Higgsed Stueckelberg Vector and Higgs Quadratic Divergence
IIZTECH/PHYS-2014-04
Higgsed Stueckelberg Vector and Higgs Quadratic Divergence
Durmu¸s Ali Demir, Canan Nurhan Karahan, and Beste Korutlu
Department of Physics, ˙Izmir Institute of TechnologyUrla, ˙Izmir, 35430 TURKEY (Dated: September 30, 2014)Here we show that, a hidden vector field whose gauge invariance is ensured by a Stueckelbergscalar and whose mass is spontaneously generated by the Standard Model Higgs field contributes toquadratic divergences in the Higgs boson mass squared, and even leads to its cancellation at one-loopwhen Higgs coupling to gauge field is fine-tuned. In contrast to mechanisms based on hidden scalarswhere a complete cancellation cannot be achieved, stabilization here is complete in that the hiddenvector and the accompanying Stueckelberg scalar are both free from quadratic divergences at one-loop. This stability, deriving from hidden exact gauge invariance, can have important implicationsfor modelling dark phenomena like dark matter, dark energy, dark photon and neutrino masses.The hidden fields can be produced at the LHC.
PACS numbers: 11.15.Ex, 12.15.-y
INTRODUCTION
With the discovery of a new resonance at the LargeHadron Collider (LHC), having a mass m h = 125 . ± . – the UVboundary of the SM. In explicit terms, one-loop quantumcorrection to Higgs squared-mass, originally computed byVeltman [4], reads as( δm H ) quad = Λ π (cid:18) λ H + 94 g + 34 g (cid:48) − g t (cid:19) , (1)where g and g (cid:48) are the SU (2) L and U (1) Y gauge cou-plings of the SM, respectively, and g t = m t /υ H ( υ H =246 GeV is the VEV of the Higgs field) is the topquark Yukawa coupling. The top quark, being the moststrongly coupled SM particle to the Higgs field, inducesthe biggest contribution and ensures a nonvanishing, un-removable coefficient before Λ . The Higgs boson massis stabilized to electroweak scale if | δm H | < m H < Λ .This is the Veltman condition (VC). The parameters in ithave all been measured, and it violates the LHC resultsfor Λ >
500 GeV [5].Having no symmetry to prevent the Higgs boson massfrom sliding to the higher scales via (1), frequently acancellation mechanism is implemented via fine-tuningof counter terms in which low and high energy degreesof freedom are mixed. This renders the whole procedureunnatural. It would be more natural, if the cancellationoccurs by means of a symmetry principle at higher scales,or if it arises by accidental cancellations of certain terms.In fact, models of new physics constructed to complete the SM beyond Fermi energies have all been motivatedby Higgs naturalness problem [5] (see also [6] for studieswithin supersymmetry). So far, however, in the 7 TeVand 8 TeV LHC searches reaching out beyond the TeVdomain, no compelling sign of evidence for new physicshas been found [7].In consequence, having no TeV scale new physics forachieving naturalness, one is forced to understand theelectroweak unnaturalness within the SM plus generalrelativity, albeit with some imperative extensions re-quired by specifics of the approach taken. In 1995, con-formal symmetry [8] was proposed as a mechanism forsolving the Higgs mass hierarchy problem (the lateststudies on the conformal symmetry as a solution to thefine-tuning problem may be found in [9]). Recently, theHiggs coupling to spacetime curvature has been found tostabilize the electroweak scale by a harmless, soft fine-tuning [10]. Furthermore, anti-gravity effects have beenclaimed to improve Higgs naturalness [11]. Alternatively,one may view the parameters chosen by nature as the ne-cessity of existence, and this leads to anthropic consider-ations [12]. In variance with all these approaches, a fine-tuning method based on singlet scalars [13] has also beenemployed. In this approach, main idea is to cancel thequadratic divergences in Higgs boson mass with the loopsof the singlet scalars that couple to Higgs field [14]. Thismethod, though a fine-tuning operation by itself, nulli-fies the quadratic divergences and accommodates viabledark matter candidates [15, 16]. Nevertheless, for realsinglet scalars with vacuum expectation value (VEV), itis not possible to kill the quadratic divergences consis-tently because there is a mixing between the CP-evencomponent of the Higgs field and the real singlet scalar,and it does not allow for simultaneous cancellation of thequadratic divergences in Higgs boson and singlet scalarmasses [17]. There are also studies on two-Higgs doubletmodels without flavor changing neutral currents, demon-strating that, although the cancellation in the coefficient a r X i v : . [ h e p - ph ] S e p of the one-loop quadratically divergent terms is possible,the parameter space is severely constrained [18]. An ad-ditional complex scalar triplet extension of the SM hasalso been studied and proven to be a solution to the fine-tuning problem [19].In the present work, as a completely new approachnever explored before, we study protection of the Higgsboson mass by a SM-singlet gauge field (not a scalar fieldas in [14]). In contrast to the attempts based on hiddenscalars [14, 15, 18, 19], which are now known to be un-able to simultaneously protect the masses of the Higgsboson and the singlet scalar [17], in the present work, weconsider a hidden U (1) gauge field V µ whose invarianceis ensured by a Stueckelberg scalar S and whose mass isspontaneously induced by the SM Higgs field. We showthat V µ and S enable cancellation of the quadratic diver-gence in Higgs boson mass with no quadratic divergencearising in their own masses. It is important that the SMHiggs boson is stabilized at one-loop along with already-stable hidden gauge and Stueckelberg scalar. This phe-nomenological advantage has important implications notonly for stabilizing the Higgs boson mass but also forcorrelating the SM Higgs field with hidden sectors.The paper is organized as follows. In Section 2 below,we construct the model starting from the basic Stueckel-berg setup. Section 3 is devoted to computation of thequadratic divergences and vanishing of the Higgs massdivergence by fine-tuning. We conclude in Section 4. THE MODEL
In this section, we consider a massive Abelian gaugefield V µ accompanied by a real scalar field S ( x ), intro-duced to preserve the gauge invariance of the theory.Originally proposed by Stueckelberg [20] and noted af-terwards by Pauli [21] that, V µ satisfies a restricted U (1)gauge invariance, with the gauge function Θ( x ) obeyinga massive Klein-Gordon equation. The mechanism pro-vides an alternative to the Higgs mechanism, where thevector boson acquires its mass with the breakdown ofthe gauge invariance of not the Lagrangian but of thevacuum. These features are encoded in the Stueckelbergmodel [22] L = − V µν + 12 m (cid:18) V µ − m ∂ µ S (cid:19) −
12 ( ∂ µ V µ + mS ) , (2)where m is the common mass for V µ and S . Despite itsmassive spectrum, this model enjoys a U (1) m invariance V µ ( x ) → V (cid:48) µ ( x ) = V µ ( x ) + ∂ µ Θ( x ) ,S ( x ) → S (cid:48) ( x ) = S ( x ) + m Θ( x ) , (3)provided that (cid:0) (cid:3) + m (cid:1) Θ( x ) = 0. Consequently, inspite of its nonvanishig hard mass, V µ enjoys exactgauge invariance, albeit with a restricted gauge trans-formation function Θ( x ) [22]. In the massless limit, m →
0, the Stueckelberg Lagrangian (2) reduces to L m =0 = − V µν + ∂ µ S ∂ µ S , which is obviously U (1) m invariant in Lorentz gauge ( ∂ µ V µ = 0) with an unre-stricted Θ( x ). Interestingly, the Stueckelberg scalar S ,transforming like the gauge field V µ in massive case, turnsinto a gauge-singlet scalar in massless limit.Inspired from the Stueckelberg model (2), we proposethe Higgsed Stueckelberg model L = − V µν + λ H † H (cid:18) V µ − √ λ a H ∂ µ S (cid:19) −
12 ( ∂ µ V µ + (cid:112) λ a H S ) , (4)where λ is a positive dimensionless constant and a H is a mass parameter. This model is manifestly gauge-invariant under both the hidden U (1) m invariance with m → √ λ a H , and the electroweak gauge group SU (2) L ⊗ U (1) Y . The Higgs potential V ( H ) = m H H † H + λ H (cid:0) H † H (cid:1) and hence the total energy is minimized atthe Higgs field configuration (cid:104) H † H (cid:105) = (cid:40) υ H if m H < , m H > , (5)where υ H = (cid:113) − m H λ H is the Higgs VEV in the brokenphase ( m H < SU (2) L ⊗ U (1) Y is spontaneously broken down to elec-tromagnetism. In unbroken phase ( m H >
0) electroweakgroup stays exact and all the SM particles but Higgs bo-son are massless.From (4) it is clear that, the two phases of the SMdirectly leave distinguishable effects on the mass of V µ and kinetic term of S . And the Stueckelberg structurein (2) is achieved properly if the mass parameter a H cankeep track of the two electroweak phases. This feature isimplemented into the Higgsed Stueckelberg model (4) bysetting a H = (cid:60) (cid:115) − m H λ H = (cid:40) υ H if m H < , m H > , (6)which obviously dogs the Higgs VEV in (5). It turnsout that (cid:104) H † H (cid:105) = a H / υ H = a H specifically in the brokenphase. This switching ability of a H ensures that, in thebroken phase of electroweak group, there arises, in ad-dition to the massive SM spectrum, a massive vector V µ with mass M V = λ υ H and a massive scalar m S = λ a H .In the unbroken phase, however, the Higgs field stands asthe only massive field. The rest, inclusing V µ and S , areall massless. In what follows, we will work in the physicalvacuum of the broken electroweak phase and necessarilyset a H = υ H everywhere.It is instructive to study the transcription of theStueckelberg U (1) m symmetry in (3) into the HiggsedStueckelberg case. To this end, one notes that the Stueck-elberg scalar S ( x ) facilitates U (1) m gauge invariance ofthe hidden sector, and also, helps keep the Hamilto-nian positive definite [20]. In this formalism, Lorentzsubsidiary condition does not follow from equation ofmotion. Imposing an operator equation of the form ∂ µ V ( − ) µ ( x ) | phys (cid:105) = 0, where V ( − ) µ ( x ) involves the freefield annihilation operators, however, gives rise to con-flict between the operator equation and the canonicalcommutation relations. This puzzle is solved via the in-troduction of an additional scalar field S ( x ), replacingthe operator equation with Φ( x ) | phys (cid:105) ≡ [ ∂ µ V ( − ) µ ( x ) + mS ( − ) ( x )] | phys (cid:105) = 0, where S ( − ) ( x ) also involves freefield annihilation operators. The operator equation de-creases the number of degrees of freedom of the La-grangian to four. The required constraint to decreaseit to three for a massive vector field comes into play withthe gauge transformation V µ ( x ) → V (cid:48) µ ( x ) = V µ ( x ) + ∂ µ Θ( x ) ,S ( x ) → S (cid:48) ( x ) = S ( x ) + (cid:112) λ υ H Θ( x ) , (7)which closely follows the Stueckelberg transformation(3). The U (1) m invariance is ensured if ( ∂ + λ υ H )Θ( x ) = 0. This restricted gauge invariance changesto an unrestricted, standard gauge invariance in the un-broken ( m H >
0) electroweak phase in which V µ and S are massless and non-interacting. Moreover, S is agauge singlet in this phase. The V µ and its Stueckelbergcompanion S do possess identical masses in broken andunbroken phases of the electroweak symmetry. In brokenphase, Stueckelberg-Feynman gauge, their propagatorsread as ∆ µν = − i g µν q − m , ∆ = iq − m , (8)where m = λ υ H is the common mass for V µ and S . PHENOMENOLOGY
In this section we study quantum corrections to massesof the Higgs boson h and Stueckelberg fields S and V µ . Note that the last term in (4) can also be written as L gf = − α ( ∂ µ V µ + α √ λ υ H S ) , where α is a real parameter, simi-lar to t’Hooft’s parametrization for Abelian Higgs model. Thechoice of α = 1 corresponds to the Stueckelberg-Feynman gauge.When α (cid:54) = 1, the restriction on the gauge function changes to( (cid:3) + αλ υ H )Λ( x ) = 0. It is also possible to choose two differentparameters α and α , to check the gauge independence of the pa-rameters. However, there is the disadvantage that the terms of theform V µ ∂ µ B survives for this choice. In the present work, we willwork in Stueckelberg-Feynman gauge. The main constraint on the model is that Higgs bosonmust weigh m h = 125 . V µ , the Stueckelberg field S , and theHiggs field h . The vertex factors are summarized in theAppendix. The Higgsed Stueckelberg hidden sector thenmodifies the Veltman condition (1) as( δm H ) quad = Λ π (cid:18) λ H + 94 g + 34 g (cid:48) − g t + λ (cid:19) , (9)wherein λ shows up as a new degree of freedom. In thephiloshopy of the original attempts in [14], one can sup-press ( δm H ) quad by choosing λ appropriately. In par-ticular, ( δm H ) quad vanishes for λ = 4 .
41. The V µ and S are degenerate in mass, and for this specific value of λ they weigh m = √ λ υ H = 517 GeV. It is possible todecrease the value of λ by simply introducing N suchfields, which in turn lowers the masses of the new fieldswhile increasing their number. In Figure 1, a schematicrepresentation of the one-loop quantum corrections toHiggs mass is shown in our extended scenario. As it isapparent from this figure, a hidden Abelian gauge sectorsplendidly cancels the quadratically divergent contribu-tions to Higgs mass from the SM fields. h W (cid:177) , Zt V, S ∆ m h (cid:144) (cid:76) m h FIG. 1. The schematic representation of the quadratically di-vergent contributions to Higgs boson mass at one-loop level.Here, h denotes the Higgs boson, W ± , Z the electroweakbosons, t the top quark, V, S the hidden gauge boson V µ andthe Stueckelberg scalar S , respectively. Higgs mass is pro-tected from destabilizing quantum effects when the hiddengauge sector is included. It is clear that suppressing ( δm H ) quad requires λ tobe finely tuned. The fine-tuning here is of the samesize as the fine-tunings required for hidden scalar sec-tors [14, 15, 17–19]. There is one big difference, how-ever. Indeed, these models based on hidden scalars suf-fer from the fact that masses of the hidden scalars andof the SM Higgs boson cannot be protected simultane-ously [17]. The hidden scalar continues to have a mass O (Λ) after suppressing the radiative contribution to theHiggs boson mass. In the Higgs-Stueckelberg model thisimpasse is overcome. To see this, one notes that massof the Stueckelberg field does actually receive quadrati-cally divergent radiative corrections from two self energydiagrams (one with Higgs boson in the loop and anotherwith both Higgs and the Stueckelberg field S in the loop).The self energy diagram with a Higgs boson and vec-tor boson V µ in the loop diverges logarithmically. Thespruceness of this scenario emerges at this point in thatthe quadratically-divergent contributions to the mass ofthe Stueckelberg field from the two loop diagrams cancelout to give ( δm S ) quad = 0 . (10)In the same manner, the mass of V µ is protected againstquadratically-divergent quantum corrections( δm V ) quad = 0 . (11)Leaving aside the logarithmic corrections, masses of V µ and S are found to be UV-insensitive. This is actuallyexpected by gauge invariance because there exists an un-broken U (1) m invariance in both broken and unbrokenelectroweak phases. The invariance protects the mass of V µ . Interestingly, it also protects the mass of S because S by itself acts like a gauge field when V µ is massiveand becomes a non-interacting U (1) m singlet when V µ is massless. Clearly, the radiative stability of the hid-den sector can have important implications for modelling‘dark phenomena’ like Dark Matter, Dark Energy, DarkPhoton and neutrino masses. CONCLUSION AND OUTLOOK
The discovery of a new scalar [1] at the LHC, con-sistent with the SM Higgs boson, has accelerated stud-ies on the UV-sensitivity of the Higgs boson. As op-posed to the physical masses of chiral fermions and gaugebosons, which are protected by chiral and gauge sym-metries, there is no symmetry principle to protect theHiggs boson mass against quadratically divergent quan-tum corrections. In the very absence of TeV-scale newphysics, one is left with a finely-tuned Higgs sector wherenature and degree of fine-tuning vary with the modelingdetails. In the presence of hidden scalars, despite theprotection of the Higgs boson mass the hidden sector it-self is UV-unstable. In case the hidden sector is formedby the spacetime curvature scalar, the fine-tuning is se-vere yet harmless because the SM fields and couplingsare immune to its presence. The fine-tuning is as severeas hidden scalars in other field-theoretic approaches.In this Letter we have shown that a hidden sectorspanned by an Abelian vector field whose mass is in-duced by electroweak breaking and whose gauge invari-ance is sustained by a Stueckelberg scalar can lead tostabilization of the Higgs boson mass by finely tuningits coupling to the SM Higgs field. In spite of this un-avoidable fine-tuning, the Higgsed Stueckelberg modelpossesses the striking property that the hidden sector isinsensitive to the UV scale. This stability, deriving from unbroken hidden gauge invariance, can have importantcollider, astrophysical and cosmological implications. In-deed, a stable hidden sector can be utilized in construct-ing viable models of Dark Matter, Dark Energy, DarkPhoton and neutrino masses. The model can be testedat the LHC (and its successor FCC) via direct produc-tions of V µ and S fields. APPENDIX
Here we list the vertex factors: λ hhVV = 2 iλ g µν ,λ hhSS = − iυ H k µ q ν g µν ,λ hVV = 2 iλ υ H g µν ,λ hSS = − iυ H k µ q ν g µν ,λ hVS = 2 (cid:112) λ k µ g µν , (12)where k µ is the momentum of S . We used a H = υ H in a H dependent vertices. Acknowledgments
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