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Nuclear Physics B 914 (2017) 201-219

www.elsevier.com/locate/nuclphysb

A radiative model of quark masses with binary tetrahedral symmetry

Alexander Natale

Korea Institute for Advanced Study (KIAS), Seoul 02455, Republic of Korea Received 6 September 2016; received in revised form 3 November 2016; accepted 8 November 2016 Available online 14 November 2016 Editor: Tommy Ohlsson

Abstract

A radiative model of quark and lepton masses utilizing the binary tetrahedral (T') flavor symmetry, or horizontal symmetry, is proposed which produces the first two generation of quark masses through their interactions with vector-like quarks that carry charges under an additional U(1). By softly-breaking the T' to a residual Z4 through the vector-like quark masses, a CKM mixing angle close to the Cabibbo angle is produced. In order to generate the cobimaximal neutrino oscillation pattern (613 = 0, 623 = n/4, Sep = ±n/2) and protect the horizontal symmetry from arbitrary corrections in the lepton sector, there are automatically two stabilizing symmetries in the dark sector. Several benchmark cases where the correct relic density is achieved in a multi-component DM scenario, as well as the potential collider signatures of the vector-like quarks are discussed.

© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction

Neutrino mass and oscillation are thoroughly established experimentally [1-9], as is the strong evidence for cosmological dark matter (DM) [10-12]; both are widely considered the best evidence for physics beyond the Standard Model (BSM), particularly given the robustness of the Standard Model (SM) at explaining the 2016 Run at the Large Hadron Collider (LHC). Neutrino mass, and subsequent oscillation, pose an interesting set of questions: why is there such a

E-mail address: alexnatale@kias.re.kr. http://dx.doi.Org/10.1016/j.nuclphysb.2016.11.006

0550-3213/© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.Org/licenses/by/4.0/). Funded by SCOAP3.

large gap in the scale of neutrino mass relative to the rest of the fermion masses in the SM, and why is the neutrino oscillation the way that it is — comparatively large angles relative to the Cabibbo-Kobayashi-Maskawa (CKM) matrix? A general framework, proposed in 2006, tries to solve these issues by generating neutrino mass radiatively through the interactions of neutrinos with DM at the one-loop level [13]. While this proposal was not the first model of radiative neutrino mass [14], or the first model that completed the loops with DM [15], these so-called sco-togenic, or Ma, models provide a comparatively simple way to connect neutrino mass and DM with a single Higgs at the one-loop level. These Ma models have also been extended to explain lepton and quark mass [16], which yields interesting signatures at colliders [17-19]. Recently, there has been interest in extending such radiative models through the addition of vector-like fermions, and in particular vector-like quarks [20,21]. Additionally, a program utilizing various non-Abelian discrete flavor symmetries, or horizontal symmetries, has been pursued to explain the particular pattern of neutrino oscillation (see Refs. [22,23] for reviews). The measurement of 9\3 = 0 [24,25], and the observation of a 125 GeV Higgs-like boson [26,27], disfavors many minimal models of horizontal symmetries that seek to explain the structure of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. However, a recent proposal for a modified scotogenic model of neutrino mass with a tetrahedral (A4) horizontal symmetry [28] is able to support the so-called co-bimaximal mixing pattern as a genuine prediction where 9\3 = 0, SCP = ±§, and 023 is maximal. While 023 being maximal is disfavored by NOvA at the 2.5 a level [29], the maximal value of 023 is still consistent with most of the neutrino oscillation data [30-32] and so a co-bimaximal mixing pattern is still well supported by the data. This extension of the Ma model has interesting features: the addition of vector-like fermions increases potential LHC signatures, and the model has the potential for multiple components of cosmological DM [28]. Additionally, any model that utilizes the A4 symmetry could just as easily utilize the double cover known as the binary tetrahedral group, or simply T'. Frequently, models that utilize a horizontal symmetry to explain lepton mixing produce a CKM mixing matrix which is diagonal in the symmetry basis (for instance Refs. [28,33-35] are models with horizontal symmetries that are either agnostic about quark mixing or assume a diagonal CKM), however flavor symmetries have been used to explain the Cabibbo angle [36,37] and in particular an angle close to the physical Cabibbo angle can be produced by utilizing the doublet representations of T' [38-49], however this flavor symmetry has never been studied before in the context of a scotogenic, or Ma model. In this paper, a model of radiative lepton and quark masses with a T' horizontal symmetry is proposed. By using the T' symmetry and using vector-like quarks to complete the quark mass loops, an angle close to the Cabibbo angle is produced. The model is based on the soft co-bimaximal A4 model from Ref. [28], however the U(1)D is modified (most notably) where the first two generations of quarks are chiral under this 'dark' gauge. The expanded particle content required to cancel anomalies yields interesting collider signatures, and it is found that the model can support multi-component DM with several distinct cases.

2. The model

The particle content for the current proposal is listed in Table 1, and is based on the recent proposal for cobimaximal A4 neutrino mixing which generates charged lepton and neutrino mass from the loops shown in Fig. 1 [28]. The SM gauge group (SU(3)c x SU(2)l x U(1)y) is expanded with a dark gauge U(1)D, and there are several additional discrete symmetries: a dark Z2, a softly broken flavor Z2, and the horizontal symmetry T'. The 'flavor' Z2 forbids tree level masses for the charged leptons, whereas the addition of two dark symmetries (Z2 and U(1)D) as

Table 1

Particle content.

Particles

SU(3)c

SU(2)l

dark Z2

SM particles

(v,l)l lr

' (c,s)l (u,d)l

-1/2 -1 1/2

3 3 10

cr = (cr ,ur) 3 2/3 1 + 22 +

sr = (sr,dr) 3 -1/3 -1 + 21 +

(t,b)l 3 1/6 0 + 10 +

tr 3 2/3 0 + 10 +

br 3 -1/3 0 + 10 +

Fermions

Nl,R 1 0 1/2 + 3 +

El,R 1 1/2 0 - 10 +

Fl 1 0 0 - 10 +

Ulrr 3 2/3 1/2 - 20 +

Dl,r 3 -1/3 -1/2 - 22 +

Tl 3 2/3 -1/2 + 10 -

Tr 3 2/3 3/2 + 10 -

Bl 3 -1/3 1/2 + 10 -

Br 3 -1/3 -3/2 + 10 -

Scalars

n 1 1/2 -1/2 + 10 +

X + 1 1 -1/2 + 10 -

s 1 0 0 - 3 +

P 1 1/2 1/2 - 3 +

a 0 1 0 -1/2 - 12 +

1 0 1 + 3 +

1 0 2 + 10 +

well as a horizontal symmetry allows for the generation of neutrino and lepton masses through the loops from Ref. [28] as discussed below. The non-Abelian discrete symmetry T' (also known as the binary tetrahedral group) is the double cover of A4, and has many of the same mulitplication rules as A4 namely [50,51]

3 ® 3 = 1o © 1i © I2 ® 3 © 3, (2.1)

1i ® 1j = 1i+j, (2.2) however T' has three doublet representations (20, 21,22) [50,51]:

2i ® 2j = 1i+j © 3, (2.3)

2i ® 3 = 2i © 2i+1 © 2i+2, (2.4)

where i, j = 0, 1, 2 mod 3. In addition to the particle content from Ref. [28] there is another scalar doublet (p), a non-trivial T' scalar singlet (a), up-like (U) and down-like (D) vector-like quarks which are T' doublets, and up-like and down-like T' singlets (T and B) which are chiral under U(1)D. The color-charged particles U and D are added to complete the loop in Fig. 2 in

vL s uL II Nr Nl lR

Fig. 1. One-loop neutrino and lepton masses generated with minimal modification to the mechanism in Ref. [28].

CiN ,<E> \ ✓ A 4

Ql Vr VL <1R

Fig. 2. One-loop quark masses for the first two generations of quark masses where V are the corresponding vector-like quarks (U ,D).

order to radiatively generate the first two generations of quark masses. Note that these vector-like quarks have dark charge and are also odd under the additional dark Z2, whereas the particles introduced to cancel anomalies carry the softly broken flavor Z2. The scalar singlets Z1 and Z2, which have integer dark charges, are a T' triplet and trivial singlet respectively and both receive non-zero vacuum expectation values (VEVs), thus spontaneously breaking U(1)D to a residual Z2, in which particles with half-integer charges are odd under the residual symmetry and all other particles are even. Even though the right-handed (RH) quarks carry dark charge while the rest of the SM does not, the proposed model is anomaly free. The [SU(2)]2U(1)D anomaly is zero since the left-handed (LH) quarks and leptons do not carry dark charge, the [U(1)Y ]2U(1)D anomaly is canceled by the contribution from T and B:

SM : 2 x 3| (1) + 2 x 3 (-^ (-1) = 2 (2.5)

(1) + 2 x 3( -i) (-1) = 2

T/B : 3

-1> - 3

2\2 /3

While the U(1)Y[U(1)d]2 anomaly is also canceled by the T and B contributions:

2 x 3(f)(1)2+2 x 3( -31

(-1)2 = 2

~ ) = -2. (2.6)

(2.7) -2. (2.8)

T -3(i) (-D2 - <3)(D2+3( - 0(02 - < T) (-c2

That is, the U(1)D anomalies are canceled between the two light generations of qR and the chiral anomaly coming from T and B in analogy to the U(1)R12 model [52]. The [SU(3)C]2U(1)D and mixed gravitational anomalies are canceled between CR - SR and T - B separately. Note that the charge assignment required to cancel the U(1)D anomaly means that the Z2 scalar must gain a VEV in order for these dark chiral vector-like quarks to gain a mass, and since these particles carry color-charge it is important that v2 is relatively large so that these particles can evade existing collider constraints on vector-like quarks.

The mixing terms between scalars are highly restricted from the dark symmetries and the horizontal symmetry such that the scalar potential terms for the fields that do not receive VEVs are generally of the form

V4>i = ^fyi + i<^i |2 + Hi^Vi |2 + X^1^112 + IZ212, (2.9)

along with quartic interaction terms between each scalar field of the form X^j |2, where 4>i = Z1, Z2, and where the structure of coefficients is fixed by the T' assignment. The mixing terms between the scalars that do not fit this pattern are Xirf$>x- and Xqpt$aZ1 which allow for the generation of quark and lepton masses, where the Xi term softly breaks the non-dark Z2, and the XDp tnsZ1 term which allows for potentially interesting DM phenomenology but is not of particular relevance for this study.

2.1. Lepton masses

The charged and neutral lepton masses are generated through the loops shown in Fig. 1, where the 'flavor' Z2 symmetry is softly-broken by the trilinear scalar term. Note that this is the mechanism used to generate the lepton masses in Ref. [28], however the horizontal symmetry is T' instead of A4, and the charge assignment under U(1)D is chosen to be half-integer values in analogue to Ref. [53], though these changes do not change the predictions for lepton mixing. Since the leptonic sector only uses T' singlets and triplets the mixing pattern is identical to a model utilizing just A4 [28], thus the binary tetrahedral model predicts the so-called cobimaximal neutrino mixing pattern (013 = 0, 023 = n/4, SCP = ±7r/2). The correct neutrino mass matrix is generated by the soft-breaking of the T' triplet representations to Z3 via the NLNR masses and to Z2 via s1s2 terms as discussed in Ref. [28] and resulting masses for the leptons are [53]

flLflR sin (0x) cos(dx)mN ,,

mi =-16^2-(F(Xi) - F(X2)), (2.10)

where F(Xi) = Xi log(Xi)/(Xi - 1), where X1,2 = m2xi 2/m2N and mx12 are the scalar masses resulting from the X^&x mixing where tan(20x) = ^V^2. The neutrino masses are given by [28] X n

22 f2m2DmF

16n2(mF - m2)

(G(xf) - G(x), (2.11)

where G(x) = j-x + x(-1l-X)x), where xF = m2F/m2E, xs = m2/m2E and mF and mE are the vector-like fermion masses and mD is the E - F mixing which is assumed to be small [28,54]. The importance of this mechanism is that the 'clashing' symmetry of the charged and neutral leptons that predicts co-bimaximal mixing (Z3 x Z2) will generically have arbitrary radiative corrections in the scalar sector if si sj terms break T' to Z2 and the scalars generating charged lepton mass break T' to Z3, as this forces Z3 breaking counter-terms to be introduced [28]. However by having the dark symmetries (Z2 and U(1)D), and allowing the fermions (Ni) to softly break T' instead of scalars, these arbitrary corrections are prevented [28].

While this neutrino mixing pattern predicts the maximal 023, it is possible that slight perturbations or corrections can move this prediction slightly away from this value, though the correlations between mixing angles that are produced from the horizontal symmetry will generally restrict any such deviation similar to the results in Ref. [54]. It is also important to note that

scotogenic mechanisms for charged leptons and quarks in general modifies the measured Higgs branching ratios, which has been studied in detail in the context of scotogenic models [55] and have potential important consequences for muon g - 2 [20,21], though such studies are beyond the scope of this work.

2.2. Quark masses

The loop for the generation of the u, d, c, and i quark masses is shown in Fig. 2, where the loop is completed by the T' triplet scalar doublet p and the non-trivial T' singlet scalar a0, and the T' symmetry is softly-broken by the vector-like quark masses which are generated by dimension three terms. The mass matrix for p is given by

Mp = I B A B I , (2.12)

\B B AJ

where A is a combination of v2, v^, and v2 and B is proportional to just v^. This mass matrix is exactly diagonalized by the tribimaximal mixing matrix:

/ 7273 1/73 0 \ Utb = I -1/76 1/73 -1/72 I , (2.13)

V-1/76 1/73 1/72 J

since B only depends on v2 then it is a reasonable assumption that if v2 > vi that the resulting masses for p are nearly degenerate. The quark mass loop is completed by the pat,\ term, so the mixing is relatively small. Given the assumption that the Mp masses are nearly degenerate after rotating with the tribimaximal matrix, then the mass matrix that spans the p - a states, Mpa, is of the form

Mpa = ( "> vz), (2.14)

p 1 Xqv^vz ma J

where the resulting mass states of Mpa are labeled as y^2. The quark mass matrix is thus given by

U _ fqLfqR sin(0y) cos(9y)^ Mq = 32^ 2 ^ (2.15)

where tan(20y) = ^-2qv22 and is a 2 x 2 matrix where the flavor structure and the loop calculations has been taken into account. Specifically, for the up-like quarks Iq is of the form

/-(F[X1 ]cos(0y)2 + F[X2] sin (0V)2) (F[X1]-F[X2]) sin (20v) \ \ -(F[X1]-F[X2])sin(20v) F[X1 ]sin(0V)2 + FX]cos(0V)2) , .

with F [Xi ] = F [Xi1] - F [Xi2], and F [Xj ] = mVlXij log (Xij)/(Xij - 1), Xj = mj/m2^, and 0y is the mixing angle that diagonalizes My (the vector-like quark mass matrix), and for the down-like quarks Iq is

^/0 F[X1] cos2 (0y) + F[X2] sin2(0yA

I 0 (F[X1] - F[X2]) cos(0y) sin(0y J , .

where the miss-match is because of the different transformations of the vector-like quarks U and D under the horizontal symmetry (T'). The quark mass matrix squared, |M|2, is diagonalized by a 2 x 2 rotation matrix where the angle is a function of Oy. The resulting CKM matrix is the miss-match between the up-like and down-like sectors. The T' symmetry is softly broken by dimension three terms of VLVR of the form

(m11+m33+2m13 i(m11-m33) \

-i(mn -m33) mn+m33-2m13 I , (2.18)

which is protected by a residual Z4 symmetry where

V1 = (V10 + V13 )/2, (2.19)

V2 = -i(V11 - V13 )/2, (2.20)

where V11, and V13 are the Z4 flavor states transforming as non-trivial singlets 11 and respectively (where 10 is the trivial singlet) and where Vp V1j = mij and m13 = m31. Note that this basis where Eq. (2.18) has complex off-diagonals can be rotated in such a way where in the T' limit the mass matrix is given by equal diagonal values. In this basis, the soft-breaking matrix can be parameterized assuming m33 = m11 + 8 which yields

2m 11 + 8 + 2m 13 8 \ (2 21)

8 2m 11 + 8 - 2m 13 I,

if m13 ^ 1 and 8 a m13 then the deviation from the T' symmetric mass matrix is small. It is convenient to parameterize the soft-breaking matrix in terms of the mass eigenstate m11 such that m13 = em11. The rotation that diagonalizes the vector-like quarks is the same for up-like and down-like V's, but the differing textures of Mq produce different dependence on Oy for the V[ that diagonalizes | M |2. For the up-like sector the angle Ou is simply tan(2Ou) — 8/(2m\3), whereas for the down-like sector is approximately tan(2Op) = 8e/(2m13). The resulting Cabibbo angle as a function of 8/m 13, and for various e < 1/2, are plotted in Fig. 3. While the exact Cabibbo angle can only be fit for very specific choices of the soft-breaking terms, a variation on the order of 30% from the physical value can be fit in a much wider parameter space. It is important to note that in order to fit all of the parameters of the physical VCKM it is well known that a perturbation to this texture from running effects or some higher level loop contributions must exist [37], or even simply additional T' doublets at a higher mass scale. However if the predicted mixing angle, Oc, is within 15 percent of the physical Cabibbo angle then the relative difference between Oc and the physical Cabibbo angle is the same size as the next largest mixing angle in the CKM; in comparison to the implementation of quark mixing in Ref. [28], the CKM after soft-breaking of the horizontal symmetry is still approximately diagonal and thus the deviation from the physical values is quantitatively smaller in this Binary Tetrahedral model.

For a proof of concept of how this model could accommodate a fully realistic CKM, consider the two-loop mixing of the first generations with t as shown in Fig. 4, where f± ~ (0,2j, -, -), f±2 - (1/2, 2j, +, -), and - (1,2k, +, +) under (U(1)D, T', dark Z2Z2), and ^ receives a non-zero VEV and each new scalar is a singlet under SU (2). This adds additional complications to the scalar sector, however f 0 is a T' doublet and only couples to the other scalars via quar-tic terms, and so these additional scalars will not contribute to the Higgs mass directly if each component of the T' doublet receive the same VEV. This consequence is easiest to determine in

16 14 12 10

• 0C±3O%

• ec±20%

• ec±10%B

• 9c±5a

0.5 0.6 0.7 0.8 0.9 1.0

Fig. 3. The first-order value of CKM mixing angle ■pjT^jj in degrees as a function of the T' soft-breaking terms 5/m 13. Banded region represents the physical Cabibbo angle (9c) ±30%, 20%, 10%, and 5a.

Pf _i_

\x+ _]_,

Ql Ur Bl

Fig. 4. Example of a two-loop extension to generate mixing between the first two generations of quarks and the top quark, where f± ~ (0, 2j, -, -), f±2 ~ (1/2, 2j, +, -), and f° ~ (1, 2k, +, +) under (U(1)D, T', dark Z2, Z2), and f 0 receives a non-zero VEV.

the Ma-Rajasekaran basis from Ref. [5 1 ]; in this basis f0f0 = ^(f0^ 12 - f02f 11), where f1i is the ith component of the T' doublet and where (%01> = (f02> = 0. This ultimately leads to only trilinear terms between the physical degrees of freedom of f0 and $ (after taking into account additional scalar mixing terms between f0 and f 1) but has no contributions to the SM-like Higgs boson mass, regardless of which of the distinct T' doublet representations are chosen for f, and so there are no clearly problematic contributions to flavor physics from these new scalars. There is non-trivial f 0 - f1 mixing as the multiplication rules of T' (2j ® 2j = 3) generate the appropriate terms in the scalar potential which are not eliminated after the new constraint equations are taken into account. In the mass matrix spanned by the scalar degrees of freedom, these mixing terms will contribute to f1 - f0 - f2 and Im(f1) — Imf) mixing, but the details of this mixing and the proper prediction of the CKM matrix depend on the exact choice of T' representations chosen for the f scalars. These additional scalars will alter the mass matrices for the first two generations of quarks in order to generate the proper CKM, thus the physical pseudoscalar couplings and masses should be correlated to the perturbation of the one-loop quark mixing angles. These scalars also allow new interactions for the vector-like quarks, however if mf is large these new decay channels may be suppressed at the LHC. In addition, these new scalars could be treated as

mediators for the dark sector if the lightest scalar resulting from the a0 - p mixing is the DM candidate for the sector that carries charge under the additional U(1) gauge group.

2.3. Scalar sector

There are three scalars with integer charges under the U(1)D gauge symmetry, where $ breaks SU(2)L x U(1)Y just as the Higgs field in the SM, where the VEVs of the scalars charged under U(1)D ((Z1i) = v1i and (Z2) = v2) break the dark gauge to a residual Z2 dark parity, where particles with half-integer charges have odd parity and all others have even parity. The scalar potential relevant to the symmetry breaking is

y = ^tf® + + M2 J2 Zîi + "2° (Zli Zn )2

+ k 1 a>2(' —1 ) z î Z 1 i)(J2°ji- 1Z î Z 1 i ) + k 1 3 (£ Z i(i+2) Z 1 i Z î Z 1 (i+2) ) i i i

+ ((ZÎZ3)2 + (Z3Z 1)2 + (Z ÎZ2)2 + H.C.) (2.22)

+ m2Z2*Z2 + y(Z2Z2)2E(Z1iZ1iZ2* + H.C.)

+ J2 kH 1&$ZÎiZ1i + kH2®t$ZÎZ2

where ^12 can be taken to be by rotating the relative phase between Zu and Z2, and m = el ~ (note that mk terms are modulo 3). If the $ - Z1 and Z2 - Z1 mixing is ignored, then minimizing the potential for Z11 yields:

1 dV 2 2 222 22 222

— —— = n! + X10 Zi(v2i) + X11 (v21 + «X2 + mv23)(v21 + mv22 + m2v2l3) Zn 9Z11 1 0 1 1 11 12 13 11 12 13 (2.23)

+ A. 13 vn(v22 + v23) + A. 13/ vn(v22 + v23), with similar terms for Z12 and Z13. These constraints are met given

U11 = ^ = ^ = U1 = V 3k1o + 2k13+ 2k 13/ ' (2.24)

and since both $ and Z2 are trivial T' singlets the corrections to Eq. (2.24) will be equal for all Z1i, so this minimization condition can be satisfied even with the mixing terms if all Z1i VEVs are equal. With v11 = v12 = v13 = v1 and Z11 = Z12 = Z13 = Z1 the full constraint equations are

0 = fi2H + kHV2 + 3kH 1v2 + "h 2 v2 (2.25)

0 = + 3k10 v2 + 2 (k 13 + k13/)v2 + M12v2 + "h 1 v2 + k^vf (2.26)

0 = M2 + k2v2 + -H12— + kH2v2 + 3k12v2. (2.27)

The resulting mass matrix for h,V2Re (Z1), and V2R e(Z2) is

( 2XhV2 6Xhivvi 2Xh2W2 ^

6Xh 1 vvi 6(Xi + 2(Xi3 + Xi3' ))v2 3(2X2V2 + Mi2)vi

y2Xn2vv2 3(2X2V2 + /12>1 2X2v2 - 2/12

(2.28)

In general, there is mixing between the Z1 and Z2 scalars, however for simplicity we take the term 3(X2v2 + /12)v1 to be negligible, and v2 ^ v22 so that , and V2Im(<f>°) become the longitudinal components of the W± and Z gauge bosons, and h is the 125 GeV SM Higgs (note that since v2 is responsible for the TR,L and BRL masses that the assumption that v2 > v2 is necessary for the masses of T and B to be much larger than the SM quarks). The mass matrix for ^/2Im(Z1,2) is

/ 6/12 v2 3/12 v1 \

U™ (229)

which is diagonal for the linear combinations of \fl(v1Im(Z1) + 2v2Im(Z2))/^Jv2 + 4v^, which is massless and becomes the longitudinal mode for the Z' boson, and \/2(v1Im(Z2) — 2v2Im(Z1 ))/^Jv2 + 4v| which becomes the massive pseudoscalar particle A. The physical scalar masses are thus

m2h - 2v2 ( XH — 3XH\ — 2-^ 2 I ■ (2.30)

V 2(X13 + X13') X2v2 — 3/121 2 2 2 2 3 v 12 2

m^R - 6v1 (X10 + 2(X13 + X13/)), m^R - v2(X2 — -/12 — )■ mA

3/12 (v2 + 4v2)

-—^^-2) (2.31)

Many of the fermions with U(1)D charge are chiral, and have no Yukawa couplings at tree-level to Z1i (or Z2) with the exception of T and B, which are given by

mT = v2yT ■ mB = v2ys ■ (2.32)

thus for Yukawa couplings on order of one, and v2 > v, these vector-like quark masses can be large.

The mass eigenvalues from p — a and n — X are given by

myo = „

y1,2 2

m2± =

x1,2 2

3mp + m2a + cos(20y)(±m2p ^ m2a) ± 6vvX sin(20y)

(2.33)

my± = 2 mp (2.34)

m2 + mX + cos(20x)(±m? ^ m2 ) + 2vXi sin(20x)

(2.35)

mxo = 2 m2 ■ (2.36)

where tan(20x) = 2, tan(20y) = mrm:, and there are three copies of y± that are nearly degenerate.

2.4. Dark gauge sector

The relevant Lagrangian terms for the Z gauge boson are

1 sin (k) 4 "V 2

where DM is the covariant derivative (DM = dM + iqDgZZ') and V is the scalar potential from Eq. (2.22). Expanding Zu and Z2 under the assumptions in the previous section, yields

L 3 --Z"VZ'Xv - -nK1Z"vB'V + (D'Z1i)H.D'Z1i) + (D»Z2)\D''Z2) - V, (2.37)

2/ ~ „2/a„,2 , ,2

g2(3v2 + 4v2), (2.38)

and where the ZZ'Z' term becomes

2g2 (3Re(Z1)v1 + 4Re(Z2)v2)Z' Z", (2.39)

and the ZZZ'Z' becomes

g2(3Re(Z1)2 + 4Re(Z2)2)Z" Z(2.40)

The structure of the U(1)D charges makes this model very similar to leptophobic Z' models since the only tree-level coupling to the SM is to quarks. There is no tree-level kinetic mixing term, however such a term can arise at the loop-level due to the coupling to quarks which generates the k term, and so there is some non-zero coupling to leptons. This kinetic mixing can have an important impact on the dark sector constraints [56,57], though for the purposes of this study the mixing is taken to be negligible.

There are some notable differences in this model compared to many common Z' models, namely the top and bottom quarks do not directly couple to the Z' so collider searches of the form pp — Zr —> tt do not apply. Mixing constraints involving the b quark do not occur at tree-level, and the first two up-like quarks carry opposite charges to the first two down-like quarks. There are additional considerations as leptophobic Z' models generically produce FCNCs in the RH quark decays [58,59] which can be heavily constraining. However, K0 - K and B0 - B° are less important since the LH quarks do not carry dark charge [60,61]. Previous studies of Z' with only RH coupling to quarks have been carried out, and indicate that a Z' with a mass on order of 1 TeV is still viable for certain mixing constraints, particularly in a case with a U(2) flavor symmetry [62] which can be accommodated by the T/ horizontal symmetry [63]. There are also additional constraints on leptophobic models from the early LHC searches (cf. Ref. [64,65]), and from the 13 TeV run [66-68], however there is still a viable range of parameter space for Z' models with a gZ ^ 0.1 and masses between 1 TeV < mZ/ < 1.5 TeV. Another interesting possibility is if the coupling constant gZ ^ 1, which allows for much lighter Z"s (albeit very weakly interacting) [69]. The addition of vector-like quarks that couple to the Z' are another source of FCNC, however mixing of T - t and B - b are forbidden at tree-level from the flavor Z2 assignment. If neither the horizontal or the Z2 flavor symmetry are broken by any hard terms, then the RH coupling to the Z and Z' are fixed, and so it is reasonable to assume the RH quark mixing to be exactly diagonal at first order which eliminates FCNCs in the gauge sector. Some amount of FCNC are unavoidable in the scalar sector but these are severely restricted by the additional symmetries, that is the gauge and T' assignments prevent tree-level FCNCs and the soft-breaking terms of T' are in the fermion sector. However, there are still contributions to the FCNCs from box and penguin diagrams. For instance, the largest source of FCNCs in the box diagrams are from the charged scalar p and the vector-like quarks U and D, which generate K0 - K0 mixing through the dimension six (sRdL)(s~LdR) operator which can be estimated as

A. Natale /Nuclear Physics B 914 (2017) 201-219 N Z'

Fig. 5. Relevant diagrams for NN annihilation contribution to (av).

f^fRmy- my)

A2 32^2(4m23 + 82) (m2V; - m2y,)(m2V; - m2y)

10-7Am2 , _y/_

Am2 Am2

iy' jy

(2.41)

where 8 and m13 are from the 0v mixing angle, my (my) are the masses of the p - a mixing and myi are components of the appropriate vector-like T' doublets, with the assumption that the couplings fqL,R ~ 0.1. The Am2yy term is constrained by the oblique parameters and other

electroweak precision tests, whereas Am2y can be quite large, and so the overall contribution to FCNCs will in general be below existing upper-limits [70]. This suppression is a general feature of the box diagrams in this model, and so similar expressions can apply to different dimension six FCNC operators. There are also penguin diagram contributions, though the loop structure is similar to the one-loop diagrams that generate the up and down-like quark masses from Fig. 2. Since these loops involve the vector-like quarks and the doublets pi, the contribution to the leptonic FCNCs decays, such as B° ^ l±P, will primarily arise from Z' kinetic mixing term which is small. That is, while n — x mixing can introduce some leptonic FCNC terms, there is no tree-level p - n mixing which means there is no direct connection between the additional scalars that couple to quarks and the scalars that couple to leptons.

2.5. Dark matter

The dark sector is potentially very complicated: in the dark-charged sector the DM candidate is the lightest mass state of any of the neutral scalars (n0, p0, and a0) or the lightest neutral fermions Ni, and for the dark-parity-only sector it is potentially the lightest neutral fermion (E0 of F0) or si. However, there are numerous constraints on scalar DM particles that interact with the electroweak (EW) gauge bosons, so to avoid these we set the mass spectrum such that N\ and s1 are the only DM candidates. The main interactions relevant for the relic density are show in Fig. 5 for the fermionic DM candidate, and in Fig. 6 for the scalar DM candidates. Even in this framework however, there are several major scenarios: either N or s are the bulk of the cosmological DM or each species significantly contributes to the cosmological DM. Before any further assumptions of masses are made, the relevant SM-DM couplings are given in the mass basis by

eLN\R(cos(0x)xl - sin(6X)x2 ) + vlNrx0

fsS1(elEr + vlEr ),

eRNiL(cos(9x)x2 + sin(0x)x- ) + vlNirx + ^77. p0

fNE(cos(0y)y0 - sin(0y)y0)N 1,2,3ER.

(2.42)

(2.43)

(2.44)

(2.45)

ï s ®,yi,2,y±Xuù

Fig. 6. Relevant diagrams for ss annihilation contribution to {ov).

There are potential collider constraints on the masses of the scalars, and of the vector-like fermions, however masses on order of a few hundred GeV for the scalars and on order of 500 GeV can be sufficient to avoid these constraints [17,71]. An interesting consequence of the proposed model is the existence of a particular mass scheme where the scalar DM species mass is on order of hundreds of MeV (the minimal Ma model can also accommodate MeV DM [72] ), but because of the multiple component nature the relic density constraints can still be met with the fermionic dark matter on order of 100 GeV. However, in this mass range there are extra constraints that need to be considered. In particular the Higgs invisible width contribution from the h ^ ss is an important constraint, where the partial-width is given by

,,2i2 /,„„2/,„„2

Th^ss = -L—-4-, (2.46)

which yields Th^ss/Th & 0.07 for ms & 100 MeV and ^sh — 0.01. Additionally, there are potential constraints from ss ^ LL via the t-channel exchange of ER,L as well as Big Bang Nucleosynthesis (BBN) constraints for such light DM [73]. In this mass range the scalar DM species only represents approximately one third of the total cosmological DM, for a particular set of masses and couplings, which reduces constraints on the annihilation cross section of DM. Unlike other models with sub-GeV DM the Z' and the DM scalar DM species have couplings to SM particles and cannot avoid BBN constraints [74], and so results from Planck on the annihilation cross-section rule out this particular mass scheme [75].

However, there are three additional mass schemes that produce thermal relic DM candidates in a mass range that can avoid the constraints from Planck:

• Model A: mE0 — 455 GeV, mE± — 450 GeV, mF0 — 600 GeV, mx± — 646 GeV, mx± — 654 GeV, mx0 — 650 GeV, my± — 247 GeV, my0 — 250 GeV, my0 — 252 GeV, with — 0.1 and mv — 1200 GeV. 1 2

• Model B: mE0 — 455 GeV, mE± — 450 GeV, mF0 — 600 GeV, mx± — 646 GeV, mx± — 654 GeV, mx0 — 650 GeV, my± — 247 GeV, my0 — 250 GeV, my0 — 252 GeV, with —

0.00033 and mz, = 2.7 GeV, ms = 100 GeV, mNl = 70 GeV. Model C: mE0 = 850 GeV, mE± = 825 GeV, mF0 = 600 GeV, mx± = 998 GeV, mx± = 1006 GeV, mx0 = 1002 GeV, my± = 646 GeV, my0 = 650 GeV, ^y0 = 654 GeV, -with gZ = 0.0025 and mZ = 20 GeV, ms = 150 GeV, mM = 9 GeV.

In each of these models the rest of the particle content is taken to be on the order of a TeV. In order to determine relic density and direct detection constraints on the spin-independent (SI) cross sec-

Relic Denisty LUX (2016) — Overlap

mN [GeV]

Fig. 7. Scan of Model A mass scheme in ms-mn plane, where plotted points produce a DM relic density between 0.11 < ^oh2 < 0.13, are lower than the bound on ogi from LUX 2016/PandaX, and the region where the DM candidates meet both constraints.

tion in an automated way, the relevant Lagrangian terms are implemented using FeynRules [76] in order to generate model files to use with MicrOmegas [77]. For the numerical analysis the relic density is allowed to vary:

For direct detection the LUX 2016 and PandaX direct detection limits on the SI cross section are used (where the reported limit is on order of 4 times stronger than LUX 2015) [78-80]. However, since this model is potentially multi-component, the direct detection constraints are fitted to the SI cross section modified by the proportion of the total relic density that each species contributes to the overall cosmological relic density for that particular mass [81].

Model A: The dark matter masses are scanned over for the case where fs = 0.22, fiL = fiR = 0.7, and XSH = 0.01 and displayed in the ms -mN mass plane in . Even taking into account the latest LUX/PandaX constraints on the SI cross section, significant regions of the thermal relic DM parameter space survive.

Model B/C: For these models a full numerical scan was not performed, however, both are able to fit the relic density range from Eq. (2.47) and are below the SI cross section upper-bound from LUX/PanadaX, and for the mass choices are able to avoid the constraints on the annihilation cross section from Planck. These parameter spaces are of potential interest for the existence of their relatively light Z' and, in Model C's case, their relatively heavy scalar masses.

2.6. Collider signatures

The proposed model has additional EW states, an additional gauge boson, and additional heavy colored states any of which could produce a novel collider signature. However, new EW scalar states that do not mix with the Higgs (i.e. inert or dark scalars) may be challenging to find at a hadron collider [17], however the phenomenology of the x±2 states are essentially identical to previously studied models and so the primary source of potential collider signatures at the LHC are from the vector-like quarks or the Z'. The U and D vector-like quarks can be pair produced at the LHC, and their subsequent decay chain is to a quark, a lepton, and both species of DM, where the flavor of the lepton species will be fixed by the horizontal symmetry, however,

0.11 < Q.H2 < 0.13.

(2.47)

Fig. 8. Pair production from gluons of T(B) and subsequent decay to SM particles and the fermionic DM species (Ni).

this decay chain involves multiple mediators which can reduce the signal and complicate the analysis. An alternative signature is the production and decay of T and B, which decays to a top or bottom quark respectively, and x± which subsequently decays to a neutral fermion Ni and a charged lepton as shown in Fig. 8. Thus, if these vector-like quarks are the lightest colored-states, then the primary collider searches for colored particles in this model are to 2 bottom (top) quarks, 2 leptons, and missing energy, in contradistinction to the typical vector-like quark searches [82]. Additionally, the Z' can be searched for at the LHC through dijet signatures [65,83], assuming the mass and coupling choices of Model A. Future precision measurement of the Higgs coupling may also be able to rule out this model from either a more precise measurement of the invisible branching fraction [84] or from deviations in various Higgs boson couplings that occur in scotogenic models generally [55].

3. Conclusion

In this work a scotogenic model of neutrino and charged lepton masses was extended to generate the first two generations of quark masses through their interaction with vector-like quarks. Additionally, the binary tetrahedral symmetry T', in lieu of A4, is utilized for the first time in the scotogenic framework. Using a particular T' assignment which is softly-broken by the vectorlike quark mass terms to a residual Z4, the model is found to produce a mixing angle close to the Cabibbo angle. The quark masses are generated through their interaction with the additional vector-like quarks, which carry both dark charge and are odd under an exactly conserved Z2. The first and second generation of quarks are allowed to transform under the U(1)D, which necessitates the addition of T' singlets that are chiral under the dark gauge, but vector-like under the SM gauge group, in order to cancel anomalies. The model thus has a leptophobic Z', and two DM candidates (one scalar s and one fermion N), and is found to successfully fit the relic density, the constraints on the annihilation cross section from Planck, and evade the latest SI

cross section limits from the direct detection experiments LUX and PandaX. In addition, the decays of the vector-like quarks T and B are shown to produce a very interesting signature which could be found at colliders in the future with an interesting final state when compared to existing vector-like quark models being investigated at the LHC.

Acknowledgements

I am grateful to Anthony DiFranzo, Kiel Howe, Bithika Jain, Corey Kowanacki, Pyungwon Ko, Takaaki Nomura, Yusuke Shimizu and Peiwen Wu for critical discussions that made this work possible.

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