Breaking Mirror Twin Color

We investigate simple extensions of the Mirror Twin Higgs model in which the twin color gauge symmetry and the discrete $Z_2$ mirror symmetry are spontaneously broken. This is accomplished in a minimal way by introducing a single new colored triplet, sextet, or octet scalar field and its twin along with a suitable scalar potential. This spontaneous $Z_2$ breaking allows for a phenomenologically viable alignment of the electroweak vacuum, and leads to dramatic differences between the visible and mirror sectors with regard to the residual gauge symmetries at low energies, color confinement scales, and particle spectra. In particular, several of our models feature a remnant $SU(2)$ or $SO(3)$ twin color gauge symmetry with a very low confinement scale in comparison to $\Lambda_{\rm QCD}$. Furthermore, couplings between the colored scalar and matter provide a new dynamical source of twin fermion masses, and due to the mirror symmetry, these lead to a variety of correlated visible sector effects that can be probed through precision measurements and collider searches.


I. INTRODUCTION
The Twin Higgs [1] and other 'Neutral Naturalness' scenarios [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] feature color-neutral symmetry-partner states which stabilize the electroweak scale, thereby reconciling a natural Higgs with the stringent direct constraints on colored states from the Large Hadron Collider (LHC). The original Mirror Twin Higgs (MTH) [1] provides the first and perhaps structurally simplest model of this kind, hypothesizing an exact copy of the Standard Model (SM) along with a discrete Z 2 symmetry that exchanges each SM field with a corresponding partner in the mirror sector. Assuming the scalar sector respects an approximate SU (4) symmetry that is spontaneously broken, the Higgs doublet arises as a pseudo-Nambu-Goldstone boson (pNGB) at low energies. The Z 2 exchange symmetry and the presence of mirror top-partners and gauge-partners shield the Higgs from the most dangerous quadratically divergent contributions to its mass. The leading contribution to the Higgs potential is only logarithmically sensitive to the cutoff, which can naturally be of order 5 TeV. Thus, the MTH offers a solution to the little hierarchy problem, and, furthermore, a variety of ultraviolet (UV) completions exist [17][18][19][20][21][22][23][24][25].
Several considerations motivate extensions of this basic framework. First, the Z 2 symmetry must be broken to achieve a phenomenologically viable vacuum, featuring a hierarchy between the global SU (4) breaking scale and the electroweak scale. From a bottom up perspective a suitable source of Z 2 breaking can be implemented 'by hand' in a variety of ways, including a 'soft' breaking mass term in the scalar potential [1] or a 'hard' breaking through the removal of a subset of states in the twin sector, as in the Fraternal Twin Higgs [26]. A second issue is that in the standard thermal cosmology the MTH predicts too many relativistic degrees of freedom at late times, clashing with observations of primordial element abundances and the microwave background radiation. The removal of the lightest first and second generation twin fermions, which are not strictly required by naturalness considerations, provides a simple way to evade this problem [26][27][28] though other methods have also been explored [29][30][31][32][33][34]. Following these successes many other cosmological topics can be addressed, including the nature of dark matter [27,29,32,[35][36][37][38][39][40][41][42][43][44], the order of the electroweak phase transition [45], baryogenesis [38,46], and large and small scale structure [47,48].
It is appealing to have a dynamical origin for these soft and/or hard Z 2 breaking mechanisms. One possibility is that the Z 2 is an exact symmetry of the theory but is spontaneously broken [49][50][51][52][53]. Such spontaneous Z 2 breaking could result from a pattern of gauge symmetry breaking in the mirror sector that differs from the SM's electroweak symmetry breaking pattern. Interestingly, such spontaneous mirror gauge symmetry breaking can dynamically generate effective soft Z 2 breaking mass terms in the scalar potential required for vacuum alignment. They can also produce new twin fermion and gauge boson mass terms, which mimic the hard breaking of the Fraternal Twin Higgs scenario [26] by raising the light twin sector states. Due to the exact Z 2 symmetry, this scenario generically leads to a variety of new phenomena in the visible sector that can be probed through precision tests of baryon and lepton number violation, quark and lepton flavor violation, CP violation, the electroweak and Higgs sectors, and directly at high energy colliders such as the LHC. 1 This approach was advocated recently in Ref. [56,57], which explored the simultaneous spontaneous breakdown of mirror hypercharge gauge symmetry and Z 2 symmetry. In this work we examine the spontaneous breakdown of the twin color symmetry. Beginning from a MTH model, with an exact Z 2 symmetry, we add a new scalar field charged under SU (3) c and its twin counterpart. A suitable scalar potential causes the twin colored scalar to develop a vacuum expectation value (VEV), spontaneously breaking both twin color and Z 2 . Depending on the scalar representation and potential, a variety of symmetry breaking patterns can be realized with distinct consequences. There are several possible residual color gauge symmetries of the twin sector which may or may not confine, and when they do at vastly different scales. The possible couplings of the scalar to fermions may also produce new twin fermion mass terms. All of these possibilities lead to very different twin phenomenology and the rich variation that can spring from an initially mirror Z 2 set up.
While the complete breakdown of twin color was explored in Ref. [57], the aim was a particular cosmology and employed two scalars that acquired VEVs. We focus on a different part of the vast span of possibilities that is in some sense a minimal set of color breaking patterns. These follow from the introduction of a single new colored multiplet (in each sector) which may transform in the triplet, sextet, or octet representation. This scalar field is assumed to be a singlet under the weak gauge group, though it may carry hypercharge.
A detailed analysis of these possibilities is presented in Sec. II. In Sec. III the couplings of the colored scalars to fermions are investigated and shown to dynamically generate new twin fermion mass terms, providing a possible way to realize a fraternal-like twin fermion 1 Other connections between Twin Higgs models and SM flavor structure have been explored in [39,54,55]. spectrum. The correlated effects of these couplings in the visible sector through a variety of precision tests are discussed in Sec. IV. The new colored scalars can also be directly probed at the LHC and future high energy colliders, and we detail the current limits and prospects for these searches in Sec. V. Finally, we conclude with some perspectives on future studies in Sec. VI.

II. SPONTANEOUS BREAKDOWN OF TWIN COLOR
Our basic starting point is a MTH model, with its exact copy of the SM called the twin sector. In all that follows the label A (B) denotes visible (twin) sector fields and the exact Z 2 exchange symmetry interchanges A and B fields. To this base we add the scalar fields, Φ A and Φ B , that are respectively charged under SM and twin SU (3) c gauge symmetries.
We consider the following complex triplet, complex sextet, and real octet representations for the scalar fields: which are singlets under SU (2) L so that the weak symmetry breaking pattern is not modified. In this subsection we analyze the symmetry breaking dynamics of the colored scalar sector in isolation. This enables us to highlight some of the differences in the color symmetry breaking for the triplet, sextet and octet cases. The investigation of the entire scalar potential including the Higgs fields and the full electroweak and color gauge symmetry breaking is carried out in subsequent subsections. Throughout we use the standard definitions for the SU (3) generators, T a = 1 2 λ a with Gell-Mann matrices λ a and a = 1, 2, . . . 8, and SU (2) generators, τ α = 1 2 σ α with Pauli matrices σ α and α = 1, 2, 3.

Color triplet scalar
First, consider triplet scalars Φ A,B ∼ (3, 1, Y Φ ), which can be represented as a complex vectors, i.e, (Φ A ) i , with color index i = 1, 2, 3. The Z 2 symmetric scalar potential for Φ A and Φ B is The µ 2 and λ terms respect a large U (6) global symmetry while the δ term preserves only a smaller U (3) A × U (3) B × Z 2 symmetry. We are often interested in the parameter regime |δ| λ. 2 When δ < 0, the vacuum spontaneously breaks Z 2 [2]. The desired vacuum is corresponding to the gauge symmetry breaking pattern Fluctuations around the vacuum are parameterized as B a doublet under [SU (2) c ] B , and ϕ B and η B being singlets. Expanding the potential in Eq. (2) about the vacuum, the scalar masses are found to be In the limit |δ| λ the global symmetry breaking pattern is U (6) → U (5), yielding 11 NGBs B , and real singlet η B ). The field φ A obtains a mass proportional to the U (6) breaking coupling δ and can be considered to be a pNGB in this limit. The fields η (2) B , η B are exact NGBs and are eaten by the five massive twin gluons, which obtain masses of order m G B ∼ g S f Φ . Since the triplet scalar is also assumed to carry hypercharge Y Φ , it gives a mass to the twin hypercharge boson. We will examine these effects below when we include the Higgs fields in the scalar potential.
Finally, there is the radial mode ϕ B with mass of order √ λf Φ .

Color sextet scalar
We next take Φ A,B ∼ (6, 1, Y Φ ) as color sextets, which can be represented as complex symmetric tensor fields, i.e, (Φ A ) ij , with i, j = 1, 2, 3. The most general Z 2 symmetric potential for Φ A and Φ B is The first line of Eq. (6) respects U (12) global symmetry. The second line explicitly breaks We focus on the regime |δ 1,2 | λ. The vacuum structure is analyzed following the techniques of Ref. [58] and is governed by the values δ 1 and δ 2 . There are two spontaneous Z 2 breaking vacua of interest, which we now discuss.
The first relevant sextet vacuum leads to the gauge sym- The orientation of this vacuum is Assuming |δ 1,2 | λ, this vacuum is a global minimum for the parameter regions δ 2 < 0 and The fluctuations around the vacuum can be parameterized as B a doublet under [SU (2) c ] B , and ϕ B and η B singlets. Inserting (8) into the potential (6), the masses of the scalar fluctuations are found to be For small δ 1,2 the symmetry breaking pattern is U (12) → U (11), producing 23 NGBs (com- B , and real singlet η B ). The field φ A is a pNGB and obtains a mass proportional to the U (12) breaking couplings δ 1 , δ 2 . But, since δ 1 respects a U (6) B symmetry, which is spontaneously broken to U (5) B , it does not contribute to the φ B mass. However, as δ 2 explicitly breaks U (6) B to U (3) B , φ B is a pNGB with mass proportional to δ 2 . The fields η (2) B and η B are exact NGBs, and are eaten by the heavy gluons. The radial mode ϕ B has a mass proportional to The second viable sextet vacuum produces the gauge symme- The orientation of this vacuum is Assuming |δ 1,2 | λ, this vacuum is a global minimum for the parameter regions δ 2 > 0 and δ 1 < −δ 2 /3. The fluctuations around the vacuum can be parameterized as where we have defined the real [SO(3) c ] B quintuplets φ B = φā B Tā and η In the |δ 1,2 | λ limit the symmetry breaking pattern is again U (12) → U (11), yielding 23 B , and real singlet η B ). The field φ A is a pNGB with mass proportional to the U (12) breaking couplings δ 1 , δ 2 . But, since δ 1 respects a U (6) B symmetry, which is spontaneously broken to U (5) B , it does not contribute to the φ B mass. The coupling δ 2 explicitly breaks U (6) B to U (3) B , however, so φ B is a pNGB with mass proportional to δ 2 . The fields η (5) B and η B are exact NGBs at this level and are eaten by the five heavy gluons and the hypercharge gauge boson.
Finally, the radial mode ϕ B has a mass proportional to √ λf Φ .
A Z 2 symmetric potential involving the colored scalars is given by The first line of Eq.
while the V 6 term contains dimension six operators, which are discussed below.
Again, the vacuum structure is obtained following the methods of Ref. [58]. We first suppose V 3 and V 6 are set to zero. The cubic coupling in V 3 can be forbidden by a parity symmetry, Φ A,B → −Φ A,B , while the higher dimension terms in V 6 are generally expected to be subleading. For δ < 0 the vacuum spontaneously breaks the Z 2 symmetry, and can be parameterized as The vacuum angle β does not appear in the potential at this level, and thus corresponds to a flat direction. Several possible dynamical effects can explicitly break the large O(8) A ×O(8) B symmetry, lifting the flat direction and generating a unique ground state. These include tree level contributions to V 3 and V 6 as well as radiative contributions to the potential.
a. Cubic term Let us first consider the cubic coupling, where A is taken real and positive without loss of generality, and we consider the A/µ 1 regime. For δ < 0 the vacuum spontaneously breaks the Z 2 symmetry and is described by The twin color gauge symmetry is broken from The scalar fluctuations are parameterized as where B is a [SU (2) c ] B doublet, and ϕ B is a singlet. Inserting (17) into the potential (13) and expanding about the vacuum, the scalar masses are found to be In the small δ, A/µ regime the symmetry breaking pattern is O ( B is an exact NGB, and is eaten to generate mass terms for the heavy gluons. Finally, ϕ B is the radial mode with mass proportional to √ λf Φ . b. Higher dimension operators Since a cubic term in the potential aligns the vacuum in the T 8 direction, it is interesting, in light of Eq. (14), to ask if the vacuum can point entirely along T 3 . To this end, we consider a dimension six operator, which, given that the MTH model should have a relatively low UV cutoff, is generally expected to appear.
Imposing the parity symmetry Φ A,B → −Φ A,B , which forbids the cubic term, we consider a simple representative dimension six operator where Λ is the UV cutoff and c is the Wilson coefficient. We work in the regime cµ 2 /Λ 2 1.
For δ < 0 and c > 0 we find the following Z 2 breaking vacuum orientation: The twin color gauge symmetry is broken from fluctuations around the vacuum are parameterized as Inserting (21) into the potential given in Eqs. (13) and (19) and expanding about the vacuum, the scalar masses are found to be In the small δ, cµ 2 /Λ 2 limit the symmetry breaking pattern is O ( . This is conveniently studied by computing the one-loop effective potential in the MS scheme: The potential has minima at β = nπ/3, which, noting Eq. (14), each lead to the gauge Each is simply an SU (3) c transformation from T 8 , so without loss of generality we consider the vacuum orientation as given by Eq. (14) with β = 0, i.e., So, the analysis mimics that of the cubic term, but with the pNGB mass of order α s f Φ .

B. Full scalar potential and nonlinear realization
The previous analysis can be adapted to realistic potentials involving both the Higgs and the colored scalar fields. We use a nonlinear parameterization of the scalar fields, working in unitary gauge and including only the light pNGB degrees of freedom to provide a simple and clear description of the low energy dynamics. The technical details of each analysis are similar to each other and to analysis of the hypercharge scalar in Ref. [56]. Therefore, we present only the triplet scalar case in detail. We do comment on how the sextet and octet models differ, but relegate much of the details to the Appendix.

Color triplet scalar
Taking the new scalars to be color triplets (see Sec. II A 1 above), we now include the Higgs fields. The Z 2 symmetric scalar potential is given by where we have defined The terms in the first line of Eq. (25) respect a U (4) × U (6) global symmetry, while those in the second line explicitly break this symmetry. We demand that the symmetry breaking quartics δ H and δ HΦ are small compared to λ H and λ HΦ , to ensure the twin protection mechanism for the light Higgs boson. Though not strictly required, if δ Φ is small compared to λ Φ the color triplet scalar in the visible sector can naturally be lighter than f Φ .
In the absence of the colored scalar fields, choosing δ H > 0 leads to a vacuum with This implies order one modifications of the light Higgs boson's couplings to SM fields, which is experimentally excluded. However, we saw in Sec. II A 1 that taking δ Φ < 0 spontaneously breaks the Z 2 symmetry, with Φ B obtaining a VEV but Φ A = 0.
Crucially, this symmetry breaking makes the δ HΦ interaction into an effective Z 2 breaking mass term for the Higgs scalars, allowing the desired vacuum alignment, with H A H B .
The nonlinear parameterization for the Higgs fields is given by (see also Ref. [56]) while for the colored scalars we have Here f H is the global U (4) breaking VEV, v H is related to the VEV of H A , h is the physical Inserting the nonlinear fields in Eqs. (26) and (27) into the scalar potential, Eq. (25), and neglecting the constant terms, we find the scalar potential for the pNGB fields: The potential (28) has a minimum with φ A = 0, v H = 0 which obeys the relation where we have introduced the vacuum Expanding the potential about the minimum and using Eq. (29), we obtain the masses of the physical scalar fields h and φ A : To ensure the Higgs mass in Eq. (30) is positive we require δ H > 0, and combining this requirement with the vacuum relation (29) leads to the condition δ HΦ < 0. We also demand that m 2 φ A > 0 in Eq. (31), which restricts the value of δ Φ once δ H , δ HΦ are specified. To make contact with the standard definition of the weak gauge boson masses, we define the electroweak VEV and its twin counterpart as where In particular, the quartic couplings may be written as Fixing the vacuum angle to be sin ϑ 1/3, the free parameters of the model can then be chosen as m φ A and f Φ . 3 We can also estimate the scale of these parameters. This follows from imposing certain restrictions on the symmetry breaking quartics, δ Φ and δ HΦ , which are related to m φ A and f Φ via Eq. (33). Since the gauge and Yukawa interactions break the U (4) × U (6) symmetry, the symmetry breaking quartics will be generated radiatively and cannot be taken too small without fine tuning. The quartic δ Φ is generated by strong interactions at one loop, implying its magnitude is larger than roughly α 2 s ∼ 10 −2 . On the other hand, δ HΦ is generated at one loop by hypercharge interactions, or at two loops due to top quark Yukawa and strong interactions, suggesting its magnitude be larger than about 10 −4 . We also take these couplings to be smaller than the U (4) × U (6) preserving quartics and thus require |δ Φ,HΦ | 1 for strongly coupled UV completions. Collectively, these conditions suggest m φ A and f Φ fall within the 100 GeV-10 TeV range. Of course, direct constraints from the LHC typically require m φ A to be 1 TeV, as we discuss later.
The potential (28) contains cubic interactions involving the Higgs and colored scalar. In Such couplings can lead to modifications of the Higgs couplings to gluons and photons, and are discussed in Sec. V.

Color sextet and octet models
A similar analysis can be carried out for color sextet or octet, and we refer the reader to the Appendix for details on their nonlinear parameterizations. One important difference in those models is the presence of additional pNGB scalar degrees of freedom φ B in the twin sector, as was already apparent in Secs. II A 2 and II A 3. Otherwise, the analyses of the sextet and octet are very similar to that of the triplet. In particular, the trilinear coupling involving the visible sector Higgs boson and colored scalar are always given by Eq. (34).

C. Twin gauge dynamics and confinement
We now discuss the gauge interactions in the various models, including the nature of the unbroken non-Abelian and U (1) gauge symmetries and confinement in the twin sector. As seen above, several twin color breaking patterns are possible depending on the representation of the colored scalar and form of the scalar potential. By accounting for both twin color and electroweak symmetry breaking, we found five distinct patterns of gauge symmetry breaking: II : III : IV : Of these, cases I-IV feature a residual non-Abelian color gauge symmetry and confinement at a low scale. In cases I, II, and IV, this non-Abelian group is SU (2) c , while in case In MTH models with unbroken color gauge symmetry the confinement scale is similar to the ordinary QCD confinement scale, Λ A ∼ 1 GeV. In models I-IV confinement naturally occurs at a much lower scale, because the number of massless gluonic degrees of freedom contributing to the running below the TeV scale is much smaller. The one-loop beta function can be written as Before estimating the confinement scale for these models, we note that additional dynamical Z 2 breaking effects, such as new twin fermion mass terms or a shift in the strong gauge coupling at the UV scale, α B s (f Φ ) = α A s (f Φ ) + δα s , may raise or lower this scale by several orders of magnitude. Nevertheless, the general expectation is that the twin confinement scale is much lower than that in the visible sector, in contrast to MTH models with where n A f (n B f ) denotes the number of active Dirac fermions in the A (B) sector at a given energy scale. The visible sector potentially contains a color triplet scalar φ A in the effective theory, with index T s = 1 2 , and n A s the number of light triplet scalars in the A sector. In the left panel of Fig. 1 we display the evolution of the strong coupling constants in the visible (red) and twin (blue) sectors. We see that the twin strong coupling becomes large near scales of order Λ B ∼ MeV. As mentioned above, this is primarily a consequence The generator of the unbroken electromagnetic symmetry for each case are II : IV : In cases I and II the twin electric charges depends on a particle's T 8 as well as the colored scalar's hypercharge Y Φ . This occurs because the triplet and sextet can carry hypercharge, which leads to mass mixing between the neutral hypercharge and color gauge bosons. On the other hand, the octet in case IV is real, so the EM generator is identical to the SM.   Table I  We emphasize here the great difference in the twin particle spectrum compared to the basic MTH model. Though much of the dynamics are determined by the Z 2 twin symmetry with the SM fields, we end up with new unconfined quarks, from the part of the field along the VEV direction, as well as new SU (2) c bound states. Insights into this bound state spectrum and dynamics of the phase transition can be found in, for example, [60][61][62][63][64][65][66][67][68][69][70][71], but a few qualitative items are worth mentioning. First, the lightest quark masses are a few MeV, which is just above the confinement scale so mesons, composed of a quark and an anti-quark, and baryons, composed of two quarks, can likely be simulated as nonrelativisitic bound states.
In the absence of additional scalar couplings to matter there is a conserved baryon number that renders the lightest twin baryon stable, which may be interesting from a cosmological perspective. In addition, the mass of the lightest SU (2) glueball is m 0 ∼ 5 Λ B [72,73] so it is likely that the glueball and meson/baryon spectrum will overlap. However, as the lightest glueball is a 0 ++ state it will decay rapidly a pair of twin photons.  (40) in each sector are given by where n A f (n

III. SCALAR COUPLINGS TO MATTER
Thus far we have only considered the dynamics of the gauge sector and scalar potential.
We now investigate the consequences of new couplings of the colored scalars to fermions.
These couplings have two primary motivations. First, they cause the visible sector colored scalar φ A to decay, explaining in a simple way the absence of stable colored relics.
Second, following spontaneous color breaking in the mirror sector, such couplings produce new dynamical twin fermion mass terms. Consequently, the spectrum of twin fermions can be deformed with respect to the mirror symmetric model, which may have important consequences for cosmology and phenomenology. We emphasize, however, that the exact and similarly for the mirror sector. The table also indicates the corresponding decays of φ A and the twin fermion mass terms generated by each coupling, which will be discussed in more detail below. We will also make a few brief remarks below regarding possible couplings beyond those in Table II. A. Decays of φ A From Table II, the visible sector colored scalars φ A can decay in a variety of ways, depending on their quantum numbers and the particular couplings allowed by gauge symmetry.
Color triplets can decay to a pair of SM quarks, a quark and a neutrino, or a quark and a charged lepton. To illustrate, consider Φ A ∼ (3, 1, 2 3 ) with general Lagrangian containing the following interactions: where in the second line we have used Eqs. (26) and (27) which lead to the decay φ A → dd.
Taking into account the various flavors of quark and lepton, there are a variety of potential collider signatures of the colored scalars, which we explore in Sec. V. Of course, the colored scalar can decay in more channels than those listed in Table II. One possibility is that φ A decays to a pair of SM bosons. For instance, the color octet may decay to a pair of gluons through the dimension five operator Tr Φ A G A G A . Another interesting possibility emerges if operators that couple fields in the two sectors are present. These are typically higher dimension operators, and can naturally arise when 'singlet' fields [75], which transform by at most a sign under Z 2 , are integrated out. As an example, taking Φ A,B ∼ (3, 1, 2 3 ), we can write the operator (Φ AūA )(Φ BūB ) ⊃ f Φ φ AūAûB3 , leading to the decay of φ A to one SM quark and one twin quark. The same operator could allow the twin quark to decay back into the visible sector via an off-shell φ A .

B. Dynamical twin fermion masses
Before considering new twin fermion masses, we first recall the ordinary mass terms originating from twin electroweak symmetry breaking: These Higgs Yukawa interactions lead to the usual mass terms that are larger than those in the SM by the factor v B /v A = cot ϑ ≈ few.
The new twin fermion masses generated by spontaneous color symmetry breaking depend on the particular scalar representation and symmetry breaking pattern. The following discussion is intended to be illustrative, with examples presented for triplet, sextet, and octet models. The full set of possible twin fermion mass terms for a given model is provided in Table II. While we restrict our analysis to the SM fermion field content, we note that additional interesting possibilities for twin fermion masses arise if new singlet fermions are present in the theory [57].

Color triplets
We first study a triplet example with quantum numbers Φ ∼ (3, 1, 2 3 ). The Lagrangian contains the following interactions coupling the scalar to pairs of fermions: where in the second line we have set the scalar to its VEV, Φ i = f Φ δ i3 (Eq. (3)), effecting the spontaneous symmetry breakdown of We have also used the quark decomposition in Eq. (45). We note that the couplings λdd, c QQ in Eq. (51) are antisymmetric in generation space.
We see that new twin fermion mass terms beyond those generated by the Higgs VEV arise from the interactions in Eq. (51). In particular, there are 'Majorana-like' mass terms for the down-type quark fields, which are allowed since these fields are not charged under the unbroken twin electromagnetic gauge symmetry; see Table I. 5 There are also mass terms which marry '3rd color' ([SU (2) c ] B singlet) quark fields with leptons. From the electric charges in Table I

Color sextet
For the color sextet scalar we focus, for concreteness, on the case Φ B ∼ (6, 1, − 2 3 ). With these quantum numbers we can add the following interactions to the Lagrangian: where the couplings λdd, c QQ in Eq. (55) are symmetric in generation space. In contrast to the triplet case, no lepton mass terms are generated from Eq. (56). There are, however, new mass terms generated for down type quarks. We examine each of the two possible gauge symmetry breaking patterns for the color sextet in turn.
For case II, the sextet scalar obtains a VEV, Φ Bij = f Φ δ i3 δ j3 (Eq. (7)), leading to the symmetry breaking pattern Eq. (45), the twin quark masses that follow from Eq. (53) are given by These are Majorana mass terms for the '3rd color' ([SU (2) c ] B singlet) down quark fields, and are consistent with the fact that these quarks are not charged under the unbroken twin electromagnetic gauge symmetry; see Table I. (10)), case III, the down type quarks obtain a mass gauge symmetries in the low energy theory.

Alternatively, if the symmetry breakdown proceeds via [SU
The new mass terms in Eqs. (54) and (55)  In models with a real octet scalar, Φ B ∼ (8, 1, 0), there are two possible couplings to quark pairs that arise from dimension 5 operators, As with the sextet, no lepton mass terms are generated from Eq. (56), while the resulting quark mass terms are similar to the standard ones arising from the Higgs Yukawa couplings (50) in that they marry SU (2) L singlet and doublet quarks. The precise form of the quark masses depend on the pattern of gauge symmetry breaking.
For case IV, the octet scalar obtains a VEV, Φ B = √ 2f Φ T 8 (Eq. (16)), leading to the Eq. (45), the twin fermion masses that follow from Eq. (56) are given by Interestingly, in this case all quark colors obtain a mass from a single interaction.
In case V, the octet scalar obtains a VEV, In this case, only the first and second quark colors are lifted, while the third color does not obtain a mass. This is consistent with the unbroken symmetry. The mass terms in Eqs. (57) and (58) can be as large as O(100 GeV) for order one Wilson coefficients and f Φ ∼ Λ.

Other sources of twin fermion masses
Thus far we have considered twin fermion masses involving a single colored scalar field, and all such possibilities of this type are shown in Table II. Additional options arise from couplings involving two colored scalars. First, there is always the possibility of coupling the gauge singlet operator |Φ B | 2 to the usual Higgs Yukawa operators, e.g., After Φ B obtains a VEV, effective Yukawa couplings are generated in the twin sector, which can exceed the SM ones by a factor of 10-100 for the light generations without spoiling naturalness; see the discussion in Ref. [56] for further details. Furthermore, we can couple two color triplet scalars to pairs of quark fields in nontrivial ways to generate new twin quark masses. As an illustration consider Φ ∼ (3, 1, 2 , which provides an additional mass term beyond those presented in Eq. (51).

IV. INDIRECT CONSTRAINTS
The previous section showed that the spontaneous breakdown of twin color and Z 2 can also dynamically generate new twin fermion mass terms, when there are sizable couplings between the colored scalar fields and matter fields. The exact Z 2 symmetry correlates these new masses to visible sector phenomena, including baryon and lepton number violation, quark and lepton flavor changing processes, deviations in electroweak probes, and CP-violation.
Indirect tests in the visible sector can limit the size and structure of the new twin fermion mass terms. Given the range of models and possible new couplings (see Table II), a complete vetting of these constraints is beyond our scope. Instead, we provide illustrative examples of the characteristic phenomena that can occur. Many of the phenomena we consider here occur in the context of R-parity violating supersymmetry; for a review see Ref. [76].

A. Baryon and lepton number violation
In triplet models with hypercharge Y Φ = 2 3 , − 1 3 , − 4 3 the proton may decay, which leads to strong constraints on certain combinations of couplings. For a comprehensive review on proton decay see Ref. [77]. For example, consider Φ ∼ (3, 1, − 1 3 ) with non-vanishing couplings to the first generation, In this case, tree level exchange of φ A allows the proton to decay into a pion and positron, p + → e + π 0 , with decay width where |α| = 0.0090 GeV 3 [78] is the nucleon decay hadronic matrix element, F + D 1.267 [79] is a baryon chiral Lagrangian parameter, and f = 131 MeV. The current limits from Ref. [80] for this channel are τ p /Br(p + → e + π 0 ) > 1.6 × 10 34 yrs at 90% C.L. The non-observation of proton decay generally places strong limits on pairs of couplings that violate B in triplet scalars models. Depending on the flavor structure of the couplings, there may be other proton decay modes and other nucleon/baryon decays allowed.
In scenarios with a single colored scalar in the visible sector, nucleon decays with ∆B = 1 are usually the most sensitive probes of B violating couplings. Processes like neutronantineutron oscillations and dinucleon decays with ∆B = 2 are expected to be less sensitive.
However, if there are additional colored scalar fields present then such ∆B = 2 processes can be observable; see e.g., Ref. [81] for a recent study.
In triplet models with Y Φ = 2 3 , − 1 3 , certain combinations of scalar-fermion couplings can violate lepton number by two units while conserving baryon number. In such cases we generally expect that neutrino masses are generated radiatively. For instance, consider again Φ ∼ (3, 1, − 1 3 ), but with with the following interactions: These interactions break lepton number by two units. Neutrino masses will be generated at one loop, with characteristic size Here we have fixed m φ A = 1 TeV and used the bottom mass for m d , which leads to the strongest constraint.

B. Quark and lepton FCNC
The interactions of the colored scalars with matter in Table II can also lead to new tree level or radiative flavor changing neutral currents (FCNCs) in the quark and lepton sectors.
A variety of rare FCNC processes are possible, many of which impose strong constraints on the new scalar-fermion couplings.
For instance, sextet and octet models can mediate new tree level contributions to ∆F = 2 transitions in the kaon system. Taking Φ ∼ (6, 1, − 2 3 ) as an example, we write the interaction If the diagonal couplings λ 11 dd and λ 22 dd are nonvanishing, then tree level sextet scalar exchange generates the effective interaction with Wilson coefficient Current constraints on such operators probe new physics scales of order 10 4 TeV [82], which, noting Eq. (65), limits the typical size of these couplings to be at the level of 10 −3 or smaller.
Octet scalars, Φ ∼ (8, 1, 0), can also induce neutral meson mixing at tree level. After electroweak symmetry breaking, the scalar-quark coupling is If, for instance, c 12 Qd is nonzero, exchange of φ A generates the effective interaction where i, j denote color indices. The Wilson coefficient is given by While color triplet scalars do not mediate tree level ∆F = 2 transitions, sizable loop contributions to these operators can arise. As an example consider Φ ∼ (3, 1, There are two types of one-loop box diagrams that generate contributions to Kaon mixing [83,84]. The first involves the exchange of two colored scalars and leads to the effective Lagrangian (64). In the limit m φ A m t , the Wilson coefficient is The second type of diagram involves the exchange of one W boson and one colored scalar, leading to the effective Lagrangian For anarchic couplings λūd and heavy scalar mass m φ A m t , the leading contribution is Thus, the typical constraints on the couplings in this case are at the 10 −2 -10 −1 level.
Color triplets can also facilitate lepton flavor violation, such as the decay µ → eγ. If Φ ∼ (3, 1, − 1 3 ), for example, the coupling λ QL in Eq. (59) is The µ → eγ branching ratio is found to be where τ µ 2.2 × 10 −6 s is the muon lifetime. The MEG experiment has placed a 90% CL upper bound on the branching ratio, Br(µ → eγ) MEG < 4.2 × 10 −13 [85]. So, for a colored triplet with mass of order 1 TeV, the couplings are typically constrained to be smaller than about 0.03.

C. Electric dipole moments
When multiple scalar-fermion couplings are present in the theory new physical complex phases to appear. These can source new flavor-diagonal CP violation in the form of fermion electric dipole moments (EDMs). To illustrate, we investigate the contribution to electron electric dipole moment coming from a triplet Φ ∼ (3, 1, Exchange of up-type quarks leads to an electron EDM at one loop, described by the effective In the case of flavor anarchic couplings, the top loop dominates and leads to the prediction The best constraint on the electron EDM comes from the ACME collaboration: |d e | < 1.1×10 −29 e cm [86]. We see that for generic complex phases the constraints on the couplings are quite severe for this scenario. We expect that the neutron EDM can also provide a promising probe of certain combinations of couplings.

D. Charged current processes
The new interactions of fermions with colored scalars can also lead to new charged current processes. To illustrate, we consider here the decays of charged pions that occur for Φ ∼ Nonvanishing (λ QL ) 11 or (λ QL ) 12 lead to a modification to the lepton universality ratio, We have neglected the effects of decays such as π − → e −ν µ , etc., which do not interfere with the SM weak contribution, retaining only the dominant coherent contributions. The SM prediction [87] and measured value [88] are where the experimental uncertainty dominates the theoretical uncertainty. We apply a 2σ C.L. bound by demanding the new physics correction in Eq. (79) is less than twice the experimental uncertainty. This leads to the constraint In addition to pion decays, such couplings may be probed in hadronic tau decays as well as tests of charged current universality in the quark sector.

E. Discussion
Evidently, interactions between the colored scalar and matter can manifest in a host of precision tests. The exact Z 2 symmetry in our scenario ties any constraints coming from these measurements to the possible form and maximum size of the new twin fermion mass terms generated by those couplings (see Sec. III). We have seen that some of these constraints can be quite stringent (e.g., from baryon number violation or FCNCs), although it is clear that they hinge, in many cases, on a particular coupling combination or flavor structure. Though it is beyond our scope, it would be interesting to explore more broadly how the various patterns of new twin fermion mass terms arising from twin gauge symmetry breaking intersect with experimental constraints.

A. Direct searches for colored scalars
The colored scalar field φ A in the visible sector can naturally have a mass near the TeV scale and could therefore be produced in large numbers at hadron colliders like the LHC.
We concentrate on pair production, p p → φ A φ * A , since as an inevitable consequence of the strong interaction it provides the most robust probe of the colored scalars. There can also be single φ A production channels provided the scalar-fermion couplings discussed in Sec. III are sizeable, e.g., qq → φ A , qg → φ A , etc, but we focus on the various signatures expected from colored scalar pair production.
• Squark Searches: Color triplet scalars with quantum numbers (3, 1, − 1 3 ), (3, 1, 2 3 ) can decay to any quark flavor and a neutrino, φ A → q ν. The resulting collider signa-tures are identical to those of squark pair production in the Minimal Supersymmetric Standard Model, in which the squark decays to a quark and a massless stable neutralino. Therefore, searches for first and second generation squarks, sbottoms, and stops can be directly applied to these scenarios. A CMS search based on 137 fb −1 at √ s = 13 TeV rules out a single squark decaying to a light jet and massless neutralino for squark masses below about 1.2 TeV [89], while comparable limits have been obtained by ATLAS [90]. Final states containing a bottom or top quark along with a neutrino resemble sbottom or stop searches, which constrain the triplet scalars to be heavier than about 1.2 TeV [89,91]. The HL-LHC and, especially, a future 100 TeV hadron collider will be able to significantly extend the mass reach for such scalars.
Taking stops as an example, the HL-LHC (3 ab −1 , √ s = 14 TeV) will be able to constrain scalar masses up to about 1.6 TeV [92], while a future 100 TeV collider can probe scalars as heavy as 10 TeV [93].
• Leptopquark Searches: The color triplet models may also feature 'leptoquark' signals if the scalar decays to a quark and a charged lepton. A number of searches have targeted various leptoquark signals, depending on the flavor of the quark and charged lepton in the decay. Searches for first-and second-generation leptoquarks focus on the signature jj, with being an electron or muon. The best limits to date exclude scalar masses in the 1.4-1.6 TeV range and below [94][95][96]. The scalar may also have a significant branching ratio into a light jet and a neutrino. To cover these scenarios experiments have searched for the νjj final state, though these tend to give somewhat weaker constraints in comparison to the jj channel. In the future, the HL-LHC will be able to probe first and second generation leptoquarks in the 2-3 TeV range, while a future 100 TeV hadron collider will be able to extend the reach to the 10 TeV range and beyond; see, e.g., Ref. [97] for a phenomenological study of the prospects in the µµjj channel.
Various searches for third generation leptoquarks exist in which the scalar decays involve one or more of τ, b, t. For example, scalars decaying to tτ (bτ ) are constrained to be heavier than about 900 GeV (1 TeV) by ATLAS and CMS searches [98][99][100].
There is also a CMS search in the tµ channel that constrains scalar masses below 1.4  [113,114] for an in-depth survey.

B. Higgs coupling modifications
A coupling between the colored scalar and the Higgs fields is an essential ingredient in our scenario. This couplings allows for viable electroweak vacuum alignment, following spontaneous Z 2 breaking by the Φ B VEV. Consequently, the physical Higgs scalar and the colored scalars are coupled, (34). Through this coupling the new colored, charged scalars generate one loop contributions to the hγγ and hgg effective couplings, which can modify the decay of the Higgs to two photons or the production of the Higgs in gluon fusion. These modifications can be expressed in terms of modifications of the Higgs partial widths. Assuming 2m φ m h , we find (see e.g., Ref. [115]): where A SM γγ ≈ 6.5, A SM gg ≈ 1.4, d Φ is the dimension of the scalar representation, T Φ is its Dynkin index, and c Φ = 1 ( 1 2 ) for complex (real) scalars. The LHC has measured the hγγ and hgg couplings with 10% precision [116,117]. For sin ϑ 1/3, we find that current measurements can only probe relatively light scalars and low symmetry breaking scales f Φ , typically below about 300 (500 GeV) for color triplet (sextet and octet) scalars. In most cases direct searches for pair produced colored scalars yield stronger limits. However, as these searches depend on the assumed decay mode, Higgs coupling measurements still offer a complementary test of light colored and charged scalars. Looking forward, the Higgs coupling measurements at the HL-LHC and at future colliders may be able to achieve percent level precision, probing smaller values of sin ϑ and/or heavy colored scalar masses. The radial modes of the color symmetry breaking will also have a small effect upon the Higgs couplings, but as shown for the analogous hypercharge case the effect is typically negligible [56].

VI. OUTLOOK
The Mirror Twin Higgs provides an elegant symmetry-based understanding of the apparent little hierarchy between the EW scale and the dynamics at the 5-10 TeV scale posited to address the big hierarchy problem. Arguments related to vacuum alignment and cosmology suggest that the mirror symmetry protecting the light Higgs must be broken, and an attractive possibility is that this Z 2 breaking is spontaneous in nature. In this work, we have investigated the simultaneous spontaneous breakdown of the twin color gauge symmetry and Z 2 . Remarkably, despite being related by an exact mirror symmetry in the UV, vast differences between the two sectors are exhibited in the low energy effective theory below the TeV scale as a result of spontaneous symmetry breaking. These difference manifest in the residual unbroken gauge symmetries, color confinement scale, and particle spectrum.
The richness of these effects is tied to the variety of possible colored scalar representations and associated symmetry breaking patterns. We have outlined five minimal possibilities for models with a single color triplet, sextet, or octet, and explored how the twin sector departs from the mirror onset. In particular, we have shown how new dynamical mass terms may be generated for the twin fermions. These effects are tied by the discrete Z 2 symmetry to precision tests in the visible sector, allowing additional handles on uncovering the twin structure without direct access to many of the states. Furthermore, the new colored states may be probed at the LHC and at future high energy colliders.
There physical colored scalar fields: The same expression for the cubic scalar coupling V ⊃ A hφ † A φ A h Tr φ † A φ A , as in Eq. (34), is also obtained.

b. SU (3) → SO(3)
In this case, the colored scalar fields can be parameterized in unitary gauge as where φ A is a complex sextet of [SU (3) c ] A , φ B is a real quintuplet under [SO(3) c ] B , and φ 2 ≡ Tr φ † A φ A + Tr φ 2 B . In particular, we represent the sextet as a symmetric tensor, (φ A ) ij with i, j = 1, 2, 3, and the real quintuplet as φ B = φā B Tā, with real components φā B and indexā = 1, 3, 4, 6, 8 running over the broken generators.
By inserting Eqs. (26) and (A5) into Eq. (A1) we can derive the potential for the pNGB scalars. Minimizing this potential leads to the same condition defining the vacuum angle as was found for the triplet scalar, Eq. (29), as well as the same expression for the physical Higgs boson mass, Eq. (30). Furthermore, we find the following expressions for the masses of the physical colored scalar fields: We also obtain the same expressions for the cubic scalar coupling V ⊃ A hφ † A φ A h Tr φ † A φ A , as in Eq. (34).
Including the Higgs fields, we will consider the following Z 2 symmetric scalar potential: where |H| 2 = H † A H A + H † B H B and |Φ| 2 = Tr Φ 2 A + Tr Φ 2 B . We have included the possibility of a cubic interaction and higher dimension operators, As discussed in Sec. II A 3, the inclusion of such terms leads to a unique ground state in In this case, the color octet can be parameterized in unitary gauge as where φ A is a real octet of [SU (3) c ] A , φ B is a real triplet under [SU (2) c ] B , andφ 2 ≡ Tr φ 2 A + Tr φ 2 B . We represent the octet as φ A = φ a A T a with a = 1, 2, . . . 8 and the triplet as φ B = φ α B τ α with α = 1, 2, 3. All components φ a A , φ α B are real scalars. Inserting Eqs. (26) and (A11) into Eq. (A8) including the cubic term V 3 (A9), we can derive the potential for the pNGB scalars. Minimizing this potential leads to the same condition defining the vacuum angle as was found for the triplet scalar, Eq. (29), as well as the same expression for the physical Higgs boson mass, Eq. (30). Furthermore, we find the