Dilaton Vs Higgs: Nearly Conformal Physics

Nuclear Physics B Proceedings Supplement Nuclear Physics B Proceedings Supplement 00 (2014) 1–6

Dilaton vs Higgs: Nearly Conformal Physics

G.A. Kozlov Joint Institute for Nuclear Research, Joliot Curie st., 6, Dubna, Moscow region, 141980 Russia

Abstract We consider the model in which the conformal symmetry can be broken spontaneously, and a light scalar dilaton could emerge in the low-energy spectrum. The contribution of the dark photon production relevant to two photons decays of a Higgs boson/dilaton is discussed. Keywords: Higgs, dilaton, conformal symmetry breaking, dark photon

1. Introduction limit f = v. The mass of the dilaton is naturally light, ε f , where the small parameter ε controls the de- The interest to theories in which the Standard Model ∼ · viations from the exact scale invariance. The dilaton (SM) is strongly coupled to a conformal sector does becomes massless when the conformal invariance is re- not become weaker; on the contrary, it grows more covered. One may have nearly conformal dynamics at a and more every year (see, e.g., [1-4] and the references scale Λ below which the scale symmetry is broken therein). In the classical gravity scheme at very high ∼ EW and one feeds into an EW sector. energies the scale invariance and its breaking is charac- We develop the theory in which the SM is coupled to terized by the scalar field, the dilatonσ ¯ , which appears scale invariant sector which is given by a dark matter in the action (DM) in terms of the spin 1 dark photon (DP). We study 4 1 2 2 2 S = d x g˜ κr + (∂µσ¯ ) ησ¯ r + ... the effects on the EW sector from the conformal sector, − 2 − and the new bounds on DM physics are predicted. The with the scalar curvature r and positive dimension- coupling to the scalar dilaton sector is most important less parameters κ and η. Once the conformal invari- in our analysis. ance is broken, the dilaton gets a vacuum expectation The results of constraints on the dilaton mass mσ and value (vev) f , and the last visible term in S yields the the conformal breaking scale f from the direct search Einstein-Hilbert action at low energies. The spectrum at LEP and Tevatron have been carried out in [5]. In of states would then contain the scalar resonance (the particular, based on the Tevatron data, there is a widely dilaton mode), associated with the pseudo-Goldstone allowed range for a light dilaton below 200 GeV even boson due to the spontaneous breaking of conformal for f O(1 TeV). ∼ symmetry at the scale f . There is a class of models in The model is based on the extended group SU (2) L × which the breaking of conformal invariance at the scale U(1)Y UB (1), where the index B in UB (1) is associated × ΛCFT = 4 π f triggers electroweak symmetry breaking with an extra gauge boson Bµ that would be the DP γ . (EWSB) at the electroweak (EW) scale ΛEW = 4 π v < The standard photon γ may oscillate into γ followed by Λ , v = 246 GeV [2,3]. The dilaton itself exists the invisible decay to neutrino-antineutrino pair, γ CFT → in scale invariant world, and its production and decay νν¯ , or by the electron-positron pair, γ e e+. As to → − in high energy experiments are model-dependent, how- the phenomenological realization we develop the model ever the dilaton can be seen as the Higgs-boson in the containing the hidden (dark) matter sector, which can / Nuclear Physics B Proceedings Supplement 00 (2014) 1–6 2 be explored in collider physics experiments. The DP at the LHC. It is supported by a) the pseudo-Goldstone mixes with γ via the kinetic term F Bµν with nature ofσ ¯ -field with finite mass due to spontaneous ∼ µν being the kinetic mixing strength; Fµν and Bµν are the breaking of the scale symmetry; b) the dilaton preserves strength tensors of electromagnetic Aµ and DP Bµ fields, a non-linear realization of the scale symmetry; c) the respectively. The coupling between Aµ and Bµ could dilaton serves as a portal to the hidden sector, and it be responsible for production of DP in the decay of the guarantees the renormalizability of the theory. scalar particle S γγ, where S should either be the → SM Higgs boson H or the dilaton σ. No deviations with the expectations from the SM 2. Couplings and constraints have been observed in two photon decays of the Higgs The couplings of a dilaton to two gluons are cru- boson in the experiments ATLAS [6] and CMS [7] at cial for collider phenomenology. These couplings can the LHC. An enhanced rate of S γγ, if observed, → be significantly enhanced under very mild assumption with respect to the SM prediction should be due to scale about high scale physics [3]. At energies below 4π f invariant breaking sector, where the contribution from the effective dilaton couplings to massless gauge bosons the DP with the small mass mγ to the branching ratio are provided by the SM quarks lighter than the dilaton: of the two photons decay H γγ of the Higgs-boson 2 a 2 a → σ[cEM(Fµν) + cs(Gµν) ]/(8 π f ). Here Gµν is the gluon with the mass mH would be significant field strength tensor; c = α 17/9 if m < m < m , EM − · W σ t c = α 11/3 if m > m ; c = α (11 2 n /3); BR(S γγ) (1 + a 2 Ω)BRSM(H γγ), (1) EM − · σ t s s · − light → → nlight is the number of quarks lighter than the dilaton; α 2 2 b and αs are EM and strong coupling constants, respec- where a is the positive constant, Ω (1 m /m ) , ∼ − γ H tively; m and m are masses of W-boson and the top- b > 1, BRSM is the branching ratio of the decay H γγ W t → quark, respectively. The second term in the effective due to SM calculations. The mass of the DP m is un- γ coupling above mentioned indicates a (33/2 n )- known, however, the mixing factor in (1) is predicted − light factor increase of the coupling strength compared to that in various models and to be in the range 10 12 10 2. − − − of the SM Higgs boson. The upper limit of f is esti- On phenomenological grounds, however, DP masses in mated on the level of O (6 TeV) for the light dilaton eV range are favored for the present work. If no ex- [13]. cess events are found, the obtained results would give In general, the ultraviolet (UV) coupling of an oper- the bounds on as a function of m . γ ator O of dimension d to a SM operator O of It is known that the characteristic feature of the gauge UV UV SM dimension d at the UV scale M (UV messenger) is quantum field theory is to exhibit an infrared (IR) singu- SM larities which are incompatible with the positivity of the 1 1 . d 4 OSM d OUV (2) metric [8]. A relevant example is related to the dipole M SM− M UV singularity. In particular, in Abelian Higgs model the No masses are allowed in the Lagrangian of the effective breaking of the gauge symmetry requires such a type of theory containing (2). All masses can be generated dy- singularity in two-point Wightman function (TPWF) of namically in IR. The hidden sector is formed when the the dipole field, where, e.g., a scalar field σ satisfies the dilaton fieldσ ¯ = f σ is coupled to a U(1) gauge theory. equation of 4th order (∂2)2σ(x) = 0 [9]. Consider the coupling ofσ ¯ (x) to DM sector In this paper DP is considered in the framework of the dipole field model [9-11], where the interacting 1 σ 2 , d 2 ¯ OUV (3) dipole fields are local, relativistic quantum fields with a M UV− | | genuinely indefinite metric on the space of states gen- which flows to coupling of the Higgs boson H to DM erated from the vacuum. They converge asymptoti- operator O of dimension d in IR cally to free dipole fields. One of the crucial points of IR IR dipole quantum fields is that the massless dipole fields ΛdUV dIR − 2 , in four-dimensional space-time have a logarithmic in- const d 2 H OIR (4) M UV− | | crease at spatially separated arguments that features the confinement-like phenomenon [12]. when the scale invariance is almost breaking. Here, Λ is The Higgs interpretation of a discovery at the LHC the strong coupling scale. Once H(x) acquires v, theory ˜ is not the only possibility. We suppose that the dilaton becomes nonconformal below the scale Λ, where

σ¯ = f σ might be considered as the possible candidate dUV dIR 4 d Λ − 2 Λ˜ − IR = . for the scalar particle of the mass 126 GeV observed d 2 v (5) ∼ M UV− / Nuclear Physics B Proceedings Supplement 00 (2014) 1–6 3

Below Λ˜ the DM sector becomes a standard particle sector. For a typical energy Q = √s of a collider experiment Λ˜ < √s < Λ, which leads to the constraint for the energy

dUV dIR 2 d /2 Λ − 2 d 2 s − IR > M − IR v . (6) M Based on the operator form (2) the mixing strength in µν the kinetic term Fµν B (as an observable) is

2(d 4) 2d 2d √s SM− √s IR Λ UV = . (7) M Λ M Then the eﬀect of DM sector on observables has no the Fig. 1: Upper limit of (8) as a function of √s and M. dependence on dUV, dIR and Λ, and is bounded by

sdSM < . (8) where ci(µ) is running coupling, the operator Oi(x) has 2 d 2 2 v M SM− the scaling dimension di. Under the scale transforma- µ ω µ ω di ω tions x e x , one has Oi(x) e Oi(e x), µ It is clear from (7) that the signals of new physics con- ω → → µ µν → e− µ. This gives for the dilatation current S = T xν taining DM increase with energy, the dimension of the SM operator d , and would be seen if the parameter M SM µ µ ∂ of heavy messenger is not too high. Assuming that the ∂µS = Tµ = ci(µ)(di 4)Oi(x) + βi(c) L , − ∂ci deviation from the SM is detected at the level of order i 3%, the DM would be visible at the LHC ( √s O(10 µν ∼ where T and βi(c) are the energy-momentum tensor TeV)) as long as M < 1000 TeV if dSM = 4. and the running β-function, respectively. At energies In the decay S γγ the gauge invariant operator 4 di below f one has ci(µ) (σ) − ci (µσ) . The theory → ¯ µ 1 → structure is OSM OIR ψγµ ψ HB M− , where ψ is would be nearly scale invariant if d = 4 and β(c) 0. ∼ i → the operator of the quark (in the loop) with the mass Thus, the breaking of chiral symmetry is triggered by m, and the relevant energy scale is Q m. Since the ∼ the dynamics of nearly conformal sector. SM is operated on the scale O (1 TeV), we expect The model is formulated in terms of the Lagrangian 5 ∼ <10− in the case of the top-quark with dSM = 4, or which features: the scalar CP even dilaton field σ(x) <6 10 2 if the fourth generation quarks with the mass · − as the local one arising as a generic pseudo-Goldstone O( 1 TeV) give the contribution to decay amplitude. ∼ boson from the breaking of conformal strong dynamics s At small energies , the mixing is almost zero, and the and from which the vector potential Aµ(x) is derived, the only SM Higgs boson decay into two photons would be µ conformal field given by the operator OU and a set of appropriate. the SM fields. The conformal invariance can be broken The result for upper limit (8) is shown in Fig.1 for √s by the couplings with non-zero mass dimension effects. = 8-14 TeV and M = 800-1000 TeV at dSM = 4. The The Lagrangian density with a small explicit breaking LHC is the very good facility where the DM physics can of the conformal symmetry is L = L1 + L2, where be tested well. 1 1 = ∂ µ + 2 ∂ σ µ L1 B µA B d 3 (Aµ µ )OU − 2ξ − Λ − − 3. Model U

+ψ¯(i∂ˆ m + gAˆ)ψ σ m + yψv ψψ¯ , We start with partition function − − ψ 1 ∞ 3 = ψ¯ γµ γµγ ψ Z = Dσ¯ i exp dτ d xL(τ, x) , L2 d 1 (cv av 5) OUµ − 0 Λ − − U q 1 a aµ α β β α L(x) = ci(µ) Oi(x), + ∂ + ∂ . d+1 WµαWβ OU OU i ΛU / Nuclear Physics B Proceedings Supplement 00 (2014) 1–6 4 1 The field B plays the role of the gauge-fixing La- = sgn(z0) θ(z2), z = x y, grangian multiplier, and it remains free. We assume that 8 π i ξ − the real positive parameter ξ , otherwise the model 0 2 ∞ where sgn(p ) δ(p ) is well-defined as the odd homo- becomes trivial. The scaling dimension d may appear as geneous generalized function from the space S ( ) of 4 a non-integer number d of invisible particles [14]. The the temperate distributions on . µ 4 vector operator OU describes a scale-invariant hidden sector that possesses IR fixed point at a high scale ΛU , ff presumably above the EW scale; cv and av are vector 4. The propagator and the e ective potential and axial-vector couplings, respectively. A dilaton ac- The propagator of the σ(x)-field in NCS in terms of quires a mass and its couplings to quarks can undergo TPWF W(z) has the form [10] variations from the standard form. In particular, since the scale symmetry is violated by operators involving 1 W(z) = Ω,σ(x) σ(y) Ω = E−(z), (10) quarks, the shifts in the dilaton Yukawa couplings to i ξ 2 2 quarks can appear. This is given by ε = mσ/ f which where E−(x) is the negative-frequency part of the parametrizes the size of the deviation from exact scale + generalized function E(x) = E (x) + E−(x) = invariance through mσ as the measure of this deviation. 1 2 0 (8 π)− θ(x ) sgn(x ), which is the only distribution The nine additional contributions to Yukawa couplings 2 y are taken into account (y are 3 3 diagonal matrices among the solutions of the equation 2 W(x) = 0 ψ ψ ∇ × obeying locality, Poincare covariance and the spectral in the flavor space); g is the coupling constant. In the case of a Higgs-boson decays into two pho- conditions. However the metric is not positive defined. tons, the W-bosons contribute more significantly. In the The vacuum Ω-vector satisfies the following conditions: + model considered here, the only SM quarks contribution σ−(x) Ω = 0, Ω, Ω = 1, where [σ−(x)]∗ = σ (x) in | + is dominated, because the W-boson loop contribution is the decomposition σ(x) = σ−(x) + σ (x). The solutions suppressed by two more powers of ΛU , and due to sig- of (9) can be classified by TPWF’s, where nificantly large value of Λ one can ignore it. U 4 d p 0 2 ipx The equations of motion are ( ∂/∂xµ) E−(x) = θ(p ) δ(p ) e− ∇≡ (2π)3 1 µ 1 ∂µσ Aµ ψ¯(cv γµ av γµγ5)ψ,∂µ A = ξ− B, 1 − Λ2 − = − ln( µ2 x2 + i x0), U (4π)2 − 1 where µ is the positive constant required for dimen- ∂ B = J + O , J = g ψγ¯ ψ, µ µ d 3 Uµ µ µ sioneless reason. The product of the generalized func- − ΛU− 0 2 tions θ(p ) δ(p ) is defined uniquely only with the basic 1 1 ∂ µ + + ε ψψ¯ = , functions u(p) which are equal to zero at p = 0. To sep- d 3 µOU (m yψ v) 0 ΛU− f arate the IR µ-dependence, E−(x) can also be given in the form ( 4 π) 2 ln µ2 x2 + i π sgn(x0) θ(x2) . σ σ − − | | i ∂ˆ m 1 + + g Aˆ ε yψ v ψ ˜ − f − f The Fourier transform of E−(x) is E−(p) = 2 πθ(p0) δ(1)(p2, µ˜2), whereµ ˜ = 2 e γ+1/2 µ, γ = Γ (1) − − 1 µ beying the Euler’s constant. The functional δ(1)(p2, µ˜2) + O (cv γµ av γµγ ) ψ = 0. Λd 1 U − 5 is defined on the space S ( ) of the complex Schwartz  U−  4   test functions on as In the nearly conformal sector (NCS) supported by 4  µ  O 2 the weakly changing operator U in the space-time and 1 ∂2 p2 the conservation of the current J , the σ(x)-field looks δ(1)(p2, µ˜2) = θ(p2) ln . µ 16 ∂p2 µ˜2 like the dipole field obeying the equation of the 4th order One can verify that p2 δ(1)(p2, µ˜ 2) = δ(p2), − 2 2 2 (p2)2 δ(1)(p2, µ˜ 2) = 0. The presence of the parameter µ lim + mσ σ(x) 0 (9) mσ 0 ∇ → in E−(x) breaks its covariance under dilatation transfor- mations x λ x (λ>0) and implies spontaneously and the canonical commutation relation [15] µ → µ symmetry breaking of the dilatation invariance of (9). 4 1 d p 0 2 ip(x y) This is one of the reasons for the special role of the dila- [σ(x),σ(y)] = sgn(p ) δ(p ) e− − ξ (2 π)3 ton field σ(x) in what follows. / Nuclear Physics B Proceedings Supplement 00 (2014) 1–6 5

The Green’s function in 4 space-time is given by The lowest order energy (potential) of a static ”charge” is given by the static Fourier transform ( x | |≡ 1 r) G(z) = Ω, T[σ(x) σ(y)]Ω = Ec(z), i ξ d3 p E(r) eipx D(p0 = 0, p; M˜ ) ∼ (2 π)3 where the causal function with D(p, M˜ ) = M˜ 2 G˜(p), M˜ has the dimension one in 0 0 + Ec(x) = θ(x ) E−(x) + θ( x ) E (x) mass units. Using the propagator G˜(p) (12) in the form − 1 2 γ 2 2 = ln( µ2 x2 + i ) 1 ∂2 ln e ( p /µ˜ i ) i (4 π)2 − G˜(p) = − − 4 ξ i ∂p2  p2 i   − −  satisfies the following equations   one can find   i 1   2E (x) = , ( 2)2E (x) = δ4(x). ∇ c 4 π2 x2 + i ∇ c M˜ 2 − µ E(r) r a + b ln(˜µr) , ∼ 8 πξ In -momentum space the propagator of the dilaton 4 field is given in terms of distributions where a and b are known constants. The energy of a dilaton in NCS is linearly rising as r. The result is stable 1 both at short and large distances in any finite order of G˜(p) = − d4 xeipx ln( µ2 x2 + i ). (4 π)2 ξ − perturbation theory. At small distances the dominant interactions between Following [10] one can calculate G˜(p) through the sec- a heavy quark and antiquark with the mass mq can be ond order derivative G˜(p) = (∂2/∂p2)H(p), where described by the effective potential in SU(3)

2 2 1 4 ipx ln( µ x + i ) 4 λ(mq,ησq) H(p) = − d xe − . (11) V (r) α (m ) exp( mσr) (4 π)2 ξ x2 + i ef f −3r s q − r − − µ Finally, the result for (11) is with m2 q 2 λ(m ,ησ ) = η , 2γ 2 2 q q 2 σq i ln e ( p i )/(4 µ ) 4 π f H(p) = − − , 4 ξ p2 + i where ησq = 1 in SM, otherwise ησq > 1. The lower bound for mq is which leads to 1/2 µ 2 2 f 16 1 ∂ p ln p /µ˜ i mq > παs ˜ − − ησq 3 G(p) = µ . (12) 2 i ξ ∂p  (p2 + i )2    which can exceed the top quark mass even for f v.     Thus, the mediator field in a (super)heavy quark sector The following equation is strightforward: ( p2)2 i ξ G˜(p2) = 1. In the propagator (12) one is evident. Once the new force is discovered, there will − finds an appearance of the parameterµ ˜ with the be the first ever seen not related to a gauge symmetry. dimension of mass which otherwise would appear The energy distribution of the emitted photon (γ) in in the theory as a renormalization mass, and which the decay of the dilaton into γ and the DP in terms of distinguishes our model from the standard EW theory the vector unparticles stuff U is as conventionally formulated. The differentiation over µ dΓ(σ γU) Ad 3 2 d 2 2 pµ in (12) with ∂/∂p being the weak derivative has to → = m E P − A(x , y ) , π 2 σ γ U | q q | be understood in the sense of distribution where for any dEγ (2 ) test function u(p) we have where [14]

2 2 5/2 i ln( p /µ˜ i) µ ∂ 16 π Γ(d + 1/2) G˜(p)u(p)d4 p = d4 p − − p u(p) A = , 2ξ (p2 + i)2 ∂pµ d (2 π)2d Γ(d 1) Γ(2d) − µ and the extra power of momentum p explicitly elimi- A(xq, yq) is the decay amplitude (see [12] for details), nates the IR divergence at p = 0. PU is the momentum of DP in terms of unparticles.. / Nuclear Physics B Proceedings Supplement 00 (2014) 1–6 6

the decay σ γγ to be compared with the LHC data in → searching for new light scalar object with the mass close to 126 GeV. The energy of the real photon E = (m2 P2 )/(2 m ) γ σ − U σ contains the information about the missing energy or the momentum PU carried out by the DP. A nontrivial scale invariant sector of dimension d may give rise to pecu- liar missing energy distributions in σ γ U that can → be treated in the experiment. In particular, this energy distribution can discriminate d and estimate ΛU . When combined with gg σ, the decay σ γ U provides → → especially valuable information regarding possible loop contributions from new particles lighter than a dilaton. The decay mode σ γ U is particular useful for a → light σ-dilaton (m 100 200 GeV) since it may have σ ∼ − a nearly comparable decay rate with the γγ discovery mode of the Higgs-boson decay. Our result implies that the decay mode σ γ U is near a border of the confor- → mal invariance breaking. The low rate Γ(σ γ U)/ΓHiggs total is compen- → − sated by the enhancement of the order (33/2 n )2 − light ∼ Fig. 2: Energy spectrum of the photon in decay σ γU for various O(100) of the gluon fusion production cross-section → values of d. compared to that of the SM Higgs-boson. In addition to discovery, the implementation of our prediction in the LHC analyses should be straightfor- In Fig. 2, we show the energy spectrum of the emitted ward and lead to more precise determination or limits photon in decay σ γU for various values of d with → of the dilaton and the DP couplings to gauge bosons, the dilaton mass mσ 0.2 TeV, c = 1, Λ = 1 TeV, ∼ v U top quarks and the quarks of 4th generation. f v. The only top quarks in the loop are included for the calculations because of the negligible contributions from lighter quarks. References

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