PoS(ICHEP2016)136 http://pos.sissa.it/ ∗ [email protected] Speaker. The couplings of the Standardpossibility Model to sector search for to dark scale photons (DP). invariant Theand degrees decay DP of of is the studied. freedom Higgs-like The scalar can latter boson open is intois a mediated the photon by provided the by derivative of spontaneous the breaking scalar dilaton, ofmixing emerging strength scale of between which symmetry. the The photon limits and DP. are set on the DP mass, the ∗ Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). 38th International Conference on High Energy3-10 Physics August 2016 Chicago, USA Joint Institute for Nuclear Research E-mail: Gennady Kozlov Dark photons in the decay of a Higgs-like boson PoS(ICHEP2016)136 S µ µν A B µν F ε , where ¯ γ ∼ γ . , a light Higgs . Recently, DP f as a function of Gennady Kozlov → qq ) ε S is accompanied by . In the low energy 2 q TeV − . The dilaton operator , the low energy spec- , f 10 H 10 ( ¯ π qq − O 4 )] 2 12 ∼ = g − ( s Λ m via the kinetic term √ γ or to light lepton-antilepton pair, γ + 1 ¯ is the standard beta-function with the ν ν [ where the latter would be a short-lived triggers EW symmetry breaking at the q ) ¯ γ g m → Λ is associated with a hidden vector sector ( have been probed (see the refs. in [3]). If ¯ q γ β ∑ B 3 − + a 10 µν − 1 7 are the strength tensors of electromagnetic field G ). DP mixes with − v µν a µν = B G gauge invarince through the dilaton mass operator. The f ) ( , where index may oscillate into Y g g B ) ( γ and 2 ) H 1 β 1 ( ( 0 µν 246 GeV is the vev of the Higgs boson U µ = F U K µ × = µ and two vector currents which are related to each other, and this is due to gluon-gluon fusion followed by the (heavy) quarks in the × L θ v ) µ S Y . In SM, the violation of conformal symmetry is understood through 2 ) = in the window 10 ¯ γ K ( 1 µ ε ( is the strength tensor of gluon field. The quark state stands for the anomalous dimension of the mass operator ¯ K SU U and ) µ , where µν a 2 ∂ × γ g Λ G L ( , ) or the scalar dilaton. m < g 2 γ of DP, respectively. The basic object is the Dalitz-like decay ( v H . The standard photon ¯ µ π γ are different except for SU B 4 v , and q = is the trace of the energy-momentum tensor, m and µ µ EW being the mixing strength; f θ is predicted in various models with the values in the range 10 . If the approximate conformal invariance is broken at scale ¯ l Λ ε l We study DP which is within the reach of the LHC energy The interest to search for Dark Matter (DM), in particular dark photons (DP) as a conformal We start with the conformal anomaly relevant to the coupling scalar-vector-vector containing ε → ¯ triggers the breaking of with containing DP scales boson decaying to invisible neutrino-antineutrinoγ pair, trum of states may contain adoublet light or dilaton both. with its In vacuumassume the expectation the value absence breaking (vev) of of the conformal invariance gaugescale at sector the that scale breaks electroweak (EW) symmetry, we and the field should either be and the resonant monophoton signatures inThe Higgs boson decays have been studied at the LHC [2]. anomaly reflects the violation oftended scale group invariance in SM by quantum effects. Our basis is the ex- portal to DM, is increasing fromin year particle to physics. year, and The stillrefs. hidden is a sector in mystery containing [1]), puzzle DP and bothto can in in test be cosmology collider the and explored experiments approximate at atIn low conformal high theories energies sector energies. where (see, an with Increasing e.g., exact its conformal oftheory couplings contains symmetry the a is to massless energy spontaneously scalar the enables boson, broken, the the Standard one by dilaton. low Model The the energy nature (SM) effective requirement of the fields. that dilaton couplingsimportant conformal is that governed at symmetry the be loop level realized the dilatoncompared non-linearly. has to potentially enhanced those To couplings collider to of gluons physics, the andbefore photons SM running it any Higgs. is SM particles In in contrast the loop to through the the latter, trace thea anomaly. dilaton scale couples (dilatation) to current gluons even Dark photons 1. Introduction experiments the values of where the mass the non-vanishing divergence of no excess events are found, the obtainedDP results mass. can The be used production to of imposeloop bounds with on final states coupling constant PoS(ICHEP2016)136 . Λ of the and IR ) d becomes 2 IR , ¯ γ d 1 γ , of a collider ω is appropriate = s UV i Gennady Kozlov is neither gauge d ( √ γγ to a dilaton oper- ¯ i γ the gauge invariant m → ¯ γ UV d γ H of dimension IR → O S ) has a continuous distribu- (from above), γ can decay into SM particles, E + ¯ γ 0 , , theory becomes nonconformal ) v → f may carry the features of an unpar- ( of dimension δ with the masses ¯ , γ O 2 IR UV 1) in the infra-red (IR). In this case, the ∼ 0, the only decay d 1 m , − O , 4 < Λ to DP operator 2. As acquires + 2 → / , UV  ) 1 δ S H ε 2 ( 2 H O IR m v m − 2 δ O d | d 2 IR and the relevant energy scale is the mass of the s 2 ¯ d σ | + − M 2 | 1 1000 TeV. In the decay − 4). If 2 1 H 2 UV − v | . Note that the unitarity constraint lower bound on d UV < − ( = γ d = into SM particles is controlled by the relation between M 2 is IR M ¯ d 1 d UV Λ γ M µ − ( < d IR ε − of the photon (with energy d  2 ε UV < 0 is M γ d UV − d v = qSB = M dE Λ 10 ˜ µ Λ · / δ γ ) 3 ¯ there is enough phase space that ¯ q γ 3 [5] does not use here because the operator of ε γ 2 < const m ≥ with ε ∼ in two-point Wightman functions (TPWF), e.g., for the scalar or the → ¯ γ ) + IR S IR singularity are the Conformal Field Theory (CFT) models [7]. 2 1 d ( p ) O has the form Γ m the constraint on ( 2 0 d p SM M δ Λ and the production threshold > ( 0 ¯ = in the loop. Since the new effects beyond the SM is expected on the scale O , we find γ ¯ γ 2. The δ < ) ¯ γ / m q γ s [8] S of ω √ m ˜ ¯ Λ γ TeV < m 3 . ˜ 0 Λ ( O is the dimension of the SM operator and there are no the dependence on at the UV scale d ∼ are not too large. If the deviation from the SM is detected at the level of order 3%, the DP v ¯ In CFT the ultraviolet (UV) coupling of an operator Because of the scale invariance, the operator relevant to In this paper, DP is considered in the framework of the gauge dipole field model which exhibits σ M > below the scale invariant, no the primary one. The decay of when the scale invariance is almost breaking. Once experiment the mass gap which flows in the IR to coupling of the Higgs boson vector operator dimension ator where the hidden sector becomes the standard particle one. For a typical energy gauge fields, satisfying the equations ofmodels motion exhibiting of a 4th order (see the refs. in [6]). Other2. classes of Couplings and constraints where IR singularities. In Abeliansingularity Higgs of model the the type breaking of the gauge symmetry implies the dipole Dark photons ticle stuff [4] with the scaling dimension more peaked at decay products. If otherwise DP may be associated with DM as a stable object. energy spectrum tion spreading from zero to half of the dilaton mass The signals of new physicsof with DP are increasingwould with be energy, visible and at the would LHC be as long seen as if the values operator structure is heavy (top) quark M within SM. PoS(ICHEP2016)136 = (3.3) (3.1) (3.2) (3.4) µν is the B σ 83 GeV , . m µ with zero 0 A ) , ν couplings to = ) x ∂ ( σ σ , m . -functions. The Gennady Kozlov σ µ − µ 2 I ∂ β ν
2 A − → f µ µ ∂ µ σ I − B f ( 2 , = µ ¯ . α σ I = 3 . ig µν − ¯ − F) [9], applied to the process corresponds to including the 2 1 σ F 4 η q qe ¯ ν # 10 γ ) 1 · − γ ˆ A − and → 6 2 , , . g
0  of corresponding 7 → q ¯ SI 2 H q i 1 6 0 − ¯ , L S σ b = q q 2 α − ) + -boson mass, we obtain ε m β is the cut off scale in the logarithmically , v 2 l 2 ν µ + Z 0 0 ν q − − B m Λ σ y Λ ˆ 2 = µ ∂ ). The parametrization of the i σ ln i µ B H ( x → b ¯  2 q ∂ + µ σ µ m 4 , λ 0 e ∂ ∑ , 3 q : ∑ 2 1 ) + l heavy − α ν m 1 ≡ ( − µ + i B − 0 -function coefficients. i 2 ∂ ν ¯ σ  q in the loop, α
b ∑ β ∂ L ¯ l + σ π 0 ( m )( σ µ µ + ∑ 2 µ light at the scale of the ∂ − B ε A √ L ν ) = µ = 3 x → Λ + ∂ ( " = ( σ by 2 µ α µ L
ξ L µ B is dimensioneless coupling constant, m  H A − , µ g ' α ∂ µν µ m F. Using B h ∂ ν 2 1 + µν are 9 additional contributions to the Yukawa couplings. If SM is embedded 1 parametrizes the size of deviations from exact scale invariance; F . The dilaton contribution to the decay is an auxiliary vector field. The subcanonical scalar field µ q ) is invariant under the restricted gauge transformations of the second kind σ ε A y < ) = µ I mixes with m 1 2 x 2 3.2 ( → f µ − SI / µ B L and 2 σ = A m ε µ obeys the equation is a real number, ) is splitted over all colored particles into sums over light and heavy states in the mass L B ) = ¯ ν x ξ is the mass of the charged lepton ( 3.4 ∂ carries either QCD or EW features of the coefficients σ , allows to estimate the DP mass l , x ¯ α i ¯ ν ξ m − ν ε The scale invariant sector of the theory contains the fields The calculation of the electromagnetic neutrino formfactor (EM The Lagrangian density (LD) of the Higgs-dilaton Abelian gauge model is ν B = → µ ¯ sum in ( where mass of the dilaton; in the conformal sector, the following condition ia evident where quarks that are relevant for collider physics is where dimension in mass unitsdivergences. is LD ( the primary dilaton field which provides a control over UV and IR scale separated by states lighter than that of the dilaton in the Here, divergent integral of EM where the field ξ ∂ Dark photons γ for an electron and muon loops, and as the result one finds 3. Model PoS(ICHEP2016)136 . is = + 0 )] Ω (3.8) (3.7) (3.5) (3.6) x = ), one 0 ( 0). x ) | −  obtained ( 3.1 = ) 0 we deal ¯ 0 σ , ε σ . The coef- ( in the form µ = 4 ν  ) is explicitly ) A ∂ µ ℜ I 2 η π Gennady Kozlov ) = [ x , = 3.7 0. One can eas- x ( ) are the conjugate ( x µ ) . If ( B x πδ , f , and the conformal µ i (+) ( ¯ ) )] = ξ ¯ µ A 0 is evident. The gen- σ A y + < A  which is Stueckelberg- is included, and one of ( · − , π 2 f ) , ) ) 1 µ ϕ 1 x ∂ x ). Hence, we have the new ) =  x , σ B ( η ) ( x  ) 3 , we choose )( µ = 2 ( ¯ x 2 2 and b 1 σ ∂ √ 3.3 x ( a b (+) b ω ) ( µ 2 + and ¯ 2 + x − , σ ∼ δ + ( C 0 2 m , where the vacuum vector [ 2  µ µ ¯ σ  x σ i 1 , and b ) m µ A B mixing does disappear ( π ε ) + − 2 / Ω ϕ ∂ 1 i ¯ x x ) )( ) ξ 1 2 b ( (   γ ) + 0 ) σ − ¯ ( 2 m µ ξ 0 and = − µ ). The solution of Eq. ( x is an arbitrary constant. The parameter − ( ¯ µ 2 ∂ ( πθ σ 2 ∂ ( ¯ x i ε ¯ γ 3 3 ) = σ ˜ ¯ θ ( m 2 3.5 ) = x ξ b − , + m 1 µ ( ' b µ , | 0 µ b ¯ C 2 σ B  − ) = σ + σ A , x µ 1 x ) ( is decomposed into negative-frequency (annihi- 2 0 ∂  ϕ (  0 2 − Ω ) = x ) 2 κ x ¯ h µ 4 x ε | σ m x ˜ ( ε ( 0, ∂ m ( i ε ¯ − of temperate generalized function on ln σ ¯ 1 2 σ is m  = 2 l µ ) = ) + + 1 ) 4 x ) = µ A , respectively. Here,  isign µ 2 + x b ( ( ) B ( x ℜ µν C 2 µ π . x − ) ω ~ ( ) that leads to ( ¯ ∂ ( 0 B σ − 2 2 C )] m 3 S )( I = m ln δ 2 3.8 · µν = i i ˜ 1, where F 1 ) + m ∂ + µ )] = Ω b ( ε A = = A 0 + ) )]  ∂ ( 0 − ( i 2 0 ( ¯  = ) ( σ / ) = 0 Ω µ ( , ¯ x , respectively. In order to find x A 1 , σ . The following equations of motion ) | ∂ ( ) ( · ) x x Ω a )] m x with ( has the form ω − ∂ ( h - mixing effect between vector fields ( 0 obeys Eq. ( ) will be defined later, while ¯ ( µ σ ( − ) ε ¯ [ σ σ ¯ µ 0, x σ [  A µ I ( 3.6 π m ξ T which is invariant under gauge transformations ( A ¯ = , σ ) , ) )[ ) 1  should be Lorentz-invariant, and the equation 2 x and µ Ω in ( Ω − can be fixed from the canonical commutation relations ( h ) ) m A ) ε 2 ¯ x x σ 2 / ), namely x b ( ( [ − f ( b ) + ) = ¯ ω − σ σ 1 3.1 x ( , has the mass µ ( ¯ and and = ( µ σ ∂ τ and ) B ( = ( 1 ¯ σ ), having the dimension in units of length, breaks the scale invariance under dilatation x 2 1 ~ b ( b ϕ  m 3 3.6 δ ε The TPWF for The commutator for dilaton field is The propagator of the dilaton field = µν in ( µ are important to find the solutions of the model considered here. The solution lation) and positive-frequency (creation) parts: for an arbitrary vector defined as the field, like [11] LD, where the where which is the distribution on the space ig momenta of term in LD ( given in the form where approaches the dipole equation for the Hermitian virtual dilaton field eral solution is given by the following expansion [10] ficients l transformation. I Dark photons The conformal symmetry is breakingwith explicitly the by dilaton the having mass the term flat direction. Within the equations of motion coming from ( The function where symmetry is restored (free massless dilaton field) if the ily find the equation of motion for dilaton field: PoS(ICHEP2016)136 , ¯ γ γ → can be , where S ¯ γ ) (1985) 3. γ p in ( 8 ˜ 2 τ → − ν S p Gennady Kozlov 10 µ · p , 3  . ) < 1 p ) = ε − ( obeying the equation of l p 4 ( ) δ ∼ p ) ( µν 2 ˜ κ ι τ 2 µν , ˜ l τ ( ) 2 . The decay mode (2007) 275. ln f , 3 2 / ) − 2 ε π i i m 10 1 · B650 + + − − 6 2 . (2007) 075004. 2 1 p ) 7 ( ε (2014) 075008. was used. In case of no mixing between the 1 2 i )( = ) 2 D76 π − x + ε 2 m ( 2 D89 1 µ 5 ι (2011) 3987. − is restricted by 3.3 GeV from above, and the results = B 2 − 2 f m (1973) 1554. 2 b (1969) 1760; P. Furlan et al., Riv. Nuovo Cim. ( A26 (1938) 225. p ) = 2 ( , x D7 ( 11 )  184 0 ε σ (2016) 071803. ) = (1977) 1. 83 GeV and µ → 2 p . + 2 (2014) 015008. lim ∂ f ι ( 0 1 55 2 117 ) (2013) 035030; ( ˜ τ (1974) 204. = ε (2007) 221601; Phys. Lett. ¯ which is just the rate of the two-photon decay of the Higgs boson ξ D89 ) + m 2 ¯ 1 98 D87 γ ) 1 19A γ field (DP) in four-momentum space is ( π 1 comes to the standard photon propagator. ¯ 2 ξ ) µ ( → p B ( S = , with ( ) = µν 1 )] p ˜ τ b p ( BR 0. We estimated the upper limit for the mixing strength 1 ( ˜ τ 2 0. We find that the DP mass ˜ τ ) = = B ) + ε · p ∂ ( F calculations gives ( 1 ν ˜ τ [  ∼ The propagator of The gauge and scale invariant model for DP solvable in 4-dimensional space-time is studied, ) p ( [2] E. Gabrielli et al., Phys. Rev. [7] A. Salam and J. Strathdee, Phys. Rev. [1] J.L. Feng et al., Phys. Rev. Lett. [3] S.N. Gninenko, Phys. Rev. [5] G. Mack, Commun. Math. Phys. [6] G. Kozlov, I.Gorbunov, Int. J. Mod. Phys. [8] P. Fox, A. Rajaraman and Yu. Shirman, Phys. Rev. [4] H. Georgi, Phys. Rev. Lett. [9] R.A. Buchl and B.P. Nigam, Phys. Rev. ˜ Dark photons where where the "strong gauge condition" where the DP field isgence massive. of the The dilaton. interaction The between latter DP is and a quarks portal is to mediated the by propagator of the DP diver- 4. Conclusions real photon and DP, motion τ where the main contribution is duethe to branching top-quark ratio in the loop.in This the can SM as be interpreted aswith the EM limit of used at the LHC towith probe measuring the of DP the missing sector of since the the recoil emitted DP. energy of theReferences single photon is encoded [10] R. Ferrari, Il Nuovo Cim. [11] E.C.G. Stueckelberg, Helv. Phys. Acta