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Some Consequences of the Majoron Being the Dark Radiation ∗ We-Fu Chang A,B,Johnn.Ngc, Jackson M.S

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Some Consequences of the Majoron Being the Dark Radiation ∗ We-Fu Chang A,B,Johnn.Ngc, Jackson M.S Physics Letters B 730 (2014) 347–352 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Some consequences of the Majoron being the dark radiation ∗ We-Fu Chang a,b,JohnN.Ngc, Jackson M.S. Wu a,b, a Department of Physics, National Tsing Hua University, No. 101, Section 2, Kuang-Fu Road, Hsinchu, 30013, Taiwan, ROC b Physics Division, National Center for Theoretical Sciences, No. 101, Section 2, Kuang-Fu Road, Hsinchu, 30013, Taiwan, ROC c TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, V6T 2A3, Canada article info abstract Article history: We discuss some phenomenological consequences in a scenario where a singlet Majoron plays the role Received 1 November 2013 of dark radiation. We study the interrelations between neutrino mass generation and the scalar potential Received in revised form 12 December 2013 arising from this identification. We find the extra scalar has to be light with a mass at or below the Accepted 29 January 2014 GeV level. The mixing of this scalar with the Standard Model Higgs impacts low energy phenomena such Available online 11 February 2014 as the muonic hydrogen Lamb shift and muon anomalous magnetic moment. Demanding that the light Editor: S. Dodelson scalar solves the puzzle in the muon magnetic moment requires the scalar to be lighter still with mass at or below the 10 MeV level. The cross-sections for the production of heavy neutrinos at LHC14 are also given. © 2014 Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3. 1. Introduction contribute less than 4/7toNeff as they will not be reheated but the neutrinos do. Decoupling in the muon annihilation era yields It is well known that correlations of temperature fluctuations in a contribution δNeff = 0.39. Weinberg further proposed that the the Cosmic Microwave Background (CMB) depend on the number U (1) global symmetry be a new one associated with a dark sector of effective relativistic degrees of freedom, Neff, which is usually with its own matter content. given in terms of the effective number of neutrinos species present In this Letter we examine the possibility of taking the global in the era before recombination. The expected value of Neff = 3is U (1) symmetry to be the lepton number. The spontaneous break- consistent with observations thus far. However, recent measure- ing of this U (1)L by singlet Higgs will give rise to a Majoron [7], ments of CMB from the Planck satellite [1] combined with that which we associate with the Goldstone boson that acts as DR. of the Hubble constant from the Hubble Space Telescope (HST) [2] Since the singlet that breaks the U(1)L will mix with the Standard Model (SM) Higgs boson, this allows us to connect Higgs physics resulted in a higher value of Neff = 3.83±0.54 at 95%CL. If one fur- ther includes data from WMAP9 [3], Atacama Cosmology Telescope and DR to neutrino physics. In particular, we are able to link con- (ACT) [4] and South Pole Telescope (SPT) [5] into the analysis, the straints on the parameters of the scalar sector to that in the seesaw = +0.50 mechanism responsible for neutrino mass generation, and to study extracted value becomes Neff 3.62−0.48 at 95%CL. This hints at a dark radiation (DR) component beyond the expected three neu- their interrelations. We illustrate this in Type-I seesaw [8] and in- trino species at a confidence level of 2.4σ . The origin and nature verse seesaw [9] scenarios in this Letter. of such DR component are not known. One possibility, as pointed The organization of this Letter is as follows. In Section 2,we out recently by Weinberg [6], is that it can be naturally associ- describe in detail the framework we use to study the interrelation ated with a massless or nearly massless Goldstone boson arising between the Majorons, the neutrinos, and the scalars. In Section 3, the spontaneous breaking of a U (1) global symmetry. A Goldstone we discuss some consequences on the scalar and neutrino param- boson will count as 4/7 of a neutrino, and this appears to agree eters from measurements of the muon magnetic moment, Lamb → with observation. However, in order for the temperature of the shift of the muonic hydrogen, and decay rate of μ eγ . In Sec- Goldstone bosons to match with that of the neutrinos, they must tion 4, we evaluate the range of values of heavy neutrino masses remain in thermal equilibrium with ordinary matter until muon and mixings that can be probed at the LHC. We end with a sum- annihilation. If Goldstone bosons decouple much earlier, they will mary in Section 5. 2. The framework * Corresponding author. E-mail addresses: [email protected] (W.-F. Chang), [email protected] A Majoron model consists of extended scalar and neutrino sec- (J.N. Ng), [email protected] (J.M.S. Wu). tors. The simplest Majoron model extends the Standard Model http://dx.doi.org/10.1016/j.physletb.2014.01.060 0370-2693/© 2014 Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3. 348 W.-F. Chang et al. / Physics Letters B 730 (2014) 347–352 λHSvvS (SM) by a singlet complex scalar, S, and three generations of sin- tan 2θ = . (9) i v2 − v2 glet righthanded (RH) neutrinos, N R [7]. Such extended neutrino λS S λ sector give rise to the standard Type-I seesaw. However, this is not We shall identify h ≡ h as the SM Higgs, which was recently the only possible extension. A phenomenologically far more inter- 1 SM discovered at the LHC to have a mass of 125 GeV. Note that for esting case is to add three more lefthanded (LH) singlet neutrinos. small mixing (which shall be the case below), m2 ≈ 2λv2 and This give rise to the so-called inverse seesaw, and as we shall see, hSM 2 ≈ 2 signatures testable at the LHC. m2 2λS v S . Below, we describe in turn the extended scalar and neutrino From Eqs. (7) and (9), the scalar quartic couplings can be writ- sectors. For the neutrinos, we will describe first the Type-I seesaw ten in terms of the mass eigenvalues and the mixing angle case because of its simplicity, and use it to lay the groundwork for 1 2 2 2 2 the more complicated inverse seesaw case. λ = m + m − m − m c2θ , (10) 4v2 1 2 2 1 1 2.1. The scalar sector λ = m2 + m2 + m2 − m2 c , (11) S 2 1 2 2 1 2θ 4v S The most general renormalizable Lagrangian involving the Higgs 2 − 2 m2 m1 doublet, H, and the complex singlet, S,reads λHS = s2θ , (12) 2vvS † μ † μ Lscalar = (Dμ H) D H + ∂μ S ∂ S − V (H, S), (1) and we define the short hand c ≡ cos x etc. Classical stability of x =− 2 † + † 2 − 2 † + † 2 the vacuum demands that V (H, S) μ H H λ H H μS S S λS S S 2 2 + † † m1m2 λHS H H S S . (2) λ,λS > 0, 4λλS − λHS = > 0, (13) 2 2 v v S This will be common to both the Type-I and inverse seesaw sce- narios we discuss below. After electroweak√ symmetry breaking and we see from above that these conditions are automatically sat- (EWSB), we can write H = (0,(v + h)/ 2 )T in the unitary gauge, isfied for m1,2 real and positive. where we take v = 246.221 GeV, and 2.2. The neutrino sector: Type-I seesaw case 1 2iα(x) S(x) = √ v S + s(x) e . (3) 2 With three generations of extra singlet RH neutrinos, the rele- vant Yukawa interactions read The kinetic term for S then takes the form 1 L ⊃− − c + † μ μ 2 μ y1 LL HNR y2 N R N R S h.c., (14) ∂μ S ∂ S = ∂μs∂ s + 2(v S + s) ∂μα∂ α 2 T ∗ where L = (nL ,lL ) is the LH SM lepton doublet, and H = iσ2 H . 1 μ 1 μ = ∂μs∂ s + ∂μχ∂ χ The generation indices have been suppressed for clarity. Note that 2 2 there is an accidental global U (1) symmetry associated with the s s2 conservation of lepton number (L)beforeEWSBifS is defined to + + ∂ χ∂μχ, (4) 2 μ have L =−2. The Yukawa interactions (14) then give rise to neu- v S 2v S trino masses, which take the form and we identify the canonically normalized Goldstone boson χ ≡ c 1 0 m n 2vsα as the Majoron. L ⊃− c D L + nL N R h.c., (15) The mixing between the Higgs doublet and the complex singlet mD M N R was already analyzed in Ref. [11]. The classical minimum is given √ = −3/2 = by where mD 2 y1 v, M y2 v S / 2, and we have redefined the −iα 2iα lepton fields, ψl → e ψl,toremovethee phase factor from 4λ μ2 − 2λ μ2 4λμ2 − 2λ μ2 the Majorana mass terms. 2 = S HS S 2 = S HS v , v S . (5) For ≡ m /M 1, the standard Type-I seesaw is operative. To 4λλ − λ2 4λλ − λ2 D S HS S HS leading order in , the mass eigenstates are then given by Using this, the scalar mass-squared matrix reads in the (h, s) basis = + c = − c νL nL N R , ηR N R nL , (16) 2 2λv λHSvvS 2 (6) with mass eigenvalues mν = mD and M respectively (after ap- λHSvvS 2λS v S propriate phase rotations).
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