Serendipity in Dark Photon Searches

Serendipity in dark photon searches

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As Published https://doi.org/10.1007/JHEP06(2018)004

Publisher Springer Berlin Heidelberg

Version Final published version

Citable link http://hdl.handle.net/1721.1/116307

Detailed Terms http://creativecommons.org/licenses/by/4.0/ JHEP06(2018)004 Springer June 1, 2018 May 22, 2018 March 5, 2018 : : : Published Accepted Received . , boson that couples directly d B Published for SISSA by https://doi.org/10.1007/JHEP06(2018)004 [email protected] and Wei Xue , c kinetic mixing, and on a vector that mediates γ – B Mike Williams [email protected] b , https://gitlab.com/philten/darkcast . 3 1801.04847 The Authors. Yotam Soreq, c Phenomenological Models

Searches for dark photons provide serendipitous discovery potential for other a , [email protected] [email protected] Center for Theoretical Physics, MassachusettsCambridge, Institute MA of 02139, Technology, U.S.A. Laboratory for Nuclear Science, MassachusettsCambridge, Institute MA of 02139, Technology, U.S.A. Theoretical Physics Department, CERN, CH-1211 GenevaE-mail: 23, Switzerland School of Physics andBirmingham, Astronomy, B152 University 2TT, of U.K. Birmingham, b c d a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: determining hadronic decay rates.on We a demonstrate vector our that approachto couples by baryon to deriving number the constraints and B-L current,a to protophobic a leptons force. leptophobic via Ourwith approach vector can couplings easily to berecasting the is generalized Standard provided to Model at any fermions, massive and gauge software boson to perform any such Abstract: types of vectorto particles. obtain constraints We on develop more a general framework theories, for which includes recasting a dark data-driven photon method searches for Philip Ilten, Serendipity in dark photon searches JHEP06(2018)004 ; ]. 0 9 A – dark m 4 1. Constraints ≈ field strength tensors [ 0 A ] to searching for a massive 3 – 1 13 – 1 – 23 24 3 10 21 16 ; and the dark photon decay branching fraction into invisible 2 ε 7 3 10 23 model has 3 unknown parameters: the mass of the dark photon, 0 17 5 18 1 A 21 22 21 22 15 24 production decays , whose small coupling to the electromagnetic (EM) current arises due to kinetic 0 hadrons C.6.2 Proton beam dumps C.6.1 Electron beam dumps X X A , → The minimal C.7 LEP C.3 Electron bremsstrahlung C.4 KLOE C.5 LHCb C.6 Beam dumps X C.1 BaBar C.2 NA48/2 3.2 Decays to invisible dark-sector final states 2.1 2.2 2.3 Efficiency ratios 3.1 Decays to SM final states the lab, and alsointo via invisible which dark-sector final they states can are decay expected into to visible be dominantthe SM if kinetic-mixing final strength, kinematically states allowed. —dark-sector though final decays states, which is typically assumed to be either 0 or 1 Introduction Substantial effort has been dedicatedphoton in recent years [ mixing between the Standard ModelThis (SM) mixing hypercharge provides and a potential portal through which dark photons may be produced in B C Experiments 4 Summary A Additional VMD details 3 Example models Contents 1 Introduction 2 Generic vector boson model JHEP06(2018)004 ], → 26 0 ] or – A 24 69 , decays in 0 68 ] parameter , A 2 56 , ε 0 A m ; develops the frame- η ], fixed-target [ 0 2 A 23 – 9 → have been used; φ π = h , and 0 , using electron and proton beams 0 π 0 and A ; ]. Dark photons have been searched for pZA Z e, µ → 62 [ ] experiments, and on invisible ω = → denotes the QCD vector mesons. 0 , ` 42 A γ – 2 – 0 0 – colliders; pZ decays, which has been done both at beam dumps A D 0 33 − ω, ρ, φ A decays. e → → 0 + and = η e ∗ A 0 , D V γ 0 , at , both at hadron colliders and at proton-beam fixed-target decays by previous beam-dump [ A 0 eZA γ 0 0 A → A A , where thus far → 0 ]. − → π → h ¯ q 67 + eZ q − – h e 53 + e → mixing, where 0 0 A A ], and rare-meson-decay [ → 32 ]. Many ideas have been proposed to further explore the [ and – V 52 − 27 ` – + and by performing bumpstate hunts to in be missing knownviding and mass sensitivity any spectra, to visible which invisible component requires of the the initial final state to be detected, pro- by performing bump hunts` in invariant mass spectra using theby visible searching decays for visible displaced and at colliders using secondary vertices; and Drell-Yan (DY), experiments; meson decays, e.g. bremsstrahlung, incident on fixed nuclear targets of charge annihilation, 43 The remainder of this article is organized as follows. section A variety of production mechanisms have been used in dark-photon searches, which Both existing and proposed searches for dark photons provide serendipitous discovery • • • • • • • • the best possible sensitivityof to hypothesized dark vector photons, particles. each also provides sensitivity towork other types required todetermining recast the these hadronic searches, decay which rates includes for a novel GeV-scale bosons. and robust We method apply for our framework While the production mechanisms and search strategies employed were chosen to achieve Proposed future searches largelyplan exploit on the using sameadditional positron production meson beams mechanisms, decays though incident suchusing some as on the fixed following techniques: targets for annihilation [ can be categorized as follows: context of a more genericfor model recasting searches is for well massive vector motivated. particlesa In from data-driven one this method model article, for to we determining another,by develop hadronic which recasting includes a decay the rates. framework existing We constraints demonstrate oncan our dark easily approach photons; be however, applied we to stress any that massive our gauge approach boson with vector couplings to the SM fermions. collider [ refs. [ space in the future [ potential for other types of vector particles. Therefore, interpreting these results within the have been placed on visible JHEP06(2018)004 0 − ). ), X 0 A e A + 2.2 τ (2.2) (2.3) (2.1) e ( current / ) = 0, and for similar Z X f ν interaction τ x x ( kinetic mix- ¯ χ 1, γ to SM fermions, – − Xχ ]. B = X , and 75 ` – x processes, which arise →F , current, a leptophobic 73 0 scenario, where the A 0 εe L B A πX / = − . , , involves solving the following 2 ) B X ¯ → mode [ χ , and predominantly invisibly 0 ) →F 2 g e F is the decay branching fraction, A χ ) X 10 GeV, we adopt the model of x τ Xχ X K ( B εe X L ( > , g →F 0 0 ( 0 χ A A X →F for all 0 , and = X,A /σ , and the form of the m A + χ B γ f X 0 B µ 0 γX m σ Xγ A A 2 → → σ fX → − − < µ kinetic mixing, e ]. For e 0 0 + – 3 – Z + , according to . ¯ e ) = e 70 fγ A , A χ σ σ searches. Furthermore, we only consider real f X – m 0 τ x γ ( = A KX f 0 X → →F For example, in the minimal X eZX eZA g , all information required to recast dark photon searches X → → 1 couples to an anomalous SM current, there are additional B u,d : a vector that couples to the eZ : also has a model-dependent coupling to the weak eZ X B 0 N.b. X σ L ⊂ σ σ 0 A 3 ), see e.g. ref. [ . A m 2 Z 4 -conserving model as well. = decays visibly if /m 0 0 X CP 2 A A m m 3. The , and its decay branching fraction into invisible dark-sector final states. ( / couplings; however, in such cases, the constraints from studies of flavor-changing neutral 1 X O X − is the coupling strength to fermion m denotes the production cross section, ]. The 0 f x production 72 3 or , X X,A / g σ X is the detector efficiency, whose lifetime dependence is made explicit. From eq. ( 71 Recasting a dark photon search that used the final state , in models where the This model is flavor-conserving due to its diagonal couplings. Of course, one could also consider = 2 boson that couples directly to baryon number and to leptons via 1 , and to invisible dark-sector particles, q flavor-violating currents are much stronger thanreasons, those making from this a due to the enhanced production rates of the2.1 longitudinal The ratio of productionannihilation cross is sections for both electron-beam bremsstrahlung and where and one can see thatN.b. what is needed arestrong the constraints ratios from the equation for each x that scales as refs. [ otherwise. The more generalboson mass, model has 14 parameters: the 12 fermion couplings, the where does not need tocoupling to be SM specified. fermions arises due to 2 Generic vector bosonIn model this section, we considerf a generic model that couples a vector boson B ing, and on aare vector that provided mediates in a section to protophobic force. any vector Finally, summaryhttps://gitlab.com/philten/darkcast model, and including discussion software to perform any such recasting, is provided at to three models in section JHEP06(2018)004 ), ω m ( 20%. (2.7) (2.8) (2.9) (2.4) (2.5) (2.6) V ≈ denote P and , V 2 | 2 | ) ) X X , and is given by i m q m ( 0 , where ( , 0 V V must be accounted for in 0 XP X A , Z u, c, d, s, b. ]BW 2 → → → ]BW 2 i } i , = = X ¯ ] q ¯ q } i Q ] i i i 2 V 0 q , q Q X ) q q 0 Q σ V σ d } , 0 from the lowest-lying nonets, where Q V 0 1 T x } V T for V s P 2 − T + ) T , ) ) ], which is successful at predicting low- ]Tr[ , x 1 u 0 ]Tr[ εe 1 4 d Tr[ m 1 for m 0 x ( Tr[ − V 77 ( and ( { V , ∗ { ( ∗ T , x (2 γ 2 T γ u V 2 P 2 { X × P x → → T vertices. The quark U(3)-charge matrices are g i { T 2 ¯ q 0 V – 4 – i X ) V DY q εe i T ≈ g 2 diag q T σ σ ) 0 1 3 x , and the VMD Breit-Wigner form factors, BW Tr[ εe X should be evaluated at the proper mass scale, though Tr[ 0 VPV = ( i = = diag g 0 pZX pZA q V,P V e in decays of the form X V P 9( T → X → Q 0 0 P XP A | = Q P A pZ pZ | = → → 0 σ σ 2 0 . Determining the fraction of SM DY production attributed X V A V ) X A Z 2 X Γ Γ → and → g εe → → m i i ( . When considering ¯ q ¯ q DY DY i X In this effective theory, external gauge fields — including the SM i q q σ σ = A 2 σ σ P ], we calculate meson-decay ratios using the hidden local symmetries 0 XP A 76 → → V V Γ Γ 10 GeV, the model-dependent mixing with the & ]. Therefore, we expect that the uncertainty of using VMD and U(3) quark symmetry is ). Furthermore, the value of 77 X m 2.6 Following ref. [ The VMD approach accurately predicts many observables at the 10–20% level, e.g., the width of the 2 VMD is valid, this reduces to meson [ and the relevant meson generators, are detailed in appendix where the sum runs over all possible framework of vector mesonenergy dominance SM (VMD) observables. [ photon — couple towidths for quarks producing via the mixingvector with and pseudoscalar the mesons, QCD respectively, is vector given mesons. by The ratio of the eq. ( this is a small effectto below each flavor requires knowledge ofthe the uncertainties parton that distribution arise functions due of to the limitations proton, in though this knowledge largely cancel in the ratios. attributed to each flavor, and the second term is the contribution from each subprocess For where the first term in the sum is the mass-dependent fraction of the SM DY production imation the ratio can be taken to be since only sub-GeV masses haveof been DY probed production using cross this sections production involves mechanism. a sum The over ratio quark flavors, For proton-beam bremsstrahlung the situation is more complicated, but to a good approx- JHEP06(2018)004 and (2.12) (2.10) (2.13) (2.14) (2.11) mixing ωη X → ω , 2 GeV, we do not contributions from 2 | 2 . | ) ¯ q ) , q m m ( 2 f 2 X ( V ρ, ω, φ, m is obtained directly from m V 4 0 = = = A , − ]BW ) V V V ]BW 0 /σ 1 X A Q X Q s V for for for are not. The ratio of widths for , σ m V ( T ! 0 T µ is SU(3) allowed, e.g. 0 2 2 f 2 X i X i A ) R 2 by summing the ωπ ]Tr[ σ d P σ ) m − m ]Tr[ 0 d V x µ V V x + → ; however, for masses + µ 0 0 QT ¯ 1 + ν − u QT i A A ω → → P ν 2 s hadrons 0 u

x σ σ P T x A x → T V X ( 9( 9 – 5 – X and Tr[ i m        Tr[ = Γ 2 for X 2 V 3 / ) × V ρη f = P 2 x | P ) π 0 | 2 X → X X A g 2 εe 12 g σ hadrons σ ( ( ω , and 1 f → 0 ¯ q X C = εe q A g 0 Γ = X A couples to the EM current, its decay rate into hadrons is simply ¯ f → → = 0 f V V decays satisfies a similar expression: A γ → σ σ 0 , 3 for γ Xγ A X − 0 Γ ` → → A + P P ` Γ Γ → production amplitudes. P φ contribute. Finally, the ratio of production cross sections due to the are allowed, whereas is chosen such that the process = 1 for V and decays is given by 0 ). Because the 0 , and f boson is assumed to decay predominantly into invisible dark-sector final states if ¯ f V C ω X ρπ f Xγ , X 2.13 The sensitivity in many dark photon searches is predominantly due to a single produc- ρ Including such interference is trivial if the relative phases of the amplitudes are known; however, this → 3 → → production mechanism is onlyknown important and in where interference hadronic effects environments, are where expected these to phases be are negligible. generally not where expect to obtain a reliableeq. prediction ( for Γ 2.2 The kinematically allowed, and into SMX final states otherwise. The partial width of the decay where a Monte Carloeach event production generator mechanism. can be used to estimate the relative importance of tion mechanism at eachone mass. of the In ratios such providedis cases, in relevant, this the the subsection. cross-section ratio ratio When is more than one production mechanism which is also calculated usingthe VMD. This approach ignores potential interference between which cannot be reducedsum into over as simple awith form the due QCD to vector the mesons fact is that multiple terms in the ω P where JHEP06(2018)004 . ) ω 2 m ) to ( → F , this V F m (2.15) (2.18) (2.17) (2.16) ( − for any A e F N.b. f -like, and + e ω ]. values above 78 hadrons hadronic cross → -like, X ρ shows that these . Taking 0, and the hadrons 1 → F B , → < − # X ] e ) and shown in figure ; i.e. this contribution is Q + , X . φ ω e 2 B ω | T m . ) ( final state is the dominant ) at low mass are fitted using φ = - m , γ , ( , m ω X hadrons into µ ) 0 V ) ( ) for specific hadronic final states F π R φ µ R m → m ( m ( |A R ( − V F )+ 2 ω µ ρ µ e components, and contributions from →F } A X + ] 9 R 2 EM X ∗ R e 2 2 α X m V } V } ( ] ] ) is known experimentally [ X Q -dependent amplitude is taken to be the = V X X φ X − T F µ R 5 GeV, the Q − Q , arising from Tr[ . ) µ ω ρ + − 0 φ + data where this decay is important, we instead T T V µ m µ – 6 – X ( µ 2Tr[ →F . + = γ " − F → mixing. → -like models, we can estimate Γ √ 0 µ e f − 6Tr[ 3 2Tr[ φ m − π + X V e { { { V e e π → + ) = 0 and only using m σ + e → ) = e 12 − 2 → m σ X ( e ( g − 0 m ) = ) = ) = γ poles, which is demonstrated to be a good approximation ( e + ≡ 0 A /σ m m m = e + F ∗ π ) ( ( ( e f A V ρ X ω X φ X -like, and m ), which accounts for ( ) is taken to be a bicubic spline with knots every 50 MeV whose ω R R R m F µ m ( hadrons ( R F F → hadrons) invariant mass. Each f itself, justifying the use of LO perturbative Γ f X -like, µ → ρ − Γ e R − + e e + , but not in the region from 1 to 2 GeV. To obtain reliable predictions for all e φ . ( σ model from m B as occur far from the ≡ . is the X µ ), where each V X m 5 GeV, as is R . m 1 ρ, ω, φ 2.15 Based on these fits, we are able to decompose The six most important hadronic contributions to The VMD approach can be used to estimate Γ and & = ∗ -like contributions, which are discussed in detail in appendix where 2 GeV. Using these low-mass estimated as above, but with φ Each of these contributions ism within 20% of its leading order (LO) perturbative value for eq. ( values are varied tofits achieve describe the all best data samples descriptiondecay well. mode. of For Due the to data.calculate a this lack Figure of contribution assuming it comes entirely from the amplitudes have Breit-Wigner forms whichbe are real provided corresponds in to appendix V the assumption that thein only appendix relevant interference effects between sum of a real function V where the minus sign applies only to sections at low mass to that of where expression already accounts for when masses, we have instead developedcross a sections. data-driven First, approach based we on normalize measured each of the most important where JHEP06(2018)004 , by X (2.19) hadrons final state), → -like contri- 0 [GeV] φ A ] (displayed as . s

KKπ ∗ 86 , √ ), can be used to ) are provided at )] 85 [ m ( 2.19 2.17 = hadrons 0 π -like and ω 3 π )–( F ω − A π 2.18 + ) + π ] (displayed as open triangles), m -like contributions are assumed is taken to be unity for invisible ( 80 φ [ 0 π =0 3 I ] A 0 f π KK )[ − π m = KKπ ( + π π F φ 3 -like, and =[ ω A F < ] – 7 – decays also have the same efficiency for the relative to the -like, quarks. Our approach reproduces Γ X 0 ρ ] (i.e. the isoscalar component of the s Q X A φ 82 [ functions defined in eqs. ( T ], high-mass 79 . F X =0 [ , and I ]Tr[ ] R d ]; and from SND, the low-mass X − , π u Q 84 [ + ω KKπ π 0 0 T 0 ) ) accounts for interference between the π π π − ], [ 0 − − 0 π π final state and is given by π π π 81 to the 2Tr[ + − [ 2.17 + + − 0 π π √ π π π π X L + + − for any vector model at low mass, where all that is needed as input are π 0.5 1 1.5 2 = = K π is smaller than the detector decay-time resolution. This is not the case π S =2( = + F F 4 K X π ) = 36 τ F ]; BaBar, for F + m 78 ( hadrons − ], and φ - → K 1 83 3 2 1

. Data used to determine the hadronic decay rates from: the PDG, for the total rate ω X

− − −

+ X

→

≡ R )[

K −

− 10

10 10 ( ) µ µ e e σ

µ − F + +

→F π

≡

− We reiterate that the approach developed here, speciﬁcally eq. (

) ( e e σ The numerical values of the + + 4 π searches. Searches forprovided visible that prompt for all models; therefore, lifetime-dependent efficiency effects musthttps://gitlab.com/philten/darkcast be considered even in approach developed here israte the of most a low-mass robust vector method boson. for determining2.3 the hadronic decay Efficiency ratios The ratio of detector efficiencies for the obtain Γ the couplings of the construction when the model parametersour are method chosen to invokes be a those few of mild the dark assumptions, photon. this While is unavoidable and we believe that the butions to the All other interference effects between the to be negligible. KK 2( filled squares). See text for discussion on the solid lines. The final term in eq. ( Figure 1 to hadrons [ JHEP06(2018)004 ] 2, 2 √ , ε 0 / ) A (2.20) (2.21) ¯ d d m − values for ¯ u u X [GeV] τ s √ -like -like) ( γ ρ values are provided -like 1 are excluded. This ρ ˜ t = < = V ]), as it makes recasting 0 V ul ex A 1 which gives r for detailed discussion on the , , τ 0 1 ≈ B -like, into ( A exclusion regions, but also the ) γ 0 0 < m A 0 boson could have and still satisfy A τ A values extracted for each [ ( τ , X = ul ex -like X X r τ φ ˜ t/τ , where the ratio of efficiencies is again yield relative to the expected number of 0 at each [ = − , regions with A 0 0 e τ →F V ul →F ex 0 A A − r A X = 1 B B 0 – 8 – X X A ≈ searches had τ σ σ 0 ) ) 0 ) A 2 X A τ τ ( , ε ( ]. For the 0 contributions. See appendix 2 A hadrons, which is of course published not only the ¯ s , ε s m 0 − → ( A − µ ul ex e + m r + µ -like) e φ → , the LHCb sensitivity for some models extends to 0 . We encourage future beam dump and displaced-vertex searches to -like A ω X N.b. 2, and ( 0.5 1 1.5 2 = √ ] for V / . search selection criteria. The experiment-dependent ) ¯ 31 . d 0 d decays at each [ A C.5 C 0 + 1 3 2 ¯ u A 1

, of the upper limit on the observed

− − . Decomposition of −

denotes the largest proper decay time that an 10

→

≡ R ul ex

− − ˜ t

10 r 10

) ( µ µ e e σ

+ +

→ →

− The published information for constraints placed on dark photons from beam-dump The efficiency ratios are more complicated in searches for long-lived bosons. The recent V hadrons) ( V e e σ + -like) ( ω which LHCb does notin report appendix results, though these regions are easily handledexperiments as is discussed not sufficient tothe rigorously Monte recast Carlo the studies results need for to other be models. redone, In and principle, the are excluded for the also publish results inthe this way results (or trivial. similarly, LHCb search [ ratio, observed facilitates recasting the resultsunity. for Regions each with where the prompt in appendix derivation of these curves and on the meaning of the dashedprompt lines. searches. All existing prompt Figure 2 ( JHEP06(2018)004 0 , , A 3 dec ] for L 1 (2.26) (2.25) (2.22) (2.23) (2.24) ˜ t , 0 ˜ t ] by solving max , ε min ε lifetime is much larger 0 A limits [ , 0 min )] ε A 2 min ε . . . ( →F ), resulting in the upper edge of the 0 ) ) 0 0 0 /τ A A 1 A A sh τ ˜ t τ [ B max 0 − ε /L e A > τ ( 0 σ value where 2 min − dec A (1) correction to limits that cover several X ε τ τ L O X ( /τ ≈ g 0 0 ≈ ˜ t – 9 – A ) )] = − e min (1 + X g 0 < τ ˜ t max X 2 max g ε ) = X ( ( = 0 τ τ →F X 1 ( A ˜ t τ τ X X [ τ B X σ 2 max ε occurring at the ]. That is beyond the scope of this project. Instead, we set can be written in terms of the lengths of the decay volume, exclusion region is typically where the lifetime is much smaller than the minimum proper decay time 0 1 31 X ˜ t are obtained at each mass from the 0 A 1 A exclusion regions. ˜ t 0 A , as and sh 0 decays within this fiducial region is given by ˜ t L τ model being considered at a given mass. For the upper edge of a long-lived exclusion region occurs where X X We provide here some simple heuristics that give nearly identical results to the more is satisfied. Wethough do they not do use give nearly thesedump identical heuristics exclusion results regions, to except where obtain nearlower the the the edges high-mass of large-lifetime results edges the approximation presented of is in the no beam- section longer valid at the than the maximum proper decayIn time this required regime, to the decay ratioof before of the exiting efficiencies the is fiducial just region. the ratio of the lifetimes, and the lower edge exclusion region for the The lower eddge of the involved approach described above, providedto that the the beam-dump experimentexclusion is region, sensitive the required to enter thenentially beam-dump suppressed (enhanced) fiducial for region. This means that the efficiency is expo- which arises from theof fact decay that time, the i.e. upperdepend the limit on experimental on the upper decay observed limits time. signal placed decays on is observed independent signal decays do not The values for This approach ignores thethe kinematical dependence spread of of thethough the efficiency production a on momentum proper spectra theorders treatment and location of amounts magnitude of to for thewith an the lifetime decay existing within beam-dump the results. decay The volume, probability that a particle approximate limits by definingeach an experiment, effective where proper-time fiducialand decay shielding, region of [ as was done at LHCb [ JHEP06(2018)004 L − B . These 3 . 3 2 p . First, we will 2 2 2 2 2 2 2 3 1 3 1 ) 2 2 g p p ) ) ) ) ) ) 2 2 2 2 2 p p p p p ]. The fermionic g g 0 3 εe g g g g g − − εe εe εe εe εe εe 4 4 ( ( ( ( ( ( 4( 87 , we do not consider and Protophobic Protophobic 2 N.b. 1. 2 2 2 ) ) ≈ ) 2 2 2 2 2 2 π εe εe ) ) 2 2 e B B ) ) ) ) B 2 B ( ( 1 3 1 3 2 2 2 2 g g B B B B ¯ χ g (4 4 4 . Using these couplings and the g εe —it is straightforward to obtain g g g g 4 4 εe εe εe εe ) ) 4 e e ( ( ( ( ]). χ 4( π π − 1 A (4 (4 88 → X ( relative to those of the dark photon, except B . 1 3 LB L ) = 0 for each of these three models, followed 2 − L 2 2 2 2 2 2 1 3 1 3 1 0 0 1 ) L L L L L L LB − ¯ ) ) ) ) ) ) χ − B − − − − − − – 10 – 0 0 2 B εe − − 2 2 2 2 2 2 χ B B B B B B − εe εe εe εe εe εe B g ( ( ( ( ( ( g g g g g g 4( 4 B → X ( B ) = 0, it is straightforward to obtain all of the necessary ¯ χ χ 0 ratios, which are summarized in tables 1 3 2 3 0 1 εe g A 0 0 − → 0 − X boson that couples directly to baryon number and to leptons via A A γ 0 0 X 0 A Xγ 0 0 A 0 }→ }→ /σ B X X b X ¯ X }→ A b A ¯ A }→ → eZX pZX → eZA pZA ( ¯ c ¯ c X → → → —including the work in appendix → → − → ¯ s,b − → → ¯ s,b → → B ρ φ e σ ω ρ φ e . Couplings to SM fermions for the models studied in section ω ¯ u,c ¯ u,c σ σ σ σ ¯ σ σ + ¯ u eZ d,s + pZ u eZ d,s pZ e { e d { d σ σ σ σ 2.1 σ { σ σ { τ σ σ σ ,ν µ searches assuming X ,ν u,c,t d,s,b e,µ,τ 0 g e Table 1 x x ν x A x Production Mechanism Coupling . Production rates for the models in table kinetic mixing, and on a vector that mediates a protophobic force [ decay branching fractions to SM final states, which are presented in figure γ – astrophysical constraints in either case (see, e.g., ref.3.1 [ Decays toFor SM the final case states where X B couplings of each ofresults these of models section are providedall in of table the necessary recast the by recasting each of them under the assumption 3 Example models We now use the framework developedsearches in to the previous obtain section constraints to oncurrent, recast the a existing following dark leptophobic photon models: a vector that couples to the Table 2 for meson-decay rates which are provided in table JHEP06(2018)004 , . . − L 7 B − 5 π – π − 0 + + 5 . We B π π ≈ . The 4 0 → 2 0 ]). The 4 ) as there ) → A 2 2 2 2 ones, are 2 ρ A | | | | ρ ) ) ) ) 0 89 m B m L ) = 0 case. . m m m . m − ( ω ( ( A ( ¯ , which was χ − 2 ω B φ φ φ φ 5 ≈ B m − χ ( g m , including the B 2 ( π ) − 4 8BW 2BW π + or εe → m − − and 2 + p π )+4BW ) )+4BW ) 4( 2.2 g L π L m m m m 4 at low masses, due − ( ( ( ( → ≈ ρ ρ ρ ρ B 2 2 2 2 2 − → p 2 p ) ) ) ) ) g 2 2 2 2 p p p p 2 2 g 0 | X | εe g g g g of appendix B εe εe εe εe εe 4 ) ) ( ( ( ( ( ( 9BW 9BW B A m m − − ( ( B = )+9BW ) )+9BW 10 ) ρ ρ m m m m ( ( ( ( Protophobic B BW ]. ω ω ω ω g − )+BW ) values for all important decay 92 BW BW BW BW m m | | | | ( ( 2 2 for ω ω ) ) 2 2 p p , see also tables →F 0 g g εe εe ( ( BW BW B relative to those of the dark photon. | | A C B 2 1 ) 2 p g εe ( ] provides much better sensitivity to the and the results of section 2 2 | | 31 ) ) 2 1 ) ) 2 B m m g ( ( – 11 – εe L (%); however, we take them to be zero, since the with equal strengths. The decays ( 2 φ φ 2 | | ) − O ) ρ ). In this region, we take ≈ m m ( B ω 2BW ( 2 , which are similar to the corresponding φ , and the protophobic model are shown in figures | ( φ search [ − ) )+4BW ) L 0 B values at small masses [ m m m m for ( ( ( , 2 − 2BW A ρ | ρ ρ 2 2 2 L ) − L L )+BW ) ) ) 2 2 B B decays considered in our study are shown in figure ) − B 2 B m − 0 0 g g m g ( − B εe εe εe m 0 4 4 ( B ( ( ( ( ω g ω g )+BW B ω A )+9BW )+9BW m ( BW BW | m m BW → | ω ( ( model, which has nonzero couplings to SM neutrinos, searches | ω ω only couples to leptons via kinetic mixing. One consequence of B BW L g | BW BW ( | | 2 B − ) 2 B . In addition, we provide the both mix with the 2 2 g ) ) 2 2 B B εe B 4 B p ( g is much larger than that of the g εe εe 4 B 4 ( ( search does not provide competitive sensitivity to X constraints on require isospin violation, making them difficult to calculate reliably. One model, the constraints bear little resemblance to those on the B 0 , including specific hadronic final states, in figure − − π A 0 0 B , , B π and + results recast for 0 γ η 0 + γ 0 η η γ π 0 π 0 0 π . Meson-decay rates for the models in table 0 0 0 0 0 0 π Xγ Xη A Xγ A Xη A Xη A Xγ Xπ A A A A Xπ A A → → → → → → → → → → → → → → → 0 → 0 0 0 → 0 0 η φ → ω 0 η φ ω 0 η 0 ρ η π , ω ρ π , ω Γ Γ Γ Decay Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ A ρ B ρ For the The The searches for visible Γ Γ Table 3 available in a small region of lifetime of the to the fact thatthis is the that the LHCb long-lived for invisible dark photonsThe also recasted provide constraints eventhe for strongest the on thisHowever, we model note in that recent mostexchange constraints could of derived compete the from with neutrino coupling-mass the experiments, region SM where considered neutral-current process, in are figure currently the strongest do not consider somesame searches production that and have decayDetailed inferior mechanisms. discussion sensitivity The on to efficiency these others ratios is are that provided experiment in employed appendix dependent. the Note that for the employed in the masssince region the near and expects these branching fractions toonly be are determined using thework couplings in in table appendix modes of the are plans to useonly hadronic some final of state these used final in states any in search future considered searches here (see, is e.g., ref. [ JHEP06(2018)004 ]. 95 , and B 4 GeV. [GeV] . L mode as 0 [GeV] − model B p B X & L m constraints m − 0 limits in the B 0 B A =hadrons =hadrons m A F F Protophobic model , (middle left) L − B boson decaying into specific B − − − [GeV] − µ µ e e ¯ + ν + + + X ν e µ µ constraints are the strongest on e m = = = 0 = = F F F A F F . , (top right) 0 0 A A

0 0.5 1 1.5 2

1 1 0 0

F → B F → −

B 0.8 0.6 0.4 0.2 0.8 0.6 0.4 0.2

L p ) X ( ) L B ( − ], which are the strongest non- B couples to an anomalous SM current, additional 74 – 12 – , . (bottom) Ratio of the branching fractions to leptons B ], which provide the strongest non- [GeV] 73 [GeV] ′ -search constraints are dominant for B B A 0 10 94 m m A , model model ′ 93 B A [ protophobic =hadrons F − µ

. Since the +

0 0.5 1 1.5 2

1 3 2 0

→ B →

B µ

− ′

− A ℓ X ( ) ℓ ℓ A ( / ) ℓ

+ + → 6 GeV. Additional indirect constraints arise from the requirement of . ]. We have added to these the constraints from the LHCb searches for 0 X − − µ e 74 + + . − , e µ − µ e + =hadrons = = B + µ 73 e with F F F = m = F F X . ) , and the protophobic model relative to the ∗ ( µ B . Decay branching fractions for the (top left) , K m

L at low masses, while the non-

0 0.5 1 1.5 2 0 0.5 1 1.5 2

1 1 0 0

F → B F →

B − → 0.8 0.8 0.6 0.4 0.2 0.6 0.4 0.2

′ ) B ( ) A ( B B u,d anomaly cancellation by new vector-likeUnder fermions, which the have assumption not that yetapply the been these discovered lack [ of constraints following discovery refs. impliesin the [ that mass such region states from dothe about not 1 to exist, 5 we GeV. The recasted boson than it does tostrong the constraints arise due to thederived enhanced in production refs. rates [ of theB longitudinal region 2 Figure 3 (middle right) protophobic models.hadronic The final branching states fractions are offor shown the in figure JHEP06(2018)004 0 ] is A 52 e 2) − decays. g 0 processes, π Xγ , except for the 0 ], and LEP [ A → 46 Z ] provide the strongest model, which couples to collisions, (magenta) meson , and 94 B , pp ], BaBar [ πX 93 lifetime and branching fractions 110 → p X K , ) searches [ KX is a factor of 4/9 smaller in the protophobic model colliders, (blue) − ] apply to this model as well; however, the µ → − + 6 GeV for the protophobic model. That said, e . 74 XBB µ + , – 13 – 0 ( e u,d A 73 X B . ) ∗ 1, only the NA64 [ p ( decays considered in this study from (red) electron beam X K ≈ 0 m A ) → ¯ χ . χ . Additional constraints on the µ ], the value of u,d 8 m → 74 B , ]. Recasting these results for the protophobic model, which also 73 X ( 74 , B ]. 73 91 , 90 model. In addition, the sizable differences in the B 5 . Constraints on visible The constraints on the protophobic model are similar to those on the In the notation of refs. [ 5 as studied in refs.couples [ to an anomalousthe SM previous current, subsection. simply involves the scale factor of 4/9than discussed in the in 3.2 Decays to invisibleFor dark-sector the final case states where searches for dark photonresults decays are to shown invisible inan final figure anomalous states SM are current, used arise in from the recasting. The For example, theconstraints LHCb in the regionover 2 most of the coupling-masssearches region are explored the thus most far, stringent. the constraints obtained from The protophobic current ismeans also that anomalous the in constraintscoupling the to from the absence refs. anomalous of current [ is additionalcouplings. weaker by fermions, a factor which of 4/9lead due to to the substantial different fermionic differences in the constraints derived from the anomalous currents. decays, and (yellow) electron onshown in fixed grey target [ experiments. The constraint derived from ( absence of the constraints based on production via proton bremsstrahlung and Figure 4 dumps, (cyan) proton beam dumps, (green) JHEP06(2018)004 ] ], 99 109 , , 92 108 [ ]. The dark Xγ 95 , → 74 Z , due to its coupling to 73 ], CHARM-II [ L 98 − , ], and ] decays, from longitudinal- B 92 ], are from the LHCb searches 107 104 , , 74 , 76 106 [ 73 [ η ], Texono [ πX 97 ] and , → 96 K 103 , ]. ], 102 105 [ – 14 – 101 decays using the same experimental color scheme as , ]. B 100 KX 94 decays to SM final states using the same experimental color , → L 93 [ − B u,d − µ B + µ ] in → 74 X , 73 . The (orange) invisible constraints also apply to 4 with X ) ∗ ( K . The grey constraints come from Υ [ . Constraints derived on visible . Constraints derived on → 4 u,d B in figure mode enhancements [ processes, and from thegrey constraints, lack which of are obtained observed infor new this work anomaly-canceling following refs. fermions [ [ Figure 6 scheme as in figure neutrinos. The greyand constraints from are SPEAR, from DORIS, Borexino and [ PETRA [ Figure 5 JHEP06(2018)004 , but recast 6 1. The grey constraints show the ≈ ) ¯ χ χ → , also recast here for the protophobic model. X B ( B ] for – 15 – 74 , 73 . The grey constraints are from the same processes as in figure 4 . Constraints derived on visible protophobic decays using the same experimental color . Constraints on all models assuming 4 Summary In summary, we have developedconstraints a on framework for more recasting genericto dark models photon the searches that Standard to contain obtain Modelhadronic a decay fermions, massive rates. which boson We includes demonstrated with our a vector approach data-driven couplings by method deriving constraints for on determining a vector Figure 8 longitudinally enhanced results of refs. [ to the protophobic model as part of this study. Figure 7 scheme as in figure JHEP06(2018)004 ] . 74 , (A.1) (A.2) 73 ) for the decay constraints from m ( ], and Iftah Galon L F − ]; however, recasting 52 K B 114 , , 113 ) m , ]). Of course, searches for dark ( } V 0 , , Γ , 112 1 } , , } 1 boson that couples directly to baryon im − } 2 , 2 V , , 1 − } 1 − 111 , , 1 B m 0 { , 0 1 2 ]. The VMD form factors are Breit-Wigner , , , 1 1 0 1 { m https://gitlab.com/philten/darkcast , { { 115 diag 1 − { 1 2 3 [ diag – 16 – 2 V / diag diag 3 = 2 m 2 6 1 diag √ ρ 1 1 √ 1 2 √ √ 2 T 2 ) = = = = ≈ ≈ ≈ m 0 0 0 η φ ω ( η π T T T T V kinetic mixing, and on a vector t