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Searching for new physics - Leaving no stone unturned (2019) Probing the Twin Higgs Mechanism at Collider Experiments

f/v =3 f1/v =3 2 2 5 5 5 1200 0.8 0.9 2 Higgs Coupling Sensitivity H Discovery

1 1 10 10 HL LHC 5 HL-LHC 2 Z 1100 ✏ ✏ HL LHC A HL LHC mis 1000 ET = 200 GeV Z 5 HL LHC 2 A

2 2 ) 10 10 200 400 600 800 1000 200 400 600 800 1000 900 m [GeV] mB [GeV] A GeV ( ATLAS DV f/v =5 f1/v =5 H 800 2 2 5 m 5 m0=25 GeV 5 2

1 1 700 10 5 10 2 ✏ ✏ 5 2 600 HL LHC HL LHC Z Z CMS ≥5σ HL LHC HL LHC Tuned A A 500 2 2 10 10 200 400 600 800 1000 200 400 600 800 1000 300 350 400 450 500 550 600 m [GeV] mB [GeV] A mT (GeV)

Figure 8: Projected contours the Z0 (blue) and A0 (red) at the HL-LHC for a 1 luminosity of 3000 fb for f(1)/v = 3(5) on the top (bottom) row. The THPM model FIG. 9: Discovery prospects for the heavy Higgs in the FTH setup at HL-LHC with 3000 is on the left and the T2HDM is on the right. The regions excluded by current LHC 1 fb of luminosity at the HL-LHC (14 TeV). The CMS reach is based on the prompt decay searches for the twin bosons are shaded in the corresponding color. H ZZ 4`, while the ATLAS reach is based on the H hh DV+X channel ! ! ! ! only slowlyCan with m , while theKılıç sensitivity to the Z0 is much weaker across the entire A0 described in section IV C. The dashed contour shows the increase in the parameter region parameter range. where 10 signal events pass all cuts when the Emiss cut is reduced to 200 GeV. The blue The result is that in the THPM model, it is typically the Z0 channel that can T be used to improve the limits on ✏ at the LHC and FCC, gaining a factor of a contours indicate the ratio of ( BR) of the light Higgs to that of the SM Higgs. few over the constraints shown in Fig. 7, whereas in the T2HDM the A0 drives the ⇥ sensitivity. This must be considered quite impressive, since for these resonances both the production cross section and the branching ratio into SM states scale roughly as mis ✏2. discovery potential to larger values of mH . The dramatic e↵ect of reducing the ET cut

In the models we consider, the vector boson sector can be speciﬁed by three to 200 GeV is also shown, with the sensitivity improving at still higher values of mH ,as parameters: f(1)/v, ✏, and either mB0 or mA0 . Therefore, measuring four or more independent observables can conﬁrm the underlying structure. The observables in well as higher values of mT . The DV-based discovery region is contained within the region question can be the masses of A0 and Z0 and their resonant dilepton production rates, which will be probed though Higgs coupling measurements indicated by the blue contours. ``,A0 and ``,Z0 . This is a goal that could be pursued at the LHC alone. Both vectors would appear as resonances in the dilepton spectrum. Then, measuring both masses Therefore, the complementary information provided by the Higgs coupling measurements and any one event rate would completely specify the parameters of the model. A and a discovery in the DV-based search can be combined to test the twin Higgs framework, and probe the twin fermion sector. –16–

ACKNOWLEDGMENTS

We are pleased to thank Zackaria Chacko, David Curtin, and Ennio Salvioni for advice and discussion. We also thank Laura Jeanty and Christian Ohm for their help understanding the ATLAS searches. The research of C.K. is supported by the National Science Foundation

21 This Talk Covers arXiv:1711.05300: Chacko, CK, Najjari, Verhaaren arXiv: 1812.08173: CK, Najjari, Verhaaren arXiv: 1904.11990: Chacko, CK, Najjari, Verhaaren

See also: arXiv: 1506.06141: Curtin, Verhaaren arXiv: 1811.05977: Bishara, Verhaaren arXiv: 1904.10468: Batell, Verhaaren Probing Naturalness

Naturalness of the electroweak breaking sector has been a major motivation for TeV scale physics.

So far, null results only → “Little hierarchy problem”

Neutral naturalness: Discovery at colliders is challenging.

Next step: If we discover new degrees of freedom, how can we probe their connection to the naturalness puzzle?

Adopt Twin Higgs Mechanism as a simple benchmark case. MODEL DETAILS Mirror TH vs Fraternal TH Chacko, Goh, Harnik Craig, Katz, Strassler, Sundrum hep-ph/0506256 1501.05310

Color x EW Color’ x EW’ Two scalar doublets 3 matter (hypercharge mixing) Twin matter generations

SM sector Twin sector

In MTH, twin matter content is identical to SM. In FTH, twin matter only contains 3rd generation. In this paper we study the prospects for the LHC and future colliders to discover the twin sector Higgs and determine the form of Higgs potential, thereby confirming the MTH framework. Discovery of the twin sector Higgs scalar at the LHC has been discussed pre- viously [18, 35, 44, 59, 60], but without the emphasis on determining the structure of the potential. We find that at the LHC, much of the range of parameter space in which the twin sector Higgs can be discovered is already disfavored by existing measurements of the couplings of the light Higgs. Only a restricted set of parameters lead to Higgs coupling deviations and a twin sector Higgs signal that can both be measured at the LHC. We find that the high energy stages of linear colliders such as the international linear collider (ILC) [61] or compact linear collider (CLIC) [62] are expected to have much greater reach for the twin sector Higgs. These colliders are also projected to measure the invisible width of the light Higgs to percent level precision [57, 58]. Combining the measurements of the twin sector Higgs with precision studies of the couplings of the light Higgs results in much greater ability to confirm the MTH construction.

The outline of this paper is as follows. In Sec. II we describe the scalar sector of the MTH in detail, and develop the notation we use in the rest of the paper. In Sec. III, we determine the reach of the LHC for the twin sector Higgs. In Sec. IV, we study the potential for the ILC and CLIC to discover the twin sector Higgs, and determine its couplings. We conclude in Sec. V.

II. THE SCALAR SECTOR OF THE MIRROR TWIN HIGGS

This section outlines the dynamics of the scalar sector in the MTH framework. We begin by analyzing the Higgs potential shown in Eq. (1). It is convenient to employ an exponential parametrization of the scalar degrees of freedom. Accordingly, we deﬁne an object H which transforms linearly under SU(4) U(1), ⇥ Basics of the Model 0 0 1 HA i 0 H = =exp ⇧ . (2) 0 1 f B 0 C HB ✓ ◆ B C B C @ A B f + C B C B p2 C Symmetry structure: Z2 (A⟷B) for the entire Lagrangian,@ A scalar sector has SU(4) global symmetry,6 with SU(2)A x SU(2)B subgroup gauged.

Symmetry breaking results in 1 heavy scalar (H), 1 physical light scalar (h), and massive W/Z/W’/Z’

The two physical scalars can mix → minimal portal GeV Higgs particle. As a consequence of the mixing it has suppressed couplings to SM ﬁelds, resulting in a production cross section that is smaller than the SM prediction. This mixing also results in a contribution to the Higgs width from decays into invisible twin sector states. Unfortunately, while these signals are robust predictions of the MTH framework, they are not unique to it. They are expected to arise in any model in which the SM communicates with a light hidden sector through the Higgs portal.

If, however, the Z2 symmetry is only softly broken, so that the Yukawa couplings in the two sectors are equal, the suppression in the Higgs production cross section and the Higgs invisible width are both determined by the mixing angle, leading to a prediction that can be tested by experiment [54]. This prediction does not apply to theories that exhibit hard breaking of Z2, such as the FTH, or MTH models in which the Yukawa couplings in the two sectors are di↵erent. The prediction can be understood as a consequence of the mirror nature of the model. Since it does not depend on the enhanced global symmetry of the Higgs sector, this prediction is not speciﬁc to the MTH construction, but applies more generally to any mirror model [55, 56] in which the discrete Z2 symmetry is only softly broken, so that the Yukawa couplings in the two sectors are equal. If the breaking of the global symmetry is realized linearly, the radial mode in the Higgs potential is present in the spectrum, and constitutes a second portal between the twin and SM sectors. We refer to this state as the twin sector Higgs. As we now explain, a study of the properties of this particle at colliders, when combined with precision measurements of the light Higgs, can be used to overdetermineSoft the Breaking form of the scalar potential, thereby conﬁrming that it possesses an enhanced global symmetry as dictated by the Twin Higgs Exact Z2 symmetry is ruled out experimentally (h couplings mechanism. SM-like).

In the case when the discrete Z2 symmetry is only softly broken, the Higgs potential of the MTH modelIntroduce takes the form soft1, breaking: divergences still under control.

2 2 V = µ H† HA + H† HB + H† HA + H† HB SU(4) and Z2 A B A B ⇣ ⌘ ⇣ 2 ⌘ 2 2 + m H† H H† H + H† H + H† H . (1) A A B B A A B B Z2 only ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ We distinguish the SM sector ﬁelds with the subscript A and the twin sector ﬁelds with B. The terms in the top line of Eq. (1) respect both the global SU(4) U(1) symmetry and the Symmetric setup means few parameters,⇥ possible to 1 We employ the notationoverdetermine of [7]. the system through measurements.

4 2 We see from the second equation that in the Z2 symmetric limit (m = 0) the mixing is Here f is the symmetry breaking VEV, and ⇧ is given, in unitary gauge where all the B maximal, with vEW = vB and ✓ = ⇡/4. This would mean that the observed 125 GeV sector pNGBs have been eaten by the corresponding vector bosons, by Higgs boson couples just as strongly to the twin sector as to the SM, which conﬂicts with current data. Therefore we need000m2 > 0,ih which1 corresponds to # < ⇡/4, to obtain realistic 0 1 000ih2 phenomenology. We have⇧ = assumed here that .> 0, which is required for(3) stable vacuum [7]. B C B 0000 C B C Combining the equations ofB motion we obtainC B ih1⇤ ih2⇤ 0 0 C B C @ A Expanding the exponential we obtain µ2 = f 2 (2 + ) . (11) h ph h f + sin † p p2 f 0 h†h ✓ ◆ ! 1 The mass eigenstates h and h+ are linear combinations of h and , H = B 0 C , (4) B C B ph†h C B f +h cos ✓ C✓ h B p2 cosf sinC B ✓ ◆ = ! C . (12) B 0 1 0 C 1 0 1 T @ h+ sin ✓ cosA✓ where h =(h1,h2) is the SM Higgs doublet. Proceeding to unitary gauge in the SM sector with h =0andh =(v + h)/p2leadsto@ A @ A @ A The1 mixing angle2 ✓ is given by 0 HA = 2f 2 sin(4#) , 4f 2 [ + cos2(2#)](5) 0 f + sin v+h 1 sin(2✓)= 2 p2 2 , pcos(22f ✓)= 2 2 . (13) @ ⇣m+ m⌘ ⇣ ⌘ A m+ m 0 HB = , (6) The mass eigenvalues m+ and0 fm+ arecos givenv+h by1 p2 p2f

@ ⇣ ⌘ ⇣ ⌘ A 2 and allowsWe us see to from write the the second potential equation as that in the Z2 symmetric limit (m = 0) the mixing is m2 =2f 2 + 2 + (2 + )cos2(2#) . (14) maximal, with vEW = vB and2 ✓ = ⇡/4. This would mean that the observed 125 GeV 2 ± 2 2 ± p2(v + h) V =f 1+ µh m cos i Higgs boson couples justp as2f strongly" to the twinp sectorf as to!# the SM, which conflicts with 2✓ ◆ We cancurrent express data. Therefore in terms we need ofm4m2 +>,0,m which,and corresponds#, to # < ⇡/4, to obtain realistic p2(v + h) + f 4 1+ + sin2 . (7) phenomenology. We have assumedp2f here" that2> 0, which isf required!# for stable vacuum [7]. ✓ 2 ◆1 2 2 2 2 2 2 Combining the equations of= motion we( obtainm+ m ) 4cot (2#)m+m . (15) It is convenientThe corresponding to define the angular couplings16 variablef of4 the# twinv/( sectorp2f). Higgs In terms are of given# and byf, the VEVs Scalar spectrum ⌘and couplings in the visible and twin sectors are given bym⇥2 2 m ⇤ 2 µ f=A f (2 + ) . fB (11) NoteThe that corresponding in order to keep couplingsh+f Af ofA0wemusthave the twinsin(# sector✓)+ Higgsf BfB are givencos(# by ✓) . (25) vEW vB p p Start with: vEW 2f sin #m,vB 2f cos # . m (8) The mass eigenstates h⌘ and h+ aref linearA ⌘ combinations of h andfB , We see from this that,h+ inf A general,fmA + sin( the# masses✓)+ off B thefB visiblecos( and# twin✓) . sector particles(25) are vEW vB The equations of motion for and h take the formcot(2#) + csc(2#) . (16) related by h m cos| ✓ sin ✓ | |h | We see from this that, in general,= the massesv2 of the. visible and twin sector(12) particles are Mass eigenstates:2 2 0 1 202 2 B 1 02 2 1 2 µ + m cos(2h#+)=fmB2(=sin+m✓A)cos2 ✓sin= m(2A#)cot, #. (9) (26) Sincerelated# < ⇡/ by4wecandroptheabsolutevaluesymbolstoobtain vEW @ 2 A @2 2 A @ A TheThe mixing couplings angle of✓ is the given light bym Higgs= hf ⇥2cos(2to visible#2). vB sector particles2 ⇤ 2 are related to(10) the corresponding mB = mA 2 = mA cot #. (26) Consistency condition m+ vEW vB mT couplings in the SM by 2f 2 sin(4#) cot # =4f 2 [ += cos2(2#.)] (17) onThe scalar couplings potential ofsin(2 the✓)= light Higgsmh7 ,tocos(2 visible✓)= sectorv particlesm are. related to the(13) corresponding 2 2 EW2 2 t m+ m g = g cos(m#+ ✓m), (27) h SM SM couplings in the SM by HereThemt massrepresents eigenvalues them+ massand m ofare the given top by quark and mT the mass of the its twin counterpart, while the corresponding couplings of the twin sector Higgs h+ are given by gh SM = gSM cos(# ✓), (27) theCouplings top partner. to SM This states: inequality2 2 places2 a lower bound2 on the mass of the twin sector Higgs while the correspondingm =2 couplingsf + g of the += twing(2 sector+sin()cos# Higgs(2✓#).) h. are given by (14) (28) ± ±h+SM SM + relative to the mass of the toph partner.p i 2 We can express in terms of m+, m ,and#, The couplings of the Higgs fields g toh+SM the= twingSM sectorsin(# are✓). also related to the corresponding(28) 8 SM couplings. The light2 Higgs1 couples2 2 2 to twin2 states2 as2 it does to SM states, but with the = (m+ m ) 4cot (2#)m+m . (15) The couplings of the Higgs16f 4 fields to the twin sector are also related to the corresponding replacement v v , and a factor of sin(# ✓). The couplings of h to twin states are EW ! B ⇥ ⇤ + NoteSM that couplings. in order The to keep light2 Higgs0wemusthave couples to twin states as it does to SM states, but with the again those of the SM, but with the replacement vEW vB, and a factor of cos(# ✓). replacement vEW vB, and a factor of sin(# ✓). The! couplings of h+ to twin states are ! m As detailed in the Appendix,+ loopcot(2 induced#) + csc(2 couplings#) . to pairs of photons or(16) gluons result again those of the SM, butm with | the replacement| | v|EW vB, and a factor of cos(# ✓). from a single tree level coupling between a Higgs and! the fermion or vector in the loop, As detailed in the Appendix, loop induced couplings to pairs of photons or gluons result Sinceleading# < to⇡/4wecandroptheabsolutevaluesymbolstoobtain the same cos(# ✓)orsin(# ✓) modification. However, the decays of the heavy from a single tree level coupling between a Higgs and the fermion or vector in the loop, m v m Higgs are also a function of the+ masscot # = of theB = Higgs.T . So, the decay widths of(17) the twin sector leading to the same cos(# m✓)orsin( # v✓EW) modification.mt However, the decays of the heavy 2 Higgs to visible sector states are those of a SM Higgs with mass m+ multiplied by sin (# ✓). Higgs are also a function of the mass of the Higgs. So, the decay widths of the twin sector Here mt represents the mass of the top quark and mT the mass of the its twin counterpart, 2 theHiggs top partner. to visible This sector inequality states places are those a lower of a bound SM Higgs on thewith mass mass of them twin+ multiplied sector Higgs by sin (# ✓). B. Decays of the Twin Sector Higgs relative to the mass of the top partner.

B.With Decays the couplings of the of Twin the twin Sector sector Higgs8 Higgs in hand, we are now in a position to compute

its branching ratios. From Fig. 1 we see that for m+

MIRROR TWIN HIGGS

DISCOVERY OF A SCALAR OR VECTOR(S) C. Fine Tuning in the Model

In the Twin Higgs framework, the global symmetry of the Higgs sector is ultimately what protects the mass of the light Higgs from large radiative corrections. Therefore, the SU(4) violating parameters m2 and in the Higgs potential, Eq. (1), must be small compared to their SU(4) invariant counterparts, µ2 and , in order for the visible sector Higgs to be naturally light. Dividing Eq. (10) by Eq. (11) we obtain

2 m cos 2# 2 = , (29) µ 2+ Discoverywhich shows thatof as longHeavy as the potentialHiggs is approximately SU(4) symmetric, and ⌧ corresponds to a Twin Higgs potential. HL-LHC

1000 0.6 0.7 0.8 0.9 0.95

σ (pp → h) Γ (h → SM) 900 SM Probe into scalar sector. 800 0.99 mh, v fix 2 out of 4

) 700 GeV parameters in the potential. ( +

m 600 h couplings to SM fermions 500 suppressed by cos(�), can be determined by precision 400

Higgs measurements. 300 300 400 500 600 700 800 900 1000

mT (GeV)

FIG. 3: The region with stable vacuum but with m+

To determine the ﬁne tuning in this model, note that the parameter µ2 receives radiative corrections from the top loop, just as the Higgs mass parameter in the SM does. Assuming

13 Parameters vs. Observables 1

0.50

B-sector 0.10 tt BR WW ZZ 0.05 hh

mT= 500 GeV

mT= 800 GeV

0.01 400 500 600 700 800 900 1000

m+ (GeV)

If Z2 symmetryFIG. 1: Branching is softly ratios broken, of of the heavy the twin same sector Higgs mixing scalar to variousangle SM also ﬁnal states appears in h couplings to twinand states. the twin sector.

Once H massof the two measured, sectors. However, potential this tuned region is doesfully not determined, correspond to a Twin rate Higgs-like is predicted.potential since it is not approximately SU(4) symmetric, as we show in Sec. IIC. Clearly, the potential in Eq. (1) is deﬁned by four parameters. The measured values

of vEW and mh = m already constrain this system. Measuring deviations in the Higgs In the SU(4) limit, H→VSMVSM (and hh) is not suppressed, good couplings to SM ﬁelds determines cos(# ✓). Currently, the LHC has measured some Higgs discovery channel. couplings to 10% accuracy [63], and is expected to reach 5% precision by the end of ⇠ ⇠ the high luminosity run [64]. Linear electron positron colliders can reduce the uncertainty

to better than 1%. Measuring the mass of the twin sector Higgs m+ would then completely determine the potential. The width and branching ratios of the heavy Higgs are then completely speciﬁed. Therefore, with the mass in hand, measuring the twin sector Higgs rate into one or more visible states constitutes a powerful test of the twin Higgs framework.

Our discussion till now has focused on the case in which the discrete Z2 symmetry is only softly broken. However, a small hard breaking of the discrete symmetry by the twin sector Yukawa couplings allows a simple resolution of the cosmological problems associated with the MTH framework. We therefore brieﬂy consider the implications of a hard breaking of the discrete symmetry in the Yukawa sector. Once the twin sector Yukawa couplings are allowed to vary, the invisible decay widths

11 H Discovery Prospects at the LHC

0.8 0.9 σ(pp→H)Γ(H→ZZ→4l) 0.99 -1 1400 L=3000 fb

s =100 TeV

1200 Higgs Coupling Sensitivity ) GeV

( 1000 s =33 TeV + M

ATLAS ≥2σ 800

s =14 TeV

600 CMS ≥2σ CMS ≥5σ

400 600 800 1000 1200 1400

mT (GeV)

FIG.Large 4: Exclusion part bound of onLHC the Twin discovery Higgs model region in the mass for plane H ofin twin tension top and twin sector Higgs forwith the Highexisting Luminosity Higgs LHC run,coupling as well as forconstraints. future hadron colliders with center of mass energies of 33 and 100 TeV. The extrapolation for the HL-LHC is made by ATLAS and CMS for the H ZZ 4` process. Blue contours denote variation in Higgs ! ! couplings, with the region to the left of 0.8 already excluded by LHC measurements. The orange region does not signiﬁcantly improve tuning compared to the SM. The extrapolation to future hadron colliders is estimated using Fig. 3 of Ref. [59]. See text for further details.

colliders, with center of mass energies of 33 and 100 TeV. In order to estimate the signal cross section in our model at future colliders, we scale up the cross section for the twin sector Higgs at 14 TeV (obtained as described above) by the appropriate ratio of PDF luminosities, using Fig. 3 of Ref. [70] for 33 TeV, and Table 2 of Ref. [71] for 100 TeV. We then compare the cross section (times branching ratio) thus obtained to Fig. 3 of Ref. [59], where the sensitivity at these colliders to heavy scalars decaying to ZZ was estimated, under the assumption that the background is primarily q-¯q initiated. As can be seen from Fig. 4, going to higher energy signiﬁcantly extends the region of parameter space where hadron colliders have sensitivity to the twin sector Higgs.

16 Future Lepton Colliders

ILC (1 TeV) and CLIC (1.5 TeV) as benchmarks

Higgs couplings can be probed to 1%

W-fusion dominates production

H→hh→4b decay mode has best S/B

+ + + e ⌫e e+ H e e W + Z H Z H

W Z

e ⌫e e e e Z FIG. 5: Dominant Higgs production mechanisms at lepton colliders. deviations in couplings of the light Higgs, completely speciﬁes the scalar potential of the MTH model. Then, a measurement of the rate of twin sector Higgs events into any SM ﬁnal state overdetermines the scalar potential, and constitutes a powerful test of this framework. For our analysis, we focus on benchmark scenarios motivated by the ILC and CLIC proposals. For the high energy ILC, which is a 1 TeV machine, we consider two benchmark 1 1 scenarios corresponding to 1 ab and 3 ab of integrated luminosity. The CLIC benchmark 1 corresponds to a 1.5 TeV machine with an integrated luminosity of 1.5 ab .Signalsare generated in MadGraph5 [70] and showered with Pythia8 [71]. We use the Delphes3

[72] detector simulator with the anti-kT clustering algorithm [73] and FastJet [74] library to simulate the detector. We use the Delphes card based on the ILC construction outlined in [75]. A simulation of the CLIC detector is not yet available, but is expected to be qualitatively similar. The branching ratios shown in Fig. 1 show that of the twin sector Higgs’ visible decays products, WW is the largest. However, the WW background is prohibitively large. Instead we focus on decays to di-Higgs, for which the background is orders of magnitude smaller, making the process h+ h h 4b very attractive. While the 4b ﬁnal state is dicult ! ! to extract from background at a hadron collider, the comparatively clean environment of a lepton machine is admirably suited to such a search. Fig. 5 displays the dominant Higgs production processes at lepton colliders. Our analysis employs the dominant Higgs production process at high energies, which is WW fusion. Our study required at least three jets, each required to have p >20 GeV and ⌘ < 2.5. In T | | addition, we demand 3 b-tags and that the jets reconstruct two on-shell Higgses, with on- shell window (75 GeV, 135 GeV). We considered invariant mass bins that contained at least 85% of the signal events in the four bins surrounding the peak that passed the previous cuts. These bins were taken to be several tens of GeV wide to accommodate the expected jet energy resolution. In Fig. 6 we see the results for both the ILC benchmarks. Comparing

17 H→hh→4b Search 1 this ﬁgure to the LHC results in Fig. 4, we ﬁnd that the reach of the ILC with 1 ab is 1 comparable to that of the HL-LHC. With 3 ab the ILC will be able to discover the twin Demand 3 b-tagged jets with pT>20 GeV, |�|<2.5 sector Higgs for a greater range of twin top and twin sector Higgs masses than the LHC. The higher energy and increased luminosity of the CLIC benchmark, allows even greater Reconstruct two pairs with 75 GeV< mjj < 135 GeV opportunity to discover the twin sector Higgs, as can be seen from Fig. 7.

1000 0.8 0.9 1000 0.8 0.9 1000 0.8 0.9 σ(ee→H veve)Γ(H→hh→bbbb) σ(ee→H veve)Γ(H→hh→bbbb) σ(ee→H veve)Γ(H→hh→bbbb) -1 s =1 TeV L=1000 fb-1 s =1 TeV L=3000 fb-1 s =1.5 TeV L=1500 fb

900 900 900 ≥2σ

800 Higgs Coupling Sensitivity 800 Higgs Coupling Sensitivity 0.99 800 ≥5σ 0.99 ) ) 0.99 ) GeV GeV GeV ( ( ≥2σ Higgs Coupling Sensitivity ( + + + M M M 700 ≥2σ 700 700

≥5σ

600 ≥5σ 600 600

500 500 500 300 400 500 600 700 800 900 1000 300 400 500 600 700 800 900 1000 300 400 500 600 700 800 900 1000 m (GeV) m (GeV) T T mT (GeV)

1 1 FIG. 7: Results for both the CLIC benchmark linear collider scenario for W fusion to FIG. 6: Results for both the ILC 1 ab (left) and 3 ab (right) benchmark linear collider heavy twin sector Higgs decaying to di-Higgs to 4 bs. As in Fig. 4, the blue contours scenarios for W fusion to heavy twin sector Higgs decaying to di-Higgs to 4 bs. As in Fig. indicate deviation in Higgs couplings, the region to the left of 0.8 excluded by current 4, the blue contours indicate deviation in Higgs couplings, the region to the left of 0.8 measurements. The gray region is does not provide a stable vacuum. excluded by current measurements. The gray region is does not provide a stable vacuum. V. CONCLUSION

Finally, we quantify the confidence with which the Twin Higgs mechanism can be con-The Twin Higgs mechanism protects the mass of the Higgs against radiative corrections firmed as follows. For a given parameter point, we calculate the uncertainty in the numberwithout requiring new particles charged under the SM gauge groups. In this framework, the light Higgs emerges as the pNGB associated with the breaking of a global symmetry, of observed events after the cuts described above (due to Poisson statistics), and we also and its mass is protected against quantum e↵ects by a combination of the global symmetry estimate the uncertainty in the expected number of events at that parameter point, whereand a discrete Z2 symmetry. If the breaking of the global symmetry is realized linearly, the leading contribution is the uncertainty in the value of sin2(# ✓) arising from Higgsthe radial mode of the Higgs potential, the twin sector Higgs, is present in the spectrum. This particle provides a new portal between the visible and twin sectors. We have shown coupling measurements. In particular, we assume that Z , the multiplicative factor that that, if the discrete Z2 symmetry is only softly broken, a measurement of the mass of the measures the deviation of the Higgs coupling to the Z-boson, can be measured with atwin pre- sector Higgs, when combined with precision precision measurements of the light Higgs, cision of 0.5% [64]. Combining the uncertainties in the number of expected and observedcompletely specifies the Higgs potential. The rates for twin sector Higgs events are then testable predictions of the Twin Higgs framework. This conclusion also applies to theories number of events, we arrive at the fractional uncertainty in the ratio of observed to ex- 19 pected events, centered around the value 1. The fractional uncertainty is plotted for the 1 1 ILC ( dt =3ab )andCLIC( dt =1.5ab ) benchmarks in figure 8. L L R R 18 Checking the Prediction

Assume that dominant uncertainty arises from Higgs coupling measurements.

900 900 0.2 σ(ee→H veve)Γ(H→hh→bbbb) -1 0.3 s =1 TeV L=3000 fb σ(ee→H veve)Γ(H→hh→bbbb) s =1.5 TeV L=1500 fb-1

800 800 ) ) 0.5 GeV GeV

( 700 ( 700 0.4 + +

m m 0.5 0.3 0.4

0.2 600 600

0.1

500 500 500 600 700 800 900 500 600 700 800 900

mT (GeV) mT (GeV)

FIG. 8: The uncertainty in the ratio of observed and expected events, centered around the 1 1 value 1, for the ILC ( dt =3ab )(left)andCLIC( dt =1.5ab )(right) L L benchmarks as a functionR of m+ and mT . See main textR for additional details.

that exhibit hard breaking of the Z2 symmetry by the twin sector Yukawa couplings, provided that this breaking is small enough that the correction to the overall width of the twin sector Higgs is small. While the high luminosity LHC can potentially discover the twin sector Higgs, linear colliders such as the ILC or CLIC have much better precision and greater reach, allowing them to test the Twin Higgs framework.

ACKNOWLEDGMENTS

We thank Jessie Shelton for helpful communication. SN would like to thank Aqeel Ahmed for useful discussions. ZC is supported in part by the National Science Foundation under Grant Number PHY-1620074, the Fermilab Intensity Frontier Fellowship and the Visiting Scholars Award #17-S-02 from the Universities Research Association. CK is supported by the National Science Foundation under Grant Number PHY-1620610. CBV is supported by Department of Energy Grant Number DE-SC-0009999. ZC would like to thank the Fermilab Theory Group for hospitality during the completion of this work. CK would also like to thank the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611, where part of this work was completed.

20 breaking of the discrete Z2 symmetry that relates the two sectors [41, 42]. An alter- native approach is to introduce hard breaking of the Z2 into the twin sector Yukawa couplings, thereby altering the spectrum of mirror states [43–46]. This a↵ects the number of degrees of freedom in the two sectors at the time of decoupling, allowing the cosmological bounds to be satisﬁed. Once this problem has been resolved, questions such as the nature of dark matter [43, 47], the origin of the baryon asymme- try [48, 49], the order of the electroweak phase transition [50], and the implications of the MTH framework for large scale structure [51], can be addressed. The cosmological challenges can also be solved by making the twin sector vector- like [52]. An even more radical approach is to simply remove from the theory the two lighter generations of mirror fermions and the twin photon, which do not play a role in solving the little hierarchy problem. This construction, known as the Fraternal Twin Higgs (FTH), gives rise to distinctive signatures at the LHC involving displaced vertices [53, 54], and admits several promising dark matter candidates [55–59].

In the MTH framework the discrete Z2 symmetry and the resulting gauge in- variance under [SU(2) U(1)]2 largely sequesters the SM fields from their twin ⇥ counterparts. The Higgs boson itself is the only low-energy portal between the two sectorsin the that framework is guaranteed in which by the the twin construction. photon is massive. However, We focus if the on UV the completion framework is weakly coupled, the radial mode associated with the breaking of the symmetry can in which the discrete Z2 symmetry is only softly broken. We determine the current beexperimental light enough bounds to probe on this at current scenario, and and future explore colliders the possibility [60–64 of]. discovering In the absence both of additionalthe twin new photon fields and [65 the], twin thereZ isat only the LHC, oneHypercharge other or at renormalizable a future 100 TeV Portal operator hadron consis- col- tentlider with such the as symmetries the Future Circular that can Collider link the in two hadron sectors; mode a (FCC-hh). kinetic mixing Although of the the SM The following term is dimension 4 and consistent with hyperchargephenomenology gauge of boson new vectorBµ with bosons its twin that mix counterpart with theB U(1)µ0 , Y of the SM has been the Z2 symmetry: throughly explored [68–76], the MTH framework introduces several novel features. A massive twin photon requires contributions✏ µ⌫ to the masses of the mirror gauge Bµ0 ⌫B . (1.1) bosons beyond those from electroweak2 symmetry breaking. This can be accomplished either by simply introducing a massHowever, term for a massless the twin hyperchargetwin photon ruled gauge out boson, (mixing or at Here Bµ⌫ is the usual Abelian field1 strength.loop). This operator has the e↵ect of giving theby SM extending quarks and the Higgs leptons sector. charges In our proportional analysis, we to consider✏ under both the twin cases. photon For the and first twin case, we simply include an explicit Proca mass term for the twin hypercharge boson, Z, which are denoted A0 and Z0 respectively.Case 1) Explicit mass term (Proca) 2 Apart from some discussion inm theB contextµ of dark matter [55, 57], previous 0 B0 B0 , (1.2) analyses have largely neglected this mixing,2 µ since it is not radiatively generated in the low-energy theory to at leastModel three-loop parameters order relevant [3, 33]. for An vectors✏ generated : f, �, mB’. at the where mB is a free parameter in the Lagrangian. Since a Proca mass term can be 0 Two observable masses and rates. four-loopobtained level from is the small Stückelberg enough to construction, remain consistent which is with unitary the very and renormalizable,strong bounds on theafter photon gauge mixing fixing, with this another theory is massless ultraviolet vector complete. [66, 67 The]; for Proca a twin mass electron can also of be mass 9 ofthought order MeV of as these arising bounds from the require VEV✏ of. a10 new . Higgs However, field that ultraviolet carries charge completions under of thetwin MTH hypercharge, based on but compositeness not under twin generically SU(2)L [22 introduce]. If this Higgs states field charged is heavy, under it will both SMdecouple and twin from gauge the low groups, energy with spectrum, the result but if that it carries✏ is expected only a small to receive charge under one-loop 2 3 correctionsU(1)Y0 , the of twin order photon, 10 –10 though . Such no longer large massless, values of will✏ can still only be light. be accommodated if the twinAlternatively, photon A0 ratheris massive. than include an explicit mass term for Bµ0 , the twin photon canIn acquire this paper a mass we consider from the the VEV collider of an signals additional arising Higgs from field kinetic in the mixing twin sector. between theFor hypercharge concreteness, gauge we consider boson of a the scenario SM and in which its twin the counterpart Higgs content in of the the MTH theory model, is extended to include two Higgs doublets in each of the SM and twin sectors [5, 18– 20]. The VEVs of the two doublets in the visible sector must be aligned to leave the photon massless, but the twin sector doublets are not so constrained. In particular, the VEVs of the two Higgs doublets in the–3– twin sector must be somewhat misaligned to ensure that the twin photon has a mass.

As a consequence of the coupling in Eq. (1.1), the massive Z0 and A0 mix with the hypercharge gauge boson. Therefore the SM quarks and leptons acquire charges under the twin photon and twin Z, which can therefore be searched for at colliders. At hadron colliders, production through light quarks followed by decay into dileptons is a particularly promising channel. We ﬁnd that values of the mixing parameter ✏ 2 as small as 10 can potentially be probed at the LHC and FCC-hh.

In the softly-broken Z2 scenario, a few physical parameters determine all the couplings in the model that are relevant for LHC searches. In particular, in the models we consider, the collider signals depend on only three parameters beyond those in the SM. Consequently, if more than three independent measurements can be performed, the parameters of the model are overdetermined, allowing these theories

–4– Figure 3: Branching ratios of the Z0 (left) and A0 (right) for the THPM model for ✏ =0.1andf/v =3(5)forthetop(bottom)row.Thelightbluecurveisforasingle SM lepton ﬂavor while the dark blue curve corresponds to all six quark ﬂavors.

tons. Note that for large values of mB0 , the branching fraction of A0 into dileptons is much smaller than the corresponding Z0 branching fraction. These hierarchies in the production cross sections and dilepton branching ratios translate into a much greater discovery potential for the Z0 at colliders in most of parameter space, as we show below.

2.2 Twin Two Higgs Doublet Model

We now turn to the twin two Higgs doublet model. We label the two visible sector

Higgs doublets as H1 and H2, and their twin counterparts as H10 and H20 . Since the photon is massless, the VEVs of H1 and H2 must2x2HDM be aligned. However, the VEVs of the doublets in the twin sector need to be misaligned if they are to give mass to both the neutral gauge bosons inCase the 2) twin Two sector. (twin) Higgs doublets. Additional physical The alignment of the two twinscalars doublet in our VEVssector can can be be made quantiﬁed heavy. Freedom in many to ways. choose degree of alignment in the twin sector. For instance, the quantity

0 2 H †H 0 | 1 2| , (2.6) 0 0 0 0 (H1†H1)(H2†H2)

Maximal alignment in gives massless dark photon. Choose maximal misalignment as benchmark. –8– Model parameters relevant for vectors: f1, f2, �. Two observable masses and rates as before. Spectrum and Limits - Proca mass

1000 mZ ✏ =0.1 5 mA ✏=0.1 10 ✏=0.1 104 f/v =3 f/v =3 800 mZ f/v =5 f/v =5 104 ) [fb] ) [fb] f/v 600 3 A Z 10

[GeV] 3 103 ! !

V 5 pp pp

m 400 ( ( 2 2 10 10 13 TeV 13 TeV 200 100 TeV 100 TeV 101 300 400 500 600 700 800 900 1000 200 250 300 350

200 400 600 800 1000 mZ [GeV] mA [GeV] mB [GeV] Figure 2: Production cross section of the Z0 (left) and A0 (right) in the THPM Figure 1: The mass eigenvalues for the twin neutral vector bosons asmodel a function at the LHC of and at a future 100 TeV hadron collider for ✏ =0.1andf/v =3 and 5. the Proca mass mB0 in THPM model for ✏ =0.1andf/v =3and5.

We find that, as expected,Define the Proca A’ mass (Z’) induces to mass be mixing the between lighterTo analyze the two the (heavier) experimental signals, mass we must transform eigenstate. from Eq. (2.1)toabasis vector bosons, diagonal in both mass and kinetic terms. This is accomplished by first performing a shift of B and then rotating into the mass basis. The details of this procedure are 2 2 2 1 c m sW cW m A0 0 0 W B0 B0 µ given in Appendix A, where we obtain the field transformation to leading order in AµInZ µ the limit2 of2 large2 2 mB’. , Z’ =(2.4) B’. A’ decouples from SM. 2 sW cW m m + s m Z0 ✏. The couplings of the diagonalized fields to SM and twin particles are provided B0 Z0 W B0 ! µ ! ⇣ ⌘ in Appendix B, again to leading order in ✏. However, this perturbative analysis 2 2 2 Here m (v0/v) m , where mZ is the mass of the Z boson in thebreaks SM. This down mass near mass degeneracies, where one of the twin sector gauge bosons Z0 ⌘ Z0 0 matrix for the neutral twin sector vector bosons can be diagonalizedbecomes by a closesimple in mass to the SM Z boson. Therefore the results we present have rotation of the fields, with the mixing angle given by been obtained by diagonalizing the system numerically. For small ✏, away from mass degeneracies, the mass eigenvalues are close to the values obtained by diagonalizing m2 s B0 2W sin 2 = . the matrix(2.5) Eq. (2.4). The mass eigenvalues as a function of mB0 , for the benchmark 2 2 2 2 2 2 values f/v =3and5and✏ =0.1, are shown in Fig. 1. We see that for large m , m m +4s m m B0 B0 Z0 W B0 Z0 the mass of the heavier Z0 is close to m , while that of the lighter A0 asymptotes to r⇣ ⌘ B0 (v0/v)mW . We label the lighter of the two mass eigenstates as Aµ0 , and the heavier one as Zµ0 . 2 2 After diagonalization the SM fermions acquire charges under the A0 and the Z0. Note that in the limit mB m , Eq. (2.5)impliesthat = ✓W . Essentially, 0 Z0 This allows the twin vector bosons to be produced and detected at colliders. The this undoes the weak mixing, meaning that for large mB0 , the mass eigenstates are, production cross sections of the A0 and Z0 at the 13 TeV LHC and at a 100 TeV to a good approximation, simply the W30 and the B0 with masses m 0 cW and mB futureZ hadron collider0 are shown in Fig. 2, for ✏ =0.1andf/v =3and5.Wesee respectively. In this limit the Z0, which has its mass set by mB0 , couples more that for small values of its mass the A0 cross section is large, but drops o↵ quickly strongly to the visible sector than the lighter A0, which decouples from the SM as as the mass increases. This occurs because, for large mB ,theA0 is almost entirely mB increases. Additional details are given in Appendix A. 0 0 composed of W , which does not mix directly with SM hypercharge. Consequently, as In the opposite limit, m2 m2 , the mixing angle tends to zero. The30 B0 Z0 ⌧ mB increases, the cross section plummets. In contrast, the production cross section 0 0 0 mass eigenstates are then, to a good approximation, simply Aµ and Zµ, with masses of the Z0 remains sizeable even for masses in the TeV range. m c and m respectively. In this limit it is the lighter eigenstate, the A0,which B0 W Z0 The corresponding branching fractions are shown in Fig. 3. As expected, decays couples more strongly to the visible sector, while the couplings of the heavier Z0 are to twin fermions dominate the width. Nevertheless, the branching fractions into SM suppressed by a relative factor of tan ✓ . W fermions can be large enough for discovery in a clean resonant channel such as dilep-

–6– –7– Spectrum and Limits - 2x2HDM

1200 mZ ✏ =0.1 ✏=0.1 m 3 6 ✏=0.1 f1/v =3 1000 A 10 10 f1/v =5 mZ f1/v 2 10 5 ) [fb] ) [fb] 10 800 3 A Z 5 101 [GeV] 4 ! 600 ! 10 V 0 pp pp 10 ( ( m 3 10 400 f1/v =3 13 TeV 13 TeV 1 10 f1/v =5 100 TeV 100 TeV 2 200 10 400 600 800 1000 150 200 250 300 350

mZ [GeV] mA [GeV] 200 400 600 800 1000 mA [GeV] Figure 5: Production cross section of the Z0 (left) and A0 (right) at the LHC and Figure 4: The mass eigenvalues for the twin neutral vector bosons asat a a function future 100 of TeV hadron collider in the T2HDM for ✏ =0.1andf1/v =3and

5. For mA0 = s2W mZ0 the alignment of Z0 is orthogonal to twin hypercharge. This the twin photon mass mA0 in the T2HDM for ✏ =0.1andf1/v =3and5. For a particular ratio of vev’s,closes the portalthe to theZ’ SM becomes sector, reducing production. W’3, which is where cos n✓ c . It is convenient to define m = m cot # where m is the Z W ⌘ nW Z0 Z0 Z0 mass in the SM and m = gf s . Expressed in terms of these variables,At the the mass intermediate value m 0 = s2W m 0 , corresponding to = ✓W ⇡/2, the orthogonalA0 2 W to the B’ and decouples fromA the ZSM. 3 matrix is given by (t tan 2✓ ) Z0 field rotates into W 0 , which means it is orthogonal to B0. As we see in Figs. 5 2W ⌘ W and 6, this causes the Z0 to completely decouple from the visible sector, suppressing 2 2 1 m m /t2W A0 the production cross section and branching into visible states. At the same point, of 0 0 A0 A0 µ Aµ Zµ 2 2 2 2 , (2.12) Unlike2 mthe/t2W Procam + m /t massZ0 case,course, the theA0 is perfectly A’ remains aligned with twin coupled hypercharge, and at so its large coupling to the A0 Z0 A0 2W ! µ ! ⇣ ⌘ visible sector is enhanced. which admits novalues massless state of for m m2 >B’0.. Bounds are stronger than the other case. A0 To analyze the experimental signals, we transform to a basis in which the mass From this point, the determination of the physical states and couplingsand kinetic in this terms of the A0 and Z0, as well as those of the SM photon and Z,are model proceeds just as in the THPM model, with an example of howdiagonal. the physical As in the THPM case, this is accomplished by first performing a shift masses depend on mA0 given in Fig. 4. The mixing angle that diagonalizesof the the SM mass hypercharge gauge boson B and then rotating into the mass basis, as matrix in Eq. (2.12)isnowgivenby discussed in in Appendix A. Once the kinetic and mass terms of the vector bosons

2 have been diagonalized, it is straightforward to calculate their couplings to fermions. m s4W sin 2 = A0 . The details(2.13) can be found in Appendix B. In Figs. 5 and 6 we plot the production 2 2 2 2 2 4 cross sections and branching fractions respectively of the A0 and Z0 in the T2HDM. m s + c4W m + s m Z0 2W A0 4W A0 r We see that the production cross section for the A0 is always larger than for the Z0. ⇣ ⌘ The branching fraction of the A to SM ﬁnal states is also greater than for the Z . In the limit m2 m2 ,themixingangle tends to ⇡/2, and the mass eigen- 0 0 A0 ⌧ Z0 These features, along with its lighter mass, enhance the prospects for A0 discovery 0 0 states become, to a good approximation, just Aµ and Zµ, with masses mA0 and mZ0 while inhibiting Z0 discovery. We also see a striking feature when mA0 = s2W mZ0 , respectively. In this limit the lighter eigenstate, the A0, couples morewhere strongly the Z0 is to orthogonal to twin hypercharge and its couplings to SM states vanish. the visible sector, while the couplings of the heavier Z0 are suppressed by a relative 2 2 factor of tan ✓W . In the opposite limit, m m , the mass eigenvalues are given A0 Z0 by m sin 2✓ and m / sin 2✓ . In this limit it is again the lighter eigenstate, the Z0 W A0 W 3 Existing Constraints A0, that couples more strongly to the visible sector, while the couplings of the heavier Z are suppressed by the same relative factor of tan ✓ . This is very di↵erent from 0 W In this section we determine the constraints on the vector boson sector of the MTH the THPM model, in which the couplings of the A0 to the visible sectormodel vanish from in the direct searches and from precision electroweak observables. These are limit that the Z0 is heavy.

–11– –10– The measured invisible width of the Z-boson is LEP =499.0 1.5MeV[77]. Inv ± The predicted value for the SM contribution is SM =501.3 0.6MeV[78]. Inv ± Therefore, the preferred central value and associated uncertainty of any potential new physics contribution are given by

= 2.2 1.6MeV. (3.1) ConstraintsInv ± In our models, there are two contributions to Inv,aLEPreduction due to the The measured invisible width of the Z-boson is Inv =499.0 1.5MeV[77]. change in the couplings to the neutrinos, and a strictly positive contribution± The predicted value for the SM contribution is SM =501.3 0.6MeV[78]. Resonantdue production, to Z decays toZ coupling kinematically deviations accessible twin states.Inv We find± that the Therefore,reduction the in preferred the SM width central dominates, value and so associated that the MTH uncertainty invisible Z ofwidth any poten- LEP:tial Z new iscoupling generically physics to contribution smallerelectrons than decreases, the are SM given prediction, byinvisible bringing it closer to the LEP decay channelsmeasurement. added. Consequently, this does not significantly constrain either model. = 2.2 1.6MeV. (3.1) The modification of the Z-bosonInv coupling ± to electrons in the MTH model, as For invisiblewell as width, to the twin first sector effect states dominates, also constrains no the parameter space. The LEP In our models, there are two contributions+ to Inv,areduction due to the significantbound bound. on the Z partial partial width width of Z to electronse e, given gives by [77 ] change in the couplings to the neutrinos,! and a strictly positive contribution bound. + Z e e =83.91 0.12 MeV. (3.2) due to Z decays to kinematically! accessible± twin states. We find that the Boundreduction Wefrom require inS and the this T SM partialparameters. width width dominates, to be within so the that 2 range the MTH from 83.67 invisible MeV toZ width is generically84.15 MeV smaller in our model, than which the SM excludes prediction, the regions bringing above the it closerblack dashed to the LEP Directmeasurement. searcheslines in Fig. for Consequently,7. dilepton resonances this does not basically significantly constrain either model. dominate over the constraints listed above. Electroweak precision data as encoded in the oblique parameters [79], in par- The• modification of the Z-boson coupling to electrons in the MTH model, as well asticular to the the twinT parameter, sector states also constrains also constrains the relevant the parameter parameter space. space. The The LEP mixing among the neutral vector bosons+ alters the relation between the Z bo- bound on the partial width of Z e e, given by [77] son mass and the W boson mass! in the SM. The leading order correction is given in Eq. (A.20). Using the+ notation of ref. [80], Z e e =83.91 0.12 MeV. (3.2) ! ↵T m2± =1 Z , (3.3) We require this partial width to be within2 the 2 range from 83.67 MeV to 1+↵T mZ0 84.15 MeV in our model, which excludes the regions above the black dashed Here mZ0 is the Z boson mass in the SM, and ↵ is the fine structure constant lines in Fig. 7. evaluated at mZ0 . ElectroweakWhile precision this e↵ect data is the as largest, encoded there in are the other oblique contributions parameters to both [79 the], in par- • S and T parameters from the reduction of the Higgs coupling to SM states. ticularThese the eT↵ectsparameter, have been also calculated constrains in the general the relevant case [81 parameter]. Applied to space. our The mixingmodels, among we theobtain neutral vector bosons alters the relation between the Z bo-

son mass and the W boson2 mass in the SM. The2 leading order correction is 3v ⇤UV v ⇤UV given in Eq. (A.20).T Using2 the2 notationln ,S of ref. [802 ],ln . (3.4) ⇡8⇡cW f(1) mZ0 ⇡ 6⇡f(1) mZ0 ↵T m2 While this contribution to T is smaller=1 than Eq. (Z3.3, ), it also has the opposite (3.3) sign, reducing the deviation. We use the current2 bound on the T -parameter 1+↵T mZ0

Here mZ0 is the Z boson mass in the SM, and ↵ is the ﬁne structure constant evaluated at mZ . 0 –13– While this e↵ect is the largest, there are other contributions to both the S and T parameters from the reduction of the Higgs coupling to SM states. These e↵ects have been calculated in the general case [81]. Applied to our models, we obtain

2 2 3v ⇤UV v ⇤UV T 2 2 ln ,S 2 ln . (3.4) ⇡8⇡cW f(1) mZ0 ⇡ 6⇡f(1) mZ0 While this contribution to T is smaller than Eq. (3.3), it also has the opposite sign, reducing the deviation. We use the current bound on the T -parameter

–13– Future Reach - HL-LHC

f/v =3 f1/v =3 2 2 5 5 5 2

1 1 10 10 HL LHC 5 2 Z ✏ ✏ HL LHC A HL LHC Z 5 HL LHC 2 A 2 2 10 10 200 400 600 800 1000 200 400 600 800 1000 m [GeV] mB [GeV] A f/v =5 f1/v =5 2 2 5 5 5 2

1 1 10 5 10 2 ✏ ✏ 5 2

HL LHC HL LHC Z Z HL LHC HL LHC A A 2 2 10 10 200 400 600 800 1000 200 400 600 800 1000 m [GeV] mB [GeV] A

Figure 8: Projected contours the Z0 (blue) and A0 (red) at the HL-LHC for a Proca (left): Z’ drives sensitivity1 at high mass while A’ decouples. luminosity of 3000 fb for f(1)/v = 3(5) on the top (bottom) row. The THPM model Possible to discoveris on the left and both, the T2HDM test is ontheory. the right. The regions excluded by current LHC searches for the twin bosons are shaded in the corresponding color. 2x2HDM (right): A’ drives sensitivity at high mass (Z’ couplings only slowly with mA0 , while the sensitivity to the Z0 is much weaker across the entire generically smaller).parameter range. Z’ discovery ruled out by constraints.

The result is that in the THPM model, it is typically the Z0 channel that can be used to improve the limits on ✏ at the LHC and FCC, gaining a factor of a

few over the constraints shown in Fig. 7, whereas in the T2HDM the A0 drives the sensitivity. This must be considered quite impressive, since for these resonances both the production cross section and the branching ratio into SM states scale roughly as ✏2. In the models we consider, the vector boson sector can be speciﬁed by three

parameters: f(1)/v, ✏, and either mB0 or mA0 . Therefore, measuring four or more independent observables can conﬁrm the underlying structure. The observables in

question can be the masses of A0 and Z0 and their resonant dilepton production rates,

``,A0 and ``,Z0 . This is a goal that could be pursued at the LHC alone. Both vectors would appear as resonances in the dilepton spectrum. Then, measuring both masses and any one event rate would completely specify the parameters of the model. A

–16– Future Reach - 100 TeV

f/v =3 f1/v =3

2 5

5 5 2 10 1 10 1 2 FCC Z ✏ ✏ 5 FCC 2 A FCC Z FCC A 5 2 2 2 10 10 200 400 600 800 1000 200 400 600 800 1000 m [GeV] mB [GeV] A f/v =5 f1/v =5 5 2 2 5 2 1 1 10 5 10 ✏ ✏ 5 2 5 2 FCC FCC Z Z FCC FCC A A 2 2 10 10 200 400 600 800 1000 200 400 600 800 1000 m [GeV] mB [GeV] A

Figure 9: Projected contours the Z0 (blue) and A0 (red) at a FCC hadron machine As before, discovery of 1both vectors possible in Proca mass for a luminosity of 3000 fb for f(1)/v = 3(5) on the top (bottom) row. The THPM case, butmodel not is onin the 2x2HDM left and the T2HDM is on the right. The regions excluded by current LHC searches for the twin bosons are shaded in the corresponding color.

measurement of the other rate then provides a test of the theory. From Fig. 8, we see that this is possible at the LHC for the THPM model. Al-

though some part of the parameter space where both the A0 the Z0 can be discovered is already ruled out by the current bounds on the Z0, a sizeable region remains. In the T2HDM, however, this is not the case. Although the LHC has excellent sensitiv-

ity to the A0, the regions where the Z0 can be discovered at the HL-LHC are already ruled out by current bounds on the A0. Before a FCC-hh machine begins taking data, it is very likely the tunnel will be used for a lepton collider. The FCC-ee is projected to measure some Higgs couplings to better than 1%, an order of magnitude beyond the HL-LHC [87]. In particular, the coupling between the Higgs and Z bosons may be measured to 0.3% at 95% conﬁdence [88]. We therefore expect that before the FCC-hh begins taking data,

precise measurements of the Higgs couplings will already have determined v/f(1) to high accuracy. In both the THPM model and the T2HDM, the hadron machine can then completely specify the twin vector boson sector by measuring the mass and

–17– H h

FRATERNAL TWIN HIGGS

DISCOVERY THROUGH DISPLACED VERTICES Removing the Lighter Quarks

Only third generation quarks: m > �QCD’

Lightest hadrons are glueballs.

12 glueball states, lightest one is 0++

(SM/Twin) top loops induce ggh vertices as usual.

G0 can mix with the Higgs - leads to DV.

Lattice: m0 ~ 6.8 �QCD’ (range of interest: 15-30 GeV) c⌧ FIG. 2: Contours of the log10 meter as a function of the glueball mass m0 and the twin top-quark mass mT . The solid (dashed) contours take the mass of the heavy Higgs to be 1000 (500) GeV. Near the lower bound of the heavy Higgs mass the decay lengths become arbitrarily long. c⌧ FIG. 2: Contours of the log10 meter as a function of the glueball mass m0 and the twin which leadstop-quark to mass mT . The solid (dashed) contours take the mass of the heavy Higgs to be 1000 (500) GeV. Near they2 lowertan bound# of the heavy Higgs mass the decay lengths become = sin(# ✓). (22) M 2 2v2 c⌧ Glueball PropertiesEW arbitrarily long. FIG. 2: Contours of the log10 meter as a function of the glueball mass m0 and the twin The dimension-6 operator in Eq. (18) allows glueballs to decay to SM particles through top-quark mass mT . The solid (dashed) contours take the mass of the heavy Higgs to be an o↵-shellGwhich Higgs.0 can leads decay The to decay through width mixing for G0 decaying with the to Higgs. two SM particles is given by [48] 1000 (500) GeV. Near the lower bound of the heavy Higgs mass the decay lengths become y2 tan # 2 =2 sin(# ✓). (22) arbitrarily1 long.y vEW2 2 B S 2 SM 2 M 2vEW (G0 SM) = 2 2 4⇡↵s F0++ h SM(m0), (23) ! 12⇡2 M 2 m m ! ✓  h 0 ◆ The dimension-6 operator in Eq. (18) allows glueballs[Juknevich to decay (2010)] to SM particles through which leads to B S B (B) (B)µ⌫ ++ 3 SM 2 where 4⇡↵ans F0 o++↵-shell=42⇡↵ Higgs.s 0 Tr TheGµ⌫ decayG width0 for G2.03mdecaying0 [39] and to twoh SM SM(m particles0)isthedecay is given by [48] y h |tan # | i⇡ ! with = sin(# ✓). (22) width of a SM Higgs with2 mass2 m0. Consequently, G0 has exactly the branching fractions M 2vEW 2 2  1 y v 2 SM 2 EW B S SM 2 of a SM Higgs with mass(G0 m0SM). Also, = because 2h SM2 (m0)2 m02 the total4⇡↵sGF00++width scalesh SM( asm0), (23) The dimension-6 operator in Eq. (18)! allows glueballs12⇡ to decay!M tom SMh ⇠ m particles0 through ! 7 ✓  4 ◆ m0. Based on Eq. (22) the width also scales like sin #. This[Curtin, means Verhaaren that the (2015)] glueball decay an o↵-shell Higgs. The decay width for G0 decaying to two SM particles is given by [48] B S B (B) (B)µ⌫ ++ 3 SM 2 lengths areandwhere quite 4 sensitive⇡↵s F0++ to=4 the⇡↵ glueballs 0 Tr massGµ⌫ G and the0 twin top2.3 mass.m0 [39] and h SM(m0)isthedecay h | | i⇡ ! 2 2 width1 of a SMy HiggsvEW with mass mB0.S Consequently,2 SM G20 has exactly the branching fractions (G TheSM)G0 =lifetime as a function of glueball4⇡↵ massF and++ twin top( massm ), is shown(23) in Fig. 2. We 0 Width depends2 2 strongly2 2 on m0 sand0 mT. h SM 0 ! 12⇡ M mh m0 SM! 2 use HDECAYof a✓ SM[56] Higgs to calculate with mass the SM◆m0. Higgs Also, width because for mh

1000 3 -1 0 -2 1 mH=1000 GeV cτ 900 Log10 Meter 5 2 800

4 mH=500 GeV ) 700 5

GeV -2 (

T 1 0 600 -1

M -3 4 2 500 -3 3

400

-4 300 5 10 15 20 25 30 35 40

m0 (GeV)

FIG. 2: Contours of the log c⌧ as a function of the glueball mass m and the twin The other glueballs10 meter are much longer-lived.0 top-quark mass mT . The solid (dashed) contours take the mass of the heavy Higgs to be 1000 (500) GeV. Near the lower bound of the heavy Higgs mass the decay lengths become arbitrarily long. which leads to y2 tan # = sin(# ✓). (22) M 2 2v2  EW The dimension-6 operator in Eq. (18) allows glueballs to decay to SM particles through an o↵-shell Higgs. The decay width for G0 decaying to two SM particles is given by [48]

2 2 1 y vEW B S 2 SM 2 (G0 SM) = 2 2 4⇡↵s F0++ h SM(m0), (23) ! 12⇡2 M 2 m m ! ✓  h 0 ◆ B S B (B) (B)µ⌫ ++ 3 SM 2 where 4⇡↵s F0++ =4⇡↵s 0 Tr Gµ⌫ G 0 2.3m0 [39] and h SM(m0)isthedecay h | | i⇡ ! width of a SM Higgs with mass m0. Consequently, G0 has exactly the branching fractions SM 2 of a SM Higgs with mass m0. Also, because h SM(m0) m0 the total G0 width scales as ! ⇠ 7 4 m0. Based on Eq. (22) the width also scales like sin #. This means that the glueball decay lengths are quite sensitive to the glueball mass and the twin top mass.

The G0 lifetime as a function of glueball mass and twin top mass is shown in Fig. 2. We use HDECAY [56] to calculate the SM Higgs width for mh < 125 GeV. Note that near the lower bound on the heavy Higgs mass the decay lengths become much longer as sin(# ✓) 10 Light Higgs Bounds

1200 1200 10 11 12 13 14 15 16 1100 1100

1000 1000

900 900 GeV GeV H H

m 800 m 800

700 700

600 600

500 500 300 350 400 450 500 550 600 650 300 350 400 450 500 550 600 650

mT GeV mT GeV recast of ATLAS [1811.07370] FIG. 3: Region of parameter space excluded by ATLAS exotic Higgs decay DV search [62] Searches for DV in muon system set the “sweet spot” for for glueball masses from 10 to 13 GeV on left and 14 to 16 GeV on the right. the decay length → glueball mass window. withSearches the experimental in tracker exclusion promising curve. Glueballs for heavier with mass glueballs. below 10 GeV have decay lengths that are too long for to be excluded, while those with masses above 16 GeV have decay lengths that are too short to produce enough events in the muon system.

With the added luminosity of the remaining LHC runs, the sensitivity to displaced exotic Higgs decays will increase. In ref. [63] these muon system searches were extrapolated through the high luminosity run (HL-LHC). In Fig. 4 we apply these extrapolations to the FTH parameter space using the same procedure as in Fig 3. Once again, since only benchmark masses of 10, 25, and 40 GeV were used by the experimental analysis, we approximate the sensitivity to intermediate glueball masses by using the experimental results of the closest mass benchmark. We ﬁnd that the heavier glueballs are not expected to be constrained for low mT . As we will soon demonstrate, these regions of lighter mT and heavier m0 can be probed by relying on the heavy Higgs.

In ref. [39], LHC projections of displaced Higgs decays were made for other search strategies. These searches use the tracker rather than the muon system, providing access to shorter decay lengths, and therefore heavier glueballs; see also [40]. While the projected sensitivity appears promising, these types of searches have not yet been implemented experimentally, and no detailed estimation of backgrounds is currently available. Of course, the introduc- tion of new instrumentation and search strategies, such as [64] may well yield even stronger results than these extrapolations suggest. The sensitivity of future lepton colliders have also been shown to have great potential to discover these displaced Higgs decays [65].

13 Light Higgs Bounds - HL-LHC

1200 10 20 21 22 1100

1000 23 900 GeV

H 24

m 800 25 700

600 Unphysical Tuned 500 300 400 500 600 700 800 900 1000

mT GeV

FIG. 4: Projected regionrecast of of parameter Coccaro space et excludedal. [1605.02742] by muon system DV searches [63] from Higgs decays to glueballs of mass 10 GeV as well as masses in the [20,25] GeV range. For the deﬁnition of the tuned and unphysical regions, see section II.

parameter space using the same procedure as in Fig 3. Once again, since only benchmark masses of 10, 25, and 40 GeV were used by the experimental analysis, we approximate the sensitivity to intermediate glueball masses by using the experimental results of the closest mass benchmark. We ﬁnd that the heavier glueballs are not expected to be constrained for low mT . As we will soon demonstrate, these regions of lighter mT and heavier m0 can be probed by relying on the heavy Higgs.

In ref. [39], LHC projections of displaced Higgs decays were made for other search strategies. These searches use the tracker rather than the muon system, providing access to shorter decay lengths, and therefore heavier glueballs; see also [40]. While the projected sensitivity appears promising, these types of searches have not yet been implemented experimentally, and no detailed estimation of backgrounds is currently available. Of course, the introduc- tion of new instrumentation and search strategies, such as [64] may well yield even stronger results than these extrapolations suggest. The sensitivity of future lepton colliders have also been shown to have great potential to discover these displaced Higgs decays [65].

14 Heavy Higgs Bounds

b

b h

H

h G0

G0 FIG. 5: Diagram for the H hh DV+X channel. One light Higgs decays visibly while ! ! mis theDirect other decayssearches: to glueballs, H → leadinggBg toB aBR DV, is visible small, objects but and highET formultiplicity triggering in the helps. Triggering is main challenge.event. No exclusion for FTH beyond existing bounds.

Prompt b-jets pT >25 GeV, ⌘ < 3 H → (h→bb)(h→gBgB) provides visible| | energy, can trigger DV n 5 with p >1 GeV, m > 10 GeV with single DV. H →Tracks hh has goodT Track BR (~1/7).DV mis Event ET > 130 GeV

TABLE I: Preselction cuts for displaced vertices in the search of [71]. In ﬁnal selection the mis ET cut is increased to 250 GeV.

FeynRules [73] and MadGraph interfaced with Pythia8[74]tosimulatetheG0 decays, reading o↵ the ﬁnal state tracks from the Pythia event record. After boosting the decaying

G0 particles according to the kinematic distributions described above, we impose the selection cuts of Table I. The Emis 250 GeV requirement can be satisﬁed, if only ineciently, T due to the second glueball that escapes the detector and carries away invisible momentum.

In Fig. 6 we show the contours (in red) of the fraction of events where the DV takes place in the tracker volume speciﬁed in [71], and where the tracks meet the selection requirements as a function of mH and m0. For nearly the entire parameter region, this fraction is above &90% for production through gluon fusion (left) and VBF (right). In plotting the blue contours, the nTracks requirement is increased from 5 to 6, which according to [71] leads to an additional background reduction by more than 50%. A potential concern for the FTH setup is a reduction in the DV reconstruction eciency due to the displaced nature of b- meson decays. However, this is expected to be a small e↵ect for the b-meson boost factors corresponding to the range of glueball mass we consider.

16 Heavy Higgs Bounds

1200 5 1 mis 1200 mis 1200 mis σGF+VBF(H)→hh→bb+ET + DV σGF+VBF(H)→hh→bb+ET + DV σGF+VBF(H)→hh→bb+ET + DV -1 -1 -1 m0=20 GeV 14 TeV 3000 fb m0=25 GeV 14 TeV 3000 fb m0=30 GeV 14 TeV 3000 fb 1100 1100 1100 b 10 1 1 1000 1000 1000

10 10 ) ) b ) 3 900 h 900 900 5 GeV GeV GeV ( ( ( H H H M 800 H M 800 M 800

700 h 700 700 3 G0 600 600 600 Tuned Tuned Tuned 5 3 500 500 500 350 400 450 500 550 600 650 700 350 400 450G0 500 550 600 650 700 350 400 450 500 550 600 650 700 FIG. 5: Diagram for the H hhmT (GeV) DV+X channel. One light HiggsmT decays(GeV) visibly while mT (GeV) ! ! the other decays to glueballs, leading to a DV, visible objects and Emis for triggering in the FIG.complementary 7: Contours of the numberto sensitive of theT signal region event of count h → afterDV all searches! acceptance and event. 1 eciency factors have been taken into account, with 3000 fb of luminosity at the

HL-LHCPrompt b-jets (14 TeV). In thep > left,25 GeV, middle,⌘ < 3 and right plots, the mass of the G0 state is taken to T | | Recast of [1710.04901] (ATLAS) DV be 20, 25,nTracks and 305 with GeVpT Track respectively.>1 GeV, mDV In> 10 the GeV orange shaded region, the tuning is not mis MET > 250 in final selection Eventimproved over the SM,ET while> 130 the GeV blue shaded region is in tension with Higgs coupling measurements. TABLE I: Preselction cuts for displaced vertices in the search of [71]. In final selection the mis ET cut is increased to 250 GeV. The escaping glueball typically has enough transverse energy to satisfy the preselection mis mis FeynRules [73]cut andETMadGraph> 130 GeV,interfaced while with thePythia full8[74]tosimulatethe selection cut of EGT0 decays,> 250 GeV presents a significant reading o↵ thechallenge, final state tracks requiring from the largerPythiaHeventmasses record. and After hence boosting lower the decaying production cross sections. In Fig. 8 G0 particles according to the kinematic distributions described above, we impose the selec- we show how relaxing the Emis cut changes the discovery contour (for 10 signal events after tion cuts of Table I. The Emis 250 GeV requirementT can be satisfied, if only ineciently, T due to the secondall cuts) glueball for that the escapes three the glueball detector mass and carries benchmarks. away invisible The momentum. outermost region corresponds to the

In Fig. 6 wepreselection show the contours cut, (in and red) theof the innermost fraction of events to the where final the DV selection takes place cut used in [71]. We see that in the trackerreducing volume specified the cut in [71], threshold and where even the tracks mildly meet can the selection substantially requirements increase the discovery reach. Of as a function of mH and m0. For nearlymis the entire parameter region, this fraction is above course, relaxing the ET cut will increase the background rate. However, since the cut value &90% for production through gluon fusion (left) and VBF (right). In plotting the blue used in [71] was optimized for a very di↵erent signal model, a lower cut may well yield an contours, the nTracks requirement is increased from 5 to 6, which according to [71] leads to mis an additionalenhanced background discovery reduction by reach more than for the 50%. FTH A potential scenario. concern As for we the saw FTH in Fig. 6, relaxing the ET cut setup is a reductioncan also in the be DV combined reconstruction with e a moreciency stringent due to the displaced cut on the nature number of b- of tracks emerging from the meson decays. However, this is expected to be a small e↵ect for the b-meson boost factors DV for reducing the background, without a significant reduction in the signal rate. corresponding to the range of glueball mass we consider.

16

D. Exploring the Twin Sector

The displaced glueball decays o↵er more than a new discovery channel for the heavy Higgs boson. They can also be used to measure parameters of the underlying microscopic physics,

18 Heavy Higgs Bounds

Sensitivity can be expanded by lowering MET cut.

This can be compensated by stronger cut on nTracks.

1200 mis 1200 mis 1200 mis σGF+VBF(H)→hh→bb+ET + DV σGF+VBF(H)→hh→bb+ET + DV σGF+VBF(H)→hh→bb+ET + DV -1 -1 -1 m0=20 GeV 14 TeV 3000 fb m0=25 GeV 14 TeV 3000 fb m0=30 GeV 14 TeV 3000 fb 1100 10 Signal Events 1100 10 Signal Events 1100 10 Signal Events

1000 1000 1000

) 900 ) 900 ) 900 GeV GeV GeV ( ( ( H H H

M 800 M 800 M 800

700 700 700 250 250 mis 250 mis 230 mis ET Cut in GeV ET Cut in GeV ET Cut in GeV 230 230 200 600 600 600 180 200 Tuned 200 Tuned 130 Tuned 180 180 130 130 500 500 500 300 400 500 600 700 300 400 500 600 700 300 400 500 600 700

mT (GeV) mT (GeV) mT (GeV)

Recast of [1710.04901]1 (ATLAS) FIG. 8: Regions with more than ten signal events with 3000 fb of luminosity at the mis HL-LHC (14 TeV) for a range of ET cut thresholds. In the left, middle, and right plots,

the mass of the G0 state is taken to be 20, 25, and 30 GeV respectively. In the orange shaded region, the tuning is not improved over the SM.

and test the self-consistency of the Twin Higgs mechanism. In ref. [38] a similar strategy was proposed that relies on extracting parameters in the scalar potential from prompt decays of the heavy Higgs. The discovery of displaced glueball decays would make it possible to extend the consistency check to the twin fermion sector.

In particular, measuring the glueball mass and lifetime allows one to ﬁt two otherwise undetermined model input parameters. The glueball mass is proportional to the twin QCD scale, which for simplicity we take to be equivalent to the Landau pole in the one-loop B A B running of ↵s . Assuming identical initial conditions for ↵s and ↵s at ⇤UV,theLandaupole is in turn determined by the twin quark masses, with the twin top mass related to the SM top mass by a factor of tan #. If measurements similar to those described in ref [38] are performed at the HL-LHC that determine the relevant parameters in the twin scalar sector, then a measurement of the glueball mass adds an indirect probe of the UV completion scale of the Twin Higgs model. Furthermore, with the glueball mass and the parameters in the twin scalar sector known, Eq. 23 can be used to predict the lifetime of the glueball, which can be tested against the experimental measurement of c⌧, providing a nontrivial check of the Z2 symmetry structure in the underlying model. The full symmetry structure of the

Twin Higgs setup is deﬁned not only by the Z2 symmetry, but also the approximate SU(4) global symmetry of the scalar sector. This latter symmetry plays an essential role in setting the H hh branching ratio, and therefore a comparison of the expected and observed ! 19 Post-Discovery

Glueball mass and lifetime measurement would allow extending consistency checks for the Twin Higgs mechanism to the fermion sector.

Glueball mass gives handle on mT and UV scale.

Combining with measurements of prompt H channels leads to one nontrivial consistency check (SU(4) as well as Z2).

Challenges: Precision in glueball mass measurement (HCAL), theory fudge factors (improve through lattice?) Summary

f/v =3 f1/v =3 2

2 1200 0.8 0.9 5 5 5 2 Higgs Coupling Sensitivity H Discovery

1 1 HL-LHC 10 10 HL LHC 1100 5 2 Z ✏ ✏ HL LHC A 1000 Emis= 200 GeV HL LHC T Z 5 HL LHC 2 ) A 2 2 900 10 10 200 400 600 800 1000 200 400 600 800 1000 GeV m [GeV] ( mB [GeV] A ATLAS DV H 800 f/v =5 f1/v =5 m m =25 GeV 0 2 2 5 5 5 2 700

1 1 10 5 10 2 ✏ ✏ 5 2 600 CMS ≥5σ HL LHC HL LHC Tuned Z Z 500 HL LHC HL LHC A A 300 350 400 450 500 550 600 2 2 10 10 200 400 600 800 1000 200 400 600 800 1000 m (GeV) m [GeV] T mB [GeV] A FIG. 9: Discovery prospects for the heavy Higgs in the FTH setup at HL-LHC with 3000 Figure 8: Projected contours the Z0 (blue) and A0 (red) at the HL-LHC for a luminosity of 3000 fb 1 for f /v = 3(5) on the top (bottom) row. The THPM model 1 (1) fb of luminosity at the HL-LHC (14 TeV). The CMS reach is based on the prompt decay is on the left and the T2HDM is on the right. The regions excluded by current LHC searchesNeutral for the twin naturalness bosons are shaded in the correspondingis an increasingly color. appealingH ZZ paradigm.4`, while the ATLAS reach is based on the H hh DV+X channel ! ! ! ! described in section IV C. The dashed contour shows the increase in the parameter region only slowly with m , while the sensitivity to the Z0 is much weaker across the entire miss A0 where 10 signal events pass all cuts when the E cut is reduced to 200 GeV. The blue parameterThe range. Twin Higgs mechanism has a small number of parameters. MultipleT contours indicate the ratio of ( BR) of the light Higgs to that of the SM Higgs. measurementsThe result is that in the THPM model,can it test is typically consistency the Z0 channel that can relations. ⇥ be used to improve the limits on ✏ at the LHC and FCC, gaining a factor of a few over the constraints shown in Fig. 7, whereas in the T2HDM the A0 drives the mis sensitivity. This must be considered quite impressive, since for these resonances both discovery potential to larger values of mH . The dramatic e↵ect of reducing the ET cut the production cross section and the branching ratio into SM states scale roughly as Opportunities: h coupling measurements,to H 200 mass GeV is also and shown, rate, with the optionally sensitivity improving dilepton at still higher values of m ,as ✏2. H

resonancesIn the models we consider, and the vector displaced boson sector can bevertices speciﬁed by three from glueballswell as higher values in ofhm andT . The H DV-based→hh discovery channels. region is contained within the region parameters: f /v, ✏, and either m or m . Therefore, measuring four or more (1) B0 A0 which will be probed though Higgs coupling measurements indicated by the blue contours. independent observables can conﬁrm the underlying structure. The observables in question can be the masses of A0 and Z0 and their resonant dilepton production rates, Therefore, the complementary information provided by the Higgs coupling measurements

``,A0 and ``,Z0 . This is a goal that could be pursued at the LHC alone. Both vectors would appear as resonances in the dilepton spectrum. Then, measuring both masses and a discovery in the DV-based search can be combined to test the twin Higgs framework, and any one event rate would completely specify the parameters of the model. A and probe the twin fermion sector.

–16– ACKNOWLEDGMENTS

We are pleased to thank Zackaria Chacko, David Curtin, and Ennio Salvioni for advice and discussion. We also thank Laura Jeanty and Christian Ohm for their help understanding the ATLAS searches. The research of C.K. is supported by the National Science Foundation

21 Backup Slides of both the light Higgs and the heavy Higgs are a↵ected. In the case of the light Higgs, by measuring the total rate into both visible and invisible ﬁnal states, it is still possible to extract cos(# ✓). In the case of the twin sector Higgs, however, without any knowledge of the branching ratio into the twin sector, it is no longer possible to predict the rate into visible states. However, for large twin sector Higgs masses, unless the hard breaking of the discrete symmetry by the Yukawas is very large, the primary decay modes are expected to be the same as in the soft breaking case. This can be seen in Fig. 2, where the invisible branching ratio of the twin sector Higgs in the FTH model has been plotted against m+ for two di↵erent values of the twin bottom Yukawa coupling, and compared against the invisible branching ratio in the MTH model. We see that once m+ >mT ,thedi↵erent curves quickly converge towards 3/7, the theoretical prediction. We conclude that for heavy twin sector Higgs bosons, if the hard breaking of the discrete symmetry is small, the predictions of the MTH continueHard to hold to aBreaking good approximation. of the Z2

1

mT =500 GeV 0.50 mT =800 GeV ) H ( 0.10 Inv BR Γ

0.05

MTH

FTH yb/3

FTH 3 yb

0.01 400 500 600 700 800 900 1000

m+ (GeV)

FIG. 2: Branching fraction of the twin sector Higgs to twin states as a function of its mass For mH>mT, MTH predictions are accurate. for the MTH (solid) and FTH (dashed and dotted) models. The blue (red) corresponds to a mirror top mass of 500 (1000) GeV. In the FTH model the twin bottom Yukawa varies from one third the SM value (dashed line) to three times the SM value (dotted line).

12 A’/Z’ Branching Fractions (Proca mass)

0 0 10 f f 10 f f f/v =3, ✏=0.1 f/v =3, ✏=0.1 ) 1 ) 1 10 10 W W XX XX 10 2 10 2 qq(all) ! ! Ah A Z 3 3 qq(all) ( ( 10 (1 ﬂavor) 10 WW (1 ﬂavor) BR BR 4 4 10 Zh 10 WW

400 600 800 1000 140 160 180 200 220 240

mZ [GeV] mA [GeV] 0 0 10 f f 10 f f f/v =5, ✏=0.1 f/v =5, ✏=0.1 ) 1 ) 1 10 10 W W XX XX 10 2 10 2 qq(all) ! ! qq(all) Ah A Z 3 3 ( ( 10 10 (1 ﬂavor) (1 ﬂavor) BR BR WW 4 4 10 Zh 10 WW Zh

500 600 700 800 900 1000 150 200 250 300 350

mZ [GeV] mA [GeV]

Figure 3: Branching ratios of the Z0 (left) and A0 (right) for the THPM model for ✏ =0.1andf/v =3(5)forthetop(bottom)row.Thelightbluecurveisforasingle SM lepton ﬂavor while the dark blue curve corresponds to all six quark ﬂavors.

tons. Note that for large values of mB0 , the branching fraction of A0 into dileptons is much smaller than the corresponding Z0 branching fraction. These hierarchies in the production cross sections and dilepton branching ratios translate into a much

greater discovery potential for the Z0 at colliders in most of parameter space, as we show below.

2.2 Twin Two Higgs Doublet Model

We now turn to the twin two Higgs doublet model. We label the two visible sector

Higgs doublets as H1 and H2, and their twin counterparts as H10 and H20 . Since the photon is massless, the VEVs of H1 and H2 must be aligned. However, the VEVs of the doublets in the twin sector need to be misaligned if they are to give mass to both the neutral gauge bosons in the twin sector. The alignment of the two twin doublet VEVs can be quantiﬁed in many ways. For instance, the quantity

0 2 H †H 0 | 1 2| , (2.6) 0 0 0 0 (H1†H1)(H2†H2)

–8– A’/Z’ Branching Fractions (2x2HDM)

100 100 f f f f

1 f1/v =3, ✏=0.1 1 f1/v =3, ✏=0.1 ) 10 ) 10

2 2 XX 10 XX 10 qq(all)

Ah ! 3 ! 3 (1 ﬂavor) 10 qq(all) 10 A Z ( ( 4 (1 ﬂavor) 4 10 10 BR BR 5 WW 5 WW 10 Zh 10

400 600 800 1000 120 140 160 180 200

mZ [GeV] mA [GeV] 100 100 f f f f

1 f1/v =5, ✏=0.1 1 f1/v =5, ✏=0.1 10 10 ) )

2 2 10 XX 10 XX qq(all) ! 3 ! 3

A h (1 ﬂavor) 10 10

Z qq(all) A ( ( 4 4 10 10 WW BR (1 ﬂavor) BR 5 5 10 WW 10 Zh Zh 500 600 700 800 900 1000 150 200 250 300 350

mZ [GeV] mA [GeV]

Figure 6: Branching ratios of the Z0 (left) and A0 (right) in the T2HDM for ✏ =0.1

and f1/v =3and5.ThebluecurveisforasingleSMleptonﬂavorwhilethedark

blue curve corresponds to all six quark ﬂavors. For mA0 = s2W mZ0 the alignment of Z0 is orthogonal to twin hypercharge. This closes the portal to the SM sector, reducing branching to visible states.

displayed in Fig. 7, the left plot showing the THPM model and the right plot the

T2HDM. The top (bottom) row is for the benchmark of f(1)/v =3(5).

LEP analyses place a strong bound on new physics contributions that a↵ect • the physics of the Z boson. Two quantities that can be measured directly from the Z line-shape are the total width of the Z, as well as the cross section for lepton pair production through the Z resonance. Given these measurements, using theory one can ﬁt other quantities in a straightforward way, such as the leptonic branching fraction of the Z, or equivalently the Z partial width to leptons. Again using theory, one can then constrain any deviation of the invisible decay width of the Z as well as any deviations of the coupling of the Z boson to leptons from their SM values. We thus begin our study of existing constraints on the MTH model with the two above-mentioned LEP line-shape measurements.

–12– Hypercharge Portal Constraints

f/v =3 f1/v =3 0.4 0.4

0.3 0.3

✏ 0.2 ✏ 0.2

0.1 0.1 EWPT Z EWPT Z Zee A Zee A 0.0 0.0 200 400 600 800 1000 200 400 600 800 1000 m [GeV] mB [GeV] A f/v =5 f1/v =5 0.4 0.4

0.3 0.3 ✏ ✏ 0.2 0.2

0.1 0.1 EWPT Z EWPT Z Zee A Zee A 0.0 0.0 200 400 600 800 1000 200 400 600 800 1000 m [GeV] mB [GeV] A Figure 7: Compilation of constraints on the MTH Model in terms of ✏ and either mB0 (THPM model on the left) or mA0 (T2HDM on the right) with f(1)/v =3(5)in the top (bottom) row. Indirect constraints are shown in black. The 13 TeV LHC dilepton resonance search bounds on the A0 (Z0)areshowninred(blue). estimate the sensitivity of future dilepton searches by a simple scaling up of existing results. As an example, if a run-II dilepton resonance search at a given mass can exclude a signal cross section of S,13 at 95% conﬁdence level, then a similar search at a 100 TeV collider will be sensitive to a cross section S,100 at the same conﬁdence level, given by = BG100 TeV L13 , (4.1) S,100 S,13 r BG13 TeV rL100 where the square roots contain the ratios of the background cross sections at the two colliders, as well as the ratio of the luminosities of the two searches. We calculate the background cross sections at 13 and 100 TeV at parton level in MadGraph.

Our results for f(1)/v = 3 and 5 are shown in Fig. 8 for the HL-LHC and Fig. 9 1 for the FCC-hh with a luminosity of 3000 fb , the THPM model on the left and the T2HDM on the right. As in the collider constraints of the previous section, we see that in the THPM model the sensitivity to A0 falls o↵ as mB0 increases, while the sensitivity to Z0 persists. Conversely, in the T2HDM the A0 sensitivity falls o↵

–15–

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