Hunting down massless dark photons in kaon physics
HHunting down massless dark photons in kaon physics
M. Fabbrichesi † , E. Gabrielli ‡† , and B. Mele ∗ † INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy ‡ Physics Department, University of Trieste and NICPB, R¨avala 10, Tallinn 10143, Estonia and ∗ INFN, Sezione di Roma, P.le Aldo Moro 2, 00185 Roma, Italy (Dated: July 13, 2017)If dark photons are massless, they couple to standard-model particles only via higher dimensionaloperators, while direct (renormalizable) interactions induced by kinetic-mixing, which motivatesmost of the current experimental searches, are absent. We consider the effect of possible flavor-changing magnetic-dipole couplings of massless dark photons in kaon physics. In particular, westudy the branching ratio for the process K + → π + π ¯ γ with a simplified-model approach, assumingthe chiral quark model to evaluate the hadronic matrix element. Possible effects in the K - ¯ K mixingare taken into account. We find that branching ratios up to O (10 − ) are allowed—depending onthe dark-sector masses and couplings. Such large branching ratios for K + → π + π ¯ γ could be ofinterest for experiments dedicated to rare K + decays like NA62 at CERN, where ¯ γ can be detectedas a massless invisible system. The clarification of the origin of dark matter (DM)might require the existence of a dark sector made upof particles uncharged under the standard model (SM)gauge group. The possibility of extra secluded U (1)gauge groups—mediating interactions in the dark sectorvia dark photons — is the subject of many experimen-tal searches (see [1] for recent reviews). These searchesare mostly based on the assumption that the secluded U (1) gauge group is broken, and the corresponding mas-sive dark photon ( γ (cid:48) ) interacts directly with the SMcharged fields through renormalizable (dimension-four)operators induced by the kinetic mixing between darkand electromagnetic photons. Experimental results arethen parametrized in terms of the dark-photon mass m γ (cid:48) and mixing parameter (cid:15) , with dark photon signaturesthat can either correspond to its decay into SM particlesor assume an invisible decay into extra dark fields. Be-cause the induced operators have dimension four, moststudies necessarily explore regions where the couplingsare very small ( millicharges ).We address instead the case of an unbroken dark U (1)gauge symmetry, with a massless dark photon (¯ γ ). Therole of massless dark photons in galaxy formation anddynamics has been discussed in [2–6]. A strictly mass-less dark photon is very appealing from the theoreticalpoint of view. Indeed, for massless dark photons it ispossible [7] to define two fields, the dark and the ordi-nary photon, in such a manner that the dark photononly sees the dark sector. In this basis, ordinary pho-tons couple to both the SM and the dark sector— thelatter with millicharged strength to prevent macroscopiceffects. Massless dark photons therefore interact withSM fields only through higher dimensional operators—typically suppressed by the mass scales related to newmassive fields charged under the unbroken dark U (1)gauge symmetry [8]—while their coupling constants cantake natural values thanks to the built-in suppression as-sociated to the higher dimensional operators. This makes the ¯ γ direct production in SM particle scattering/decaysmall and unobservable, consequently evading most ofthe search strategies for dark photons currently ongoingin laboratories. A possible exception is provided by theHiggs boson decay into dark photons in the nondecou-pling regime. This scenario has been considered in [9],where observable ¯ γ production rates mediated by theHiggs decay H → γ ¯ γ have been found at the LHC inrealistic frameworks [10, 11]. Flavor-changing-neutral-current (FCNC) decays of heavy flavors into a masslessdark photon, f → f (cid:48) ¯ γ , can offer other search channelswith potentially observable rates [8, 12].Here we focus on FCNC effects induced by masslessdark photons ¯ γ in kaon physics, and discuss the changeof picture with respect to the massive case.The kaon system can be studied with great accuracy,allowing us to probe indirectly energy scales as large astens of TeV, hence crucially constraining possible SM ex-tensions. The detection of massive dark photons in K de-cays is presently under scrutiny [1, 13]. One can consider radiative K decays where the (off-shell) SM photon γ isreplaced by a γ (cid:48) , and look for resonances at m γ (cid:48) for either e + e − ( µ + µ − ) final states, or (in case γ (cid:48) decays into darkparticles) for invisible final systems with a peak struc-ture at m γ (cid:48) in the missing mass distribution. Particularemphasis has been given to the decays K + → π + γ (cid:48) and K + → µ + ν γ (cid:48) [14–18]. However, if the secluded U (1)gauge group is unbroken, these two channels are not vi-able. Indeed, K + → π + ¯ γ violates angular momentumconservation, while K + → µ + ν ¯ γ would require unsup-pressed ¯ γ couplings.Because K + decays into a dark photon ¯ γ must nec-essarily proceed through short-distance effects, we ar-gue that the most interesting channel to look for mass-less dark photons in kaon physics could be the decay K + → π + π ¯ γ . This decay can be mediated by the FCNCtransition s → d ¯ γ , prompted by a magnetic-dipole-typecoupling generated at one loop by the dark-sector de- a r X i v : . [ h e p - ph ] J u l Q , ˜ Q Q , ˜ Q Q , ˜ Q Q Q ¯ d αL γ µ s αL ¯ d βL γ µ s βL , ( L ↔ R ) ¯ d αR s αL ¯ d βR s βL , ( L ↔ R ) ¯ d αR s βL ¯ d βR s αL , ( L ↔ R ) ¯ d αR s αL ¯ d βL s βR ¯ d αR s βL ¯ d βL s αR / m K f K B ( µ ) − / X K m K f K B ( µ ) 1 / X K m K f K B ( µ ) 1 / X K m K f K B ( µ ) 1 / X K m K f K B ( µ ) − / C / C / C / C TABLE I: In the first two rows, relevant operators are numbered according to the notation in [20, 21]. The matrix el-ements (cid:104) K | Q i | ¯ K (cid:105) (in the vacuum insertion approximation for the renormalized operators Q i at the low energy scale µ = 2 GeV) are given in the third row multiplied by the respective bag factors B i ( µ ) [21] evaluated at same scale, with X K ( µ ) = ( m K / ( m d ( µ ) + m s ( µ ))) . The fourth row gives the Wilson coefficients at the matching scale (the common factor atthe matching being C = ξ / (16 π Λ ) [9], where ξ = g L g R / m d ( µ ) = 7 MeV, m s ( µ ) = 125 MeV, m K = 497 MeV, f K = 160 MeV, and B , , , , ( µ ) = 0 . , . , . , . , .
73, respectively. grees of freedom. The dark photon gives rise in thiscase to a massless missing-momentum system inside thefinal state. Recently, the sensitivity of the NA62 ex-periment at the CERN SPS [19] to two-body K decaysinto a light vector decaying invisibly [ K + → π + +( γ (cid:48) → E miss )] has been emphasized [13]. For the three-body K + → π + π ¯ γ channel, whose kinematics is less charac-terized, the detection efficiency is expected to be less fa-vorable. Nevertheless—since the K + → π + π ¯ γ channelhas a unique potential to unveil the existence of a mass-less dark photon—we think that the NA62 Collaborationshould consider search strategies aiming at detecting thisnewly proposed process, whose branching ratio (BR) canreach 10 − in a simplified model of the dark sector, as weestimate in the following. A simplified model of the dark sector. —We estimateBR( K + → π + π ¯ γ ) in a simplified model that makes asfew assumptions as possible, while providing the dipole-type transition we are interested in.The minimal choice in terms of fields consists of a SMextension where there is a new (heavy) dark fermion Q ,singlet under the SM gauge interactions, but charged un-der an unbroken U (1) D gauge group associated to themassless dark photon. SM fermions couple to the darkfermion by means of a Yukawa-like interaction in the La-grangian LL ⊃ g L ( ¯ Q L q R ) S R + g R ( ¯ Q R q L ) S L + H.c. , (1)where new (heavy) messenger scalar particles, S L and S R , enter as well. In Eq. (1), the q L and q R fields arethe SM fermions [ SU (3) triplets and, respectively, SU (2)doublets and singlets]. Flavor indices are implicit, andwe assume common ( i.e. flavor blind) couplings g L and g R . The left-handed messenger field S L is a SU (2) dou-blet, the right-handed messenger field S R is a SU (2) sin-glet, and both are SU (3) color triplets. These messengerfields are charged under U (1) D , carrying the same U (1) D charge of the dark fermion.In order to generate chirality-changing processes wealso need in the Lagrangian the mixing terms L ⊃ λ S S (cid:16) S L S † R ˜ H † + S † L S R H (cid:17) , (2)where H is the SM Higgs boson, ˜ H = iσ H (cid:63) , and S ascalar singlet. The Lagrangian in Eq. (2) gives rise to themixing after both the S and H scalars take a vacuum ex-pectation value (VEV), respectively, µ S and v —the elec-troweak VEV. After diagonalization, the messenger fields S ± couple to both left- and right-handed SM fermionswith strength g L / √ g R / √
2, respectively. We canassume that the size of this mixing—proportional to theproduct of the VEVs ( µ s v ) —is large and of the sameorder of the masses of the heavy fermion and scalars.The SM Lagrangian plus the terms in Eqs. (1)–(2)(supplemented by the corresponding kinetic terms) pro-vide a simplified model for the dark sector and the ef-fective interaction of the SM degrees of freedom withthe massless dark photon ¯ γ . SM fermions couple to ¯ γ only via nonrenormalizable interactions, induced by loopsof the dark-sector states. Two scales are relevant: thedark fermion mass M Q , which parametrizes the chiralsymmetry breaking in the dark sector, and the lightest-messenger mass scale m S . Since we are considering thecontribution to the magnetic dipole operator (assumingvanishing quark masses), the dominant effective scale as-sociated with it will either be chirally suppressed (be-ing proportional to M Q /m S , for m S (cid:29) M Q ), or scaleas 1 /M Q (for m S (cid:28) M Q ) due to decoupling. In orderto have only one dimensionful parameter, in our analy-sis we assume a common mass for the dark fermion andthe lightest scalar field, which we identify with the new-physics scale Λ. This choice corresponds to the maximumchiral enhancement.This scenario is a simplified version of the model in[10–12] (possibly providing a natural solution to the SMflavor-hierarchy problem), as well as a template for manymodels of the dark sector. Bounds from K - ¯ K and astrophysics. —A most strin-gent limit to the mass scale and couplings of the abovesimplified model comes from its extra contributions to the K - ¯ K mixing in the kaon system (related to the massdifference ∆ M K of the neutral mass eigenstates K L and K S , assuming CP T ).In order to compute the dark-sector effects on ∆ M K ,we need to evaluate the dark-sector contribution to theeffective Hamiltonian for the ∆ S = 2 transitions, H ∆ S =2 eff ∆ M K = 2Re [ (cid:104) K |H ∆ S =2 eff | ¯ K (cid:105) ] . (3)The scalar-fermion interaction in Eq. (1) induces a newset of operators, which are reported in Table I, then ob-taining H ∆ S =2 eff = (cid:88) i C i Q i + (cid:88) i =1 ˜ C i ˜ Q i . (4)The Wilson coefficients at the matching scale are com-puted by considering the exchange of the lightest mes-senger state in the loop, which provides a good estimateof the dominant contribution in the large-mixing limit ofthe messenger mass sector.We compute the corresponding Wilson coefficients C i ( µ ) at the O ( α s ) next-to-leading order, after runningthem from the matching scale down to the low energyscale µ ∼ M K (in TeV) is∆ M K = 8 . × − ξ Λ , (5)where ξ = g L g R /
2, and Λ is in TeV units. We thenassume that the above contribution of the new operatorsto Eq. (3) does not exceed 30% of the measured ∆ M K value [22]. Eq. (5) turns then into an upper bound forthe allowed values for the ξ / Λ ratio.While the flavor-changing dipole operator induced inthe simplified model (see Eq. (6) below) per se is onlybounded by kaon physics, if we make the (very conserva-tive) assumption that the model also gives flavor-diagonaldipole operators and these are the same size in the quarkand lepton sectors, a bound can be derived from stellarcooling carried out by the emission of massless dark pho-tons. Under these assumptions, the limit from K - ¯ K mixing in Eq. (5) falls between the current astrophysi-cal bounds [23]—with the most stringent one from whitedwarves being 1 order of magnitude stronger and thatfrom the Sun 1 order of magnitude weaker. Amplitude and decay rate. —The K + → π + π ¯ γ decayoriginates from the dimension-five magnetic dipole oper-ator ˆ Q = (¯ s σ µν d ) ¯ F µν , where ¯ F µν is the ¯ γ field strength, σ µν = [ γ µ , γ ν ], and color and spin contractions are un-derstood. ˆ Q enters the effective Hamiltonian for ∆ S = 1transitions as H ∆ S =1 eff = e D π ξ Λ ˆ
Q , (6) where α D = e D / (4 π ) is the ¯ γ coupling strength. TheWilson coefficient multiplying the magnetic operator inEq. (6) is obtained by integrating the vertex functionin our simplified model (see Fig. 1). We have checkedEq. (6) by means of Package X [24]. s R d L QS ± ¯ γs R d L Q ¯ γS ± FIG. 1: Vertex diagrams for the generation of the dipole op-erator in the simplified model of the dark sector (same for thespecific model in [10–12]).
The operator in Eq. (6) contributes only to the mag-netic component of the process K + ( p ) → π + ( q ) π ( q ) ¯ γ ( k ) , (7)while its contribution to the process K + → π + ¯ γ identically vanishes. The amplitudeˆ M ≡ (cid:104) ¯ γ π + π | H ∆ S =1 eff | K + (cid:105) in the momentum spacecan be written asˆ M = M ( z , z ) m K ε µνρσ q ν q ρ k σ ε µ ( k ) , (8)where ε µ ( k ) is the ¯ γ polarization vector. The correspond-ing differential decay rate isd Γd z d z = m K (4 π ) | M ( z , z ) | { z z [1 − z + z ) − r − r (cid:3) − r z − r z (cid:9) , (9)where z i = k · q i /m K and r i = M π i /m K [25]. ¯ γ ( k ) π ( q ) π + ( q ) K + ˆ Qsu d d ¯ γ ( k ) π + ( q ) π ( q ) K + ˆ Qsu du
FIG. 2: χ QM diagrams for the process K + → π + π ˜ γ . Thecrossed circle stands for the insertion of the magnetic dipoleoperator ˆ Q in Eq. (6). The matrix element in Eq. (8) can be estimated bymeans of the chiral quark model ( χ QM) [26]. In thismodel quarks are coupled to hadrons by an effective in-teraction so that matrix elements can be evaluated byloop diagrams (see Fig. 2). In general there are severalfree parameters, but in the present case only M , the massof the constituent quarks, and f , the pion decay constant,enter the computation. The model has been applied tokaon physics in [27], where a fit of the CP preservingamplitudes of the nonleptonic decay of neutral kaons hasyielded a value M = 200 MeV [28] with an error of lessof 5%. According to the χ QM we obtain that the magneticcomponent generated by the dipole operator in Eq. (6)is given by M ( z , z ) m K = e D π ξ Λ M π f (cid:104) M D (0 , m π , m π , m K ; 2 m K z + m π , m K (1 − z − z ); M, M, M, M ) − D (0 , m π , m π , m K ; 2 m K z + m π , m K (1 − z − z ); M, M, M, M ) + ( z ↔ z ) (cid:105) . (10)where D and D are four-point Passarino-Veltman co-efficient functions (see [29] for their explicit form) to beevaluated numerically [24].Inserting the amplitude in Eq. (10) in the differentialdecay rate in Eq. (9) yields, after integration and by nor-malizing Γ by the total K + width Γ tot = 5 . × − MeV [22],BR( K + → π + π ¯ γ ) (cid:39) . α D η ξ Λ , (11)where we assumed M = 200, f = 92 . m K = 494, and m π + = m π = 136 MeV. The coefficient η accounts forthe renormalization of the Wilson coefficient of the dipoleoperator in going from the Λ scale to approximately m K .We assume it equal to 1, and discuss the impact of pos-sible uncertainties below.BR( K + → π + π ¯ γ ) is proportional to ξ / Λ , just as∆ M K in Eq. (5). By taking for ξ / Λ the value that sat-urates the ∆ M K constraint, we find an upper bound forthe BR which is, for the representative value α D = 0 . K + → π + π ¯ γ ) ∼ < . × − . (12)Fig. 3 shows the BR( K + → π + π ¯ γ ) contour plot versusthe scale Λ and the coupling ξ , for α D = 0 .
1. We seethat a rather large range of parameters is allowed forwhich the BR is sizable. The upper bound—given byEq. (12)—is represented in Fig. 3 by the boundary of thegray area.There are three main sources of uncertainties in theresult in Eq. (12): • The matrix element estimate computed in the χ QMdepends on the parameter M . The result in [28]seems to indicate a rather small uncertainty on thisparameter but one must be aware of the depen-dence. We find an increase by a factor 2.5 in theBR when going from M = 200 to 250 MeV; • Even though there are O ( p ) chiral perturbationtheory corrections to K + → π + π ¯ γ , these havebeen shown to be small [30]; • By taking the QCD leading-order multiplicativevalue η = 0 . µ = 2 GeV) [31], we find a BR smaller by a factor 1/4. However, it is knownthat nonmultiplicative corrections go the oppositedirection, and we thus need the (not yet avail-able) complete evolution before trusting this cor-rection. Moreover, the QCD renormalization intro-duces a strong dependence on the low-energy scale µ , because the matrix element computed within the χ QM is scale independent.On top of these uncertainties, we have the overall de-pendence on the α D strength on which the BR dependslinearly. There exist cosmological relic density boundson the ratio α D / Λ [3]. Our choice of α D = 0 . ���� ���� ���� ���� ���� ���������������������� ξ Λ [ T e V ] BR ( K + → π + π γ ) �� × �� - � �� × �� - � �� × �� - � �� × �� - � �� × �� - � α � = ����������� �� � � - � � FIG. 3: BR( K + → π + π ¯ γ ) as a function of the effective scaleΛ and coupling ξ = g L g R /
2, for a representative choice of thecoupling strength α D = 0 . Similar predictions can be obtained in the specific fla-vor model of [10–12]. In particular, for α D = 0 .
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