Digamma, What Next?

Home , Digamma, Pi (letter), Sigma, Stigma (letter), Tau

CERN-TH-2016-090

Digamma, what next?

Roberto Franceschinia, Gian F. Giudicea, Jernej F. Kamenika,b,c, Matthew McCullougha, Francesco Rivaa, Alessandro Strumiaa,d, Riccardo Torree

a CERN, Theoretical Physics Department, Geneva, Switzerland b JoˇzefStefan Institute, Jamova 39, 1000 Ljubljana, Slovenia c Faculty of Mathematics and Physics, University of Ljubljana, Slovenia d Dipartimento di Fisica dell’Universitàdi Pisa and INFN, Italy e Institut de Théoriedes PhénomènesPhysiques, EPFL, Lausanne, Switzerland

Abstract If the 750 GeV resonance in the diphoton channel is confirmed, what are the measurements necessary to infer the properties of the new particle and understand its nature? We address this question in the framework of a single new scalar particle, called digamma (z). We describe it by an effective field theory, which allows us to obtain general and model-independent results, and to identify the most useful observables, whose relevance will remain also in model-by-model analyses. We derive full expressions for the leading-order processes and compute rates for higher-order decays, digamma production in association with jets, gauge or Higgs bosons, and digamma pair production. We illustrate how measurements of these higher-order processes can be used to extract couplings, quantum numbers, and properties of the new particle. arXiv:1604.06446v2 [hep-ph] 11 May 2016 Contents

1 Introduction3

2 pp z: single production4 2.1→ Experimental status ...... 4 2.2 Theoretical framework ...... 5

3 z decays 11 3.1 Two-body z decays ...... 11 3.2 Three-body z decays ...... 13 3.3 Four-body z decays ...... 15

4 pp zj, zjj: associated production with jets 16 → 4.1 CP of z from pp zjj ...... 17 → 5 pp zV, zh: EW associated production 21 → 5.1 CP of z from pp zZ, zW ...... 25 5.2 EFT expansion and→ associated production ...... 26

6 pp zz: pair production 28 6.1→ Eﬀective theory parametrisation ...... 28 6.2 Model computation in Low Energy Theorem approximation ...... 29 6.3 Full computation beyond the LET approximation ...... 35 6.4 Pair production of a pseudo-scalar resonance ...... 36 6.5 Decorrelating single and pair production ...... 36 6.6 Resonant pair production ...... 38 6.7 Pair production phenomenology ...... 38

7 Summary 40 7.1 Identifying the weak representation ...... 40 7.2 Identifying the initial state ...... 41 7.3 Measuring z couplings ...... 42 7.4 Identifying the CP parity ...... 42 7.5 Pair production ...... 43

A Eﬀective Lagrangian in the unitary gauge 44

B z decay widths including the mixing with the Higgs 45

2 1 Introduction

Preliminary LHC data at √s = 13 TeV show a hint for a new resonance in pp γγ (thereby → denoted by the letter1 digamma, z) at invariant mass of 750 GeV [1], which stimulated intense experimental and theoretical interest. On the experimental side, dedicated analyses strengthen the statistical significance of the excess [2]. New measurements, which are underway, will tell us whether the excess is real and, if so, a thorough exploration of the new particle’s properties will start. Superficially, the situation looks similar to the discovery of the Higgs boson h, which first emerged as a peak in γγ at 125 GeV. Various computations and considerations can be readapted today from the Higgs case. However, h has large couplings to SM massive vectors, unlike z. Furthermore, in the Higgs case, the Standard Model (SM) predicted everything but the Higgs mass. Theorists made precision computations, and experimentalists made optimised measurements of Higgs properties such as its spin and parity, which did not lead to any surprise. Today, with the digamma, we are swimming in deep water. Many key issues related to the new resonance remain obscure. Does it have spin 0, 2, or more? Is it narrow or broad? Or, more generally, how large are its couplings? To which particles can it decay? Do its couplings violate CP? If not, is it CP-even or CP-odd? Is it a weak singlet or a weak doublet or something else? Is it produced through gg, qq¯ or weak vector collisions? Is it elementary or composite? Is it a cousin of the Higgs boson? Is it related to the mechanism of electroweak breaking or to the naturalness problem? What is its role in the world of particle physics? Who ordered that? Answering each one of these questions could point to different theoretical directions, which at the moment look equally (im)plausible. The observation of γγ could be only the tip of an iceberg. Here we take a purely phenomenological approach. The goal of this paper is discussing and reviewing how appropriate measurements could address some of these questions. So many possibilities are open that, not to get lost in a plethora of alternatives, we will focus

on the simplest ‘everybody’s model’. The model involves a new scalar z with Mz 750 GeV and eﬀective interactions to photons and other SM states.2 ≈ The paper is structured as follows. In section2 and appendixB we describe and ﬁt the experimental data and present the theoretical framework that can account for pp z γγ. →2 →2 In section3 we provide full expressions (including terms suppressed by powers of v /Mz) for z decays into SM vectors and we study multi-body z decays. In section4 we discuss z production together with one or more jets. In section5 we discuss production of z together

1Digamma (z) is a letter of the archaic Greek alphabet, originating from the Phoenician letter waw. The digamma was present in Linear B Mycenean Greek and Æolic Greek, but later disappeared from classical Greek probably before the 7th century BC. However, it remained in use as a symbol for the number 6, because it occupied the sixth place in the archaic Greek alphabet and because it is made of two gammas, the third letter of the Greek alphabet. As a numeral it was also called episemon during Byzantine times and stigma (as a ligature of the letters sigma and tau) since the Middle Ages. In our context, the reference to the number six is ﬁtting, as the mass of the digamma particle is 6, in units of the Higgs mass. Moreover, the historical precedent of the disappearance of the letter z is a reminder that caution is necessary in interpretations of the particle z. 2The following list of references consider this model and its collider phenomenology [3–75]. For more model- oriented studies focussing on other phenomenological aspects, see [76–214]. Studies focussing explicitly on a pseudo-scalar version include [215–242]. For alternative interpretations of the excess, see [243–270].

3 z couples to z }| { √s = 13 TeV eq. b¯b cc¯ ss¯ uu¯ dd¯ GG

σzj/σz (20a) 9.2% 7.6% 6.8% 6.7% 6.2% 27.% σzb/σz (20b) 6.2% 0 0 0 0 0.32% σzjj/σz (20c) 1.4% 1.0% 0.95% 1.2% 1.0% 4.7% σzjb/σz (20d) 1.2% 0.18% 0.19% 0.34% 0.31% 0.096% σzbb/σz (20e) 0.31% 0.17% 0.18% 0.34% 0.31% 0.024% −6 σzγ/σz (28b) 0.37% 1.5% 0.38% 1.6% 0.41% 10 −6 σzZ /σz (28b) 1.1% 1.1% 1.3% 2.0% 1.9% 3 10 −5 −6 σzW + /σz (28c) 5 10 1.7% 2.4% 2.6% 4.1% 10 −5 −6 σzW − /σz (28d) 3 10 2.3% 1.2% 1.0% 1.7% 10 −6 σzh/σz (28e) 1.0% 1.1% 1.2% 1.9% 1.8% 1 10

Table 1: Predictions for the associated production of the resonance z, assuming that it couples to diﬀerent SM particles, as more precisely described by the the eﬀective Lagrangian of eq. (5).

For production in association with jets we assume cuts η < 5 on all rapidities, pT > 150 GeV on all transverse momenta, and angular diﬀerence ∆R > 0.4 for all jet pairs, while for photon- associated production we impose ηγ < 2.5 and pT,γ > 10 GeV.

with EW vectors or the Higgs boson. Table1 summarises the predictions for these cross sections. In section6 we discuss pair production of z. Finally, in section7 we summarise how the above processes can be used to gather information on the main unknown properties of z, such as its couplings, CP-parity, production mode(s), and quantum numbers.

2 pp z: single production → 2.1 Experimental status We briefly summarise the experimental status, updating the results of [3] in light of the new pp z γγ results presented at the Moriond 2016 conference [2], which increase the statis- → → tical significance of the excess around mγγ 750 GeV (up to 3.9σ in ATLAS and 3.4σ in CMS, locally) but do not qualitatively change the≈ main implications. The LHC collaborations presented different analyses: we focus on the one dedicated to spin 0 searches (spin 2 searches give similar results). In fig.1 we fit the energy spectra, extracting the favoured values of the mass of the resonance, of its width and of the number of excess events. ATLAS and CMS data at √s = 13 TeV are consistent among themselves. In data at √s = 8 TeV the hint of an excess is too weak to extract useful information. The best fit values for the excess cross section depend on both the mass and the width of the resonance, which, within statistical uncertainties, can be anything between 0 to 100 GeV. The main lesson is that it is too early to extract detailed properties from these preliminary data. We will use the reference values listed in table2, considering the two sample cases of a narrow

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Figure 1: Fit of energy spectra obtained in spin 0 analyses. In the left plot we show the best-

ﬁt regions in the (mass, width) plane. In the right plot we ﬁx Mz = 750 GeV and show the favoured values of the width and of the excess number of events.

resonance (Γ 10 GeV, which is the experimental resolution on mγγ) and a broad resonance with Γ 45 GeV. Table2 also summarises the bound on other possible decay channels of the ≈ z resonance. In fig.2 we show the results of a global fit of signal rates and bounds for √s = 8 and 13 TeV data assuming that the 750 GeV excess is due to a new resonance z that decays into: 1) hypercharge vectors; 2) gluons; 3) a third channel which could be tt¯, b¯b, cc¯, uu¯, or invisible particles (such as Dark Matter or neutrinos). In the left (right) panel we assume a broad (narrow) resonance. A message that can be indirectly read from fig.2 is that production from gluons or from heavy quarks remains mildly favoured with respect to production from photons [69–75] or light quarks, which predict a too small 13 TeV/8 TeV cross section ratio.

2.2 Theoretical framework

The cross section for single production of a scalar z, σ = σ(pp z), can be written in the z → narrow-width approximation in terms of its decay widths into partons ℘,Γ℘ = Γ(z ℘)[3]: → 1 X Γ℘ σ(pp z) = C℘ . (1) s M → ℘ z

5 σ(pp γγ) √s = 8 TeV √s = 13 TeV → narrow broad narrow broad CMS 0.63 0.31 fb 0.99 1.05 fb 4.8 2.1 fb 7.7 4.8 fb ± ± ± ± ATLAS 0.21 0.22 fb 0.88 0.46 fb 5.5 1.5 fb 7.6 1.9 fb ± ± ± ± final σ at √s = 8 TeV σ at √s = 13 TeV state f observed expected ref. observed expected ref. e+e−, µ+µ− < 1.2 fb < 1.2 fb [3] < 5 fb < 5 fb [271] τ +τ − < 12 fb < 15 fb [3] < 60 fb < 67 fb [272] Zγ < 11 fb < 11 fb [3] < 28 fb < 40 fb [273] ZZ < 12 fb < 20 fb [3] < 200 fb < 220 fb [274] Zh < 19 fb < 28 fb [3] < 116 fb < 116 fb [275] hh < 39 fb < 42 fb [3] < 120 fb < 110 fb [276] W +W − < 40 fb < 70 fb [3] < 300 fb < 300 fb [277] tt¯ < 450 fb < 600 fb [3] invisible < 0.8 pb - [3] 2.2 pb 1.8 pb [278] b¯b < 1 pb < 1 pb [3] jj <∼2.5 pb∼ - [3] ∼ Table 2: Upper box: signal rates. Lower box: bounds at 95% confidence level on pp cross sections for various final states produced through a resonance with M = 750 GeV and Γ/M 0.06. z z ≈

Here we extend the list of parton luminosity factors C℘ given in [3] by including massive SM vectors, which can be either T ransverse or Longitudinal3

√s Cb¯b Ccc¯ Css¯ Cdd¯ Cuu¯ Cgg Cγγ CZLZL CZT ZT CZT γ CWLWL CWT WT 8 TeV 1.07 2.7 7.2 89 158 174 11(8) 0.01 0.3 3.1 0.03 0.8 13 TeV 15.3 36 83 627 1054 2137 54(64) 0.14 2.8 27 0.4 8

The gauge boson parton luminosity functions in the table are obtained convoluting the WL,T , ZL,T , and photon leading order splitting functions with the quark pdfs (“NNPDF30_lo_as_0118” set [279]), evaluated at factorisation scale µW = MW , µZ = MZ and µγ = 10 GeV. The two numbers for the Cγγ correspond to the photon luminosities obtained using the photon pdfs in the “NNPDF30_lo_as_0118” set (outside parentheses) and the number obtained with the aforementioned procedure (inside parentheses). These numbers come with a signiﬁcant uncertainty, due to the sensitivity on the aforementioned choice of renormalisation scale. We have checked that they are able to reproduce, within a factor of two, the relevant processes computed with MadGraph5 [280]. We consider this precision suﬃcient for our study, but we stress that going to higher order splitting functions for the gauge bosons can make this error smaller, which may be needed in the future. From the Table above we see that the C-factors for longitudinal vector bosons are highly suppressed. Longitudinal vector boson fusion (VBF)

3We omit mixed LT contributions since they are suppressed by an additional power of M 2 /M 2 , see W,Z z eq. (B.10).

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Figure 2: Global ﬁt of √s = 8, 13 TeV data for the 750 GeV excess assuming that it is due to a new resonance z that decays into 1) hypercharge vectors; 2) into gluons; 3) into a third channel considering those listed in the legend. In the left (right) panel we assume a broad (narrow) resonance.

can never become relevant compared to photon-fusion, and can therefore be neglected. The situation is diﬀerent for the transverse VBF, which can give a sizeable contribution to the total production.

From eq. (1) we obtain, at √s = 13 TeV

h Γgg Γuu¯ Γdd¯ Γss¯ Γcc¯ Γb¯b σ(pp z) = 4900 + 2400 + 1400 + 190 + 83 + 35 + (2) → Mz Mz Mz Mz Mz Mz Γ Γ Γ Γ Γ Γ i +150 γγ + 62 Zγ + 18 WT WT + 0.92 WLWL + 6.5 ZT ZT + 0.32 ZLZL pb . Mz Mz Mz Mz Mz Mz We do not consider production from a loop of t quarks because it cannot reproduce the diphoton excess without predicting, at the same time, a Γ(F tt¯) above the bound in table2. Assuming → that z decay to a single parton channel saturates the z decay width at Γ/Mz 0.06 implies −3 ' BR(z γγ) 0.018, 0.70, 1.6, 3.8 10 for ℘ = gg, ss,¯ cc,¯ ¯bb in order to reproduce the → ≈ { } × observed σ(pp z γγ). → → In eq. (2) we omitted QCD K-factors describing higher order corrections, since they are not known for all channels. In the case of the gluons and quarks contributions they are given at

NLO by Kgg 1.5 and Kqq¯ 1.2 (see for instance [3, 94]). In the rest of the paper we will ' ' systematically avoid including any K factor, since they are not known for the majority of the processes we consider.

7 Eﬀective Lagrangian up to dimension 5

While the above framework captures the physics of the simplest pp z process, a more → systematic parametrisation is needed to describe z production in association with other SM particles. This can be done with an Eﬀective Field Theory (EFT) approach4, which also provides an ideal language to match to explicit microscopic models. We assume that the underlying theory is broadly characterised by a mass scale Λ and that the light degrees of

freedom are the SM ﬁelds and z, so that Mz Λ. The renormalisable interactions of z are encoded in the Lagrangian

2 (∂µz) L4 = LSM + V (z,H) , (3) 2 − 5 where LSM is the SM part, while the scalar potential can be written as

2 mz 2 3 4 2 2 2 2 2 V (z,H) = z + κ m z + λ z + κ H m z( H v ) + λ H z ( H v ) , (4) 2 z z z z z | | − z | | −

with generic couplings κz,zH and λz,zH . A possible tadpole term in eq. (4) can be eliminated with a shift of z, while we have absorbed an EWSB contribution from λzH to the z mass into a redeﬁnition of eq. (3). The mass eigenstate Mz is slightly diﬀerent from the mass parameter mz, as discussed in appendixB.

For Λ in the TeV range, the leading non-renormalisable interactions between z and the SM are phenomenologically important, as we will discuss later. In full generality, the dimension-5 eﬀective Lagrangian can be written as6

2 2 2 even z g3 a a µν g2 a a µν g1 µν ¯ L = cgg G G + cWW W W + cBB BµνB + cψ HψLψR + h.c. 5 Λ 2 µν 2 µν 2 2 2 0 4 4 cz3 z(∂µz) +cH DµH c ( H v ) + , (5) | | − H | | − Λ 2

for CP-even z. In the CP-odd case, we find 2 2 2 odd z g3 a ã µν g2 a ˜ a µν g1 ˜µν ¯ L = c˜gg G G +c ˜WW W W +c ˜BB BµνB +c ˜ψ iHψLψR + h.c. , (6) 5 Λ 2 µν 2 µν 2 ˜ while both structures can co-exist if CP is explicitly broken by z interactions. Here Xµν = 1 (5) µναβXαβ. The real coefficients ci c involve different powers of couplings in the underlying 2 ≡ i 4See also [17] for an alternative parametrisation of resonant di-photon phenomenology. 5 For κ = κ H = 0 the Lagrangian acquires a Z2 symmetry z z that might or might not be identified z z → − with CP, depending on the higher order interactions of z. 6It is interesting to note that the anomalous dimensions of the operators in eq. (5) exhibit a peculiar structure ¯ with several vanishing entries [281–283]. In particular, the zVV structure only renormalises the zHψLψR and 4 ¯ 4 z H operators, while the zHψLψR operators only induce z H . This implies that, for instance, if some | | | | selection rule forbids the cH structure in the UV, then Renormalisation Group Effects, from the scale Λ to the energy at which these interactions are used, will not generate it.

8 theory and, for most of our discussion, can be taken arbitrary. Field redefinitions of the form 2 2 ψ ψ(1 + c1z/Λ), H H(1 + c2z/Λ) and z (z + c3z /Λ + c4 H /Λ) leave the leading → → → | | Lagrangian eq. (3) unaltered and the freedom of the coefficients c1−4 can be used to eliminate four combinations of higher-dimensional operators proportional to the (leading) equations of motion (see [284] for a discussion in this context). Using these redefinitions we have eliminated from eq. (5) the structures izψ¯Dψ/ + h.c., z3 H 2, zH†D2H + h.c. and z5. An equivalent | | choice, adopted in [284], is to eliminate from L5 all operators involving derivatives. For our purposes, our choice is preferable because it allows for a more direct matching of the operators in eq. (5) to explicit models [3]. With this notation, eq. (2) for the 13 TeV single z production takes the form 2 TeV 2 2 2 2 2 2 σ(pp z) = 613cgg + 7.6cu + 4.7cd + 0.44cs + 0.30cc + 0.13cb + → Λ2 2 2 2 +0.01cH + 0.02cBB + 0.007cBBcWW + 0.13cWW pb, (7) where the only interference between the contributions of different operators concerns cBB and cWW . CP-odd terms proportional toc ˜i contribute the same amount to the cross section as their CP even counterparts ci. An important observation is that the couplings cψ have a non-trivial flavour structure [6,36], and can be regarded as spurions transforming as (3, 3)¯ under the SU(3)L SU(3)R flavour ⊗ symmetry. If the matrices cψ are not aligned with the Yukawa couplings, F mediates flavour- changing neutral currents via four-fermion interactions given by

2 v ¯i j ∗ ¯i j 2 2 2 cψijψLψR + cψjiψRψL . (8) Λ Mz

In table3 we list the most stringent bounds on off-diagonal elements of the couplings cψ, evaluated in the quark mass eigenbasis. We see that off-diagonal elements must be smaller than 10−(3÷4)(Λ/ TeV), while at least one diagonal element must be of order unity to obtain a sizeable z production cross section, as can be derived from eq. (7). Since this seems to correspond to a fine tuning of parameters, we conclude that z production from quark initial states is not compatible with a generic flavour structure. There are ways to circumvent the problem. One way is to embed z in a weak doublet which gives mass to down-type quarks only, while the EW vev resides primarily in the SM-like Higgs doublet, which gives mass to up-type quarks. Different solutions, more relevant in our context, can be found for a singlet z. This can be done [6, 36] with appropriate flavour symmetries, alignment mechanisms, or by imposing a condition of minimal flavour violation (MFV) [285], which implies that cψ is a matrix proportional to the corresponding SM Yukawa couplings. Consider first the case of only cuHq¯LuR with cu proportional to the up-type Yukawa matrix, where the z production is dominated by the light quarks. In this case the coupling to top quarks is large, leading to an unacceptable decay width in z tt¯ . More interesting is the case → of couplings to down-type quarks, cdHq¯LdR. The Yukawa structure implies that the dominant z coupling is to bottom quarks, while flavour violations are kept under control either by an

9 Observable Bound p 2 ∗2 ∗ −3 ∆mK Re (csd + cds 8.9csdcds) < 1.1 10 (Λ/TeV) p| 2 ∗2 − ∗ | × −5 K Im (csd + cds 8.9csdcds) < 2.8 10 (Λ/TeV) p| 2 ∗2 − ∗ | × −3 ∆mD Re (ccu + cuc 7.0ccucuc) < 2.7 10 (Λ/TeV) p| 2 ∗2 − ∗ | × −4 q/p , φD Im (ccu + cuc 7.0ccucuc) < 3.2 10 (Λ/TeV) | | p| 2 ∗2 − ∗ | × −3 ∆mBd Re (cbd + cdb 6.3cbdcdb) < 3.3 10 (Λ/TeV) p| 2 ∗2 − ∗ | × −3 SψKs Im (cbd + cdb 6.3cbdcdb) < 1.8 10 (Λ/TeV) p | 2 ∗2− ∗ | × −2 ∆mB Abs (c + c 6.1cbsc ) < 1.4 10 (Λ/TeV) s | bs sb − sb | ×

Table 3: Bounds on off-diagonal elements of the coefficients cψ, defined in eq. (5), computed in the quark mass eigenbasis. approximate MFV or by a flavour symmetry of the underlying theory. So, while z production fromcc ¯ ,ss ¯ , or light quarks can be obtained with special flavour structures, the case of ¯bb can be more easily justified under the MFV assumption or with the implementation of an appropriate flavour symmetry.

Eﬀective Lagrangian: dimension 6

It is instructive to extend our analysis of interactions between z and the SM to the next order in the 1/Λ expansion: at dimension-6 the first contact contributions to z pair production appear. The SM field content is such that no dimension 5 operators exist, with the exception 2 of the lepton number breaking Weinberg operator (LH) /ΛL, which we will assume to be associated with a much larger scale ΛL Λ and can be ignored for our present purposes. Under this assumption, there are no dimension-6 operators linear in z. This means that the 2 2 single z production computed from eq.s (5), (6) receives corrections only at (Mz/Λ ) and µ O 2 µ not (Mz/Λ). Moreover, structures of the form z∂µzJSM are proportional to z ∂µJSM, up to aO total derivative, and can be eliminated using arguments analogous to those employed for eq. (5). We thus find that the most general dimension-6 effective Lagrangian is

2 2 2 2 z (6) g3 a a µν (6) g2 a a µν (6) g1 µν (6) ¯ L6 = c G G + c W W + c BµνB + c HψLψR + h.c. Λ2 gg 2 µν WW 2 µν BB 2 ψ (6) 2 (6) 2 (6)0 4 4 cH2 (∂µz) 2 2 4 +cH DµH cH ( H v ) + H v + (z ) , (9) | | − | | − Λ2 2 | | − O where we ignore terms with at least z4 that do not have any phenomenological impact in our (6) analysis. With the exception of the last term cH2, the terms in eq. (9) share the structure of the dimension-5 Lagrangian. Note that in this case the CP-even and CP-odd states have the same interactions, since z appears quadratically while CP violation could generate interactions 2 ˜ µν a a (6) of the form z VµνV (V = B,G ,W ) and a complex phase for cψ . Whether or not eq.s (5) and (9) provide an adequate description of the processes under study, and whether higher-order terms in the eﬀective Lagrangian can potentially play a role in the study of speciﬁc processes, depends on a number of assumptions about the underlying

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Figure 3: Predictions for ΓZZ /Γγγ as a function of ΓγZ /Γγγ and ΓWW /Γγγ in the general eﬀective ﬁeld theory up to dimension 7 operators for a CP-even scalar. Note that the prediction does not depend on assumptions about the mixing with the Higgs boson. Two sets of predictions (left and right) are possible due to a sign ambiguity relating couplings to widths. The shaded

regions and the region above the dashed line are excluded. If ΓγZ /Γγγ and ΓWW /Γγγ were measured in the future, the prediction for ΓZZ /Γγγ could be used to unambiguously test the eﬀective theory description.

dynamics. The validity of the EFT cannot be determined entirely from a bottom-up perspective. We will comment on this issues in the appropriate sections below.

3 z decays The eﬀective Lagrangian expanded in the unitary gauge can be found in appendixA.

3.1 Two-body z decays If z is CP-even, it can mix with the Higgs boson h. The mixing angle is given in eq. (B.7) of appendixB (see also [94]), and the mass eigenvalues in eq. (B.8). Equations (B.10) provide the z two-body widths ΓX Γ(z X) taking into account the full dependence on Mh,t,W,Z . We ≡ → 2 2 ignore higher order operators that give corrections suppressed by Mz/Λ . The mixing angle θ is experimentally constrained to be small, given that after mixing with h, z acquires the decay ∗ widths of a Higgs boson h with mass Mz: 3 M M ∗ 2 ∗ h z M Γ(z X) = Γ(h X) sin θ + , e.g. Γ(h ZZ) z (10) → → ··· → ' 32πv2 where v = 246 GeV is the Higgs vacuum expectation value. Imposing the experimental bound Γ(z ZZ) < 20 Γ(z γγ) we obtain → → ∼ s Γ( γγ) < z sin θ 0.015 −→6 (experimental bound on the z/h mixing angle). (11) | | ∼ 10 Mz

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Figure 4: Predictions for ΓZZ /Γγγ and ΓWW /Γγγ as a function of ΓγZ /Γγγ in the general effective field theory up to dimension 7 operators for a CP-odd scalar. In each case the two solutions correspond to a sign ambiguity relating couplings to widths. The shaded regions are excluded. If ΓγZ /Γγγ were measured in the future, these predictions for ΓZZ /Γγγ and ΓWW /Γγγ could be used to test the effective theory description.

Using the complete expressions for the widths of appendixB we see that in the CP-even case the four decay widths Γγγ,ΓγZ ,ΓZZ ,ΓW +W − , are controlled by only three parameter combinations involving cBB, cWW , and vcH cos θ + 2Λ sin θ. This means that once the rates relative to diphoton production ΓγZ /Γγγ and ΓW +W − /Γγγ have been measured, the ratio ΓZZ /Γγγ is predicted, up to a sign ambiguity in the relation between the operator coefficients and the diphoton width. This means that, without making any assumptions on the size of the mixing with the Higgs and up to operators of dimension 7, we obtain one prediction that can be tested in future measurements. This is illustrated in fig.3 (see also [18]). The z decays into EW vectors for a CP-odd scalar are described by only two parameters. Thus, once the relative rate ΓγZ /Γγγ is measured, both ΓW +W − /Γγγ and ΓZZ /Γγγ are predicted, again up to a sign ambiguity. These two predictions can be tested, allowing for the determination of the z properties in a very model independent manner [18, 45]. This is illustrated in fig.4. Finally, we provide compact expressions for the widths by expanding the full expressions of eq. (B.10) for θ 1 and Mh,W,Z Mz (correct up to 10% approximation): the widths reduce to the expressions of [3]: ∼

2 3 πα Mz 2 2 Γ(z γγ) = (cγγ +c ˜γγ) , (12a) → Λ2 2 3 8πα3Mz 2 2 Γ(z gg) = (cgg +c ˜gg) , (12b) → Λ2

12 2 ¯ NψMzv 2 2 Γ(z ψψ) = cψ +c ˜ψ , (12c) → 16πΛ2 z z 3 Mz 2 Γ(z hh) = cˆH , (12d) → 128πΛ2 2 3 3 πα Mz 2 2 Mz 2 Γ(z ZZ) = 2 4 4 cZZ +c ˜ZZ + 2 cˆH , (12e) → Λ sWcW 128πΛ 2 3 3 + − 2πα Mz 2 2 Mz 2 Γ(z W W ) = 2 4 cWW +c ˜WW + 2 cˆH , (12 f ) → Λ sW 64πΛ 2 3 2πα Mz 2 2 Γ(z γZ) = 2 2 2 cγZ +c ˜γZ . (12g) → sWcWΛ

Here sW and cW are sine and cosine of the weak mixing angle and Nψ is the ψ multiplicity (e.g. Nψ = 3 for an SU(2)L singlet quark). We have defined 2 2 cγγ = cBB + cWW , cγZ = sWcBB cWcWW , 4 4 − cZZ = sWcBB + cWcWW , cˆH = cH + 2κzH Λ/Mz . (13) 2 2 In the Mh,W,Z M limit, κ H m z H and cH z DH /Λ are the only two operators that z z z | | | | contribute to the z decays into Higgs or longitudinal vector bosons, and appear only in the combinationc ˆH . This is because, after the field redefinitions discussed below eq. (5), combina-

tions of cH and κzH orthogonal toc ˆH can be eliminated in favour of other operators in eqs. (3), (5) and a combination of z5 and z3 H 2, which do not contribute to 2-body z decays. Keeping | | instead terms suppressed by Mh,W,Z /Mz, more operators contribute to the decay widths. In the left panel of ﬁg.5 we show the allowed values of ( cWW , cˆH )/cBB (white region) together with the various bounds. In the right panel of ﬁg.5 we show the relative contributions of the

γγ, γZT , ZT ZT and WT WT channels to VBF production cross section as a function cWW /cBB: in the allowed range photon fusion is the dominant VBF production mechanism only in the

neighbourhood of cWW 0, while the other channels become relevant, or even dominant for ∼ cWW cBB . This shows that in the eﬀective theory describing the interactions of a scalar | | ∼ | | singlet in an SU(2)L U(1)Y invariant way, it is generally not possible to give a meaning to the photon-fusion production⊗ mechanism without considering also the other relevant VBF

channels, unless cWW cBB (see also related discussion in [18,231]). | | | |

3.2 Three-body z decays z decay modes into more than two particles carry information about z couplings. For example, they allow us to access vector polarisations, and to deduce in this way the structure of z interactions with gauge ﬁelds. However these processes, occurring at higher-order, have relatively small branching ratios. We focus on two classes of special enhanced processes.

First, the zHψψ¯ couplings lead to a two-body z ψψ¯ width suppressed by v/M , while → z the three-body z Hψψ¯ width is unsuppressed. In the limit v M we ﬁnd → z 3 2 ¯ 2 NψMzcψ Γ(z ηψψ) M Γ(z ηψψ¯ ) = i.e. → = z 0.98% (14) → 1536 π3Λ2 Γ(z ψψ¯ ) 96 π2v2 ≈ → 13 Γ(ϝ → �� γ�� )/Γ(ϝ → γγ) � �� =�

� ��-� ��

/ � � / σ �  �

�� γγ σ -� �� γ��

� �� -� -� � � �� /�� /��

Figure 5: Left: iso-contours of Γ(z X)/Γ(z γγ) for X = ZZ (red continuous → → curves), X = γZ (green dashed), WW (blue dashed), hh (black dot-dashed) as a function of (cWW , cˆH )/cBB. Shaded regions are excluded. Right: ratio of the production cross section by 0 Vector Boson Fusion (VBF) in the channels VV = γγ, γZT ,ZT ZT ,WT WT respectively, divided by the total VBF production cross section as a function of the ratio cWW /cBB for cH = 0.

where η is any of the 4 components of the Higgs doublet H, namely the Higgs boson h and the longitudinal polarisations of Z and W ±. Taking into account their masses and assuming that the fermion ψ is a quark with negligible mass we ﬁnd

Γ(z huu¯ ) Γ(z hdd¯ ) → = → = 0.62%, (15a) Γ(z uu¯ ) Γ(z dd¯ ) → → Γ(z Zuu¯ ) Γ(z Zdd¯ ) → = → = 0.57%, (15b) Γ(z uu¯ ) Γ(z dd¯ ) → → Γ(z W +ud¯ ) Γ(z W −du¯ ) → = → = 0.89% (15c) Γ(z qq¯ ) Γ(z qq¯ ) → → If z is produced from qq¯ partonic scattering, one expects a sizeable three body decay width as well as associated processes discussed in section5. Present data could already provide signiﬁcant bounds, if the relevant searches are performed.

The second enhanced higher-order decay rate arises because collinear and/or soft emission of particles with mass m is enhanced by infra-red logarithms lnn M /m, where n = 1, 2 when ∼ z a vectors splits into two vectors, and n = 1 when it splits into fermions or scalars. At leading

order in ln(Mz/m), such phenomenon can be approximated as radiation.

14 decay I1 I2 I3 I4 I5 I6 Γ4`/Γγγ κ1 eeee 84.3 84.4 169. 137. 137. 0.0556 3.63 10−4 0.235 × ∓ µµµµ 29.0 29.1 58.1 52.7 52.8 0.0556 1.36 10−4 0.216 × ∓ ττττ 9.45 9.51 19.0 19.6 19.6 0.0555 0.49 10−4 0.195 × ∓ eeµµ 45.4 45.5 90.9 78.8 78.8 0.0556 4.12 10−4 0.224 × ∓ eeττ 24.5 24.5 49.0 50.9 51.0 0.0555 2.56 10−4 0.194 × ∓ µµττ 16.6 16.6 33.2 32.2 32.2 0.0555 1.64 10−4 0.205 × ∓

Table 4: Coeﬃcients that deﬁne the z `+`−`0+`0− distributions. In the last column, the → negative (positive) sign of κ1 corresponds to the CP-even(odd) case.

The QCD eﬀects is hidden into hadronisation. Considering for example the zGG or zGG˜ couplings, we ﬁnd the total rates

σ(pp z ggg) → → = 11% (16) σ(pp z gg) → → having imposed the cuts on jets described in the caption of table1. After averaging on the gluon polarizations, the zGG or zGG˜ couplings produce the same z ggg distributions. → The most interesting such effect concerns off-shell photons γ∗ (see also [237]), while for massive electro-weak bosons the contribution to 4-fermion final states is anyway dominated by

the on-shell VV production (with ln(Mz/MW ) 2.2 off-shell effects account for approximately 20% of the on-shell production [286]). ≈ From an off-shell photon, we find

X − + Γ(z γ℘ ℘ ) 22% Γ(z γγ) (17) ℘ → ≈ × → where ℘ denote ﬁnal-state particle species and the sum is dominated by ℘ = W (5%, thanks to double IR logarithms), ℘ = u (4%) and ℘ = e (4%). Splitting into electrons and muons is particularly important, given their small mass and given that collider experiments can precisely measure their energy and direction.

3.3 Four-body z decays Four-body z decays are interesting because they allow to reconstruct the CP-parity of z. The largest of these decays is into gluons: we ﬁnd σ(pp z gggg) 0.3% σ(pp z → → ≈ × → → gg) after imposing the cuts on jets described in the caption of table1. The z g+g+g−g− → amplitude (where denotes the gluon helicity) depends on whether z is scalar or pseudo- ± scalar [287–289]. However, for kinematical reasons, pp zjj scatterings (section 4.1) allow us → to discriminate the CP parity much better than z jjjj decays [290]. → The kinematical distributions of pp z γ∗γ∗ `+`−`0+`0− decay allow us to measure → → → whether z is scalar or pseudo-scalar, in analogy with pion π0 physics [291,292]. Note that these techniques ﬁnd little prospects of realisation in the context of Higgs physics, due to the large

15 di-photon background; at 750 GeV, the situation is more favourable. In our case, the rate of z into 4 leptons is 2 Γ4` 2α S I1 2 I2 2 = 2 R,R = 2 2 + I4 cγγ + + I5 + I6 c˜γγ (18) Γγγ 3π cγγ +c ˜γγ 2 2 where the numerical factors Ii are reported in table4, and S is a symmetry factor equal to 1/4 when identical leptons are present (` = `0), and 1 otherwise; in the former case, the 4` rate 2 grows as ln Mz/m`. The total rate is independent of whether z is a scalar or a pseudoscalar (up to terms suppressed by m`/Mz) and one relevant distribution to access this information follows deﬁning φ as the relative angle between the planes of the two `+`− pairs in the centre-of-mass frame (such that for φ = 0 the two pairs lie in a common plane with the same-sign leptons adjacent to each other). Then, one has

2 2 2π dΓ4` I2c˜γ I1cγ SI3 cγc˜γ = 1 + κ1 cos 2φ + κ2 sin 2φ with κ1 = S 2− 2 , κ2 = 2 2 cos δ , Γ4` dφ 2R(cγ +c ˜γ) 2R cγ +c ˜γ (19) and the sign of κ1 discriminates the scalar case (κ1 0.2) from the pseudo-scalar case ≈ − (κ1 +0.2); their precise values are reported in table4. The κ2 term violates CP and is ≈ present only when both couplings are present, and δ is the phase diﬀerence between the scalar and pseudo-scalar coupling. More details and distributions can be found in [291]. The z gggg amplitude also →

4 pp zj, zjj: associated production with jets → In the previous section we have discussed examples where more complicated processes involving z, albeit having small rates, contain important information about the nature of z. Associated production with additional hard jets falls in the same category. The relevant cross sections

σzj,zjj for producing z together with one or two jets (including b jets) at the 13 TeV LHC are

2 TeV 2 2 2 2 2 2 σ(pp zj) = [164cgg + 0.51cu + 0.30cd + 0.03cs + 0.022cc + 0.012cb ] pb (20a) → Λ2 2 TeV 2 2 σ(pp zb) = [1.95cgg + 0.008cb ] pb (20b) → Λ2 2 TeV 2 2 2 −3 2 2 2 σ(pp zjj) = [29cgg + 0.088cu + 0.05cd + 10 (4.2cs + 3cc + 1.8cb )] pb (20c) → Λ2 2 TeV 2 −3 2 2 2 2 2 σ(pp zjb) = [0.59cgg + 10 (26cu + 15cd + 0.84cs + 0.52cc + 1.6cb )] pb (20d) → Λ2 2 TeV 2 −3 2 2 2 2 2 σ(pp zbb) = [0.15cgg + 10 (26cu + 15cd + 0.8cs + 0.5cc + 0.4cb )] pb . (20e) → Λ2 We ignore interferences in zjj, zjb, zbb cross sections. The operators coupling z to two EW vector bosons, both longitudinal and transverse, have not been considered here, because they

16 contribute to the VBF topology, which already contains two forward jets. Here we implemented

the following cuts to single out hard jets: η < 5 on all rapidities, pT > 150 GeV on all transverse momenta, and angular diﬀerence ∆R > 0.4 for all jet pairs. These results are summarised in

the first five lines of table1, shown in units of the leading results σz from eq. (7). A measure of σzj/σz, which in some cases is expected to be relatively large (see table1), can discriminate between different initial states: the zGG operator leads to more initial-state jet radiation than the zqq¯ operators. This was discussed in Ref. [37] which proposed the average pT of z as a good discriminator. In this analysis, and throughout the whole article, we are implicitly assuming that higher order terms in the EFT expansion are under control also for processes that can potentially probe the high-energy region, such as zj or zjj associated production. We shall discuss this in more detail in section 5.2, but here we mention that these effects are associated with operators of dimension-7 or higher that can be in the form of direct a b ν c ρµ abc contact contributions, such as zGµνGρ G (in a microscopic model with loops of heavy coloured states , this corresponds to emission of the jet directly from ), or higher derivative terms; in both casesQ they are suppressed by two powers of the large scaleQ Λ.

4.1 CP of z from pp zjj → The diﬀerential distribution of the zj and zb cross sections does not allow us to discriminate a scalar z from a pseudo-scalar z. For example the gluonic and quark operators contribute as 6 8 4 4 4 dσ 3g3 2 2 Mz + s + t + u (gg zg) = (cgg +c ˜gg) (21) dt → 128πs2Λ2 stu 2 4 2 2 2 4 2 dσ g3 2 2 g3(t + u ) v 2 2 Mz + s (qq¯ zg) = (cgg +c ˜gg) + (cq +c ˜q) (22) dt → 36πs2Λ2 s 2 tu 2 4 2 2 2 4 2 dσ g3 2 2 g3(s + u ) v 2 2 Mz + t (gq zq) = − (cgg +c ˜gg) + (cq +c ˜q) (23) dt → 96πs2Λ2 t 2 su

2 with s + t + u = Mz (see also the analogous Higgs cross sections [293]). On the other hand, production of z in association with two jets provides kinematic distributions that are sensitive to the CP nature of z. A well known variable that is sensitive to the CP nature of z is the azimuthal angle between the two jets ∆φjj [290, 294, 295]. In principle other jet distributions are also sensitive to the CP nature of z. For instance, [296] has examined a set of jet shape variables for the determination of the CP nature of a SM-like Higgs boson, which are potentially interesting for z as well. In the following we will examine the sensitivity to the CP nature of z of the thrust of the hard jets in the event P i∈ jets n pi T = max P | · | . n pi i∈ jets | |

This variable, unlike ∆φjj, exploits both transverse and longitudinal momentum of the jets, hence carries independent information on the CP nature of z which can be in principle combined with that carried by the ∆φjj distribution. Furthermore ∆φjj and the thrust are expected to have diﬀerent sensitivities to QCD aspects such as hadronization or soft and collinear emissions,

17 so that it is useful to cross-check the impact of these effects. Similar considerations apply to different experimental effects. Given the differences between the SM Higgs boson and z, it is worth reassessing the validity of the choices that are standard for studies of the SM Higgs boson, keeping in mind that z is significantly heavier than the Higgs boson. Hence, all effects related to the velocity of z or the recoil of the two jets against the scalar are less useful. Another important difference is that for the case of the Higgs boson two contributions, one from gluon fusion and one from vector boson fusion, are normally considered and often selection cuts are imposed to reduce the former and retain the latter. For z this could be a meaningful choice if it will be demonstrated that the production mechanism is mainly from photon or electroweak boson fusion processes, so that z is produced in a hard process without accelerated colour charges and features such as a rapidity gap in the distribution of hadrons is expected. For the time being this situation is not favoured and it seems more likely that z production involves accelerated colour charges, requiring a reassessment of the strategy to isolate the signal from the background. This consideration is further reinforced by the fact that the possible decays of z are not known yet. Relying on the existence of the diphoton decay mode, the relevant final state would be

pp zjj γγjj , → → which is usually not considered for the SM Higgs (however see [297] and references therein for early studies of this final state for the SM Higgs boson). On top of the above differences with respect to the SM Higgs boson, another crucial aspect is the rate of signal events, which for the γγ final state might be a fraction of fb, once two extra jets are required. This forces a careful choice of the strategy to distinguish the two CP hypothesis. The irreducible background from SM processes arises from pp γγjj and in general appears to be not negligible compared to the expected signal rate. As a→ matter of fact our calculations below show that the overall signal-to-background ratio for the γγjj final state is smaller than the one for the observed resonant γγ bump. Background and signal differential cross-section are computed at leading order with MadGraph5 without improvements beyond the fixed order at which we compute each process. The jets are defined as quarks or gluons around which no other quark or gluon is found in a region of angle ∆R = [(∆φ)2 + (∆η)2]1/2 = 0.4. Furthermore a quark or gluon considered as jet must lie in the geometrical and pT acceptance

ηj < 5, pT > 75 GeV . (24) | | For photons we require

pT,γ1 > 40 GeV, pT,γ2 > 30 GeV, ηγ < 2.37, 700 GeV < mγγ < 800 GeV . (25) With these deﬁnitions of hard jets and photons we ﬁnd σ (jjγγ) σ (jjγγ) sig = 0.16, bck = 0.30 , (26) σsig(γγ) σbck(γγ) where the larger fraction of background diphoton events with jets arises in part by the collinear enhancement for obtaining photons from quark fragmentation in large invariant mass dijet

18 events as well as multiplicity factors for jet emissions and “internal bremsstrahlung” from off- shell intermediate states of the diphoton background process.7 In principle one can devise selections to increase the signal-to-background ratio, e.g. by requiring harder isolation between jets and photon to reject the background from jet fragmentation. However, we do not find this useful in view of the limited amount of signal events that we can anticipate. At this stage the two CP hypothesis can already be distinguished as demonstrated in fig.6 by the

∆φjj distribution. Still the distinction between the two CP hypotheses can be ameliorated by imposing selections that affect the shape of the distributions. For instance we note that in the low ∆φjj region the distribution is heavily influenced by the isolation requirements for the jets, which are not CP-sensitive, and at large ∆φjj, where the two distributions are most different, the background is larger, and steeply varying. For this reason it is worth exploring possible further selections to make the differences between the distributions expected for the two CP hypothesis visible in a region of ∆φjj where the background is low and possibly flat. To this end we identified ∆ηjj and mjj as possible variables on which to impose cuts. We remark that, unlike the SM| Higgs| analyses aimed at isolating VBF Higgs production, selections on these variables do not necessarily increase the inclusive signal-to-background ratio. Nonetheless, we find them helpful to identify the CP nature of z. For instance requiring

mjj > 500 GeV and ∆ηjj > 2.5 , (27) | | a fraction about 20% of both signal events and background events are retained and the probability density of ∆φjj distribution is shown in ﬁg.6. The main eﬀect of these selections is to eliminate the constraints on the jet ∆φjj from jet isolation requirements, hence they can be relatively mild compared to standard VBF Higgs analysis. In order to estimate the luminosity needed to identify the CP nature of z we use the expected distributions to draw sets of Nev pseudo-events. We compute the likelihood ratio

Y pdf(CP-odd, ∆φi) = 2 ln , L − pdf(CP-even, ∆φi) i=1...Nev where ∆φi are the ∆φ values of each pseudo-experiment. Performing a large number of pseudo- experiments, as customary in these analyses [298], we take the likelihood ratio above as our test-statistics to distinguish the two CP options for z. The distribution of the test-statistics for the baseline selection with the extra cuts in eq. (27) are reported in the two panels in the middle row of fig.6 for Nev = 100 events and 20 events, respectively. Given the efficiency of the cuts in eq. (27) the two panels correspond to the same integrated luminosity −1 L ∼ 100 fb 6 fb/σsig(γγ). Considering the area of the tail of the CP-even distribution above the CP-odd× median, we find that the CP-even hypothesis can be rejected with 90% C.L. and, adding the cuts in eq. (27), above 95% C.L. Similar results hold for the converse exclusion. In the bottom row of fig.6 we show the distribution of the test-statistics when in the pdf for each CP hypothesis we add the pdf of the SM background with rate twice that of the signal,

7Additional backgrounds can arise from jets being misreconstructed as isolated photons. In pp z analyses, → such backgrounds have been found to constitute less than 10% of the total background [1].

19 ��

��

�� -�� -�� - - ��

�� / �� γγ �� / �� γγ ��

�� Δϕ(��) Δϕ(��) ��

��

�� CP-even CP-even

�� CP-odd �� CP-odd

��

�� -�� -� � � �� -�� -�� ℒ ℒ

��

�� CP-even CP-even �� CP-odd �� CP-odd ��

��

�� -� -� -� -� � � � -� -� -� -� � � � � ℒ ℒ

Figure 6: Upper row: Normalised ∆φjj distributions in pp γγjj events for the CP-even → (blue) and CP-odd (yellow) hypothesis as well as for the irreducible SM background γγjj (green). In the left panel we impose only the minimal selection to have z γγ and two jets, while in the right panel we impose the extra requirements in eq. (27) to enhance→ the diﬀerence between the two CP hypothesis. Middle row: distribution of the test-statistics in absence of background. Bottom row: distribution of the test statistics for a total background rate twice the signal rate, as indicated by eq. (26).

as suggested by eq. (26). The inclusion of background deteriorates the exclusions, which drop

to 85% C.L. and 95% C.L., respectively. Results on an observable similar to ∆φjj have been discussed in [231], which claims similar results. For the thrust we ﬁnd similar results, which are illustrated in ﬁg.7 and are obtained with

the same procedure as for ∆φjj. With the same number of events as above we expect an exclusion at 88% C.L. for the analysis without the cuts in eq. (27) and above 95% C.L. adding

these cuts. Including the background in the same way as for the study of ∆φjj we expect the exclusion to drop at 65% C.L. and around 75% C.L. for the two cut options, respectively.

The combination of the results from ∆φjj and the thrust is meaningful once one takes into account their correlation. For illustration we show the doubly diﬀerential distribution in the

20 �� -�� -�� -�� -�� / �� γγ �� / �� γγ ��

��

��-�� -��

�� -�� -��

�� -�� -� � � �� -�� -�� -� � � �� ℒ ℒ ��

�� -�� -��

�� -� -� -� � � � � -� -� � � � ℒ ℒ

Figure 7: Upper row: Normalised thrust distributions in pp γγjj events for the CP-even → (blue) and CP-odd (yellow) hypothesis as well as for the irreducible SM background γγjj (green). The inset in each panel shows the cumulative distribution, which highlights the diﬀerences between the shapes of the distributions. In the left panel we impose only the minimal selection to have z γγ and two jets, while in the right panel we impose the extra requirements in eq. (27) to→ enhance the diﬀerence between the two CP hypothesis. Middle row: distribution of the test-statistics in absence of background. Bottom row: distribution of the test statistics for a total background rate twice the signal rate, as indicated by eq. (26).

plane (T, ∆φjj) for the CP-even and CP-odd hypotheses as well as for the background. If z couples to quarks, rather than to gluons, the diﬀerence between CP-odd and CP-even distributions gets suppressed by small quarks masses, and is not observable.

5 pp zV, zh: EW associated production → Production of z in association with EW bosons provides an additional handle to distinguish diﬀerent initial states and the structure of their couplings to z. The cross sections σ(pp → zV ) σ V for producing z together with an SM vector (see also [55]) or with the Higgs ≡ z

21 ��

��

�� Δϕ Δϕ Δϕ ��

�� -�� -�� -��

Figure 8: Double diﬀerential (T, ∆φjj) probability distribution of CP-even (left), CP-odd (middle) and background (right) after the cuts eq. (27).

f ̥ f ̥ f ̥

f V ′ V f VL f V

Figure 9: Diagramatic representation of partonic processes contributing to zV associated production due to z couplings to EW gauge bosons (left-hand diagram) and SM fermions (middle and right-handed diagram). The z interaction vertices derived from eq. (5) are marked with a box, while the gauge interaction vertices of SM fermions are marked with a disk.

boson, receive contributions from diagrams such as those in ﬁg.9. At the 13 TeV LHC, for the CP-even case, we ﬁnd

2 TeV 2 −2 2 −3 2 −3 2 σ(pp zγ) = [0.12 cu + 1.9 10 cd + 1.6 10 cs + 4.4 10 cc + (28a) → Λ2 × × × −4 2 −5 2 −4 2 −5 + 4.9 10 cb + 8.5 10 cBB + 6.6 10 cWW + 3.2 10 cBBcWW ] pb ×2 × × × TeV 2 −2 2 −3 2 −3 2 σ(pp zZ) = [0.15 cu + 9.1 10 cd + 5.5 10 cs + 3.3 10 cc + → Λ2 × × × −3 2 −5 2 −3 2 + 1.4 10 cb + 2.7 10 cBB + 2.3 10 cWW × −5 × −3 2 × −6 2 3.2 10 cBBcWW + 1.9 10 cgg + 8.2 10 cˆH ] pb (28b) − ×2 × × + TeV 2 2 −2 2 −3 2 σ(pp zW ) = [0.2 cu + 0.19 cd + 1.0 10 cs + 5.1 10 cc + → Λ2 × × −6 2 −3 2 −5 2 + 4.9 10 cb + 4.7 10 cWW + 1.1 10 cH ] pb (28c) ×2 × × − TeV −2 2 −2 2 −3 2 −3 2 σ(pp zW ) = [7.7 10 cu + 7.8 10 cd + 5.1 10 cs + 7.0 10 cc + → Λ2 × × × × + 4.2 10−6 c2 + 1.8 10−3 c2 + 4.5 10−6 c2 ] pb (28d) × b × WW × H 22 2 TeV 2 −2 2 −3 2 −3 2 σ(pp zh) = [0.14 cu + 8.5 10 cd + 5.2 10 cs + 3.3 10 cc + → Λ2 × × × + 1.4 10−3 c2 + 6.6 10−4 c2 + 0.12 10−6 c2 ] pb × b × gg × H TeV −6 −6 2 0.35 10 cH κ H pb + 0.4 10 κ pb , (28e) − Λ · z · zH

We imposed the cuts ηγ < 2.5 and pT,γ > 10 GeV for the photon, and no cut for the massive vectors. The numerical values have been obtained using MadGraph5 and the NNPDF LO pdf p 2 2 set with a running factorization scale µF = Mz + pT . Higher order QCD corrections can be important for these processes, but are not expected to change our results by more than (1) O factors. For the top quark loop contribution to gg hz production we have used the automatic → loop calculation available with MadGraph5. In the massless fermion limit, the helicity structure of the amplitude proportional to cVV differs from the cψ one, due to the chiral-breaking nature of these scalar-fermion interactions. For this reason, the only interference between the different dimension-5 interactions occurs for cBB and cWW in their contributions to vertices with photons and Z-bosons. Eq.s (28) hold for the CP-even case and become slightly different in the CP-odd case, as discussed below. In eq. (28) (and the analogous results summarised in table1) we can identify several interesting features. In the limit where z is principally produced through quarks, associated EW production is always dominated by the center and right diagrams of fig.9 (contributions from the first diagram could be larger only when prompt z production is dominated by the γγ z channel). Then, the ratio σ V /σ , and the production rates pp V (z γγ), are → z z → → independent of the total z width and only depend on the q flavour. Processes like this one (or z γγ γ`−`− discussed in section 3.2), whose amplitudes are constructed from the main → ∗ → amplitude pp z γγ with the addition of a SM vertex, are important since their rates can → → −3 be determined model-independently. For example, we obtain σ γ/σ = 4, 16, 4 10 , for z z { } × q = s, c, b , a prediction that could be used to single out qq production channels. { } Another handle for discriminating between different parton initial states is zW ± associated production. Assuming flavour diagonal new physics, in the case of pure b¯b annihilation, zW production is suppressed, since the contribution from initial state top quarks is negligible, while contributions from lighter initial state quarks are CKM suppressed. On the other hand, for the ss¯ (cc¯) cases, the zW + (zW −) channel is expected to be the dominant mode because the production process can be initiated by valence quarks at the price of only Cabibbo angles.

In fig. 10 we show a combination of the results of eq. (28) with present collider constraints, in a way that makes the expectations for EW associated production more manifest. The figure shows the maximum cross section for a given final state (different colours in the legend) produced by z in association with a vector (horizontal axis in the plot), under the following conditions: i) the total z decay width is Γ . 45 GeV; ii) the z partial widths in each channel are constrained by 8 TeV and 13 TeV data (as specified in table 1 of [3]); iii) we maximise over 13 TeV 8 TeV different production channels, but require σpp→z /σpp→z & 4, to ensure compatibility between 8 TeV and 13 TeV data.

23 10-1

gg L

pb b b H L - f 10 2 c c

® s s ¸ H ΓΓ

BR WW V + ¸ -3 ZZ Σ 10 ΓZ

- Γ Z W+ W

Figure 10: Maximal zV associate production cross-section σzV at the 13 TeV LHC times the z f branching fraction BR(z f) for various possible V = γ, Z, W +,W + (on the horizon- → → tal axis) and f = gg, b¯b, cc,¯ ss,¯ γγ, W +W −, ZZ, γZ (diﬀerent coloured bands as speciﬁed in the legend) states, subject to current experimental constraints. Values below the individual contours are possible and allowed by current data. See the main text for details.

We observe that the largest allowed rates are σ +W,Z BR(z jj) 0.1 pb and are satu- z × → . rated when z is predominantly produced from ss¯ and/or cc¯ annihilation. This signature might however be challenging to access experimentally due to the W, Z+jets (and tt¯) backgrounds. Recent studies in the bbZ channel, searching for resonances both in the bb invariant mass and in the Zbb invariant mass spectrum, place bounds at the level of 0.5 pb [300]. Finally, among + − the purely EW final states, σ +W,Z BR(z W W ) could still reach (10 fb), while all z × → O photonic signatures of zV production are already bounded below (fb). O Concerning EW-induced associated z production (the left diagram of fig.9), the largest possible rates are actually expected when prompt single z production is dominated by b¯b annihilation. Then, the photonic contribution to the first diagram in fig.9 gives σ γ 0.21 1 fb z ' − depending on whether the width is saturated by Γb¯b or by other channels (in this latter case, ¯ a minimum Γb¯b & 10Γγγ is still necessary to guarantee dominance of bb production over γγ production, which would be in tension with the 8 TeV/13 TeV comparison). On the other hand, contributions of vertices involving W, Z to the left diagrams in fig.9 can lead to one

order of magnitude larger rates for all σzV , for the simple reason that the constraints on these couplings are an order of magnitude weaker, see table2. We close this discussion by noting that EW associated production is also one of the few model-independent z production process which can be probed at e+e− colliders [14] or photon [56,57] colliders.

24 10 10 0.7 Θ

Β 1 1 d 0.6 s dcos Σ p H p -1 CP -1 d even -

` - even L s Σ 10 10 Β=0 CP L d Σ L 0.5 Β= ` 0.1 1 Σ -2 odd -2 H 10 - 10 1 Β=0.3

H CP 0.4 Β=0.5 -3 -3 Β=1 HCP-oddL 10 10 -1.0 -0.5 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 cosΘ Β

Figure 11: Left panel: angular dependence of the partonic zZ production cross-section in the centre-of-mass frame for the case of the CP-even operator (cγZ ) for diﬀerent values of the Z velocity β. The prediction for the CP-odd operator (c˜γZ ) coincides with the curve at β = 1. Right panel: normalised diﬀerential zZ associated production cross-section at 13 TeV as a function of β, the Z velocity in the centre-of-mass frame. We also show, on the right axis, the ratio s/p of s-wave to p-wave contributions to the rate.

5.1 CP of z from pp zZ, zW → EW associated production also allows us to test the CP nature of z interactions. In the case of the zZγ coupling, measuring the Z polarization in z Zγ decays is not enough to disentangle → its CP nature [301]. On the other hand, in zZ associated production, the intermediate photon is virtual and the longitudinal polarisation of the Z is accessible close to threshold and can be used to probe the CP nature of this interaction. We will consider this in the context of µν ˜µν the CP-even (odd) operators zZµνF (zZµνF ) in the EW broken phase, see eq. (13). The angular dependence of the differential partonic cross-sections in the CP-even and CP-odd cases for pp zZ are → 1 dσˆ 3 CP−odd = (1 + cos2 θ) , (29) σˆCP−odd d cos θ 8 2 2 2 2 1 dσˆCP−even 3 1 + cos θ + 8MZ s/λˆ (ˆs, MZ ,Mz) = 2 2 2 , (30) σˆCP−even d cos θ 8 1 + 6MZ s/λˆ (ˆs, MZ ,Mz) where λ(a, b, c) a2 + b2 + c2 2(ab + bc + ca),s ˆ is the partonic invariant mass squared of ≡ − the system and θ is the angle between the direction of the Z relative to the beam direction in the centre-of-mass reference frame. The angular dependence of the CP-odd case is purely p 2 p-wave, as illustrated in eq. (29), independently of the Z velocity β 1 (MZ + M) /sˆ. For this reason the largest CP-even/CP-odd discrepancy is close to threshold,≡ − where the angular dependence is flat in the CP-even case, corresponding to s-wave dominance, as illustrated by the left panel of fig. 11. The dependence on β of the ratio between s and p-wave contributions to the CP-even cross section is illustrated in the right panel of fig. 11. At the same time the pp zZ production cross-sections for CP-even and CP-odd cases →

25 D 0.200 13TeV

GeV 0.150 33TeV 100 1 @

T 0.100 100TeV dp

Σ 0.070 d L Σ

1 0.050 H 200 400 600 800 1000 pT @GeVD

Figure 12: Transverse momentum distribution of zV associate production at 13 TeV (highest peaked spectra), 33 TeV (middle peaked spectra) and 100 TeV (lowest peaked spectra) pp collisions due to cVV 0 interactions. The overlapping lines of diﬀerent colours correspond to various EW bosons V,V 0 = γ, Z, W . close to threshold diﬀer,

2 2 σˆCP−even cγZ 6MZ sˆ = 2 1 + 2 2 . (31) σˆCP−odd c˜γZ λ(ˆs, MZ ,Mz)

For a given Γ(z Zγ) this corresponds, after parton luminosity integration, to a 7% enhancement of the CP-even→ cross-section over the CP-odd one at the 13 TeV LHC. This is shown in the right panel of ﬁg. 11. The same angular dependence and cross-section ratio also appears in zZ production from the zZZ couplings, with the replacement cγZ cZZ andc ˜γZ c˜ZZ , as well as in zW + − → → production from the zW W couplings, with the replacement cγZ cWW andc ˜γZ c˜WW. The CP properties of these interactions can also be probed using the→ angular distributions→ in z ZZ 4f and z W +W − 4f decays [231], or z γ∗γ∗ 4f as discussed above. → → → → → → 5.2 EFT expansion and associated production

An important aspect of associated production is that, contrary to resonant z production, the centre-of-mass energy of the parton process is not fixed and can vary in a wide range. In this context the question of the validity of our EFT expansion can become important and is complicated by the difficulty, contrary to resonant production, of associating a precise energy scale to the process. In fact, from an EFT perspective, operators of dimension 7 or higher can also contribute to these processes (from our discussion before eq. (9) it is evident that there are no dimension-6 operators linear in z). Their effect (which has been ignored in our analysis) (7) 2 2 4 grows as ci sˆ /Λ in the amplitude squared for pp zV, zh, and has to be compared with ∼ | | (5) 2 2 → the leading dimension-5 contribution ci s/ˆ Λ . While both effects grow (and eventually even cease making physical sense, when∼ perturbative | | unitarity is violated [302]), their relative

26 ys ¸ 1 yc ¸ 1 yb ¸ 1 5.00 ¸ + + W @ ¸ - 1 singletD + W @ ¸ singlet + ZHsinglet D L ¸ + ZHsinglet ¸ + + L D D ¸ + W - D W @singlet @singlet 1 ¸ + D 1.00 D ZHsinglet L 0.1 GeV GeV 0.50 GeV ¸ + ¸ + Z 10 10 10 H W + ¸ doublet + @ doublet W - fb fb ¸ + @ fb L @ @ doublet @ W - 0.01 T T D T ¸ + @doublet 0.1 Γ 0.10 ¸ + D dp dp dp

Z H ¸ ¸ + doublet D + Γ

Σ Σ ¸ 0.05 + W + Σ d d ZH @ d ¸ + L doubletdoublet 0.001 ¸ + WH Γ singletL L D + H L 0.01 0.01 ¸ W doublet 100 200 300 400 500 100 200 300 400 500 100 200 300 400 500 yb ¸ pT @GeVD pT @GeVD pT @GeVD

Figure 13: Transverse momentum distribution of z + V production at the 13 TeV LHC due to ¯ ss¯ (left panel), cc¯ (center panel) and bb (right panel) annihilation induced by the cs,c,b couplings of eq. (5), that we rewrote in terms of yqz cψv/Λ. The EW doublet results are obtained in the limit of degenerate doublet components. ≡

(5) (7) size crucially depends on Λ and the Wilson coeﬃcients ci and ci and cannot be determined without explicit UV assumptions. Fortunately most of our analyses rely on the use of total cross sections, where the rapidly falling PDF distributions imply that the bulk of the EFT contributions are near threshold, 8 as illustrated in ﬁg. 12 for cBB and cWW interactions. In this kinematic region the centre- of-mass energy is close to that of single z production where the EFT description holds by construction, and the question of EFT validity can be expressed transparently in terms of the

Mz/Λ expansion. An illustrative example where the above-mentioned anomalous energy growth can be used to learn about the underlying theory, is the following. We compare a simple, renormalisable model where z = H0 is the neutral CP-even component of an additional EW doublet (and its couplings to SM fermions are dimensionless), with the scenario of eq. (5) where z is a singlet and its interactions cψ are in fact non-renormalisable. We further assume that prompt z production is dominated by heavy quark annihilation. In the singlet model, scattering amplitudes for qq¯ zVL (V = Z,W ) are dominated by the contact interaction (last diagram → 2 in ﬁg.9) and grow as s/ˆ Λ at large energy. This can be seen in the hard pT spectra in ∼ hadronic collisions as shown in ﬁg. 13. In a renormalisable SU(2)L invariant theory, on the other hand, this anomalous UV behaviour is regulated by the presence of additional degrees of freedom. In the case where z is the neutral component of a SU(2)L doublet, the zZ production now receives additional contributions from s-channel exchange of the other neutral components of the doublet, while zW production is regulated by the exchange of the associated charged scalars. 8Note that even the discussion of the previous section, which relies on certain kinematic distributions, is most powerful near threshold β 0. ≈

27 6 pp zz: pair production → 6.1 Eﬀective theory parametrisation

In the eﬀective Lagrangian description of section 2.2, z pair production receives contributions at diﬀerent orders in the 1/Λ expansion. In fact, starting already at the renormalisable level, the

coupling λzH (which survives also in the limit of large separation of scales Λ Mz) generates ∗ VLVL zz and pp h zz via SM Higgs production channels. We have already shown → → → in the table below eq. (1) that the VLVL contribution to single production is small, and this result does not change substantially for pair-production so that this channel can be neglected. On the other hand, for pp h∗ zz we ﬁnd → → −4 2 σ(pp zz) = 1.7 10 λ H fb , (32) → z where we neglected the subleading contribution coming from mixing with the Higgs and from dimension-6 operators.

The presence of the relevant coupling κz in the renormalisable part of the Lagrangian eq. (3), implies that the rate for pp z∗ zz, with z∗ produced by dimension-5 operators, can be thought to be formally of the→ same→ order in the EFT expansion; we write it as

TeV2 2 2 2 2 2 2 2 (33) σ(pp zz) = κ (270 cgg + 1.9cu + 1.4cd + 0.07cs + 0.04cc + 0.017cb ) fb. → z Λ2

This implies that pair production can be reasonably large for realistic values of κz  (κ /57)2 gg production σ  z zz = (κ /63)2 uu¯ production . (34) σ z z  2 ¯ (κz/88) bb production Note, however, that arguments based on vacuum stability restrict the coeﬃcient of the cubic coupling to κz < 6λz in the limit of large λz, while vacuum meta-stability allows only for a small violation| | of this upper bound [54].

There are a number of reasons why this description in terms of an effective Lagrangian trun- cated at dimension-5 might be incomplete in certain cases, and the next-order in 1/Λ becomes necessary. First of all, it is plausible that the separation between Mz and Λ is mild, as already suggested by the relatively large rates necessary to accommodate the observed excess. Sec- ondly, it is possible that the z couplings to the underlying dynamics (e.g. additional particles in the loop) are sizeable and larger than the typical SM couplings. Finally, approximate global symmetries, preserved only by higher (in this case dimension-6) order interactions, can lead to natural situations where the Wilson coefficients of the leading effects in the EFT expansion are actually suppressed. Examples of this are models where z is odd under an approximate Z2 symmetry, explicitly broken only by small effects, so that both κz and the full L5 in eq. (5)

28 are suppressed. In any of these cases, the contribution from dimension-6 operators in eq. (9) (and in some cases of two insertions of dimension-5 operators), can be relevant. These give 4 TeV 4 2 (6) (6)2 2 2 −3 4 σ(pp zz) = 1.1cgg + 2.1cggcgg + 0.52cgg + 73c 3cgg + 10 (0.0074cγγ + → Λ4 z 2 (6) (6)2 −6 4 4 4 4 4 +0.099cγγcγγ + 0.38cγγ ) + 10 (11cu + 6cd + 0.25cs + 0.14cc + 0.05cb ) + −3 (6)2 (6)2 (6)2 (6)2 (6)2 +10 (4.4cu + 2.4cd + 0.1cs + 0.06cc + 0.02cb ) pb (35)

Interference in the quark diagrams is suppressed by the small quark masses and we have neglected for clarity the interference between these eﬀects and those of eq.s (32), (33), assuming

that either an approximate symmetry or a coupling hierarchy can account for a small κz. In the limit where one production mode dominates we get the results shown in table5 using σ(pp zz) σ(pp zz γγz) = 2σ(pp z γγ) → (36) → → → → σ(pp z) → and 2 σ(pp z γγ) σ(pp zz 4γ) = σ(pp zz) → → (37) → → → σ(pp z) → and having fixed σ(pp z γγ) = 3 fb, which is the experimentally favoured value as extracted from a fit to the→ preferred→ cross sections of table.2, under the assumption of production from gluon fusion. The present experimental bounds on pp zz pair production at √s = 8 TeV are listed in → table6. Using present data, the 4 γ limit can easily be improved down to 0.1 fb or better with a dedicated search. The 4j bound implies 2 Γγγ Γγγ σ(pp zz jjγγ) < 0.2 pb, σ(pp zz γγγγ) < 0.1 pb. (38) → → Γjj × → → Γjj × We see that, unless z is produced from γγ partons, detectable cross sections for zz production (6) (6) need c℘ c℘ and a not too large Λ. Large c℘ are in some cases rather plausible, in particular for initial-state quarks ℘ = q where these couplings can be generated at tree level in a UV- complete underlying model. An explicit realization of this is a model with additional heavy ¯ 0 ¯ vector-like quarks with couplings yzz q and yzz larger than the Yukawa couplings ¯ Q Q QQ (6) (5) 0 yH H q. In such theories the ratio of Wilson coefficients cq /cq yz (neglecting a small Q SM ∼ contribution proportional to the SM Yukawa yq ), can be large.

6.2 Model computation in Low Energy Theorem approximation

To better appreciate the prospects oﬀered by z pair prduction, it is instructive to consider an explicit renormalizable model, where z γγ is mediated by a loop of heavy vector-like → fermions r coupled to z as Q X ¯ LQ = r(iD/ Mr yrz) r. (39) r Q − − Q

29 z couples to σzz/σz = σγγz/2σγγ σ4γ/σγγ ¯ 2 (6) 2 −6 2 (6) 2 bb 0.015% ( TeV/Λ) (cb /cb) 3.6 10 ( TeV/Λ) (cb /cb) 2 (6) 2 × −6 2 (6) 2 cc¯ 0.021% ( TeV/Λ) (cc /cc) 2.1 10 ( TeV/Λ) (cc /cc) 2 (6) 2 × −6 2 (6) 2 ss¯ 0.023% ( TeV/Λ) (cs /cs) 1.5 10 ( TeV/Λ) (cs /cs) 2 (6) 2 × −6 2 (6) 2 uu¯ 0.058% ( TeV/Λ) (cu /cu) 0.23 10 ( TeV/Λ) (cu /cu) ¯ 2 (6) 2 × −6 2 (6) 2 dd 0.050% ( TeV/Λ) (cd /cd) 0.31 10 ( TeV/Λ) (cd /cd) 2 (6) 2 × −6 2 (6) 2 GG 0.13% ( TeV/Λ) (cgg /cgg) 0.006 10 ( TeV/Λ) (cgg /cgg) 2 (6) 2 × −3 2 (6) 2 γγ 1.9% ( TeV/Λ) (cγγ /cγγ) 2.9 10 ( TeV/Λ) (cγγ /cγγ) × Table 5: Predictions for leading order contributions to pair production of the resonance z at √s = 13 TeV.

σ(pp zz jjjj) σ(pp zz γγjj) σ(pp zz γγγγ) → → → → → → Bound at LHC, √s = 8 TeV < 0.1 pb [303] 26 fb [304] − . Background at √s = 8 TeV see [303] 0.07 fb 4 ab ∼ ∼ Background at √s = 13 TeV 5 [303] 0.2 fb 8 ab × ∼ ∼

Table 6: Summary of pp zz searches. The 4γ search [304] was not optimized for double production of resonances. →

Throughout we will consider these fermions as coloured and/or carrying hypercharge, but for simplicity we do not consider fermions with SU(2)W charge. pp zz is unavoidably obtained → by attaching twice z to the loop as well as by a possible cubic z3 term, as depicted by the Feynman diagrams in ﬁg. 14. In the limit that the new fermions are relatively heavy,

2Mr & Mz, we may employ the low energy theorem (LET) to determine the dominant coupling to gluons and photons [305–311]. To see this we may write the contribution of any new massive 3 2 coloured field to the QCD β-function as ∆β3 = ∆b3g3/16π where, as an example, for Nr new coloured fermions of Casimir Ir (normalised such that Ir = 1/2 for the fundamental representation) we have ∆b3 = 4NrIr/3. Writing the mass of this field as M(z) (which explicitly includes z as a background field) and running the gauge coupling from a high scale Λ to some low scale µ, the gauge kinetic terms pick up a correction at the mass threshold, given by aµν a LLET = ∆b3α3/8π ln(M(z)/µ)G Gµν. The case for QED is analogous. This derivation is general for any field whose mass depends on z. For our simple example case it gives X α3 aµν a 2 0 α µν z LLET = IrNr G G + q N F Fµν ln 1 + . (40) 6π µν r r 6π v r r

0 where the loop contribution also includes Nr fermionic vector-like components with electric charge qr, and we have deﬁned Mr vr . (41) ≡ yr

30 � Ϝ � Ϝ Ϝ   Ϝ � Ϝ �

Figure 14: Feynman diagrams contributing to pair production of the 750 GeV resonance.

Expanding the logarithm provides the low energy theorem (LET) description of multiple scalar production from gluon or photon fusion.9 In fact, we can see that in the absence of a scalar self coupling the pair production amplitude is related to the single production amplitude simply by

a factor of 1/vr. To make our expressions more transparent, we limit our discussion to the case of NQ copies of identical electrically-neutral coloured fermions with Casimir IQ and NL copies of colourless fermions with charge qL. We also take masses and couplings universal in the two sectors, which are then described by the two scales vQ MQ/yQ and vL ML/yL. The extension to general fermion representations is completely straightforward≡ and can≡ be expressed in terms of eﬀective

vQ and vL. In particular, heavy fermions with both colour and electric charge simultaneously contribute to both vQ and vL. Using this description the decay widths of the particle z into gluon and photon pairs are α2 N 2 I2 M 3 2 4 2 3 3 Q Q z α qLNL Mz Γgg = 3 2 , Γγγ = 3 2 . (42) 18π vQ 144π vL The corresponding single production cross section σ(pp z), initiated by gluon and photon annihilations, is → 2 2 1 Mz Mz σ(pp z) = Γgg Cgg + Γγγ Cγγ , (43) → s Mz s s where, as deﬁned in [3], 2 Z 1 Z 1 sˆ π dx sˆ sˆ 2 dx sˆ Cgg = g(x)g ,Cγγ = 8π γ(x)γ , (44) s 8 s/sˆ x sx s s/sˆ x sx and s (ˆs) is the proton (parton) squared centre-of-mass energy.10 The pair-production cross section pp zz also depends on the value of the possible 3 → cubic interaction, κF Mzz in the potential of eq. (4). It is convenient here to rewrite it as κF = κMz/2vQ.

9 2 (6) 2 2 2 A translation to the operators in eq.s (5) and (9) is c /Λ = I N /(12π v ), and cgg /Λ = I N /(24π v ). gg r r r − r r r 10The parton distribution functions also depend on the factorisation scale, however we have suppressed this variable in the equations above and taken the factorisation scale as µ = √sˆ throughout.

31 MQ =0.8 TeV MQ =1.2 MQ =1.6 LET ] [

- -

Figure 15: The function ζ(κ) that determines the ratio between the pair and single production cross section for gluon-initiated processes, as deﬁned in eq. (47). The solid line refers to the LET result, while the other lines show the result when ﬁnite fermion mass form factors are included in the calculation, as discussed in section 6.3.

Higher order terms, such as the quartic coupling, are not relevant for this study. In the LET limit, after partonic integration, the colour and spin averaged total pair production cross section at the LHC is

" 2 2 # 1 Γgg 0 Mz Γγγ 0 Mz σ(pp zz) = 2 2 Cgg + 2 Cγγ , (45) → 8π Mz vQ s vL s where the weighted partonic luminosities, including the kinematic dependence from the two interfering diagrams and the phase space factors, are

Z 1 r 2 0 y 4z 3z C (z) = dy Cgg(y) 1 1 κ , (46a) gg 4z − y − y z 4z − Z 1 r 2 0 y 4z vL 3z Cγγ(z) = dy Cγγ(y) 1 1 κ . (46b) 4z − y − vQ y z 4z − If we assume that gluon-initiated production dominates over photon-initiated production, in the LET limit the ratio of the cross sections of double to single z production depends on the new fermion content and quantum numbers only through the scale vQ, and is simply given by 2 C0 (M 2 /s) σ(pp zz) Mz s gg z → = 2 ζ(κ), ζ(κ) 2 2 2 . (47) σ(pp z) vQ ≡ 8π M Cgg (M /s) → z z The solid line in ﬁg. 15 shows ζ(κ) as a function of the z self coupling, using the LET result in eq. (47). For comparison we also show the values of ζ(κ) determined by the full one-loop

32 calculation for various masses MQ of the new fermion (with NQ = 1 and IQ = 1/2). A good ﬁt of the LET result is ζ(κ) 3.2 10−4 1 κ + 0.3κ2 . (48) ≈ × − Equations (47) and (48) allow for a quick estimate of the pair production cross section, assuming that the single production cross section is known. For example, if we want to reproduce the experimentally favoured value σ(pp z γγ) 3 fb for gluon-initiated production, we ﬁnd p → → ≈ vQ = NQIQ BRγγ 3.5 TeV and eq. (47) leads to

2 −5 1 κ + 0.3κ σ(pp zz) = 4.7 10 fb − 2 . (49) → × (NQIQ BRγγ) Taking into account the branching ratios we have

2 −5 1 κ + 0.3κ σ(pp zz γγγγ) = 4.7 10 fb − 2 , (50) → → × (NQIQ) 2 2 −5 1 κ + 0.3κ Γgg σ(pp zz gggg) = 4.7 10 fb − 2 , (51) → → × (NQIQ) Γγγ 2 −5 1 κ + 0.3κ Γgg σ(pp zz γγgg) = 9.4 10 fb − 2 . (52) → → × (NQIQ) Γγγ having assumed that production is dominated by gluon fusion. In ﬁg. 16 we take into account diphoton-initiated pair production, and show the results of the low energy theorem prediction for σ(pp zz γγjj) and σ(pp zz γγγγ), → → → → assuming a vanishing z3 coupling for a benchmark model of two triplets of coloured fermions and three leptons with unit charge. This choice of benchmark parameters is motivated by ﬁg. 5

of [3], such that the required ranges of Γγγ and Γgg may be found for reasonably perturbative couplings, particularly in the narrow width scenario. This can be seen by comparing with the

required values of vQ and vL, which show that for MQ and ML in the range of 100’s GeV, the required Yukawa coupling becomes non-perturbative only at the extreme ranges of the parameter space, when the width is becoming large. For this benchmark model, experimental constraints on σ(pp zz 4g) already place relevant bounds on the parameter space. In some regions of allowed→ parameter→ space the −1 σ(pp zz γγgg) final state may be observable in the future with 300 fb of integrated luminosity.→ However,→ in much of the parameter space the cross section∼ for this process is too small. In the upper left hand plane for much of the parameter space σ(pp zz 4γ) > 0.1 fb, also suggesting that this channel could be observable. However, in most→ of the region→ where this is observable the dominant production mode is from photon fusion, and this region is disfavoured due to the reduced increase in single production cross section going from 8 TeV to 13 TeV. In fig. 17 we project the different cross sections along the narrow width line (lower edge of the green region in fig. 16) and along the Γ/M = 0.06 line (upper edge in fig. 16). There are a number of interesting features. As before, it is clear that dijet pair production places an

33 [] = - -

]

/ -

[ -

- > - - - - - - - /

Figure 16: Within the green region one can reproduce the di-photon excess σ(pp γγ) 3 fb → ≈ at the 13 TeV LHC. Contours of constant σ(pp zz) are shown as solid lines, σ(pp zz → → → 4g) as dashed, σ(pp zz γγgg) as dotdashed, and σ(pp zz 4γ) as dotted. These cross sections are computed→ using→ LET with two triplets of coloured→ fermions→ and three leptons 3 with unit charge, and the z coupling is set to zero. The required scales vQ = MQ/yQ and vL = ML/yL are also shown.

interesting constraint on the parameter space. Second, as Γgg is reduced, then Γγγ must be increased as we go along either boundary. Correspondingly, the diphoton contribution to single and pair production increases. Since partonic gluons are softer than partonic photons, the photon fusion contribution to pp zz is relatively more important, with respect to the gluon → fusion, than the photon fusion contribution to pp z. This means that there are regions of → parameter space where pp z is dominated by gluon fusion, giving good consistency between → 8 TeV and 13 TeV data, and at the same time pp zz is dominated by photon fusion. → Another feature worth highlighting is that as one goes to very small Γgg and Γγγ is increased, the inclusive pair production cross section may become larger than the single production cross section, while, even in a strongly-coupled model, one expects that it should be 10 50 times smaller because of the reduced parton luminosity. This is particularly noticeable for− the Γ/M = 0.06 assumption. This is not, however, physical. It is rather signalling the breakdown of perturbation theory since the value of Γγγ required to explain the excess is becoming so large that for the benchmark parameters chosen and for ﬁxed vL the implied Yukawa coupling to charged fermions are becoming too large. Thus in the region where pair production is com- parable or larger than single production the predicted rates for either should not be trusted. More speciﬁcally, the ratio of pair to single production cross sections scales approximately as

2 σ /σ (yM /4πMQ) . (53) zz z ∝ z

For the LET description to remain valid we require MQ & Mz: in the strongly coupled limit,

34 [] [] = =+ < < = = ] ] [ [ - - - - - - > - - - - - - - - - - - - / /

Figure 17: Cross sections as a function of Γgg/M once the requirement of σ(pp z 2γ) = 3 fb has been imposed. The narrow width assumption is shown on the left panel (corresponding→ → to travelling along the lower boundary of the green region in ﬁg. 16) and a broader resonance is shown on the right panel (travelling along the upper boundary in ﬁg. 16).

y 1, it is possible to have vz . Mz while the LET description remains valid, at the cost of approaching the non-perturbativity limit, as can be seen in fig. 17. Finally we note that in regions of parameter space where gluon fusion dominates the production by far the largest observable final state is pp zz 4g. The cross section for → → pp zz γγgg stays approximately in the region 10−3 10−1 fb. →In summary,→ pair-production is experimentally interesting.→ For the benchmark scenario −5 considered here we find that σ(pp zz 4g) & 1 fb provided that Γgg/M & 9.5 10 → → × −4 (vQ 290 GeV), and for σ(pp zz 2g2γ) 0.1 fb provided that Γgg/M 7 10 . → → & & × (vQ . 100 GeV). For other representations these numbers will be different, however it is clear that for (1) Yukawa couplings the model may accommodate the observed excess while O predicting pp zz 4g and pp zz 2g2γ rates within reach of the LHC. Whether these signals are observable→ → will depend→ on the→ SM background, which is discussed in section 6.7.

6.3 Full computation beyond the LET approximation For large portions of the relevant parameter space the LET description may not be valid as either very large Yukawa couplings may be required (especially when the vector-like fermions are very massive MQ Mz) or the low-energy approximation breaks down, because MQ . Mz. This second problem can be solved by including the full one-loop result, which is at ﬁrst order in perturbation theory, so will still break down for large Yukawa couplings, but is all orders in the heavy fermion masses, allowing the study of scenarios with MQ Mz. The pair production of Higgs-like scalars at one loop from virtual fermions has been studied∼ for all fermion masses for some time [312,313] and later with QCD corrections [314–316]. As the full one loop expressions are lengthly we refer the reader to [313] where the relevant formulæ are conveniently presented. In ﬁg. 18 we show contours of constant pair production cross section for the benchmark

35 [] [] = = = =+ = = = =+

] ] [ [

- - - - - - > > - - - - - - - - - - - - - - / /

Figure 18: As in the narrow width case in ﬁg. 17, but for massive fermion loops with the full form factor included. The left panel assumes vanishing self coupling (κ = 0), and the right a Higgs-like self-coupling (κ = 1).

scenario, now with lepton mass ML = 400 GeV and quark mass MQ = 800 GeV, for the case of a narrow width. We show the results for vanishing self coupling (κ = 0), and for a Higgs-like self-coupling (κ = 1). As can be seen, including the full one loop fermion mass dependence leads to relevant quantitative differences from the LET approximation of fig. 16, while the qualitative aspects are similar. When it occurs, the breakdown of the description only occurs as large Yukawa couplings are required, as all mass effects are included.

6.4 Pair production of a pseudo-scalar resonance We may also consider the single and pair production of a pseudoscalar at one loop due to interactions with heavy vector-like fermions ¯ ¯ =y ˜z iγ5 + MQ . (54) L Q Q QQ In this case the single production cross section only differs from the scalar case by an additional factor of cP = 9/4. For pair production the cross section is identical to pair production of the scalar, with the additional simplification that the cubic coupling κ vanishes in the CP-symmetric limit. An example plot for the pseudoscalar is shown in fig. 19. In composite models z can be a pseudo-scalar analogous to the η in QCD: a Goldstone boson of an accidental global symmetry spontaneously broken by the new interaction that ˜ 2 2 becomes strong at ΛTC. Its linear and quadratic couplings are zGG and z G : the latter 2 2 operator breaks the global symmetry and thereby its coefficient is suppressed by Mz/ΛTC.

6.5 Decorrelating single and pair production From section 6.2 it may appear that in complete models a precise correlation between single and pair production operators is generically expected. On the other hand, we saw in section 6.4

36 [] = = =- =+

] [

- - - > - - - - - - - /

Figure 19: As in ﬁg. 18, but for a pseudo-scalar resonance. In the CP-symmetric limit the self-coupling vanishes.

that for a pseudoscalar the correlation between the two changes. This is related to the fact that, based on CP symmetry, one would only expect odd powers of z coupled to GGe, and even powers coupled to GG. However, the decorrelation of single and double production may be even greater in the presence of other global symmetries. To illustrate this let us consider a model where z is odd under a Z2 symmetry. In this case one would only expect even powers of z in the eﬀective theory, thus single production is forbidden. One can introduce single production, but this would be controlled by a parameter which breaks the Z2 and may thus be small. In this way, pair production may be enhanced relative to single production in the presence of additional approximate global symmetries.

As an example, consider the model of eq. (39) with two flavours of heavy quark Q1,2 with interactions ¯ ¯ ¯ ¯ LQ1,2 M1Q1Q1 + M2Q2Q2 + y1,2zQ1Q2 + y2,1zQ2Q1 . (55) ⊃ Using the LET and keeping the dependence of the fermion masses on z we obtain a contribution to the GG coupling (log(M+(z)/Λ)+log(M−(z)/Λ)) where M± are the two mass eigenstates, ∝ in the presence of a background z field value, and Λ is a high energy scale that drops out when expanding in powers of z. In the end the effective coupling to the GG operator is 2 y1,2y2,1z 4 α3 aµν a LLET = IrNr 0 z + (z ) G Gµν (56) × − M1M2 O 6π

Notably, the linear term is absent. This is not surprising, since eq. (55) exhibits a global Z2 ¯ ¯ symmetry under which z z and Q1Q2 Q1Q2. Had we included some couplings → − ¯ → −¯ which break the symmetry, such as κ1zQ1Q1, κ2zQ2Q2, then a linear term could be generated proportional to κ1,2. As these terms break a symmetry they may be naturally small. In this way, in more involved models for the z excess, it may be that pair production is larger than expected based on a na¨ıve extrapolation of the single production rate.

37 = = = ] [

[]

Figure 20: Single production cross section for a heavy resonance R for a different values of the effective operator scale ΛR, as defined in eq. (57). For reasonable values of ΛR, if the branching ratio BR(R zz) is not too small, then R could lead to significant contributions to the z pair production→ rate.

6.6 Resonant pair production Finally, a scenario in which pair production may be considerably enhanced is given by a heavy resonance R which is produced and subsequently decays to pairs of z, pp R zz. This possibility was discussed briefly in [3]. There are many different model possibilities,→ → thus for simplicity we will only consider R coupled to gluons and z as 2 g3 2 1 2 R RGµν + ARzzRz . (57) L ⊃ 2ΛR 2 In fig. 20 we show the single R production rate as a function of the resonance mass and the effective scale ΛR. This simple analysis demonstrates that, if the branching ratio BR(R zz) → is not too small, z pair production may be significantly enhanced in the presence of new heavy resonances. Pair production is also implied if z has SM gauge interactions (see e.g. [317]).

6.7 Pair production phenomenology Although we will not attempt a thorough collider analyses of the pair production signature, it is useful to consider the typical character of pair production events. In ﬁg. 21 we show the invariant mass distribution of the resonance pairs and the pT spectrum of each resonance when produced from gluon fusion for the LET result as well as for two benchmark masses for heavy vector-like quarks. Although the location of the peak of these distributions lies in approximately the same place, regardless of the mass of the vector-like fermions in the loop, we see that further details such as the height of the peak and the tails of the distributions can vary signiﬁcantly depending on the mass of the fermions in the loop. Thus, if we may be fortunate enough to see pair production in the next few years at the LHC, then with additional events it may be possible to gain some additional information on the nature of the coloured particles

38 M =0.8 TeV MQ =0.8 TeV Q

] ] LET - - LET [ [ M =1.2 TeV

Q MQ =1.2 TeV

[] []

Figure 21: Normalised invariant mass and pT spectra for pairs of resonances produced from gluon fusion. Spectra are calculated using the LET result (solid black), and for two benchmarks,

MQ = 800 GeV (dotted) and MQ = 1200 GeV (dashed). A vanishing self-coupling is assumed. The invariant mass spectrum is peaked around 200 GeV above the pair production threshold. The

pT spectrum is peaked near to 400 GeV, indicating that in pair production events this typical boost should be expected for the diphoton or digluon system which may be useful for additional background reduction.

in the loop. However, by the stage that pair production of a 750 GeV resonance had been observed one would expect any additional coloured particle to be observed directly, thus from this context pair production would provide a complementary probe of the coloured particles. So far we have considered only production cross sections and distributions. In order to observe pair production it is necessary to be able to discriminate the ﬁnal state over any SM background. The 4j ﬁnal state, already explored at √s = 8 TeV in the search [303], has a good chance to probe the region of models parameters space where double production is enhanced.

In fact we find QCD jet production dσ/dmjj 200 fb/GeV in the region of phase-space where ∼ mjj 750GeV for candidate resonances mass mjj defined as in [318]. This means a background ' of about 40 pb in a mass window mjj m < 100 GeV. | − z| Also the jjγγ final state, presently not investigated by the experiments, has very good chances to constrain the models. In particular the latter might have very low background rate and, in view of eq. (38), we estimate that a search in this channel at 13 TeV would yield a useful bound in interesting regions of parameters space already with present luminosity. For this process we find an irreducible background from pp jjγγ around 0.2 fb for mγγ m < → | − z| 50 GeV and mjj m < 100 GeV. Given that the SM background is dominated by q γ | − z| → radiation, we implemented the following cuts: ∆R > 0.4 for any final state partons pair; ηj < 2.5, ηγ < 2.37; pT,j > 150 GeV, pT,γ > 40 GeV, pT,γ > 30 GeV. | | | | 1 2 Finally, we remark that the 4γ final state is interesting, as it is prominent in scenarios in which the production from electroweak boson initial states is not negligible. Signals in this final state can reach and exceed fraction of fb, while the irreducible background is negligible for projected luminosity of the LHC. If other decay modes exist then additional signatures are possible. For example, if z can

39 decay to dark matter then a search for γγE/ T or jjE/ T may reveal evidence for pair production. Once again the visible ﬁnal state particles should reconstruct the resonant mass. In any case the kinematic characteristics described in this section may be used to aid the observation of pair production.

7 Summary

In this paper we have discussed a variety of observables that can be used to learn more about the nature of the digamma resonance z and extract its properties. Methods to discriminate a spin-0 from spin-2 diphoton resonance are already well established. In fact, the ATLAS and CMS collaborations already present analyses for both cases. Thus, from a phenomenological perspective, there is little to add on the topic of spin discrimination. For this reason we have focussed on the spin-0 hypothesis and on different open phenomenological questions. While our presentation was organised in terms of physical processes, in this section we summarise our results in terms of what can be learned from the different measurements and how these can be used to infer the properties of z. Our basic assumption is that z is a scalar particle and our conclusions are based on an Effective Field Theory (EFT) approach. Preliminary LHC data suggest that the neutral particle

z must have a rather large coupling to photons, corresponding to an eﬀective scale Λ/Mz < (10 20)cγγ. Taking into account that cγγ is induced at loop level, at least in a weakly-coupled∼ − theories the constraint on Λ/Mz restricts the applicability of the EFT theory and points towards the expectation that z is not an isolated particle, but part of a new-physics sector in the TeV domain. Whenever the EFT expansion breaks down, one must necessarily turn to a model-by- model analysis. Nevertheless, we believe that our EFT description can catch the main features of the underlying theory and that the observables discussed here are likely to bear fruit also in the context of complete models.

7.1 Identifying the weak representation

In this paper we have focused primarily on the case in which z is an electroweak singlet, but an important issue is to establish empirically if it is a singlet, a doublet with hypercharge

1/2, or belongs to some other higher representation of SU(2)L U(1)Y containing a neutral component. This can be tested as follows. ⊗

1. Higher representations will contain other components, some of which will be electrically charged, with mass around 750 GeV and small mass splittings induced by electroweak breaking effects. Their single production is model dependent and can affect the profile of the bump at 750 GeV in a measurable way. Pair production, via Drell-Yan processes, is universal and depends only on the electroweak quantum numbers.

2. Identifying the initial state of the production process will give indirect hints about the electroweak nature of z. While a singlet can couple at dimension 5 to all SM particles, a doublet is more likely to be produced from quarks, to which it may have renormalisable

40 couplings, than gluons or electroweak gauge bosons, to which can couple only through dimension-6 operators. However, this conclusion is model dependent.

3. The different dimensionalities of the EFT couplings to quarks (dimension 5 for singlet z and 4 for doublet) and gauge bosons (dimension 5 for singlet and 6 for doublet) imply different pT spectra in z associated production. Although the applicability of the EFT is limited for the calculation of pT distributions, the EFT can describe the onset of the different behaviours of singlet and doublet z. These results must then be compared with specific models.

Hereafter we will assume that z is an electroweak singlet.

7.2 Identifying the initial state

The identiﬁcation of the parton process that produces z can be done using the following considerations.

1. Employing the dependence of parton luminosities on √s, one can use the values of σ(pp → z) at diﬀerent energies as discriminators of the initial process. In particular, the ratio between σ(pp z) at 13 TeV versus 8 TeV, as summarised in section 2.1, is already → favouring production from gg, b¯b, cc¯, and ss¯ with respect to production from light quarks or photons.

2. Measuring decay channels other than z γγ will give information on possible production → mechanisms, since any initial state of the production process contributes to z decays. Di- jet measurements already constrain some parameter space for gluon production [3]. Other decay modes are illustrated in ﬁg.2.

3. The rapidity distribution and the transverse momentum spectrum of the diphoton system retain features of the initial parton state and can be used to discriminate between light- quark and gluon or heavy-quark initiated productions [37].

4. The rate of z+jet production is a useful discriminator of the initial state. The ratios σzj/σz, with the same acceptance cuts applied across all production channels, are shown in table1. This ratio is (27%) for gluon initiated production and (6 9%) for heavy and light quark initial states.O O −

5. For the b-quark initial state the ratio σzb/σz is 6.2%, whereas for all other initial states it is less than 1%, see table1. Thus, zb associated production is an excellent indicator of b-quark initial states [37].

6. z production in association with a gauge or Higgs boson is a useful discriminator, see table1. In particular, no vector bosons accompanying z are expected from gluon initial states, and no W from b initial states. Taking ratios of zW , zZ, and zh provides us with additional handles to identify the production subprocess.

41 7. If z is a singlet produced from quarks, the particular structure of the operator zqqH¯ implies a sizeable three-body decay width, Γ(z qqH¯ ) 1% Γ(z qq¯) where → ∼ × → H = h, Z, W ± (see section 3.2). { }

7.3 Measuring z couplings z couplings are crucial ingredients needed to understand the particle’s nature. 1. z couplings can be extracted from production rates, as functions of the total width. If z is suﬃciently broad that its width can be measured, than absolute determinations of its couplings are possible.

2. Furthermore, if the z resonance will turn out to be broad, by measuring its shape one could observe interference with the SM background qq¯ γγ amplitude in a way which, → in principle, allows us to probe the structure of z couplings to quarks. 3. In an EFT approach valid up to operators with dimension 7 or higher, the 4 decay channels z γγ, Zγ, ZZ, WW are described by 3 parameters (up to a discrete ambiguity) → for a CP-even z. This allows for one consistency condition that can be tested experimentally. For a CP-odd z, one needs only 2 parameters (up to a discrete ambiguity) and 2 consistency conditions are obtained. The results are shown in ﬁg.3 and ﬁg.4.

4. The z invisible width can be derived by tagging z production with an extra jet.

5. Associated production processes (z+ jet or z + V ) can be used to probe interactions at different momenta, testing the derivative structure of effective couplings. Moreover, they can test the substructure of the effective couplings, beyond the domain of the EFT. This is because the extra jet, gauge or Higgs boson can be attached to the internal particles generating the effective couplings.

6. Pair production gives direct information about the properties of the UV completion of the EFT and the couplings of z to the new sector. In the strong coupling regime, and in the presence of approximate symmetries, pair production becomes especially relevant.

7.4 Identifying the CP parity

We discussed ways of measuring the CP properties of z, in the three possible cases: CP even, CP odd, or undeﬁned CP in a theory with explicit CP violation (see also [231] for a recent thorough study aimed at addressing this question). The following three measurements rely only on z γγ decays, which are guaranteed if the discovery is conﬁrmed, and whose rates can then be→ unambiguously and model-independently predicted.

1. The z γγ decay guarantees the existence of z γ∗γ∗ `+`−`0+`0− with `, `0 = e, µ , → → → { } with a rate about 10−3 smaller than the γγ rate. The distributions have been computed in section 3.2 and they allow us to disentangle the cγγ andc ˜γγ contributions.

42 2. The z γγ decay guarantees the existence of z γγ e+e−e+e− events, where the → → → γ convert to electron-positron pairs in the detector matter, giving access to the photon polarisations [319]. The small angle between each e+e− pair however makes this measurement very diﬃcult.

3. The z γγ decay guarantees the existence of pp jjz events, where the jets are → → emitted at least from initial state partons. The rate is larger if z is produced via gg collisions. As discussed in section 4.1, the angular distribution of the two jets is sensitive to the CP parity of z.

The following measurements rely on z ZZ, Zγ, one of which (at least) is always guaranteed, although at the moment we cannot tell→ how large the corresponding rates will be.

4. The parity of z can be measured from the distribution of z ZZ `+`−`0+`0− events, similarly to what is done for the Higgs. → →

5. The z Zγ decay allows for a measurement of the z parity using Z `+`− and → → provided that γ converts into `+`− either virtually or in matter, as discussed above.

6. The zZZ and zZγ couplings imply a rate for pp zZ whose angular distribution → allows for a reconstruction of the CP-parity of z as discussed in section 5.1.

Finally we have a possibility which is not guaranteed by the diphoton decay, but which would be completely decisive.

7. If z decays into hh, its observation would immediately imply that z is a CP-even scalar, provided that CP is conserved.

7.5 Pair production In this paper we have emphasised pair production as a new tool for investigating the properties of z. It is not possible to make deﬁnitive model-independent predictions for pair production. The reason is that while single production already gives some information on possible dimension 5 couplings of z to SM states, pair production involves not only the same dimension-5 operators but also a priori uncorrelated dimension 6 operators and an z3 self interaction. Pair production cross sections are expressed in terms of a well-motived set of operators in section 6.1. A particularly simple, but representative, class of models provides a correlation between operators in terms of model parameters, see eq. (40). Furthermore, the model assumptions combined with the required single production rate allow for the pair production rates to be predicted. In addition, one is no longer constrained to an EFT description and rates may also be calculated when the new states are light. In section 6.2 a popular model of z coupled to charged and coloured fermions is studied. In both the low-energy theorem approximation and in the full calculation employing the one loop form factors, a number of qualitative conclusions can be drawn. Constraints on the production of pairs of di-jet resonances are already relevant,

43 excluding some of the parameter space. This suggests that in the future this final state may reveal evidence for the pair production of z. The γγgg final state is more challenging, with smaller cross sections. This is also true for the 4γ final state.

Acknowledgments This work was supported by the ERC grant NEO-NAT. JFK acknowledges the financial support from the Slovenian Research Agency (research core funding No. P1-0035). The work of RT is supported by Swiss National Science Foundation under grants CRSII2-160814 and 200020-150060. RT also acknowledges, for computing resources, Heidi, the computing support of INFN Padova, and the grant SNF Sinergia no. CRSII2-141847. We thank Clifford Cheung, JoséRamon Espinosa, Stefano Frixione, Riccardo Rattazzi, Michele Redi, Javi Serra for useful discussions.

A Eﬀective Lagrangian in the unitary gauge

The eﬀective Lagrangian for the scalar singlet in eq.s (3), (4), (5), (6) and (9) can be expanded in the unitary gauge as follows:

1 2 2 3 4 2 V (z,H) = m z + κ m z + λ z + κ H m zh(h/2 + v) + λ H z h(h/2 + v) , (A.1) 2 z z z z z z z

2 2 2 2 2 even z g3 a a µν g2 + − µν e µν e µν e µν L = cgg G G + cWW W W + cZZ ZµνZ + cγγ γµνγ + cγZ γµνZ 5 Λ 2 µν 2 µν 2 2 2 2 cψ ¯ cH e 2 2 2 † 2 cH 2 + (h + v)ψψ + 2 2 h + 2hv + 2v 2cWWµWµ + Zµ + ∂µ(h) √2 8cWsW 2 2 1 0 3 2 2 3 c 3 z(∂µz) c h h + 4h v + 6hv + 4v + z , −4 H Λ 2 (A.2)

2 2 2 2 odd z g3 a a µν g2 + − µν e µν e µν L = c˜gg G G˜ +c ˜WW W W˜ +c ˜ZZ ZµνZ˜ +c ˜γγ γµνγ˜ + 5 Λ 2 µν 2 µν 2 2 2 e µν c˜ψ 5 +˜cγZ γµνZ˜ + i (h + v)ψγ¯ ψ , (A.3) 2 √2

2 2 2 2 2 2 z (6) g3 a a µν (6) g2 + − µν (6) e µν (6) e µν (6) e µν L6 = c G G + c W W + c ZµνZ + c γµνγ + c γµνZ Λ2 gg 2 µν WW 2 µν ZZ 2 γγ 2 γZ 2 (6) c (6) 2 (6) ψ ¯ cH e 2 2 2 † 2 cH 2 + (h + v)ψψ + 2 2 h + 2hv + 2v 2cWWµWµ + Zµ + ∂µ(h) √2 8cWsW 2 (6) 2 1 (6)0 3 2 2 3 cH2 (∂µz) 4 c h h + 4h v + 6hv + 4v + h(h/2 + v) + (z ) , (A.4) −4 H Λ2 2 O

44 B z decay widths including the mixing with the Higgs

In this section we write down the complete formulæ for the decay widths of z including the eﬀect of 0 its mixing with Higgs boson h. The operators proportional to κzH and cH induce, after EWSB, a mixing between z and the Higgs given by 0 2 1 2 2 1 2 2 cH v Lmix = mH h m z hzv κ H m + , (B.5) −2 − 2 z − z z Λ where mH is the Higgs mass parameter (not yet the physical Higgs mass Mh). The mass matrix is diagonalised by the rotation

h h cos θ + z sin θ, z z cos θ h sin θ , (B.6) → → − with mixing angle 0 2 2v(m κ H + c v /Λ) tan 2θ = z z H . (B.7) m2 m2 z − H The masses of the physical h and z eigenstates are q 2 1 2 2 2 2 2 2 0 2 2 M = m + m (m m ) + 4v κ H m + c v /Λ , h 2 z H − H − z z z H (B.8) q 2 1 2 2 2 2 2 2 0 2 2 M = m + m + (m m ) + 4v κ H m + c v /Λ , z 2 z H H − z z z H where we have used uppercase letters to indicate the physical masses. After diagonalising the mass 0 mixing, all the couplings of z to SM particles acquire corrections of order cH v/Λ. Moreover, after mixing with the Higgs, z inherits the Higgs couplings to SM particles, suppressed by sθ sin θ. ≡ In order to include these contributions in the z interactions and decay widths, we parametrise the loop-induced Higgs couplings to gg, γγ, γZ as

2 2 2 g3ch,g 2 e ch,γ 2 e ch,γZ µν Lh−loop = hGµν + hFµν + hFµνZ . (B.9) v v vsWcW The value of the c coeﬃcients in the SM (and its dependence on the loop function) can be found, for instance, in [320]. The complete expressions of the widths taking into account all the subleading corrections are given by " # πα2M 3 2 z 2 2sθch,γ Λ 2 Γ(z γγ) = 2 cγγ cθ + +c ˜γγ , (B.10a) → Λ cγγ v " # 8πα2M 3 2 3 z 2 sθch,gΛ 2 Γ(z gg) = 2 cgg cθ + 2 +c ˜gg , (B.10b) → Λ cggv

2 " # ¯ NψMzv 2 2 ˜ Γ(z ψψ) = (cψcθ yψsθ) fψ (xψ) +c ˜ fψ (xψ) , (B.10c) → 16πΛ2 − ψ " # 2πα2M 3 2 z 2 ch,γZ Λ 2 Γ(z γZ) = 2 2 2 cγZ cθ sWcWsθ +c ˜γZ fγZ (xZ ) , (B.10d) → sWcWΛ − cγZ v πα2M 3 3vc c c (vc + 2Λs /c ) z 2 2 (TT ) 2 ˜(TT ) θ H ZZ θ θ H (LT ) Γ(z ZZ) = c c f (xZ ) +c ˜ f (xZ ) f (xZ ) s4 c4 Λ2 θ ZZ ZZ 4m2 → W W − z

45 2 2 2 v cH (vcθ + 2Λsθ/cH ) (LL) + 4 f (xZ ) , (B.10e) 128MZ 2πα2M 3 3vc c c (vc + 2Λs /c ) z 2 2 (TT ) 2 ˜(TT ) θ H WW θ θ H (LT ) Γ(z WW ) = c c f (xW ) +c ˜ f (xW ) f (xW ) s4 Λ2 θ WW WW 4m2 → W − z 2 2 2 v cH (vcθ + 2Λsθ/cH ) (LL) + 4 f (xW ) , (B.10 f ) 128MW " M 3 z 4 3 (6)0 2 4 2 (6)0 2 3 2 0 Γ(z hh) = 2 2v sθcH 4v cθsθcH 12Λv cθsθcH → 128πΛ4v2 M 2 2M 2 − − z − H +6Λv3c3c0 + v2c(6) 2s M 2 s2 2c2 2M 2 s2 + Λvc c M 2 c2 2s2 2c2M 2 θ H H θ z θ θ H θ θ H z θ θ θ H 2 2 2 2 2 − −2 2 2 2 − − +6Λ c M s 2Λvc c 3M s + 3ΛM vc c 3s + 12Λ M vc s κ θ H θ − θ z H θ z θ z θ z θ θ z #2 4Λ2M vc s2κ + 2Λ2M vc3κ 8Λ2v2c2s λ + 4Λ2v2s3λ f (x ) , (B.10g) − z θ θ zH z θ zH − θ θ zH θ zH h h

2 2 ˜ where xP = MP /Mz and the phase space functions f and f are given by

3/2 fψ(x) = (1 4x) , ˜ − fψ(x) = √1 4x , − 3 fγZ (x) = (1 x) , − f (TT )(x) = 1 4x + 6x2 √1 4x , − − f˜(TT )(x) = (1 4x)3/2 , (B.11) − f (LL)(x) = 1 4x + 12x2 √1 4x , − − f (LT )(x) = (1 2x) √1 4x , − 2 − fh(x) = (1 2x) √1 4x . − −

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