UCI-TR-2019-23, arXiv:1909.02029, Phys. Rev. D100 (2019) 095015 The Majoron at two loops Julian Heeck1, ∗ and Hiren H. Patel2, y 1Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA 2Department of Physics and Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, CA 95064, USA We present singlet-Majoron couplings to Standard Model particles through two loops at leading order in the seesaw expansion, including couplings to gauge bosons as well as flavor-changing quark interactions. We discuss and compare the relevant phenomenological constraints on Majoron pro- duction as well as decaying Majoron dark matter. A comparison with standard seesaw observables in low-scale settings highlights the importance of searches for lepton-flavor-violating two-body decays ` ! `0+Majoron in both the muon and tau sectors. CONTENTS is Weinberg's dimension-five operator (LH¯ )2=Λ [1]. This operator violates lepton number (∆L = 2) and leads to 2 I. Introduction 1 Majorana neutrino masses of order hHi =Λ after elec- troweak symmetry breaking, which gives the correct neu- II. Majoron couplings at tree level and one loop 2 trino mass scale for a cutoff Λ ∼ 1014 GeV. Besides ex- A. Neutrino couplings 3 plaining neutrino oscillation data, an EFT scale this high B. Charged fermion couplings 3 has little impact on other observables and thus nicely ac- C. Couplings to gauge bosons 4 commodates the absence of non-SM-like signals in our experiments. III. Majoron couplings at two loops 4 An ever-increasing number of renormalizable realiza- A. Coupling to two gluons 5 tions of the Weinberg operator exist in the literature, B. Coupling to two photons 5 the simplest of which arguably being the famous type-I C. Coupling to Z and photon 6 seesaw mechanism [2] that introduces three heavy right- D. Coupling to two Z bosons 6 handed neutrinos to the SM field content. While the E. Coupling to two W bosons 7 Weinberg operator explicitly breaks lepton number, un- F. Coupling to γ-Higgs and Z-Higgs 7 derlying renormalizable models could have a dynamical G. Flavor changing quark couplings 8 origin for ∆L = 2 via spontaneous breaking of the global U(1)L symmetry. This leads to the same Weinberg oper- IV. Phenomenology 8 ator and thus Majorana neutrino masses, but as a result A. Light Majorons 8 of the spontaneous breaking of a continuous global sym- B. Comparison with seesaw observables 10 metry also a Goldstone boson appears in the spectrum. C. Majoron dark matter 11 This pseudo-scalar Goldstone boson of the lepton number symmetry was proposed a long time ago and was dubbed V. Conclusion 11 the Majoron [3, 4]. Acknowledgements 12 The Majoron is obviously intimately connected and coupled to Majorana neutrinos, but at loop level also References 12 receives couplings to the other SM particles. This makes it a simple renormalizable example of an axion-like par- ticle (ALP), defined essentially as a light pseudo-scalar I. INTRODUCTION with an approximate shift symmetry. Although not our focus here, by coupling the Majoron to quarks it is even The Standard Model (SM) has emerged as an incredi- possible to identify it with the QCD axion [5{9], thus bly accurate description of our world at the particle level. solving the strong CP problem dynamically. The main Even its apparently accidental symmetries, baryon num- appeal of the Majoron ALP is that its couplings are not arXiv:1909.02029v2 [hep-ph] 14 Nov 2019 ber B and lepton number L, are seemingly of high quality free but rather specified by the seesaw parameters, which and have never been observed to be violated. One could, opens up the possibility to reconstruct the seesaw La- however, argue that the established observation of non- grangian by measuring the Majoron couplings [10]. This zero neutrino masses is not only a sign for physics beyond is aided by the fact that the loop-induced effective op- the SM but also for possible lepton number violation by erators that couple the Majoron to the SM are only two units. This argument is based on an interpretation suppressed by one power of the lepton-number break- of the SM as an effective field theory (EFT) and the ob- ing scale Λ, whereas right-handed neutrino-induced op- servation that the leading non-renormalizable operator erators without Majorons are necessarily suppressed by 1=Λ2 [11{17], rendering it difficult to reconstruct the see- saw parameters in that way. In this article we complete the program that was ∗ [email protected] started in the inaugural Majoron article [3] and derive y [email protected] all Majoron couplings to SM particles. The tree-level 2 6 and one-loop couplings were obtained a long time ago; gw X here we go to the two-loop level in order to calculate the LZ = niZ= [CijPL − CjiPR] nj ; (3) 4 cos θW remaining couplings, which include the phenomenologi- i;j=1 cally important couplings to photons as well as to quarks 6 3 gw X X − ∗ + of different generations. Armed with this complete set LW = p `αBαjW= PLnj + njB W= PL`α ; 2 αj of couplings we then discuss various phenomenological j=1 α=1 consequences and constraints on the parameters. This 6 h X includes a discussion of Majorons as dark matter (DM). L = − n [C (m P + m P ) h 2v i ij i L j R The rest of this article is structured as follows: in i;j=1 Sec. II we introduce the singlet Majoron model and re- +Cji(mjPL + miPR)] nj ; produce the known tree-level and one-loop couplings. In Sec. III we present results of our novel two-loop cal- where gw = e= sin θW with Weinberg angle θW and culations necessary for the Majoron couplings to gauge 3 3 bosons and to quarks of different generations. The phe- X ∗ X ` ∗ ∗ nomenological aspects of all these couplings are discussed Cij ≡ VkiVkj ;Bαj ≡ VαkVkj = Vαj : (4) in Sec. IV. Finally, we conclude in Sec. V. k=1 k=1 ` In the last equation we used Vαk = 11αk since we work in the basis where the charged-lepton mass matrix is diag- II. MAJORON COUPLINGS AT TREE LEVEL onal. AND ONE LOOP The 6 × 6 matrix C and the 3 × 6 matrix B satisfy a number of identities [20, 21] that are particularly impor- In this article we consider the minimal singlet Majoron tant in order to establish ultraviolet (UV) finiteness of model [3], amplitudes involving neutrino loops: c L = −LyN H − 1 N λN σ + h.c. − V(H; σ); (1) R 2 R R C = Cy = CC; which introduces three right-handed neutrinos NR cou- BBy = 11 ;CM CT = 0 ; pled to the SM lepton doublets L and Higgs doublet n y T H, and one SM singlet complex scalar σ carrying lep- B B = C ; BMnC = 0 ; (5) ton number L = −2, minimally coupled to the right- T BC = B ; BMnB = 0 : handed neutrinos proportional to the Yukawa matrices y and λ. We do not specify the scalar potentialp V(H; σ) So far we have not made any assumption about the but simply assume that σ = (f + σ0 + iJ)= 2 obtains scale of MR. In the following we will work in the a vacuum expectation value f, which then gives risep to seesaw limit MD MR, resulting in a split neutrino the right-handed Majorana mass matrix MR = fλ/ 2. spectrum with three heavy neutrinos with mass matrix J is the Majoron, σ0 is a massive CP-even scalar with MR and three light neutrinos with seesaw mass matrix −1 T mass around f, assumed to be inaccessibly heavy in the Mν ' −MDMR MD, naturally suppressed compared to following. Both MR and the charged-lepton mass ma- the electroweak scale v. This hierarchy permits an ex- trix are chosen to be diagonal without loss of generality, pansion of all relevant matrices in terms of the small 3 × 3 matrix A ≡ U yM M −1, where U is the unitary effectively shifting all mixing parametersp into y. Elec- D R T 3 × 3 Pontecorvo{Maki{Nakagawa{Sakata (PMNS) ma- troweak symmetry breaking via hHip= (v= 2; 0) yields trix. Parametrically this corresponds to an expansion in the Dirac mass matrix MD = yv= 2. The full 6 × 6 c the scale hierarchy v f which we refer to as the seesaw neutrino mass matrix in the basis (νL;NR) = V nR is then expansion. To leading order, the matrices take the form y 1 c T 0 MD 11 − AA A L = − n¯RV T V nR + h.c. C ' ; 2 M MR y y D (2) A A A 1 c B ' U(11 − 1 AAy) UA ; (6) ≡ − n¯RMnnR + h.c. ; 2 2 T −AMRA 0 where V is the unitary 6 × 6 mixing matrix to the states Mn ' : 0 MR nR, which form the Majorana mass eigenstates n = nR + nc . The diagonal mass matrix M = diag(m ; : : : ; m ) T R n 1 6 Note that AMRA is diagonal, which imposes constraints consists of the physical neutrino masses arranged in as- on A and provides an implicit definition of U. These con- cending order. Throughout this article, we denote mass straints may be automatically satisfied using the Casas{ matrices with capital letters Mx and individual mass Ibarra parametrization [22]; however, more useful for eigenvalues with small letters mi. In the mass eigenstate our purpose is the Davidson{Ibarra parametrization [23], − basis, the tree-level neutrino couplings to J, Z, W , and T y which uses Mν = −AMRA and MDMD as the indepen- h, take the form [18, 19] dent matrices containing all seesaw parameters.