Singleton Portals to the Twin Sector

The mirror twin Higgs framework allows for a natural Higgs mass while being consistent with collider bounds on colored symmetry partners to standard model quarks. This mechanism relies crucially on a discrete symmetry which relates each standard model field to a mirror partner. These partners are charged under gauge groups identical to, but distinct from, those in the standard model. The minimal twin Higgs scenario provides only one low-energy connection between the visible and twin sectors, the light Higgs boson. We present a new class of portals connecting the two sectors, using fields that have 'no twin' partner under the discrete symmetry. Scalar, fermion, and vector states may provide such 'singleton' portals, each with unique features and experimental signatures. The vector portal, in particular, provides a variety of renormalizable interactions relevant for the LHC. We provide concrete constructions of these portals and determine their phenomenology and opportunities to probe the twin sector at the LHC. We also sketch a scenario in which the structure of the twin sector itself can be tested.


I. INTRODUCTION
While the standard model (SM) is the dominant paradigm in particle physics it is not the complete theory of nature. Its limitations (ignoring gravity) fall into three classes. The first concerns its structure, such as the number of fermion generations, the hierarchical pattern of quark mixing, the anarchic pattern of lepton mixing, and the apparent minimality of electroweak symmetry breaking (EWSB) and fermion mass generation. The second class pertains to its failure to explain experimental facts such as neutrino masses and mixing, dark matter, and the observed baryon asymmetry of the Universe. The final class relates to apparently finely chosen parameters, like the smallness of the effective QCD θ angle and the hierarchy between the electroweak and Planck scales.
Extending of the SM to account for these shortcomings is a driver of theoretical particle physics research. While the motivations can appear theoretical, success is claimed only when theory and experiment agree. To this end, the Large Hadron Collider (LHC) continues to search for new particles and interactions that might explain the outstanding puzzles in particle physics. Following the discovery of the SM-like Higgs boson [1,2], one of the top priorities of the LHC is to examine its properties. This includes measuring Higgs couplings to SM fields, as well as using the Higgs itself to probe beyond the SM (BSM).
The hierarchy between the electroweak scale, which sets the Higgs mass, and the Planck scale is arguably a strong link between Higgs and BSM physics. The LHC has completed many powerful and sophisticated searches for new particles that might begin to explain this hierarchy through Higgs compositeness [3][4][5][6] or a new symmetry [7][8][9]. So far, however, no hints have appeared.
This further motivates the neutral naturalness framework, of which the twin Higgs [10] is the prime example, where the Higgs mass is naturally light while being consistent these null results. This class of symmetry based solutions to the hierarchy problem employ a discrete symmetry to ensure that the symmetry partners of SM fields do not carry SM color.
The MTH model has only one guaranteed low-energy portal between the two sectors, the Higgs itself. Kinetic mixing between SM and twin hypercharge can provide another portal and has been explored in the context of dark matter signals [20,52]. The heavy twin Higgs can be another portal when the UV completion is weakly coupled, and its collider signals have also been explored [72,73].
These particles can connect the two sectors because they carry no gauge charges. In this case the SM field ψ A can mix with its twin partner ψ B without violating any symmetries This mixing allows the physical eigenstates to be a linear combination of the SM and twin fields thereby linking the sectors.
However, there is another class of BSM states that can connect the two sectors. These particles have no partner under the discrete symmetry, but change by at most a phase. The Z 2 symmetry of the twin Higgs model requires the phase to be either 0 or π Because the particles have no twin partner under the Z 2 we refer to them as singletons.
Schematically, their coupling is The simplest type of singleton portal is a scalar, φ. The only gauge invariant renormalizeable couplings of φ to SM and twin fields that preserved the discrete symmetry are where the minus sign applies when φ is odd under the Z 2 . If φ is odd, and acquires a vacuum expectation value (VEV), the first term in Eq. (4) spontaneously breaks the Z 2 symmetry. 1 This provides an attractive origin of the soft Z 2 breaking required by Higgs coupling measurements [27].
If φ transforms nontrivially under any symmetry other than the twin Z 2 , then only the second operator in Eq. (4) is allowed. This includes a simple φ → −φ symmetry that might stabilize φ against decay, making it a possible dark matter candidate. 2 However, this |φ| 2 interaction is also SU (4) symmetric in the Higgs fields, and consequently does not contribute to the potential of pseudo-Nambu-Goldstone bosons (pNGBs) like the physical Higgs. Therefore, in this case φ and the physical Higgs have no renormalizable interactions.
Similarly, when φ is even under the twin Z 2 the first term in Eq. (4) respects the global symmetry of the Higgs potential, providing no interactions between the physical Higgs and φ.
If φ is odd under the twin Z 2 , and is not protected by some symmetry from large quantum corrections, we expect its mass to be near the cutoff of a few TeV. To leading order a heavy φ merely modifies the terms in the Higgs potential and we can simply integrate it out of preserving Higgs potential, without including φ and using the notation of [16], as where the potential has an approximate SU (4) symmetry and leads to nearly equal VEVs for H A and H B if we take 0 < δ 1.
We use a nonlinear parameterization of the Higgs field, see [27], neglecting the heavy radial mode. We work in unitary gauge where are all but one the pNGBs of the broken SU (4) have been eaten, and find Here f is the SU (4) breaking VEV, v parameterizes how much of the VEV is in either sector, and h is the physical Higgs boson. By defining ϑ ≡ v/(f √ 2) we find the VEVs in where v A = v EW = 246 GeV. In this language the Higgs potential is By requiring that there be no tadpole term we find sin 4ϑ = 0, which since must ϑ = 0 to have nonzero v A implies ϑ = π/4, and hence v A = v B . We also find the Higgs mass is which, as expected, is controlled by δ, the global symmetry breaking parameter of the potential.
In the twin Higgs set-up Higgs couplings g h satisfy where g hSM is the SM coupling [27]. Therefore, having equal VEV in each sector implies the Higgs' couplings to visible fields are half the SM prediction, which is ruled out by LHC measurements. This result is modified by introducing a Z 2 -odd singleton, with VEV v φ and potential The interaction between the scalars in Eq. (4) become This modifies the Higgs' no-tadpole constraint to Experiment requires ϑ 1/3, so the above relation requires κ < 0. The Higgs mass becomes The constraint that ϕ have no tadpole and the mass of ϕ are There is also a mass mixing term between the scalars This is diagonalized by a simple matrix where and the mass eigenvalues are From Fig. 1 we find that our estimate of ϕ having a few hundred GeV mass is borne out for larger ϑ, but as this angle decreases, m ϕ increases. The right plot makes clear that the mixing angle θ is always quite small.
Including the effects of φ we find the couplings of h and ϕ to SM fields g h = g hSM cos ϑ cos θ, g ϕ = −g hSM cos ϑ sin θ.
This means that production of the ϕ proceeds exactly as the Higgs, but with the production rate reduced by sin 2 θ 10 −4 . Unfortunately, this reduction in rate makes ϕ very challenging to discover, even at next generation colliders. The most promising channel at a hadron collider would likely be production through gluon fusion and decay to ZZ. However, even at 100 TeV the projected bounds for such a scalar [74] remain several orders of magnitude above the ϕ cross section. A high energy lepton collider can exploit the ϕ → h h → bbbb final state [73,75], but here too, the cross section is prohibitively small. Therefore, while the neutral singleton provides a simple origin of the soft Z 2 breaking in the Higgs potential, its experimental signatures may be too faint to see in the near future.

B. Fermionic portal
The most obvious fermionic singleton is the gauge singlet right-handed neutrino ν R . This can couple as where the minus sign applies if ν R is odd under the Z 2 . The Yukawa couplings Y A,B are complex rectangular matrices in flavor space, which are forced to be identical by the discrete symmetry, so we simply write them as Y ν . The ν R 's large Majorana mass term, m R , leads to a seesaw [76][77][78][79][80] in both sectors.
However, ν R also induces mixing between the SM and twin neutrinos, creating the mass where we see the magnitudes of Y ν and m R have factored out. The physical eigenstates are where the mixing angle is defined by The mass eigenvalues are where v i is the VEV of H i given in Eq. (7).
No matter the values of Y ν and m R , the mixing angle is completely determined by the hierarchy of VEVs in the two sectors. Collider bounds on Higgs couplings imply that v B 3v A , while requiring the tuning be no worse than 10% implies v B 6v A [27]. Therefore, the interesting range of mixing is As pointed out in [23] this mixing between the A and B neutrinos can be used to reduce the contribution to ∆N eff from the twin sector, which CMB measurements constrain to ∆N eff 0.3 at 95% confidence [81]. However, the bounds on neutrino mixing can be quite constraining. In particular, meson decays bound neutrino mixing in these cases, see for instance [82], often an order of magnitude beyond the prediction given in Eq. (29). The only exception is if the singleton only connects the τ neutrinos. The limits here are weaker, and this level of mixing can be accommodated if the twin neutrinos are not heavier than a few hundred MeV. This is easily accomplished by M R ∼ TeV and Y ν order one.
One might hope to avoid this large mixing by using the inverse seesaw mechanism [83,84].
In the usual inverse seesaw set-up, a Dirac partner N is given a TeV scale mass M with the right-handed neutrino, and a small Majorana mass µ, The mass of the left-handed neutrino is then where v EW = 246 GeV is the Higgs VEV. This gives a neutrino mass in the tens of meV range for M ∼ TeV, µ ∼ keV, and Y ∼ 10 −2 . The heavy neutrinos have M scale masses.
Because both ν R and N are gauge singlets, either can be singletons under the twin Z 2 .
However, as we saw in the standard seesaw, taking ν R as a singleton leads to large mixing between the SM and twin neutrinos. This remains true in the inverse seesaw, but now the SM neutrino states are exactly masses and the twin neutrinos have mass By choosing Y ν and µ appropriately, since (v 2 A + v 2 B ) 1 TeV both Y ν and µ need to be smaller than the standard case, the twin neutrinos can be made light enough to escape the bounds from meson decays. These neutrinos are light enough to lead to tension with ∆N eff bounds, but this can be overcome in other ways such as asymmetric reheating of the two sectors [47]. In this regime future precision β-decay experiments will probe down to eV masses through combination of endpoint [85] and kink [86] measurements of the β-decay spectrum. The values of the mixing given in Eq. (29), at least to the electron neutrino, are projected to be probed by these experiments.
Finally, we can also choose the right-handed neutrinos to be twin partners ν RA and ν RB which have Dirac masses with the singleton N . The Lagrangian is with the sign of the Dirac mass term determined by the twin parity of N . The lightest state is mostly SM neutrino, and has a mass Though the mixing of this neutrino with the M mass states is small, there is large mixing with the two states with mass which is at or just below the TeV scale. This again leads to the mixing in Eq. (29) and because the mixing states are heavy, leads to the same constraints as in the standard seesaw.
In summary, constraints on neutrino mixing limit the possible fermionic singleton portals.
However, there are few cases, like τ flavored singletons or inverse seesaw with singleton righthanded neutrino that remain viable. The latter scenario offers signals that will be probed soon by the measurement of the β-decay spectrum, at least in the electron flavored case.

C. Vector portal
A singleton vector X µ can link the SM to the twins through kinetic mixing with the where B µν is the field strength of the hypercharge gauge boson B µ and similarly for X µ .
The sectors can also be connected by direct coupling of X µ to matter currents In each case the sign between the two terms is determined by action the discrete symmetry on X µ . Though we have shown a general coupling here, in what follows we assume vector-like couplings, or C A = 0.
The bounds on the SM photon mixing directly by some parameter with a massless twin photon require 10 −9 [87,88], for a twin electron with MeV scale mass. The most stringent result from bounds from supernova 1987A [89], which applies to energy scales in the few to few hundred MeV range.
With X µ as intermediary, however, the two photons have tree level mixing at order ε 2 , as shown in Fig. 2. Thus, the supernova bound becomes ε 10 −4.5 , which is too small a coupling to play a role in LHC searches. Even if the X boson is massive, these bounds persist since the kinetic mixing operator can be generated by the running above m X if the sum of fermion charges is nonzero as discussed below.
The kinetic mixing of X with the other U (1)s from above the cutoff depends strongly on the field content, and so we take it as a free parameter ε UV . This serves as a boundary condition for the running of the mixing operator which runs from the cutoff Λ UV down to m X . The running can be calculated from the top diagram in Fig. 3, Here the sum runs over all the fermions which couple to X and B and have masses below m X . If some fermion has a mass above m X , then it only contributes to the sum down to the mass of the fermion.
Taking the this sum to be order one, g X ∼ g, and m X a few hundred GeV we find ε ∼ 10 −4.5 . So, it seems we have already saturated the cosmological bound. But, it may be In which case there is no contribution from this running. A well known realization of this behavior is the gauged L µ − L τ current. It is also anomaly free, and so does not require any new states at the cutoff, making a vanishing ε UV seem natural also.
For vectorial x f couplings, the cancellation can also occur within each generation, But, this is only satisfied for x L = x Q , which makes U (1) X anomalous in the low energy theory. In particular, it violates the relation as required for the vanishing of the mixed SU (2) 2 L U (1) X anomaly. Resolving this anomaly requires new states charged under SU (2) L in the UV, which is difficult to reconcile with precision EW measurements.
We must now consider the running below m X . In this case we integrate X out of the spectrum and calculate by inserting the generated four-fermion operator in the lower diagram of Fig. 3. This yields where again the sums over the A and B sector fermions include all the "active" fermions over that mass range. The important thing to notice is the q 2 dependence. This diagram generates a higher dimensional operator, not the usual kinetic mixing.
This result can also be understood from the equations of motion. The fermion loops in where we have not written the terms involving fermions. This can be inserted back into the Lagrangian to obtain terms likeε which agrees with the explicit loop calculation.
Therefore, below m X the running of ε can only occur at some higher loop order. If we assume vector-like couplings that have the same charge conjugation behavior as hypercharge, then only even numbers of U (1) boson legs can connect to the fermsion loops. In this case the leading contribution to ε would have to be at least four-loop, with three X propagators integrated out. If such a diagram does generate kinetic mixing, it is likely well below the level of experimental constraint.
In short, it is consistent for ε to be small enough to agree with cosmological constraints, but this greatly restricts the collider phenomenology. If, however, the twin hypercharge, B B , also gets a mass m B B 1 GeV, these bounds are much weaker [90] and the kinetic mixing can be much larger. This gives both X and both the massive neutral bosons in the twin sector Z B and A B large enough couplings to SM field to potentially produce them at colliders. The singleton X can also keep the two sectors in thermal contact.
In particular, it can put the decoupling temperature in between the QCD scales of the two sectors, reducing (though not eliminating) the tension with ∆N eff bounds. The origin of m B B is associated with a soft breaking of the Z 2 , the origin of which we take to lie above the low energy degrees of freedom. It could be arise from a Stückelberg mechanism or from a scalar field getting a VEV, and spontaneously breaking the discrete symmetry and twin hypercharge. 3

Anomalous Singletons
The MTH construction only addresses the little hierarchy problem, its cutoff, Λ UV , is a few TeV. Consequently, there is little obstruction to considering anomalous vector couplings.
Such couplings merely require that new states appear below a scale [92] Λ Anomalous ∼ m X g X , 3 The combination of lifting the twin neutrinos to a few GeV and lowering the decoupling temperature of the two sectors so that it is between the QCD phase transitions of either sector can reduce the MTH ∆N eff to about 0.7, which may agrees with the BBN constraint which can be ∆N eff 1 [91].
to cancel the anomalies. As long as Λ Anomalous ≥ Λ UV there is no inconsistency. However, as shown below resolving the mixed anomalies in the UV requires particles charged under both X and B A,B . This generically generates kinetic mixing in the low energy EFT, so we find that anomalous currents lead to the kinetic mixing of Eq. (36).
As a concrete example, suppose X is the force carrier of gauged L µ , muon number. Then the U (1) 3 X anomaly is proportional to When X is odd under the discrete symmetry there is a relative sign between the terms and the anomaly vanishes. Both the A and B sector muons are crucial to the cancellation.
The mixed U (1) X anomalies are more subtle. For instance, the U (1) X U (1) 2 Y anomaly does not vanish. In the A sector we have Which generates a term of the right hand side of the current conservation equation, where A B is constant. By including the twin sector couplings we obtain the full equation While A B is identical for the A and B sectors, the anomaly does not cancel, because each pertains to a distinct gauge field, either SM or twin hypercharge. In general, we cannot cancel A sector anomalies by identical terms in the twin sector.
An anomalous X µ has additional interactions among the gauge bosons [93] which can lead to stringent bounds on light vectors [94,95]. These follow from the Wess-Zumino terms in the low-energy EFT. For heavy vectors, these interactions are typically negligible. They introduce new decays like X → Z A,B γ A,B , but the branching fractions are order 10 −4 [96].
While these decays are very interesting and distinct, they are subdominant to other signatures, so we neglect them in what follows.

III. EXPERIMENTAL BOUNDS ON VECTOR SINGLETONS
In this section we determine the indirect and direct experimental bounds on vector singletons. We assume the usual MTH construction and add a new gauge group U (1) X with gauge boson X µ . These new gauge interactions may or may not be anomaly free. In particular, is used as an anomaly free case study, as well as the restriction to the third generation. We also consider the anomalous L µ A ± L µ B current.
We take X µ massive, through the Higgs or Stückleberg mechanism. In general, we assume whatever dynamics gives mass to X µ decouples from the low-energy dynamics and treat m X as a free parameter. The Lagrangian pertaining to X µ is where J i is the particular current of interest. Note that this Lagrangian is invariant under the Z 2 symmetry if X µ → ±X µ .
Appendix A includes a perturbative form of the couplings and other formulae for these analyses. However, these results are only valid when m X is not too close to either m Z Z or m Z B . For quantitative results we use a numerical diagonalization procedure, while the pertubative analysis provides qualitative understanding of those results.

A. Indirect Bounds
Indirect bounds on this scenario arise primarily from Z physics at LEP. These include the T parameter, the invisible Z width, and the partial widths of the Z into light leptons.
As shown in Fig. 4, these restrict the viable (m X , ε) space.
From [97] the T parameter satisfies where α is the fine structure constant at the Z mass, and in the last equality we have used the perturbative result in Eq. (A9). From [98] T = 0.09 ± 0.13 so, at 95% confidence we As the blue shaded region in Fig. 4 makes clear, this bound is most constraining for m X close to m Z A , and drops off steeply for larger masses.
In this model, the Z branching fractions into individual leptons are more constraining than its invisible width. We use the PDG [99] values BR(Z → ee) = (3.363 ± 0.004)% and BR(Z → µµ) = (3.366 ± 0.007)% to determine 95% confidence bounds. Unlike the T parameter, these bounds are sensitive to the strength of the new gaugue coupling g X .
To indicate the range of the bounds, we use two benchmarks on r X = g X /g: r X 1 and r X = 1. Here g is the SM weak gauge coupling. The bounds are computed for the global B − L case, as the larger number of direct couplings of X µ to twin states changes the Z A width more, giving stronger bounds. The two regions, red and orange, in Fig. 4 correspond to these stronger and weaker bounds, respectively.

B. LHC Bounds
Both the singleton vector boson X and the twin massive neutral gauge boson, Z B , and in some cases A B , are potentially accessible at the LHC. Since the X can couple directly to 4 Note that when the masses are degenerate a more careful treatment is required. A and B sector states, its couplings may not be suppressed by the kinetic mixing parameter ε. However, when it has no direct quark couplings, production arises through ε suppressed mixing with the Z A . In that case its production cross-section scales as ε 2 .

Model
On the other hand, Z B and A B only couples to SM quarks through kinetic mixing, and so their production at the LHC is always suppressed by at least ε 2 . Their visible decays are similarly ε 2 suppressed. Therefore, the most promising discovery channel appears to be be jets + MET [100], thereby avoiding the extra suppression. However, even this least suppressed scenario gives sensitivity far below direct X searches.

Discovering the X
The X couplings depend on which combination of lepton and baryon numbers are gauged and, as discussed above, this significantly affects its signature at the the LHC. Here we explore the phenomenology of three gauged combinations of lepton and baryon numbers: In this case X couples directly to SM quarks and leptons, so the cross-section satisfies Here the charges, x A , are given in Tab. I and the only free parameter (to leading order in ε) is r X = g X /g, i.e. the ratio of the U (1) X gauge coupling to the SM weak isospin gauge coupling. Note the sign of the B-sector charges is fixed in the Lagrangian Eq. (51). the bb initial state, with the corresponding PDF suppression, or through light quarks with ε 2 suppression. The latter is suppressed because the couplings to light quarks are only generated by mixing with Z A . The dominant final states in this case are bb, tt, and τ + τ − all of which are covered by current searches, see for example [103][104][105]. Currently, the strongest bound is from τ + τ − searches and is shown in Fig. 6. The plot on the right shows the bounds on r X as a function of m X for two ε benchmark values. We see that for m X ∼500 GeV the new gauge coupling cannot be more than half the SM weak-isospin gauge coupling, but this bound relaxes as the X mass increases.
Finally, we gauge the difference of A and B sector muon numbers. Now production must proceed via order ε couplings to light quarks, which are generated through mixing with Z A . As a result, the cross-section is suppressed by ε 2 . Further, because only the dimuon final state carries no additional powers ε, the strongest bound comes from dimuon searches.
Fixing r X = 1, the bound on ε is shown in Fig. 7 and is O(10 −2 ).

Discovering the twin Z and A
The kinetic mixing mediated by X leads SM quarks and leptons to obtain small couplings to the neutral bosons in the twin sector. The mass of Z B is controlled by ratio of Higgs VEVs in the two sectors, v B /v A . This ratio is also tied to limits on deviations in Higgs couplings [27] and imply m Z B 3m Z A , while requiring the model be no more than 10% tuned ensures m Z B 6m Z A . The A B mass is much less constrained, as long as it is at least a few GeV to agree with cosmological bounds. This means that both neutral twin gauge bosons can be considerably lighter than X.
The same search strategies used to detect X apply to Z B and A B , with a few important differences. Because Z B and A B couple to A sector fields through mixing with X, their couplings are ε suppressed relative to the X coupling. From Eqs. (A18)-(A22) in App. A, which apply in the massless twin photon limit, In addition, when X is heavier than Z B this mixing further reduced, This same behavior is inherited by the massive twin photon, since its couplings come from an additional mass mixing with the twin Z.

Testing the Twin Higgs Structure
Discovering the X would give us access to a new sector but it may be difficult to distinguish a twin sector from something generic. A high energy lepton collider might measure the X width, and show that full width is twice the visible width. This would at least hint at the twin structure. Otherwise, the X introduces many new parameters, and so its discovery does not probe the constrained nature of the twin Higgs framework.
Discovering both X and a twin boson, say Z B does not significantly improve the prospects of testing the model. This is because the X couplings depend on g X and the leading couplings to Z B depend on g X . Consequently the production rates of the two states are related by a free parameter.
However, discovering both Z B and A B is different. Finding both states gives access to four experimental results, the two masses m Z B and m A B and the rates into the final states used to discover them. Both of these rates depend on the combination g X ε in the same way.
In this case much of the dependence on X factors out.
The set of unknown parameters of the model thus are: f , g X , m X , m A B , ε, and the set of x f . If the discoveries occur in the same channel, di-leptons say, the combination g X εx f will be common to the two rates. Furthermore, the dependence on each vector's total width on m X is higher order in ε -see Eqs. (A20) and (A21) -and can thus be neglected to leading order. The only remaining uncertainty is that X might couple directly to other states, for instance if the discovery were made in di-leptons the X might also couple to quarks directly.
This uncertainly could likely be reduced by checking other search channels directly at the m Z B and m A B masses. These additional excesses would fill out the number of X couplings. If no additional excess is seen in those channels, the corresponding x f charges can be bounded.
Then the twin Higgs structure would make a precise prediction of the widths of each vector.
The measured rate of one vector could then be compared with the other, testing the twin Higgs structure.

IV. CONCLUSION
The mirror twin Higgs has become one of the most natural ways to resolve the hierarchy problem in the wake of LHC searches for colored symmetry partners. However, the minimal form of the twin Higgs along with the discrete symmetry seem to make connecting the two sectors, and hence probing the new states, difficult. In this work we have introduced a new type of portal between the SM and twin sectors, which has no twin under the discrete symmetry, transforming by at most a phase. These singleton portals provide new opportunities to probe the twin sector and may provide the means to produce novel phenomena within the twin Higgs framework.
At the renormalizable level, scalar singletons primarily affect the Higgs potential. Though they can easily explain the origin of soft Z 2 breaking, their collider signals may be beyond current experimental capabilities. However, their mixing with the radial mode of the symmetry breaking, the heavy twin Higgs, could well be stronger and could lead to measurable effects.
Right-handed neutrinos are interesting fermion singletons. However, the variety of con- These vectors are discoverable at the LHC and future colliders. These next generation machines may also be able to use the singleton set-up to probe the twin sector. They may even be able to distinguish the twin Higgs framework from more generic new physics.

ACKNOWLEDGMENTS
We are grateful to Zackaria Chacko for many valuable comments and advice regarding the In this Appendix we calculate the masses and couplings of the massive vectors to leading order in the kinetic mixing parameter ε. We begin by writing the total kinetic mixing in the Lagrangian as Then, by making the field redefinitions the kinetic terms are diagonalized. Note, however, that this parameterization is only valid Now, the photon A µ and neutral weak boson Z µ are defined by where c W ≡ cos θ W and similar for s W . We then make the definitions 5 This bound on ε is simply the requirement that there be not ghosts, as can arise if the mixing is too large.
It is straightforward to prove that if X µ had mixed with N separate gauge bosons then only ε ≤ 1/ √ N keeps keeps the determinant of the mixing matrix positive.
which lead to mass terms m 2 To give the twin photon a mass, the 2 × 2 mass matrix of (Z B , A B ) must be rank 2. To do this, we add another mass term for the hypercharge gauge boson B B of the form Finally, we diagonalize the mass matrix to determine the physical eigenstates. For simplicity we give the results with ∆m 2 B B = 0 as they illustrate the qualities pointed to the in the main text. The eigen vales of this matrix can be obtained in powers of ε: while the A A,Bµ fields are massless. Here we have used the notation Note that when the masses are close to degenerate the perturbative expansion breaks down.
We find the mass eigenstates are related to the interaction fields by The couplings to the SM Higgs h and a second vector spring from the interactions where ϑ = v/(f √ 2) where f is the VEV the breaks the global SU (4) symmetry in the Higgs sector and v is the amount of the VEV that is in the SM sector. In this parameterization v EW = f √ 2 sin ϑ. It is also worth noting that m Z B = m Z A cot ϑ, so that the couplings of the Higgs to Z bosons in each sector are equal, as expected from the gauge structure. The couplings are Appendix B: Useful Formulae The pp → X cross section is obtained by where S is the center of momentum energy of the hadron collider and where y B is the boost of the dijet ( L.O. partonic) rest frame and Υ = min{y cut , − 1 2 log τ }. In either case the f (x) are parton PDFs. We use the MSTW2008 PDFs [106] with factorization scale taken to be m X . As our vector are often a little too wide for the narrow width approximation, we use the results from [107] for the partonic cross section σ(ŝ) =ŝ m 4 z g 2 q L + g 2 . (B4) The decay width of a vector V into fermions is, For SM and twin quarks we multiply this by the factor 1 + αs π to account for the leading QCD correction. Thus, the production of Z B will go like ε 4 or x 2 A ε 2 . The decay of a vector V into a pair of W s is, The decay width of a vector V to Z A h, This is modified to other final states in the obvious way.