which do not depend on (cid:104) A (cid:105) f ( z, (cid:104) A (cid:105) ) = Ω( z ) ˆ f ( z, g ( z, (cid:104) A (cid:105) ) = Ω( z )ˆ g ( z, . (38)The differential equations for ˆ f , ˆ g are now the stan-dard bulk EOMs without the A vev, which have solu-tions in terms of Bessel functions J ν and J − ν . It is moreconvenient to work in the basis of the warped analog offlat space sines and cosines [54] satisfying C c ( R ) = 1, S c (0) = 0, C (cid:48) c ( R ) = 0, and S (cid:48) c ( R ) = m where c is thebulk mass parameter, leading to a simpler quantizationcondition. The explicit expressions for these C c , S c func-tions are S c ( z ) = πmR (cid:18) Rz (cid:19) − c − × (39) (cid:16) J c + ( mR ) Y c + ( mz ) − Y c + ( mR ) J c + ( mz ) (cid:17) C c ( z ) = πmR (cid:18) Rz (cid:19) − c − × (40) (cid:16) Y c − ( mR ) J c + ( mz ) − J c − ( mR ) Y c + ( mz ) (cid:17) One needs to multiply by z − c in order for these functionsto satisfy the bulk EOMs.The boundary conditions are modified by the field re-definition in Eq. (38). The UV boundary conditions areunchanged since Ω( z = R ) = 0. On the IR brane how-ever, we must apply Wilson line transformation Ω( R (cid:48) )relating f to ˆ f at z = R (cid:48) , which was worked out in [21].Applying the IR boundary conditions to ΩΨ and ΩΨ m produces two 4 by 4 systems of equations involving theneutrino and twin neutrino wave functions. There existsa solution if and only if the determinant of the coefficientmatrix is 0. Thus the determinant of each IR boundary condition matrix gives the spectral function of the theory,resulting in f ν ( p ) = − | ˆ m ν | ( C − + M ν R S − ) S − [ C − ( S − M ν R C ) + | ˆ m ν | S ( C − + M ν R S − )](41)where ˆ m ν = m ν (cid:0) RR (cid:48) (cid:1) c − c and the twin form factor hasthe same form with c → c m , c → c m , M ν R → M ˜ ν R .Expanding in small parameters R/R (cid:48) and v/f to leadingorder reproduces the masses of the lightest modes ob-tained from taking into account only the would-be zeromodes and neglecting the KK modes. In the limit thatthe Majorana masses M ν R , M ˜ ν R →
0, the form factors re-produces the the top sector form factors in the CTH [21].The neutrino sector contribution to the Coleman-Weinberg potential for the Higgs can be written in termsof the spectral functions as V eff ( h ) = − π ) (cid:90) ∞ dpp Re log[ ρ ν ( ip ) ρ mν ( ip )] . (42)In principle, the potential can be calculated numericallyby inserting the form of the spectral functions. However,we obtain an analytical approximation for the potentialby expanding in powers of sin ( hf ).If there were no Z breaking, the Twin Higgs Z wouldguarantee that the quadratically divergent pieces cancel,see [21]. However, we are interested in breaking the Z in the neutrino sector. The Higgs mass correction can befound by differentiating with respect to x = sin ( hf ) andis given by δm h = 1 f ∂V∂x (0) (43) (cid:39) − π ) f (cid:90) ∞ dpp (Re( f ν ) − Re( f mν )) + O ( f ν )where terms quadratic and higher in the form factorshave been neglected since the form factors are exponen-tially suppressed for p (cid:38) M KK .Without Z breaking, Re f ν − Re f mν = 0 and thelargest possible contribution to the potential from theneutrino sector vanishes up to terms higher order in theform factors. However, if we allow the Z to be broken,Re f ν − Re f νm (cid:54) = 0, and we can obtain potentially largecorrections to the Higgs mass term. ∗ Electronic address: [email protected] † Electronic address: kufl[email protected] ‡ Electronic address: [email protected][1] Z. Chacko, H.-S. Goh, and R. Harnik, Phys. Rev. Lett. , 231802 (2006), hep-ph/0506256. [2] G. Burdman, Z. Chacko, H.-S. Goh, and R. Harnik,JHEP , 009 (2007), hep-ph/0609152.[3] H. Cai, H.-C. Cheng, and J. Terning, JHEP , 045(2009), 0812.0843.[4] N. Craig, S. Knapen, and P. Longhi, Phys. Rev. Lett. , 061803 (2015), 1410.6808.[5] N. Craig, S. Knapen, and P. Longhi, JHEP , 106(2015), 1411.7393.[6] Z. Chacko, Y. Nomura, M. Papucci, and G. Perez, JHEP , 126 (2006), hep-ph/0510273.[7] B. Batell and M. McCullough, Phys. Rev. D92 , 073018(2015), 1504.04016.[8] N. Arkani-Hamed et al. , Phys. Rev. Lett. , 251801(2016), 1607.06821.[9] R. Barbieri, T. Gregoire, and L. J. Hall, (2005), hep-ph/0509242.[10] N. Craig, A. Katz, M. Strassler, and R. Sundrum, JHEP , 105 (2015), 1501.05310.[11] R. Barbieri, L. J. Hall, and K. Harigaya, JHEP , 172(2016), 1609.05589.[12] Z. Chacko, N. Craig, P. J. Fox, and R. Harnik, (2016),1611.07975.[13] N. Craig, S. Koren, and T. Trott, (2016), 1611.07977.[14] V. Prilepina and Y. Tsai, (2016), 1611.05879.[15] M. Farina, JCAP , 017 (2015), 1506.03520.[16] M. Farina, A. Monteux, and C. S. Shin, Phys. Rev. D94 ,035017 (2016), 1604.08211.[17] I. Garca Garca, R. Lasenby, and J. March-Russell, Phys.Rev. Lett. , 121801 (2015), 1505.07410.[18] N. Craig and A. Katz, JCAP , 054 (2015),1505.07113.[19] I. Garca Garca, R. Lasenby, and J. March-Russell, Phys.Rev.
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