IFT-UAM-CSIC-19-73 FTUAM-19-10

Non-Resonant Searches for Axion-Like Particles at the LHC

M. B. Gavela,1, 2, ∗ J. M. No,1, 2, † V. Sanz,3, ‡ and J. F. de Troconiz1, §

1Departamento de Fisica Teorica, Universidad Autonoma de Madrid, Cantoblanco, 28049, Madrid, Spain2Instituto de Fisica Teorica, IFT-UAM/CSIC, Cantoblanco, 28049, Madrid, Spain

3Department of Physics and Astronomy, University of Sussex, BN1 9QH Brighton, United Kingdom

We propose a new collider probe for axion-like particles (ALPs), and more generally for pseudo-Goldstone bosons: non-resonant searches which take advantage of the derivative nature of theirinteractions with Standard Model particles. ALPs can participate as off-shell mediators in the s-channel of 2 → 2 scattering processes at colliders like the LHC. We exemplify the power of thisnovel type of search by deriving new limits on ALP couplings to gauge bosons via the processespp → ZZ, pp → γγ and pp → jj using Run 2 CMS public data, probing previously unexploredareas of the ALP parameter space. In addition, we propose future non-resonant searches involvingthe ALP coupling to other electroweak bosons and/or the Higgs particle.

I. INTRODUCTION

Axion-like particles (ALPs) [1, 2], and more generallypseudo-Goldstone bosons, often appear in extensions ofthe Standard Model (SM). They may be connected tosolutions to the strong CP problem [3–19] and/or to theexistence of new spontaneously-broken global symmetriesin Nature. In the following the term ALP will be usedindistinctly to denote all such pseudo-scalars.

ALPs are being searched at high-energy colliders [20–27, 30, 31], beam dump experiments [32, 33], via theireffects in flavour physics [29, 34–38] and through theirastrophysical signatures [39–42] (see Ref. [43] for a re-view).

In this work we propose a novel approach to probethe existence of ALPs at high-energy colliders, namelynon-resonant searches where the ALP is an off-shell me-diator in the s-channel of 2 → 2 scattering processes.The ALP pseudo-Goldstone nature implies that its in-teractions with SM particles are dominantly derivative,enhancing the cross sections for center-of-mass energiess � m2

a, where ma denotes the mass of the ALP a. Inthis kinematical regime, the processes tailored to searchfor ALPs at the Large Hadron Collider (LHC) includethose with two SM bosons in the final state: electroweakgauge bosons (W , Z, γ), gluons g and/or the Higgs par-ticle h. For ma � 100 GeV, the gluon-initiated 2→ 2 di-boson scattering processes pp (gg) → ZZ, WW , Zγ andZhmay be mediated by a virtual ALP, as shown in Fig. 1.This can also occur for the processes pp (gg)→ jj (gg) orpp (gg)→ γγ in the regime where a large invariant massmjj or mγγ is required in the final state.

The theoretical framework used throughout this workis the model-independent approach of effective field the-ories (EFT). If the Higgs particle is considered to be partof an exact SU(2)L doublet at low energies, as predictedin the SM, the putative beyond the Standard Model(BSM) electroweak physics may then be described by

an EFT linear expansion [44, 45] in terms of towers ofgauge invariant operators ordered by their mass dimen-sion. Alternatively, since a non-doublet component ofthe Higgs particle is at present experimentally allowed(at the ∼ 10% level [46]), a non-linear EFT (also calledchiral) [47–53] based on a momentum expansion is alsoof interest. In the following we concentrate on the lin-ear EFT for the SM plus an ALP [1, 2, 24], and discusswhen pertinent the comparison with a chiral EFT, no-tably for the interactions between the ALP and the Higgsboson [24, 25].

a∗

Z

Z

a∗

Z

h

FIG. 1. Feynman diagrams for the processes gg → ZZ (left)and gg → Zh (right) via an off-shell ALP in the s-channel.

II. BOSONIC ALP LAGRANGIAN

Linear expansion. In the linear ALP EFT, the newphysics scale to be considered is the ALP decay constantfa, which will weight down the higher-dimensional opera-tors built from the SM fields and a. The most general CP-conserving effective Lagrangian describing bosonic ALPcouplings contains – up to next-to-leading order (NLO)– only four independent operators of mass dimensionfive [1, 2, 24, 54],

δLeff ⊃ cGOG + cB OB + cW OW + caΦOaΦ , (1)

where

OG ≡ −a

faGµνG

µν , OW ≡ −a

faW aµνW

µνa ,

OB ≡ −a

faBµνB

µν , OaΦ ≡ i∂µa

faΦ†←→D µΦ .

(2)

arX

iv:1

905.

1295

3v2

[he

p-ph

] 1

0 Se

p 20

19

2

In these expressions Gµν , Wµν and Bµν denote respec-tively the SU(3)c×SU(2)×U(1) field strengths, and thedual field strengths are defined as Xµν ≡ 1

2εµνρσXρσ,

with ε0123 = 1. The ci constants are real operator co-efficients and Φ denotes the SM Higgs doublet, with

Φ←→D µΦ ≡ Φ†

(DµΦ

)−(DµΦ

)†Φ. The first three oper-

ators in Eq. (1) induce physical agg, aγγ, aγZ, aZZ andaW+W− interactions,

δLeff ⊃ −gagg

4aGµνG

µν − gaγγ4

aFµν Fµν − gaZγ

4aFµνZ

µν

− gaZZ4

aZµνZµν − gaWW

4aWµνW

µν , (3)

where

gagg =4

facG , gaγγ =

4

fa

(s2w cW + c2w cB

)(4)

gaWW =4

facW , gaZZ =

4

fa(c2w cW + s2

w cB) (5)

gaγZ =8

faswcw(cW − cB) , (6)

and sw and cw denote respectively the sine and cosineof the Weinberg mixing angle. The Feynman rule forthe interaction aV1V2 (with V1,2 being SM gauge bosons)stemming from these operators is given by

− i gaV1V2pρV1

pσV2εµνρσ . (7)

The last operator in Eq. (2), OaΦ, induces a mixing be-tween a and the would-be Goldstone boson eaten by theZ. Its physical impact is best illustrated via a Higgs fieldredefinition, Φ→ Φ eicaΦa/fa [1], which trades OaΦ for

ia

fa

[QYuΦuR −QYdΦdR − LY`Φ`R

]+ h.c. , (8)

where Yu,d,` denote the SM Yukawa matrices. We focusin this Letter on experimental signals involving ALPs andSM bosons (W, Z, γ, g and h), yet we briefly comment onsignatures involving the OaΦ fermionic coupling in Sect.IV-4.1

Chiral expansion. The operators OG, OW and OBin Eq. (2) also appear in the chiral expansion at NLO.In addition, and at variance with the linear EFT, novelALP-Higgs couplings are present in the chiral expan-sion already at leading order (LO), namely the operatorA2D(h) [24] which is a custodial breaking two-derivativeoperator with mass dimension three:

L LOa ⊃ c2DA2D(h) = c2D

[iv2Tr[TVµ]∂µ

a

faF(h)

],

(9)

1 A complete –bosonic and fermionic– ALP basis can be obtainedsubstituting OaΦ in Eq. (2) by general flavour-changing fermionicoperators e.g. [1, 2, 24, 54]. The physical effects of the latter areproportional to the Yukawa couplings of the fermions involved.

where v = 246 GeV denotes the electroweak scale asdefined from the W mass, Vµ(x) ≡ (DµU(x)) U(x)†,

T(x) ≡ U(x)σ3U(x)† and U(x) = eiσjπj(x)/v , with

πj(x) corresponding to the longitudinal degrees of free-dom of the electroweak gauge bosons and σj are the Paulimatrices. The physical Higgs particle h is introduced inthe chiral expansion via polynomial functions [55] of h/v,F(h) = 1 + a2Dh/v + b2D(h/v)2 + O(h3/v3), with a2D,b2D constant coefficients. A2D is the chiral counterpart(“sibling”) of the linear operator OaΦ in Eq. (2), witha key difference: in addition to ALP-fermion couplingsanalogous to those in Eq. (8), A2D induces interactionsbetween the ALP, the electroweak gauge bosons and anynumber of Higgs particles, e.g. a trilinear a−Z − h cou-pling (see Fig. 1). The associated experimental signa-tures at the LHC will be discussed in Sect.IV-4. Notethat such couplings can be found in the linear expan-sion only at next-to-NLO (NNLO ), i.e. mass dimensionseven [25, 56], and are thus expected to yield subleadingeffects in that case.

III. NON-RESONANT ALP-MEDIATEDDI-BOSON PRODUCTION

The key observation is that, due to the derivative na-ture of the ALP interactions under discussion, the ALPmediated scattering processes gg → a∗ → V1V2 exhibita harder scaling with the invariant mass of the event√s = mV1V2

than usual s-channel mediated exchanges.In all generality, the contributions from bosonic ALP

couplings in Eq. (3) interfere with the absortive part ofSM 2 → 2 di-boson amplitudes. Nevertheless, given thepresent loose bounds on ALP couplings, pure ALP ex-change dominates the cross section for some LHC chan-nels. A quartic dependence on ALP couplings results inthis case,

σV1V2∝ g2

agg g2aV1V2

s ∼ s

f4a

, (10)

in the ALP off-shell regime s � m2a,m

2Vi

.2 The sametype of energy behaviour holds for gg → a∗ → Zh fromEq. (9). Such energy dependence is valid only as far asthe energies probed in the scattering process are smallerthan the cut-off scale of the EFT,

√s < fa.

The behaviour in Eq. (10) is to be compared with theenergy dependence for a usual 2→ 2 s-channel mediatedprocess, which scales instead as 1/s far above from the s-channel resonance. Factoring in the proton parton distri-bution functions (PDFs), which tame the energy growthin Eq. (10), the differential cross section for the ALP-mediated process pp → a∗ → V1V2 diminishes –at ener-gies much larger than the resonance’s mass– more slowly

2 This has been noted in a different setup in Ref. [27].

3

with the invariant mass than for a usual s-channel res-onance whose couplings do not depend on the momentainvolved. The momentum dependence of the ALP inter-action in Eq. (7) significantly smooths out the decreaseof the cross section at large

√s, allowing to distinguish

ALP-mediated processes from the SM background.

For sufficiently small ALP couplings, the size of the in-terference with the SM background becomes comparableto the pure ALP-signal and must be taken into account.The value of the couplings for which this happens de-pends on the specific final state V1V2. For γγ and theother electroweak di-boson final states, the ALP signalinterferes with one-loop SM processes. The interferenceis constructive or destructive depending on the relativesign of gagg and gaV1V2 ; in any case, it can be disregardedat present, given the size of ALP couplings that can becurrently probed at the LHC (see section IV). FutureLHC analyses will need to include them, though, andthis exploration is deferred to a future work [74].

The situation is different for di-jet (gg) final states:the SM contributes at tree-level (via gluon exchange),and the interference already dominates at present overthe pure ALP signal, for the gagg values at reach. It isdestructive and will be taken into account in the gg → gganalysis in Sect. IV-3.

It is illustrative to compare the shape of the s-channelinvariant mass distribution stemming from an off-shellALP of negligible mass, with that from a heavier (andbroad) mediator with derivative couplings, e.g. a KK-graviton with large width. This is illustrated in Fig. 2for an electroweak diboson (ZZ) final state; the SM ex-pectation is depicted in addition. The different genericpatterns of decelerated decrease highlight that it is pos-sible to distinguish the presence of a light ALP from thatof a heavy BSM resonance, even if the heavy resonance’scouplings are derivative as well, via the study of mV1V2

distributions.

A supplementary handle to discern whether a puta-tive BSM signal would correspond to an ALP is givenby the angular distribution of the final states [57]: therich Lorentz structure of ALP couplings in Eq. (7) in-duces a distinctive angular pattern3 which may be usedto infer the pseudoscalar nature of a. We leave such anexploration for a future work.

The non-resonant s-channel ALP signatures exploredin this work have several further attractive features:(i) In the regime under discussion with s � m2

a, thesignal cross section and distributions are essentially inde-pendent of the value of ma. This implies that the searchis equally sensitive to any ma significantly below the en-ergy range probed by the search. In particular, for the

3 See e.g. Refs. [58–61] for analogous studies in Higgs physics.

200 400 600 800 1000 1200 1400

-210

-110

1

1

�

d�

dmZZ

mZZ (GeV)

Light ALPSM diboson

KK-graviton (broad)

FIG. 2. 13 TeV LHC normalized differential cross section fordi-boson production as a function of the di-boson invariantmass mZZ =

√s. The signal of an s-channel ALP with ma �

mjj (solid black line) is compared to the SM prediction (greenarea) and a broad KK-graviton with mass 500 GeV and 30%width (dashed red line).

LHC searches considered in Sect. IV, the derived sensi-tivity can be safely applied to any ALP mass below 100GeV. (ii) Being a non-resonant process, no hypothesis isneeded on the value of other possible couplings which donot contribute to the process under consideration. Thisis at variance with on-shell analyses, for which the de-pendence on other ALP couplings may appear throughthe partial decay widths.4 In this sense, non-resonantsearches are more model-independent and thus more ro-bust.

From the theoretical point of view, the gaV1V2couplings

depend only on the ratio ci/fa (see Eqs. (4)-(6)), but thevalue of fa is relevant to assess the validity of the EFT,which limits the energy range that can be safely consid-ered in an LHC search (e.g. to bins satisfying

√s < fa)5,

and discuss this for specific LHC searches in Sect. IV. An-other pertinent question is the possible impact of radia-tive corrections and of higher dimensional operators. Forthe former, self-energy corrections to the s-channel ALPpropagator only become non-negligible close to the EFTvalidity boundary, and we do not consider their effect

4 Most present ALP limits based on resonant processes have con-sidered only one independent gaV1V2

coupling at a time amongthe set in Eq. (3) (see e.g. Refs. [21–23, 32, 34]), with some re-cent analyses considering the simultaneous presence of two or atmost three independent couplings [26, 28, 29].

5 If the underlying BSM theory were in the weak coupling regime,and led at one-loop to the operators in Eq. (2), their coefficientscould plausibly be suppressed by an additional αi/(8π) factor.This would drastically reduce the set of valid energy bins in LHCsearches. We stick here instead to the general and widespreaddefinitions in Eqs. (2) and (3).

4

here. Higher dimensional operators, e.g. those weighteddown by the same O(1/f2

a ) factor than the amplitudesdiscussed above, can also contribute only at loop level,as fa must intervene as powers of a/fa and no ALP ispresent in the final states considered here. Furthermore,only by engineered fine-tuned cancellations could suchoperators impact significantly on the results of this work.

IV. NON-RESONANT LHC SEARCHES

In this section we derive new limits on gaV1V2couplings

through the non-resonant ALP-mediated processes dis-cussed above, using public data from LHC Run 2 (

√s =

13 TeV) CMS searches. ALP production in the s chan-nel is dominated by gluon-gluon fusion, as the qq inducedALP production amplitude is proportional to the quarkmasses – see Footnote 1– and thus highly suppressed.Possible final states to be considered include gg, ZZ,WW , Zγ, γγ or Zh. While it is of high interest to exploreall of them, since they probe different operator combina-tions within the EFT, we focus below on the processespp → a∗ → ZZ, pp → a∗ → γγ and pp → a∗ → gg.For these channels, the CMS collaboration has recentlypublished new results, providing explicit calculations ofthe corresponding SM backgrounds. We use those pub-lic data to compute approximate limits on gagg × gaZZ ,gagg × gaγγ and gagg, respectively. In all three analyses,the ALP mass is fixed to ma = 1 MeV (i.e. effectivelymassless at LHC energies) and the ALP width Γa is as-sumed to respect Γa � ma.

For the pp → a∗ → ZZ and pp → a∗ → γγ channels,our sensitivities are estimated from a simplified binnedlikelihood ratio analysis. The likelihood function is builtas a product of bin Poisson probabilities

L(µ) =∏k

e−(µsk+ bk) (µsk + bk)nk

nk!, (11)

where nk, bk and sk denote respectively the observeddata, SM background and ALP signal prediction in agiven bin k, and the signal strength modifier µ is takenas the only floating parameter in the likelihood fit (seeRef. [24] for details), with no systematic uncertaintiesconsidered for simplicity. For the pp → a∗ → gg chan-nel we perform a χ2 fit to the data including systematicerrors but no bin-to-bin correlations.

Other important search channels are also briefly dis-cussed below, albeit their analysis is left for the future:pp→ a∗ → Zγ, pp→ a∗ → Zh (which provides a uniquewindow into the chiral EFT via the operator A2D inEq. (9)) and pp → a∗ → tt (which would yield accessto the operator OaΦ in Eq. (2)).

1) pp→ a∗ → ZZ

The process pp → a∗ → ZZ → ``qq is studied next,following the semi-leptonic di-boson CMS analysis atLHC

√s = 13 TeV with 35.9 fb−1 [62]. We focus on

the “low-mass merged” CMS analysis category target-ing the invariant mass region mZZ ∈ [450, 2000] GeV,with one Z boson decaying leptonically, Z → `` and theother decaying hadronically. The boosted hadronic Zdecay products are required to merge into a single jet,Z → J . The jet is reconstructed via the anti-kT algo-rithm with R = 0.8 (AK8). Our signal process is simu-lated in MadGraph aMC@NLO [63], with a subsequent par-ton showering and hadronization with Pythia 8 [64] anddetector simulation via Delphes 3 [65], including the useof jet-substructure variables as discussed in detail in Ap-pendix A1. Following Ref. [62], the analysis is dividedinto b-tagged and untagged categories, targeting respec-tively the Z → bb and Z → qq (with q = u, d, s, c) decays.The b-tagging of the merged jet J provides a strong back-ground suppression, yielding a further increase in sensi-tivity.

600 800 1000 1200 1400 1600 1800 2000m``J (GeV)

10−1

100

101

102

103

Eve

nts

/50

GeV

CMS-B2G-17-013Untagged

ALP signal (ci = 1, fa = 2 TeV)

Z+ jetst t , tW , WWZV (V = Z , W )Data

600 800 1000 1200 1400 1600 1800 2000m``J (GeV)

10−1

100

101

102

103

Eve

nts

/50

GeV

CMS-B2G-17-013Tagged

ALP signal (ci = 1, fa = 2 TeV)

Z+ jetst t , tW , WWZV (V = Z , W )Data

FIG. 3. m``J distributions for the ALP ZZ signal with ci = 1,fa = 2 TeV (dashed black line) and SM background from Z+jets (yellow), ZV (red) and tt (cyan) after CMS event selec-tion, in the untagged (top) and b-tagged (bottom) categories.The experimental data are shown as black dots.

As an illustration of the impact of the derivative na-ture of ALP interactions, the

√s = 13 TeV cross-section

5

σ(pp → a∗ → ZZ) for cG = cW = cB = 1 and fa = 1TeV is 81 pb. The CMS event selection is discussed indetail in Appendix A1. Fig. 3 shows the invariant massm``J distribution resulting for the signal after the CMSevent selection, for ci = 1 and fa = 2 TeV (correspond-ing to the largest value of m``J in the CMS analysis),together with the SM background publicly available inRef. [62] (and dominated by Z+ jets), both for the un-tagged (top plot) and b-tagged (bottom plot) categories.A binned likelihood analysis of the m``J distribution af-ter CMS event selection combining the untagged and b-tagged categories is then performed, which allows to seta 95% C.L. exclusion limit on the signal cross section ofσ = 25 fb. This corresponds to fa > 4.1 TeV for ci = 1,and is valid for any value of the ALP mass up ma ∼ 200GeV without significant modifications of the signal prop-erties. Note that, since the “low-mass merged” CMSanalysis uses data up to mZZ = 2 TeV, our derived limiton fa for ci = 1 lies within the region of validity of theEFT. In Fig. 4 (top) the corresponding new limit on gaZZ(see Eq. (5)) resulting from our non-resonant analysis isdepicted as a hatched area, for a fixed value g−1

agg = 1TeV.

For comparison, Fig. 4 (top) depicts as well previousbounds in the literature for gaZZ , which also assume theadditional presence of gagg, albeit obtained from on-shellALP searches. For ma . 0.1 GeV, the ALP is stableon LHC scales, resulting in constraints on gaZZ frommono-Z searches (in violet), see Ref. [24]. The radiative(2-loop) contribution of gaZZ to gaγγ allows to obtainfurther constraints for certain ranges of ALP masses forwhich strong constraints on gaγγ exist (see the discussionin Refs. [25, 28]). For ALP masses below the GeV scale,limits on gaZZ are thus set by beam dump searches (inyellow) [70–72] (we adapt here the bounds compiled inRef. [32]), and by energy-loss arguments applied to thesupernova SN1987a [41, 42] (in blue), both through ab-sence of extra cooling (labelled “length” in Fig. 4 andthrough absence of a photon burst from decaying emit-ted axions (labelled “decay” in Fig. 4. Furthermore, theradiative contribution of gaZZ to gaγγ is also constrainedby LHCb [73] (see Ref. [35]) in the small region 4.9 GeV< ma < 6.3 GeV (in dark grey) and by ATLAS/CMSsearches for γγ resonances (in red) for ma > 10 GeV (weadapt here the bounds from Refs. [20, 26]). We stressthat the latter limits are from LHC Run 1 (

√s = 7 and

8 TeV), and as such√s = 13 TeV Run 2 analyses should

significantly improve on those. Next, although LHC tri-boson searches for ma � 100 GeV have yielded veryweak constraints [30], the radiative contribution of gaZZto gaγγ provides as well sizeable constraints. We do notinclude here, though, the expected tree-level bounds ongaZZ from ZZ resonance searches by ATLAS and CMS(e.g. from Ref. [62]) for ma > 200 GeV. To our knowl-edge, these have not yet been obtained and are com-plementary to the non-resonant search presented in this

10−3 10−2 10−1 100 101 102 103 104

ma (GeV)

10−4

10−3

10−2

10−1

100

101

102

103

g aZZ

(TeV−1

)

LHC(γγ resonance)

SN1987a(decay)

SN1987a(length)

LHC mono-Z

LHCb

Beam Dumps

g−1agg = 1 TeV

Non-resonant LHC (this work)

10−3 10−2 10−1 100 101 102 103 104

ma (GeV)

10−5

10−4

10−3

10−2

10−1

100

101

102

g aγγ

(TeV−1

)

LHC(various)

LEPL3

SN1987a(length)

Babar

LHCb

Beam Dumps

g−1agg = 1 TeV

Non-resonant LHC (this work)

FIG. 4. Top: Bounds on the ALP coupling gaZZ as a func-tion of ma. The hatched region corresponds to the limit fromnon-resonant LHC searches derived in this work using CMSdi-boson data [62]. Also shown are limits from LHC mono-Z searches (violet), beam-dump experiments (yellow), super-nova SN1987a (blue), LHCb (dark grey) and LHC resonantγγ searches (red), see text for details. Bottom: Bounds onthe photonic couplings gaγγ , with color code as for the topfigure. Limits from BaBar (dark grey), L3 (cyan) and LEP(green) are also depicted, see text for details.

work. The study of such ZZ resonant searches is left fora forthcoming work [74].

2) pp→ a∗ → γγ

Non-resonant ALP searches are also possible for finalstates to which a light ALP could decay, such as γγ, byselecting events with a large invariant mass mγγ � ma.The recent CMS search for non-resonant new physics

6

in γγ final states [75] with 35.9 fb−1 of 13 TeV LHCdata is used here. In analogy with the previous sec-tion, we simulate the signal process pp → a∗ → γγ withMadGraph aMC@NLO, Pythia 8 and Delphes 3, obtaininga signal cross-section σ(pp → a∗ → γγ) = 47 pb forcG = cW = cB = 1, fa = 1 TeV and with the ini-tial requirement mγγ > 500 GeV. The subsequent CMSevent selection applied here is detailed in Appendix A2,with the main SM backgrounds [75] being γγ and γ + j(with the jet j mis-identified as a photon). After theevent selection, we perform a binned likelihood analy-sis of the mγγ distribution for the two selection cate-gories discussed in Appendix A2 according to the rapid-ity of the photons. This leads to a combined 95% C.L.observed exclusion limit on the signal cross section ofσ ' 1.2 fb. This limit corresponds to fa > 14.2 TeV forci = 1, which we find to be valid up to ma ∼ 200 GeVwithout significant modifications of the signal properties.The resulting bound on gaγγ is depicted in Fig. 4 (bot-tom) for g−1

agg = 1 TeV as a hatched area. Bounds fromresonant searches at the LHC, beam dump experimentsand astrophysical constraints (supernova SN1987a) arealso shown, see Sect. IV-1 for details. For comparison,the figure also shows bounds from resonant searches byBaBar [76] (in dark grey) (as obtained from Ref. [35]),from L3 [77] (in cyan), as well as from LEP searches (ingreen) for new physics in e+e− → 2γ, 3γ processes (seeRefs. [21, 22] for a detailed discussion). Regarding thelatter, Refs. [21, 22] assume a vanishing gluonic couplinggagg, and thus apply in our case only for ma < 3mπ; forma > 3mπ the ALP can decay into hadronic final statesin the presence of a non-vanishing gagg coupling, whichcould significantly weaken the LEP bounds, and we re-frain to claim an exclusion in that region. We also do notinclude here projected limits on gaγγ from light-by-lightscattering at the LHC in proton-proton [27] and Pb-Pbcollisions [23] (the latter is also significantly weakened inthe present scenario by the presence of gagg), as they arenot competitive with the search presented here.

3) pp→ a∗ → gg

As discussed previously, the gluonic coupling gagg canbe constrained independently, using di-jet searches at theLHC. In this work we use the 13 TeV CMS search on di-jet angular distributions [78] for this purpose. The detailsof our selection procedure are given in Appendix A3. A95% C.L. excluded region of 1.9 TeV > fa > 3 TeV forcG = 1 follows from our analysis. The latter includes theinterference between the ALP signal and the SM back-ground, which dominates over the pure ALP signal inthe region considered, as already mentioned. Note thatthe limit obtained is weaker than the benchmark valueg−1agg = 1 TeV used in Fig. 4 (corresponding to fa/cG = 4

TeV), as it should be. This bound is to be taken onlyas a qualitative estimate, as the analysis uses data in the2.4 to 3 TeV range, and thus in the limit of validity of

the EFT.

4) Further non-resonant ALP searches

(i) pp→ a∗ → Zh. This process yields a powerful probeof the chiral EFT through the operator A2D in Eq. (9).For Z → `` and h → bb this signature is similar to thatanalyzed in Sect. IV-1 for the b-tagged category, sincethe process pp → a∗ → Zh has similar m``J kinemat-ics and expected cross section than pp → a∗ → ZZ, forc2D ' cW , cB . This suggests that the analysis performedabove could be adapted to probe very efficiently the ALP-mediated Zh signal. Furthermore, there are several ad-vantages in performing a dedicated Zh search along thelines in Ref. [62]: the SM background distribution for themerged jet mass mJ is smaller around mh than aroundmZ (as shown in Ref. [62]), and the SM backgrounds af-ter the CMS event selection are significantly smaller inthe b-tagged category, as shown in Fig. 3; iii) h decaysdominantly to bb.

(ii) pp→ a∗ → tt. This channel allows to probe theALP-fermion couplings induced by the operator OaΦ inthe ALP Linear EFT. Because the amplitude of any phys-ical ALP-fermion coupling is proportional to the fermionYukawa couplings (see Eq. (8) and Footnote 1), ALP pro-duction via gluon fusion with tt in the final state is anoptimal channel which deserves detailed future study.6

(iii) pp→ a∗ → Zγ. This channel provides a key com-plementary probe to the ZZ and γγ searches discussedin Sect. IV-1) and -2, given its clean signature. The non-resonant analysis of this channel using public informa-tion (e.g. from Ref. [79]) requires however further as-sumptions w.r.t. the ZZ and γγ analyses. The study ofALP-mediated Zγ signatures is thus left for a forthcom-ing work [74].

V. CONCLUSIONS & OUTLOOK

In this work we have proposed a new approach to probethe existence of ALPs (and more generally of pseudo-Goldstone bosons), via non-resonant searches at the LHCwhere the ALP can be produced as an s-channel off-shellmediator. The search takes advantage of the derivativenature of the ALP interactions with SM particles. Us-ing CMS 13 TeV public data, we have derived new limitson ALP couplings to SM gauge bosons via the processespp → ZZ, pp → γγ and pp → jj (gg). These providethe most stringent bounds on ALPs over a wide regionof masses, in the presence of an ALP-gluonic coupling

6 Signals sensitive to the ALP-top coupling have been briefly dis-cussed in Ref. [24] in the context of the non-linear operatorA2D(h). Nevertheless, the bound obtained followed from aZhinteractions contained in A2D(h), which do no apply for OaΦ

(i.e. in the linear EFT expansion at NLO).

7

gagg. They have the advantage of being equally sensi-tive to light ALPs with masses up to the kinematical en-ergy scale of the LHC analyses considered∼ O(100) GeV.Possible extensions of the analysis to other final statessuch as Zγ, Zh and fermionic final states (tt) have beendiscussed as well, altogether highlighting the power ofnon-resonant searches for ALPs at colliders.

Acknowledgements

We thank Gonzalo Alonso-Alvarez, Jesus Bonilla,Pablo Quilez and Stefan Pokorsky for useful discussionsand comments on the manuscript. The work of V.S. isfunded by the Science Technology and Facilities Coun-cil (STFC) under grant number ST/P000819/1. M.B.G.and J.F.T. acknowledge support from the “Spanish Agen-cia Estatal de Investigacion” (AEI) and the EU “FondoEuropeo de Desarrollo Regional” (FEDER) through theprojects FPA2016-78645-P and FPA2017-84260-C3-2-R,respectively. The work of J.M.N. was supported bythe Programa Atraccion de Talento de la Comunidadde Madrid via grant 2017-T1/TIC-5202. M.B.G. andJ.M.N. acknowledge support from the European Union’sHorizon 2020 research and innovation programme un-der the Marie Sklodowska-Curie grant agreements 690575(RISE InvisiblesPlus) and 674896 (ITN ELUSIVES), aswell as from the Spanish Research Agency (Agencia Es-tatal de Investigacion) through the grant IFT Centro deExcelencia Severo Ochoa SEV-2016-0597. In addition,they warmly thank Venus Keus for hospitality at theUniversity of Helsinki during the very last stages of thiswork.

Appendix A: Details on LHC analyses

1. CMS ZZ [62]. The analysis requires the leading (sub-leading) lepton from the event to have pT > 40 (30) GeVand |η| < 2.1 (2.4), and the invariant mass of the di-lepton pair is required to fall within 70 GeV < m`` < 110GeV and have p``T > 200 GeV. In addition, the “low-mass merged” category contains an anti-kT jet with alarge radius R = 0.8 (AK8) and pJT > 200 GeV. Themerged jet mass is required to be in the range 65 GeV< mJ < 105 GeV. The analysis further makes use ofthe information on the subjettiness variables τ1 and τ2for the reconstructed AK8 jet to build τ21 = τ2/τ1 [80].The τ21 distribution for the signal (from Delphes 3), SMbackgrounds and experimental data (obtained from [62])which pass the above selection criteria is shown in Fig. 5for the combination of untagged and b-tagged selectioncategories, and the CMS event selection then requiresτ21 < 0.4 for the AK8 merged jet. Overall, the CMSanalysis translates into an average ALP signal selectionefficiency of ∼ 8%.

2. CMS γγ [75]. The analysis requires two photons withpT > 75 GeV each, an invariant mass mγγ > 500 GeV

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

τ21

0

100

200

300

400

500

600

700

800

900

1000

Eve

nts

/0.0

5

CMS-B2G-17-013 ALP signal (ci = 1, fa = 2 TeV)

Z+ jetst t , tW , WWZV (V = Z , W )Data

FIG. 5. τ21 differential distribution for the ALP ZZ signalobtained with Delphes 3 (dashed black line), together withthe main SM backgrounds and experimental data from [62],after CMS event selection (without the τ21 < 0.4 cut).

and a distance ∆Rγγ > 0.45. One of the photons has tobe detected in the electromagnetic calorimeter (ECAL)barrel (EB) region, corresponding to |η| < 1.44. Theother photon can either be detected in the EB region orin the ECAL endcap (EE) region, 1.57 < |η| < 2.5, re-spectively defining two distinct analysis regions (labelledEBEB and EBEE) for the search. The CMS reconstruc-tion efficiency for EB (EE) photons in the signal regionis approximately 0.90 (0.87) [75]. We find the CMS anal-ysis yields an average ALP signal selection efficiency of∼ 72% for a signal sample with mγγ > 500 GeV. Thedi-photon invariant mass mγγ distribution after event se-lection for the ALP signal (with cG = cW = cB = 1 andfa = 5 TeV) and SM backgrounds is shown in Fig. 6 forthe EBEB (top) and EBEE (bottom) categories. Com-bining both EBEB and EBEE categories we obtain anobserved 95% C.L. exclusion limit of fa > 14.2 TeV forci = 1, quoted in Sect. 4-2.

3. CMS di-jet [78]. The analysis selects two R = 0.4anti-kT (AK4) jets with |ηj | < 2.5, pjT > 200 GeV, mjj >2.4 TeV and also imposes angular cuts on the di-jet sys-tem, yboost = (yj1 + yj2)/2 < 1.11 (where yj1,2 denote the

rapidities of the two jets) and χjj = exp(|yj1 − yj2|) < 16.The analysis is restricted in this work to the first invari-ant mass bin considered in Ref. [78], mjj ∈ [2.4, 3.0] TeV.After event selection, the CMS analysis provides the nor-malized χjj distribution for the experimental data andfor the SM background prediction. We computed the nor-malized χjj signal distribution after CMS events selectionusing MadGraph aMC@NLO, Pythia 8 and FastJet [81],including both the pure ALP signal, scaling as (cG/fa)4,and the interference with the SM background, scalingas (cG/fa)2. The size of the pure ALP signal and that

8

600 800 1000 1200 1400 1600 1800 2000 2200 2400

mγγ (GeV)

10−1

100

101

102

103

Eve

nts

/100

GeV

CMS-EXO-17-017

Barrel-Barrel (EBEB)ALP signal (ci = 1, fa = 5 TeV)

γγ

γ+ jetsData

600 800 1000 1200 1400 1600 1800 2000 2200 2400

mγγ (GeV)

10−1

100

101

102

103

Eve

nts

/100

GeV

CMS-EXO-17-017

Barrel-Endcap (EBEE)ALP signal (ci = 1, fa = 5 TeV)

γγ

γ+ jetsData

FIG. 6. mγγ distributions for the ALP γγ signal with ci = 1,fa = 5 TeV (dashed black line) and SM background from γγ(light blue) and γ+ jets (dark blue) after CMS event selec-tion, in the EBEB (top) and EBEE (bottom) categories. Theexperimental data are shown as black dots.

of the (negative) interference with the QCD backgroundare approximately equal for fa/cG ∼ 1.5 TeV, and theinterference dominates for larger values of fa (those rel-evant for our analysis, since we look at invariant massesmjj ∈ [2.4, 3.0] TeV). A χ2 fit to the data was performedusing a linear combination of the normalized QCD back-ground and ALP signal, with relative weights (1−q) andq respectively. Our procedure results in a 95% C.L. ex-cluded region of 1.9 TeV > fa > 3 TeV for cG = 1 (corre-sponding to an excluded ALP signal weight q < −0.016).Fig. 7 shows the normalized χjj distribution in the invari-ant mass bin mjj ∈ [2.4, 3.0] TeV, after the CMS eventselection for the QCD background (with its theoreticaluncertainty), as well as a normalized combination of theALP signal for fa/cG = 3 TeV and the QCD background.

2 4 6 8 10 12 14 16χjj

0.05

0.06

0.07

0.08

0.09

0.10

1 N×

dN dχ

jj

CMS-EXO-16-046 mjj ∈ [2.4 TeV, 3.0 TeV]

ALP Signal (cG = 1, fa = 3 TeV) + SM Background

SM Background (NLO QCD+EW)SM Background Theoretical UncertaintyData

FIG. 7. Di-jet differential distribution as a function of theangular variable χjj in the bin with di-jet invariant masses 2.4TeV - 3.0 TeV. The dotted black line corresponds to the QCDSM background (with its theoretical uncertainty shown as agrey band). The dashed red line corresponds to a normalizedcombination of the ALP signal for fa/cG = 3 TeV and theQCD background. The experimental data from the CMS di-jet analysis [78] are shown in black. The SM background andexperimental data have been taken from HEPDATA [82].

∗ belen.gavela@uam.es† josemiguel.no@uam.es‡ v.sanz@sussex.ac.uk§ jorge.troconiz@uam.es

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