W + W − H Production at Lepton Colliders: A New Hope for Heavy Neutral Leptons
EEur. Phys. J. C manuscript No. (will be inserted by the editor) W + W − H production at lepton colliders: A new hope for heavy neutralleptons J. Baglio a,1,2,3 , S. Pascoli b,3 , C. Weiland c,3 Institut f¨ur Theoretische Physik, Eberhard Karls Universit¨at T¨ubingen, Auf der Morgenstelle 14, D-72076 T¨ubingen, Germany Institute for Advanced Study, Durham University, Cosin’s Hall, Palace Green, Durham DH1 3RL, United Kingdom Institute for Particle Physics Phenomenology, Department of Physics, Durham University, South Road, Durham DH1 3LE, United KingdomOctober 4, 2018
Abstract
We present the first study of the production of aStandard Model Higgs boson at a lepton collider in associa-tion with a pair of W bosons, e + e − → W + W − H , in the in-verse seesaw model. Taking into account all relevant exper-imental and theoretical constraints, we find sizable effectsdue to the additional heavy neutrinos up to −
38% on thetotal cross-section at a center-of-mass energy of 3 TeV, andeven up to −
66% with suitable cuts. This motivates a de-tailed sensitivity analysis of the process e + e − → W + W − H as it could provide a new, very competitive experimentalprobe of low-scale neutrino mass models. PACS
Neutrino oscillations, as discovered by the Super-Kamiokan-de experiment in 1998 [1] and subsequently confirmed by aplethora of results [2], imply that at least two neutrinos havea non-zero mass. This cannot be explained in the StandardModel (SM) and calls for an extension of this framework.One of the simplest possibilities is the addition of new ferm-ionic gauge-singlet states that play the role of right-handedneutrinos, leading to the type-I seesaw mechanism and itsvariants [3–18]. Amongst the various seesaw realizations,one of particular interest is the inverse seesaw model (ISS)[10–12]. It was proved in [19] that, in any model that onlyadds fermionic gauge singlets to the SM field content withno cancellation between the contributions to the light neu-trinos masses from different orders of the seesaw expansionor different radiative orders, requiring the light neutrinos tobe massless is equivalent to requiring lepton number to be a email: [email protected] b email: [email protected] c email: [email protected] conserved. The inverse seesaw verifies all these conditionsand we indeed observe that in the lepton-number-conservinglimit of this model, light neutrinos are massless, indepen-dently from the seesaw scale or the size of the neutrino Yu-kawa coupling. In this renormalizable, testable, low-scaleseesaw model, light neutrino masses are suppressed not by asmall-active sterile mixing as in the high-scale type I seesaw.Instead, this model relies on an approximately conservedlepton number in agreement with the theorem [19], thus al-lowing to generate the light neutrino masses while havinglarge neutrino Yukawa couplings and heavy sterile neutrinosat the TeV scale, opening the exciting possibility of detect-ing the latter in current or future planned high-energy collid-ers, see for example Refs. [20–22] for reviews. It is partic-ularly worth noting that this model provides a prototype offermionic low-scale seesaw, making our results applicableto a wide range of models.As the neutrino Yukawa couplings in the ISS can belarge, the properties of the Higgs boson, the remnant of theelectroweak symmetry-breaking mechanism [23–26] gener-ating the masses of the other fundamental particles in theSM and that was discovered at the Large Hadron Collider(LHC) in 2012 [27, 28], can be sizeably affected. This opensnew search strategies which rely on the Higgs boson, for in-stance Higgs decays [29–32], searches in Higgs productionat lepton colliders [33, 34], or lepton flavour violating Higgsdecays [35, 36]. We also investigated recently the heavy neu-trino impact on the triple Higgs coupling [37, 38].Based on the idea that t –channel fermions coupled toa Higgs boson can give sizeable contributions to a cross-section, see for example the case of b ¯ b → W + W − H at theLHC [39], we investigate in this paper, for the first time,the impact of heavy neutrinos on the production of a Higgsboson in association with a pair of W bosons at a lepton col-lider, e + e − → W + W − H . This process has been studied inthe SM and has been found to have good detection prospects a r X i v : . [ h e p - ph ] O c t [40]. We describe the ISS model and discuss the relevanttheoretical and experimental constraints. We present our cal-culational setup before a numerical analysis of our resultsis carried out. Performing a scan over the relevant param-eters of the model, we find deviations up to −
38% on thetotal cross-section at 3 TeV, that can be enhanced to − The ISS model [10–12] is an appealing low-scale seesawmodel that extends the SM with fermionic gauge singlets.We consider here a realisation where each generation is sup-plemented with a pair of these right-handed gauge singlets, ν R and X , which have opposite lepton number. This pro-vides a realistic realisation of seesaw models close to theelectroweak scale that can reproduce low-energy neutrinomasses and mixing while being in agreement with all exper-imental bounds. The additional mass terms to the SM La-grangian are L ISS = − Y i j ν L i (cid:101) Φν R j − M i jR ν CRi X R j − µ i jX X Ci X j + h . c . , (1)where Φ is the SM Higgs field and (cid:101) Φ = ı σ Φ ∗ , i , j = . . . Y ν and M R are complex matrices and µ X is a complex sym-metric matrix. A major characteristic of the ISS is the pres-ence of a naturally small lepton-number-breaking parameter µ X to which the light neutrino masses are proportional. In-deed after block-diagonalising the full neutrino mass matrix,the 3 × M light (cid:39) m D M T − R µ X M − R m TD , (2)at leading order in the seesaw expansion parameter m D M − R ,where m D = Y ν (cid:104) Φ (cid:105) . This decouples the light neutrino massgeneration from the mixing between active and sterile neu-trinos (that is proportional to m D M − R ) and allows for largeYukawa couplings even when the seesaw scale is close to theelectroweak scale. It is worth noting that in this model, theheavy neutrinos form pseudo-Dirac pairs where the splittingis controlled by µ X as can be seen from diagonalizing the1-generation neutrino mass matrix, which gives m N , N = ± (cid:113) M R + m D + M R µ X ( M R + m D ) , (3)in the seesaw limit µ X (cid:28) m D , M R [36].Since one of the main motivations of our model is toexplain neutrino oscillations, we reproduce low-energy datafrom the global fit NuFIT 3.0 [42] by usingthe µ X -parameterisation adapted to include next-order terms in the seesaw expansion that are relevant for large active-sterile mixing [38] µ X (cid:39) (cid:18) − M ∗− R m † D m D M T − R (cid:19) − × M TR m − D U ∗ PMNS m ν U †PMNS m T − D M R × (cid:18) − M − R m TD m ∗ D M † − R (cid:19) − . m ν is the diagonal light neutrino mass matrix and U PMNS isthe unitary Pontecorvo-Maki-Nakagawa-Sakata (PMNS) [43,44] that diagonalises M light . We have chosen δ = U PMNS for simplicity. We fix the light-est neutrino mass to 0 .
01 eV, in agreement with the Planckresults [45]. The strongest experimental constraints for thisstudy come from a global fit [46] to electroweak precisionobservables (EWPO), tests of CKM unitarity and tests oflepton universality. Since we choose all mass matrices andcouplings in the neutrino sector to be real and consider di-agonal Yukawa couplings Y ν in our study, we do not expectelectric dipole moment measurements and lepton-flavour-violating processes to provide relevant constraints in thisscenario. Finally, we require that the Yukawa couplings Y ν remain perturbative, namely | Y i j | π < . . (4) The cross-section is calculated at leading order (LO), bothin the SM and in the ISS. Next-to-leading order electroweakcorrections have been calculated in the SM [47] and arefound to be negligible for center-of-mass (c.m.) energiesabove 600 GeV and of the order of −
2% at √ s =
500 GeV,that would correspond to the lowest International Linear Col-lider c.m. energy that would be relevant for our process [48].Given the size of the ISS deviation we obtain (of the orderof −
5% at √ s =
500 GeV and down to −
38% at higher c.m.energies, see later), we will not take these electroweak cor-rections into account in our analysis.The charged leptons are taken massless and their cou-pling to the Higgs boson is neglected. The calculation isdone in the Feynman-’t Hooft gauge. The Feynman diagramsat LO include s –channel exchanges of a Z boson or a photon,as well as t –channel diagrams involving the neutrinos forwhich a generic selection is displayed in fig. 1. The remain-ing t –channel diagrams are obtained with a flipping of the W and charged Goldstone boson contributions from the W − line to the W + line. We have used our own ISS model filedeveloped for the packages FeynArts 3.7 and
FormCalc7.5 [49, 50] to generate a Fortran code, and the numerical e + e + W + e + e − W + HW − W − n i e + He − W + W − n i n j HW − G − n i Fig. 1
Generic Feynman diagrams representing the ISS neutrino con-tributions to e + e − → W + W − H in the Feynman-’t Hooft gauge. Theindices i , j run from 1 to 9. Mirror diagrams are obtained by flippingall the electric charges. integration has been performed with BASES 5.1 [51] in or-der to obtain a selection of kinematic distributions.Similar to the SM calculation, the interference terms aresignificant and destructive. The dominant contribution to theISS amplitude comes from the first two diagrams in fig. 1with heavy neutrinos and which go as | Y ν | v / M R ( a + bv / M R ) , and from the third diagrams withone heavy neutrino and one light neutrino in the t –channelwhich goes as | Y ν | v / M R , in terms of the seesaw param-eters. In order to enhance the cross-section we have alsoperformed a calculation with polarised beams. More specifi-cally, we have chosen, based on the Compact Linear Collider(CLIC) baseline [52], an unpolarised positron beam, P e + =
0, and a polarised electron beam with P e − = − σ LR ( RL ) as the cross-section for a completely polarisedleft-handed (right-handed) positron with P e + = − (+ ) and a completely polarised right-handed(left-handed) electron with P e + = + ( − ) , the po-larised cross-section for arbitrary polarisation fractions P e + / e − can be written as [53] σ pol = (cid:104) ( − P e + )( + P e − ) σ LR + ( + P e + ) (5) ( − P e − ) σ RL (cid:105) , since the LL and RR cross-sections are identically zero inour process. The calculation is done in the G µ scheme (see e.g. Ref. [54],and Ref. [55] in the context of neutrino mass models) and theinput parameters are the Z mass M Z , the W boson mass M W and the Fermi constant G F . Including the Higgs mass M H , the parameter values are chosen as M W = .
385 GeV , M Z = . , M H =
125 GeV , G F = . × − GeV − . (6)Based on our previous analysis on the triple Higgs coupling[38], we use the µ X -parameterisation with a degenerate Yu-kawa texture, Y ν = | Y ν | I , with hierarchical heavy neutrinos, M R = diag ( M R , M R , M R ) . To illustrate our results we se-lect the same hierarchy as in [38], M R = . M R , M R = . M R , M R = M R . (7)From now on, M R is to be understood as a number as wellin a slight abuse of notation. These specific heavy neutrinomass ratios are related to our choice of Y ν = | Y ν | I sincethey make the constraints of the global fit [46] impact everygeneration similarly.We present in fig. 2 the variation of the total produc-tion cross-section σ ( e + e − → W + W − H ) as a function of thec.m. energy √ s , using a benchmark scenario with | Y ν | = M R = . O ( ) Yukawa couplings, is allowed by current ex-perimental and theoretical constraints. √ s [GeV] σ ( e + e − → W + W − H ) [fb] SM unpolarizedISS unpolarizedSM polarizedISS polarized
Fig. 2
LO total W + W − H production cross-section at an electron-positron collider (in fb) as a function of the c.m. energy √ s (in GeV).The solid curves stand for the SM predictions, the dashed curves standfor the ISS predictions using the benchmark scenario described in thetext. The red (blue) curves are for an unpolarised ( −
80% polarisedelectron beam) cross-section. The ratio of the ISS cross-section withrespect to the SM prediction is shown in the insert.
The gain by going from an unpolarised cross-sectionto the polarised electron beam is illustrated by the factor-of-two difference between the red curves (unpolarised) andthe blue curves (polarised). The behaviour of the ISS con-tribution in the polarized cross-section is the same as that of the unpolarized one, meaning that the use of a polar-ized beam will lead to more events thus increasing the sen-sitivity to the large deviations coming from the ISS. Themaximum of the cross-section is obtained at c.m. energiesaround 500 GeV, for which the ratio of the ISS cross-sectionwith respect to the SM cross-section, shown in the insert,is around 0.95. The negative contribution from the ISS cor-rection increases with higher c.m. energies, reaching already20% for √ s ∼ . −
38% ata c.m. energy close to 3 TeV, from which the ISS correctionstarts to decrease for increased c.m. energies.In order to get insights into the dependence of the ISScorrection on the parameters of the ISS, we have performedin fig. 3 a scan of the ISS deviation with respect to the SMproduction cross-section, ∆ BSM = ( σ ISS − σ SM ) / σ SM , as afunction of the seesaw scale M R and of the parameter | Y ν | forthe diagonal Yukawa texture we have chosen and still usingheavy hierarchical neutrinos with the parameters of eq.(7).The c.m. energy is fixed to √ s = −
80% polarised electronbeam. The grey area is excluded by the constraints appliedto the ISS, the global fit to EWPO and low-energy data [46]being the dominant constraint. . . . .
54 2 4 6 8 10 12 14 16 18 20 ∆ BSM [ % ] | Y ν | M R [TeV] ∆ BSM map for σ ( e + e − → W + W − H ) √ s = 3 TeV − − − − − − − − - % - % - % - % - % - % - % - % E x c l ud e db y t h ec o n s t r a i n t s Fig. 3
Contour map of the neutrino corrections ∆ BSM (in percent) tothe W + W − H production cross-section at a 3 TeV electron-positroncollider, using a −
80% polarised electron beam, as a function of theseesaw scale M R (in GeV) and | Y ν | in the µ X -parameterisation, usinga diagonal Yukawa texture and a hierarchical heavy neutrino mass ma-trix with the parameters defined in eq.(7). The grey area is excluded bythe constraints. The ISS contribution vanishes, as expected, for a largeseesaw scale M R and for vanishing Yukawa couplings. For alarge fraction of the parameter space, deviations of at least −
20% are allowed, and they reach a peak of − | Y ν | ∼ ∆ BSM = − ∆ BSM in the region allowedby the experimental constraints and for M R > A ISSapprox = ( ) M R Tr ( Y ν Y † ν ) (cid:18) . − .
79 TeV M R (cid:19) , ∆ BSMapprox = ( A ISSapprox ) − . A ISSapprox . (8)The coefficients (calculated here for √ s = M R > M R < / M R that we have not includedfor simplicity and clarity give sub-leading corrections thatdegrade the agreement between our fit and the full calcula-tion. For example, we find for our benchmark scenario with | Y ν | = M R = . M R < . ±
10% or moreand we advise not to use it: We get for example for | Y ν | = . M R = . ∆ BSM = − .
4% to becompared to the result of our fit ∆ BSMapprox = − . W + (inblack) and W − (in red) distributions are identical for boththe pseudo-rapidity (left) and the energy (right) observables,while the Higgs distributions are displayed in blue.For both W ± and Higgs boson, the pseudo-rapidities inthe central region have a different behaviour in the SM andin the ISS. More specifically, the ISS corrections are sub-stantial in the region | η | <
1. In the case of the energy spec-trum, depicted on the right-hand side of fig. 4, the ISS cor-rection is distributed over the whole range for the W ± bosons,while it starts to be more significant above 1 TeV for theHiggs boson. We have then considered the following twocuts on the cross-section, in order to enhance ∆ BSM : | η H / W ± | < E H > σ SMpol = .
96 fb and σ ISSpol = .
23 fb, giving ∆ BSM = − σ SMpol , cuts = .
42 fb and σ ISSpol , cuts = .
14 fb, resulting in ∆ BSM = − | Y ν | ∼ M R = . d σ / d η X [ f b ] η X e + e − → W + W − H √ s = 3 TeV P e − = − , P e + = 0% X = W + , SM X = W + , ISS X = W − , SM X = W − , ISS X = H , SM X = H , ISS d σ / d E X [ f b / G e V ] E X [GeV] e + e − → W + W − H √ s = 3 TeV P e − = − , P e + = 0% X = W + , SM X = W + , ISS X = W − , SM X = W − , ISS X = H , SM X = H , ISS Fig. 4
Pseudo-rapidity (left) and energy (right) distributions of the W + (black), W − (red) and Higgs (blue) bosons in the process e + e − → W + W − H at a c.m. energy of 3 TeV, using a −
80% polarised electron beam. The solid curves stand for the SM predictions, the dashed curves stand for theISS predictions using the benchmark scenario described in the text. the same set of cuts we get a deviation of −
34% instead of −
26% for the cross-section without cuts. The level of en-hancement is reduced compared to the benchmark scenariowith | Y ν | = η distributionswhich is closer to that of the SM prediction. In this article we have investigated the effects of heavy neu-trinos on the production of a pair of W bosons in associationwith a Higgs boson at a lepton collider, e + e − → W + W − H .After taking into account the constraints on the model wehave found sizeable deviations that are maximal at a c.m. en-ergy of 3 TeV corresponding to the last stage of the CLICbaseline, reaching a 38% decrease of the cross-section withrespect to the SM prediction, in regions of the parameterspace with Yukawa couplings | Y ν | ∼ − W bosons in association with a Higgs boson at a lepton col-lider have been investigated and our results highlight thepotential of this observable to beat future LHC measure-ments which lose sensitivity in the high mass regime [22].They also demonstrate the ability of this process to probethe coupling to the Higgs boson which is common to allsee-saw type I and III and their extensions, and motivatea detailed sensitivity analysis of e + e − → W + W − H [56] asthis could provide a new, very competitive, and complemen-tary observable to probe neutrino mass models, especially in O ( ) TeV mass regimes with diagonal and real Y ν that aredifficult to probe otherwise. Acknowledgements
J. B. acknowledges the support from the Institu-tional Strategy of the University of T¨ubingen (DFG, ZUK 63), the DFGGrant JA 1954/1, the Kepler Center of the University of T¨ubingen, aswell as the support from his Durham Senior Research Fellowship CO-FUNDed between Durham University and the European Union undergrant agreement number 609412. S.P. and C.W. receive financial sup-port from the European Research Council under the European UnionsSeventh Framework Programme (FP/2007-2013)/ERC Grant NuMassAgreement No. 617143. S.P. would also like to acknowledge partialsupport from the European Unions Horizon 2020 research and innova-tion programme under the Marie Skłodowska-Curie grant agreementsNo 690575 (RISE InvisiblesPlus) and No 674896 (ITN ELUSIVE),from STFC and from the Wolfson Foundation and the Royal Society.
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