Gauge Coupling Unification with Hidden Photon, And

Physics Letters B 768 (2017) 30–37

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Gauge coupling uniﬁcation with hidden photon, and minicharged dark matter ∗ Ryuji Daido a, Fuminobu Takahashi a,b, Norimi Yokozaki a, a Department of Physics, Tohoku University, Sendai, Miyagi 980-8578, Japan b Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa 277-8583, Japan a r t i c l e i n f o a b s t r a c t

Article history: We show that gauge coupling unification is realized with a greater accuracy in the presence of a massless Received 27 October 2016 hidden photon which has a large kinetic mixing with hypercharge. We solve the renormalization group Received in revised form 25 January 2017 equations at two-loop level and find that the GUT unification scale is around 1016.5 GeV which sufficiently Accepted 26 January 2017 suppresses the proton decay rate, and that the unification is essentially determined by the kinetic mixing Available online 16 February 2017 only, and it is rather insensitive to the hidden gauge coupling or the presence of vector-like matter fields Editor: J. Hisano charged under U(1)H and/or SU(5). Matter fields charged under the unbroken hidden U(1)H are stable and they contribute to dark matter. Interestingly, they become minicharged dark matter which carries a small but non-zero electric charge, if the hidden gauge coupling is tiny. The minicharged dark matter is a natural outcome of the gauge coupling unification with a hidden photon. © 2017 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 introduce unbroken hidden U(1)H gauge symmetry with a large kinetic mixing χ with U(1)Y [8]; the kinetic mixing with unbro- The Standard Model (SM) has been so successful that it ex- ken hidden U(1)H modifies the normalization of the hypercharge plains almost all the existing experimental data with a very high gauge coupling in the high energy, thereby improving the gauge accuracy. The lack of clear evidence for new particles at the LHC coupling unification. In this paper we focus on this simple resolu- experiment so far began to cast doubt on the naturalness argu- tion and argue that GUT with a hidden photon naturally leads to ment which has been the driving force of search for new physics minicharged dark matter. at TeV scale. On the other hand, there are many phenomena that In Ref. [9], the two of the present authors (F.T. and N.Y.), to- require physics beyond the SM, such as dark matter, baryon asym- gether with M. Yamada, recently studied the gauge coupling unifi- metry, inflation, neutrino masses and mixings, etc. Among them, cation with unbroken hidden U(1)H by solving the RG equations at the gauge coupling unification in a grand unified theory (GUT) is one-loop level, including the effect of extra matter fields charged an intriguing and plausible possibility, which has been extensively under U(1)H , and discussed a possible origin of the required large studied in the literature. kinetic mixing as well as phenomenological and cosmological im- The running of gauge couplings are obtained by solving the plications of the extra matter fields. Those hidden matters are renormalization group (RG) equations, which depend on the matter stable and contribute to dark matter. In particular, they acquire contents and interactions among them. Assuming only the SM par- fractional electric charge through the large kinetic mixing, and ticles, the SM gauge coupling constants come close to each other such fractionally charged stable matter has been searched for by as the renormalization scale increases. If we take a close look at many experiments [10–21]. the running, however, they actually fail to unify unless rather large In this paper we study the GUT with a hidden photon in a threshold corrections are introduced. The gauge coupling unifica- greater detail and argue that minicharged dark matter is its natural tion is realized with a greater accuracy in various extensions of outcome. First of all we refine the analysis of Ref. [9] by solving the the SM, such as supersymmetry [1–5], introduction of incomplete RG equations at two-loop level, and determine the GUT unification multiplets (see e.g. Refs. [6,7]), etc. One simple resolution is to scale as well as the required size of the kinetic mixing precisely. The GUT unification scale turns out to be about 1016.5 GeV which is high enough to suppress the proton decay rate, and the required * Corresponding author. kinetic mixing is χ 0.37 at the scale of the Z-boson mass. Sec- E-mail address: [email protected] (N. Yokozaki). ondly, we find that the unification is almost determined by the http://dx.doi.org/10.1016/j.physletb.2017.01.085 0370-2693/© 2017 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. R. Daido et al. / Physics Letters B 768 (2017) 30–37 31

kinetic mixing, but it is rather insensitive to the size of the hidden gY gH χ gY = , gmix =− . (6) gauge coupling or the presence of vector-like matter fields charged 1 − χ 2 1 − χ 2 under U(1)H and/or SU(5). As a consequence of the kinetic mixing, Here, g is the gauge couplings of U(1) , and g remains un- the hidden matter fields carry a non-zero electric charge, and they Y Y H changed by the transformation from the original basis to the become minicharged dark matter if the hidden gauge coupling is canonical one. One can see that the field now acquires a frac- sufficiently small. Thus the minicharged dark matter is a natural i tional hypercharge, gmixqH i/gY , which is a renormalization scale outcome of the GUT with a hidden photon. We will give concrete dependent quantity [28–30]. The U(1) coupling with a prime, g , examples of such minicharged dark matter.1 Y Y is the gauge coupling in the original basis (see Eq. (1)), and is The rest of this paper is organized as follows. In the next sec- smaller by 1 − χ 2 compared to g . Thus, the kinetic mixing with tion we explain how the gauge coupling unification is improved by Y unbroken U(1) modifies the normalization of the hypercharge adding U(1) , and show the results of solving RG equations at two- H H coupling constant, and the unification of the gauge couplings can loop level. In Sec. 3 we discuss implications of the hidden matter 5 fields for minicharged dark matter. The last section is devoted for be improved by choosing χ so that 3 gY at the GUT scale is equal discussion and conclusions. to the unified gauge coupling determined by the running of g2 2 and g3. In other words, the gauge coupling unification is realized 2. Gauge coupling unification with hidden photon in the original basis where the kinetic mixing is manifest. We shall return to the origin of such kinetic mixing later in this section. 2.1. Preliminaries In the canonical basis, i has the fractional U(1)Y charge and its effect is captured by the beta-functions of the gauge couplings. One way to improve unification of the SM gauge couplings is The actual calculations to be given in the next subsection are based on the two-loop RG equations, but let us give the one-loop RG to modify the normalization of the U(1)Y gauge coupling at high energy scales. This can be realized by introducing unbroken hid- equations below to get the feeling of how the gauge couplings evolve. den gauge symmetry U(1)H , which has a large kinetic mixing with The one-loop beta-functions of the gauge couplings in the U(1)Y [28]. The relevant kinetic terms of the hypercharge and hid- canonical basis are given by [31] den gauge fields, Aμ and A Hμ, are given by

dgY 1 3 2 2 1 1 μν χ = (bY g + bH gY g + 2bmix g gmix), L =− F F μν − F F − F F μν, (1) dt 16π 2 Y mix Y 4 μν 4 Hμν H 2 Hμν dgH 1 = 3 bH gH , where Fμν and F Hμν are gauge field strengths of U(1)Y and U(1)H , dt 16π 2 respectively. In this basis which we call the original basis in the dg 1 mix = 2 + 2 + 3 following, the gauge fields and field strengths are indicated with a (bY gmix g 2bH gmix g bH g dt 16π 2 Y H mix prime symbol. For later use, we also introduce pairs of vector-like + 2 + 2 fermions, 2bmix gY gH 2bmix gY gmix), (7) = ¯ where t ln μR (μR is a renormalization scale) and L − M ii, (2) 41 4 4 4 i 2 2 bY = + Q , bH = qH , bmix = Q iqH . 6 3 i 3 i 3 i i i i where i has a hypercharge of Q i and a U(1)H charge of qH i . The gauge interaction terms of the matter field i are written as (8) μ On the other hand, the beta-functions of the gauge couplings and ¯ γ (g Q A μ + g q A ) , (3) i μ Y i H H i H i the kinetic mixing parameter in the original basis take a surpris- ingly simple form. By using Eq. (7), the beta-functions in the orig- where gY and gH are the gauge couplings in the original basis. We assume that hypercharges of vector-like fermions are rational inal basis are written as numbers in the original basis such that they can be embedded into dgY 1 3 the SU(5) GUT multiplet. = bY g , dt 16π 2 Y The canonically normalized gauge fields, Aμ and A Hμ, are ob- dgH 1 3 tained by the following transformations: = bH g , dt 16π 2 H Aμ χ = = − dχ 1 2 2 Aμ , A Hμ A Hμ Aμ, (4) = χ(bY g + bH g ) − 2bmix g gH . (9) 1 − χ 2 1 − χ 2 dt 16π 2 Y H Y L =−1 μν − Note that the RG running of gY does not depend on gH nor χ at and then, the kinetic terms become canonical, 4 Fμν F μν the one-loop level, although its normalization is fixed by χ . At the 1 F F . In this canonical basis, gauge interaction terms of the 4 Hμν H two-loop level, this property does not hold and the RG running matter field are written as of gY depends on gH and χ (see in Appendix B). However, the dependence is still weak due to the loop suppression factor, and ¯ μ + μ iγμ(gY Q i A gH qH i A H )i we have confirmed this numerically. Therefore, because of rather μ = ¯ [ + μ + ] weak dependence on gH and χ , the gauge coupling unification is iγμ (gY Q i gmixqH i)A gH qH i A H i, (5) essentially determined by the size of χ . where

2 We emphasize here that the kinetic mixing with hypercharge is only able to 1 See e.g. Refs. [22–26] for recent works on minicharged dark matter. The suppress the gauge coupling gY compared to gY . On the other hand, introducing minicharged dark matter is often considered in a context of the mirror sector; see extra matter ﬁelds with hypercharge has the opposite effect on gY and does not Ref. [27] for a comprehensive review on mirror dark matter. improve the uniﬁcation. 32 R. Daido et al. / Physics Letters B 768 (2017) 30–37

Fig. 1. The RG runnings of gauge couplings. We take χ = 0.365 at mZ . (In the case with hidden matter ﬁelds, we further set gH = 1.1.) The black solid (green dashed) lines 16 17 show the result using two-loop (one-loop) RG equations. On the right panel, the region of μR around 10 –10 GeV is zoomed. Here, αs(mZ ) = 0.1185 and mt (pole) = 173.34 GeV. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

2.2. Numerical results of solving RG equations Ref. [8], there is no phenomenological implications for low-energy physics (except for suppressed proton decay rates), since the mass- Now we study the RG runnings of the gauge couplings using less hidden photon is decoupled from the SM sector, and the hid- 3 two-loop beta-functions, in order to see if the SM gauge couplings den U(1)H simply changes the normalization of the U(1)Y gauge unify at the high-energy scale. The two-loop beta-functions are ob- coupling. In particular, from the low-energy physics point of view, tained by utilizing PyR@TE 2 package [32,33], and the numerical there is no way to determine the correct basis at the high energy results shown in this section are based on solving the RG equa- except for requiring the successful gauge coupling unification, since tions at two-loop order unless otherwise stated. In the numerical any basis appears to be on an equal footing. calculations, we have included threshold corrections to the SU(3)C Next we introduce a vector-like fermion, which is charged only gauge coupling and top Yukawa coupling. under U(1)H , Let us first study the case without extra matter fields by solving L − ¯ the RG equations at the two-loop order. The one-loop analysis in M000, (10) = this case was studied in Ref. [8]. The beta-functions of the gauge with (Q 0, qH 0) (0, 1). The beta-functions of the SM gauge cou- couplings at the one-loop level are given by plings at the one-loop level are same as above and the beta- function of the U(1)H is given by dg 1 41 Y = 3 gY , dgH 1 4 dt 16π 2 6 = g3 . (11) dt 16π 2 3 H dg2 1 19 = − g3, = = dt 16π 2 6 2 In the numerical calculations we set M0 1 TeV, gH 1.1 and χ = 0.365. In this case, the results have turned out to be essen- dg3 1 = (−7) g3, tially same as Fig. 1. We have confirmed that the RG runnings of dt 16π 2 3 the gauge couplings as well as the unification scale are rather in- where g and g are the gauge couplings of SU(2) and SU(3) , sensitive to gH even at the two-loop level, by varying gH from 2 3 L C −4 respectively. The hidden gauge coupling does not run in this case. 10 to 1.1. Note that, in this scenario with hidden particles, the −1 −1 −1 basis where the gauge coupling unification occurs is manifest, be- In Fig. 1, we plot the RG running of α1 , α2 and α3 , where cause the hypercharges of hidden matter fields need to be quan- α = 5 g 2/(4π), α = g2/(4π) and α = g2/(4π). The black solid 1 3 Y 2 2 3 3 tized (including zero) so that they are consistent with the SU(5) (green dashed) lines show the result computed using two-loop GUT gauge group. (one-loop) beta-functions. We take χ = 0.365 at the scale of the Although the RG running of g is almost insensitive to g even Z boson mass, m . (The value of g shown in the figure is for Y H Z H at the two-loop level, the running of χ depends sensitively on the the next case we study below.) As one can see, the difference be- size of gH . We show RG running of χ for different values of gH tween the one-loop and two-loop results are not large, but the 17 in Fig. 2. For a large gH as 1.0, χ at 10 GeV becomes large as expected unification scale with the two-loop calculation is around 17 0.7, while if gH is smaller than 0.5, χ at 10 GeV remains around 1016.5 GeV, which is slightly smaller than that with the one-loop 0.45–0.5. calculation. Next we consider the case where there are Nbi pairs of bi- The above case without any extra matter fields captures the charged vector-like fermions: essence of how the kinetic mixing between hypercharge and hid- N den U(1)H improves the unification. However, as emphasized in bi L =− ¯ + ¯ MV (L,iL,i D¯ ,iD¯ ,i), (12) i=1 3 Apart from the gauge couplings, we only take into account the top-Yukawa cou- ¯ = pling and Higgs quartic coupling. In the numerical analysis, we use one-loop RG where L,i (D¯ ,i ) is 2 of SU(2)L (3 of SU(3)C ); (Q L,i, qH L,i) − = equations for these couplings. ( 1/2, 1) and (Q D¯ ,i, qH D¯ ,i) (1/3, 1). Here, L,i and D¯ ,i form R. Daido et al. / Physics Letters B 768 (2017) 30–37 33

a complete SU(5) multiplet. In this case, bmix vanishes. (See Ap- pendix A for a case where and ¯ have different q and do L,i Di H not form a complete multiplet.) The one-loop beta-functions of the gauge couplings are dgY 1 41 10 3 = + Nbi g , dt 16π 2 6 9 Y dg2 1 19 2 3 = − + Nbi g , dt 16π 2 6 3 2 dg3 1 2 3 = −7 + Nbi g , dt 16π 2 3 3 dgH 1 20 3 = Nbi g . (13) dt 16π 2 3 H In Fig. 3 and 4, we show the RG runnings of the gauge couplings and the mixing parameter for Nbi = 1, 3, 4, based on the two-loop beta-functions. We set gH and χ at mZ as gH = 0.2 and χ = 0.37. 10 In Fig. 3 (Fig. 4), we take MV = 1 TeV (10 GeV). One can see that − Fig. 2. The RG running of the kinetic mixing for g = 10 4, 0.5and1.0atm from H Z the uniﬁcation scale does not depend on Nbi nor MV . Also, the bottom to top. required value of χ at mZ for the uniﬁcation remains almost intact

Fig. 3. The RG runnings of the gauge couplings (left) and mixing (right) with Nbi bi-charged fields. The cases of Nbi = 1, 3, and 4are represented by black solid, blue dashed, and red dotted lines, respectively. We take gH = 0.2and χ = 0.37 at mZ . The mass of the bi-charged field, MV , is set to be 1TeV. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

10 Fig. 4. Same as Fig. 3 but for MV = 10 GeV. 34 R. Daido et al. / Physics Letters B 768 (2017) 30–37

4 for different choices of Nbi and MV . On the other hand, χ at the 21] and the direct detection constraint [34] on the minicharged 17 high-energy scale (e.g. 10 GeV) is sensitive to the change of Nbi particle exclude the region where the thermal relic abundance is and MV . consistent with the observed dark matter abundance. To obtain Thus, we have found that, once χ (mZ ) is fixed to be around the correct thermal relic abundance without running afoul of as- 0.37, we can freely choose Nbi, MV and gH without affecting the trophysical and direct detection constraints, one needs to include gauge coupling unification. In particular the unification is realized additional interactions between the minicharged particles and the even with a tiny hidden gauge coupling. This feature is suitable to SM sector. identify the hidden matter as dark matter, which can have a tiny To be concrete, we consider the following Higgs portal interac- electric charge (in the canonical basis). Interestingly, the stability of tions [36–40]: the dark matter is ensured by the unbroken hidden U(1)H gauge L −m2 |X|2 + λ |H|2|X|2 or symmetry. X X − ¯ λY 2 ¯ 1 γ5 −mY Y Y + |H| Y Y + h.c., (19) 2.3. Origin of the kinetic mixing Y 2 where H is a SM Higgs doublet, is some mass scale, X and Finally, let us comment on the origin of the kinetic mixing Y Y are respectively a complex scalar and a Dirac fermion with a unit χ and a possible modification of the unified gauge coupling. We U(1) charge, and m and m are their masses. In the canonical have seen that a relatively large χ ∼ 0.37 is necessary to improve H X Y basis, X and have an electric charges of the gauge coupling unification. Such a large kinetic mixing may be Y generated via an operator, χ gH qe =− . (20) 1 − χ 2 gY k0 μν L Tr( 24 F5μν)F , (14) 0 M∗ H If mX,Y H , the correct abundance is obtained for λX 0.6 and − − − m 1 TeV or λ / 10 3.5–10 3.25 GeV 1 and m 1–10 TeV = X Y Y Y where 24 is√ a GUT breaking Higgs and 24 (2, 2, 2, [41], avoiding the constraint from the LUX experiment [42]. As 17 −3, −3)v24/(2 15) with v24 ∼ 10 GeV; M∗ is a cut-off scale. long as the electric charge |qe| is sufficiently small, this allowed Then, the kinetic mixing is given by region does not depend on |qe|. Therefore, the minicharged dark matter can explain the observed dark matter if it has a Higgs por- v24 χ =−k0 . (15) tal coupling. M∗ The minicharge of dark matter is constrained by various obser- For M∗ ∼ 1018 GeV, the induced kinetic mixing is naturally of vations. First, if the minicharged dark matter is tightly coupled to O(0.1), as required for the successful unification. the photon–baryon plasma during recombination, the power spec- We also note that there could also exist an operator, trum of the Cosmic Microwave Background anisotropies is modi- fied and it becomes inconsistent with observations [14,16,18,21]. k1 μν Requiring that the dark matter is completely decoupled from the L Tr(24 F5μν F ), (16) 2M∗ 5 plasma at the recombination epoch, an upper-bound on qe is obtained as [16] which modifies the unified gauge coupling. It leads to the follow- 1/2 ing deviations, −4 mX,Y |qe| 10 , (21) 1TeV −1 −1 −1 α 1 k α 3 k α 1 k −4 1 = 1 χ, 2 = 1 χ, 3 =− 1 χ, which requires gH 10 . Direct detection experiments give even −1 60 k −1 20 k −1 15 k tighter constraints. From the LUX experiment [42], the constraint α5 0 α5 0 α5 0 on qe is [34] (17) m 1/2 | | × −10 X,Y where α5 is the squared of the unified coupling divided by 4π . qe 3.6 10 . (22) −1 − −1 −1 1TeV Therefore, if k1/k0 6, the deviation, ( α2 α3 )/α5 , is within 5% level while obtaining the large kinetic mixing of 0.5. Note that this LUX constraint can be applied to very heavy Note that it is not possible to achieve gauge coupling unifica- minicharged dark matter. tion only with the operator in Eq. (16), due to the proton decay Finally, let us comment on an alternative production of the constraint: we need a sizable kinetic mixing for gauge coupling minicharged dark matter. The minicharged particle may have an unification. interaction with a heavy particle (e.g. inflaton), and the correct relic abundance may be obtained by non-thermal productions via the interaction. For instance, if the inflaton has a quartic coupling 3. Minicharged dark matter with X, and oscillates about the origin where the X becomes (almost) massless, a preheating process could take place. For certain Let us suppose that there is a field, H , which is charged only coupling and the mass of X, the right abundance of X can be gen- under U(1)H in the original basis. Then, the field acquires an elec- erated. One way to suppress the overproduction of X is consider tric charge qe proportional to the hidden gauge coupling gH , and so, if gH is tiny, it becomes a minicharged particle. Since the stability of H is ensured by the U(1)H , such minicharged particle 4 If the minicharged particle is not a dominant component of dark matter, the can be a good candidate for dark matter. constraint from the direct detection experiment may be avoided in the following In order to account for dark matter, such minicharged particles range of qe [16]: must be somehow produced in the early Universe. As long as we 1/2 −10 mX,Y −2 mX,Y 5.4 × 10 |qe | 1.1 × 10 , (18) assume thermal production through electromagnetic interactions, 1TeV 1TeV however, they can not be a dominant component of the dark mat- since the minicharged particle may be evacuated from the Galactic disk by the su- ter. This is because the astrophysical constraints [11,12,14,16–19, pernova shock waves and Galactic magnetic fields [35]. R. Daido et al. / Physics Letters B 768 (2017) 30–37 35 a minicharged dark matter of a heavy mass. Such scenario is also consistent with the phenomenological constraints as long as the electric charge is small enough.

4. Discussion and conclusions

We have investigated the gauge coupling unification in the presence of an unbroken hidden U(1)H symmetry, which mixes with the U(1)Y of the SM gauge group. By solving the two-loop RG equations, we have found the gauge coupling unification is achieved with a better accuracy if the size of the kinetic mixing is χ 0.37 at the Z boson mass scale. The unification scale is around 1016.5 GeV, which is large enough to avoid the rapid proton decay. Interestingly, the unification behavior is essentially determined by the kinetic mixing parameter only, and it is rather insensitive to the size of the hidden gauge coupling or the presence of the vector-like fermions charged under U(1)H and/or SU(5). This im- plies that the vector-like masses can be arbitrarily light (or heavy) without affecting the gauge coupling unification. Fig. 5. The lower bound on g from Weak Gravity Conjecture is shown. The region The above findings imply that the Peccei–Quinn mechanism to H below the dashed line is conflict with WGC. We also show the upper bound on gH the strong CP-problem [44–46] can be easily embedded into our from LUX 2016 (solid) and the projected sensitivity of XENON 1T (dotted). setup: if the vector-like fermion of the SU(5) complete multiplet is coupled to a Peccei–Quinn scalar, it induces the required color anomaly [47,48]. The gauge coupling unification is preserved ir- respective of the axion decay constant, which is typically around the intermediate scale. Interestingly, if the vector-like fermion has a hidden U(1) charge, the axion is coupled to the hidden photon, and the axion will be a portal to the hidden photon. In this case, the dark matter could be composed of both the QCD axion and the minicharged dark matter. We have shown that the U(1)H gauge coupling can be arbitrarily small while keeping the successful unification. In this case, a hidden particle charged under U(1)H has a tiny electric charge due to the kinetic mixing. The U(1)H charge ensures the stability of the particle; therefore, the minicharged particle is a natural candidate for dark matter. The minicharged dark matter can have a correct relic abundance through the Higgs portal interac- tions5 while avoiding known phenomenological constraints. Thus, the minicharged dark matter naturally arises from GUT with a hidden photon. Finally, let us comment on the possible lower bound on the hid- = 10 den gauge coupling. From the Weak Gravity Conjecture (WGC) [49], Fig. 6. The RG runnings of the gauge couplings for MV 1 TeV (solid) and 10 GeV = = which claims that the gravity is the weakest force, the hidden (dashed) with flipped hidden charges. Here, gH 0.1and χ 0.37 at mZ . gauge coupling must satisfy the constraint, gH (mX,Y ) > mX,Y /M PL, 6 where M PL is the Planck mass. and 26287039 (F.T.), and by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan (F.T.). Acknowledgements Appendix A. A case with flipped hidden charges F.T. thanks K. Kohri for useful discussion on the cosmological In Eq. (12), L,i and ¯ form a complete SU(5) multiplet, effects of charged massive particles. This work is supported by To- D,i and bmix vanishes. Here, we consider a difference case: L,i hoku University Division for Interdisciplinary Advanced Research = − and D¯ ,i have flipped charges as (Q L,i, qH L,i) ( 1/2, 1) and and Education (R.D.); JSPS KAKENHI Grant Numbers 15H05889 and = − (Q D¯ ,i, qH D¯ ,i) (1/3, 1) leading to non-vanishing bmix. In Fig. 6, 15K21733 (F.T. and N.Y.); JSPS KAKENHI Grant Numbers 26247042 3 10 we show the results for MV = 10 GeV (solid line) and 10 GeV (dashed line). Again, the unification does occur with χ(mZ ) 0.37. 5 Similarly, the minicharged dark matter is expected to have the correct relic abundance through the axion portal [43]. The benefit of this case is that the non- Appendix B. Two loop RG equations renormalizable interaction (Eq. (19)) is not needed for the fermionic dark matter. 6 Here, we adopt a version of the conjecture that the mass of the lightest charged Here we give the relevant RG equations at two-loop level fol- particle ml should satisfy ml < gH (ml)M PL. lowing to Ref. [50]. This constraint is shown in Fig. 5, together with the upper bound from the LUX experiment. The WGC also claims that the cut-off scale of U(1)X , , is smaller B.1. Hidden vector-like fermion than gH ( )M PL. Requiring that be larger than the GUT scale, gH needs to sat- 17 isfy gH (MGUT) > MGUT/M PL with MGUT ∼ 10 GeV. This condition predicts many hidden particles with masses between mX,Y and MGUT, leading to large enough In the case with Nvec hidden vector-like fermions, the two-loop ∼ −2 gH (MGUT) of 10 . RGEs are given as follows. Here, qH,i denotes U (1)H charge of 0,i . 36 R. Daido et al. / Physics Letters B 768 (2017) 30–37 ⎛ ⎞ 3 N dgY gY 41 3 bi = dgH gH ⎝ 20 2 ⎠ 2 = q dt 16π 6 2 Hi dt 16π 3 = 3 2 i 1 gY 199 gY 9 2 ⎡⎛ ⎞ + + g N (16π 2)2 18 1 − χ 2 2 2 g3 10 bi g 2 + H ⎣⎝ q2 ⎠ Y 44 17 (16π 2)2 3 Hi 1 − χ 2 + g2 − y2 i=1 3 3 6 t ⎛ ⎞ ⎛ ⎞ Nbi 2 Nbi 3 Nvec ⎝ 4 ⎠ gH ⎝ 2 ⎠ 2 dgH g 4 + 20 q + 6 q g = H q2 Hi 1 − χ 2 Hi 2 dt 16 2 3 Hi i=1 i=1 π = ⎛ ⎞ ⎤ i 1 Nbi Nvec 3 2 2 2 g 4 g + ⎝ ⎠ ⎦ + H 4 q H 16 qHi g3 2 2 Hi 2 (16π ) 1 − χ i=1 i=1 ⎡ ⎛ ⎞ ⎤ Nvec Nbi dχ χ 41 2 4 2 2 dχ χ ⎣ 41 10 2 ⎝ 20 2 ⎠ 2 ⎦ = g + q g = + Nbi g + q g 2 Y Hi H dt 16 2 6 9 Y 3 Hi H dt 16π 6 3 = π = i 1 ⎡ i 1 4 1 199 gY χ 9 2 2 4 + + χ g g 1 ⎣ 199 35 gY χ 2 2 − 2 Y 2 + + Nbi (16π ) 18 1 χ 2 (16π 2)2 18 54 1 − χ 2 Nvec 4 ⎛ ⎞ ⎛ ⎞ 44 g χ 17 2 2 4 H 2 2 Nbi Nbi 2 2 + χ g g + 4 q − g y χ 10 1 + χ 2 g g χ 3 Y 3 Hi 1 − χ 2 6 Y t + ⎝ ⎠ 3 + ⎝ 2 ⎠ Y H = qHi gY gH 20 qHi i 1 9 1 − χ 2 1 − χ 2 3 i=1 i=1 dg2 g2 19 = − 9 3 44 16 dt 16π 2 6 + + 2 2 + + 2 2 Nbi χ gY g2 Nbi χ gY g3 2 2 3 9 g3 3 g 2 35 3 ⎛ ⎞ ⎛ ⎞ + 2 Y + 2 + 2 − 2 N N g2 (12) g3 yt bi − bi (16π 2)2 2 1 − χ 2 6 2 ⎝ ⎠ 2 ⎝ 32 ⎠ 2 + 6 qHi g gH g + qHi g gH g Y 2 3 Y 3 3 i=1 i=1 dg3 g = 3 (−7) ⎛ ⎞ ⎛ ⎞ 2 dt 16π Nbi 4 Nbi 4 gH χ 2 2 2 3 2 + ⎝ ⎠ + ⎝ ⎠ 20 q 6 q gH g χ g3 11 gY 9 2 2 2 Hi − 2 Hi 2 + + g + (−26) g − 2y = 1 χ = (16π 2)2 6 1 − χ 2 2 2 3 t ⎛ i 1 ⎞ i 1 ⎤ N bi 2 ⎝ 2 ⎠ 2 2 17 2 2 ⎦ dyt yt 17 gY 9 2 2 9 2 + 16 q g g − g y = − − g − 8g + y Hi H 3χ Y t χ 2 2 3 t 6 dt 16π 12 1 − χ2 4 2 i=1 2 3 dλ 1 g dg2 g2 19 2 = 2 − Y − 2 + 2 = − + Nbi 24λ 3λ 9λg2 12λyt 2 dt 16π 2 1 − χ 2 dt 16π ⎡6 3 ⎛ ⎞ 3 2 Nbi 2 4 2 g 3 1 g g 3 gY 3 gY 2 9 4 4 + 2 ⎣ + Y + ⎝ 2 ⎠ H + + g + g − 6y Nbi 2 qHi 2 2 2 2 2 t 2 2 − 2 − 2 8 (1 − χ ) 4 1 − χ 8 (16π ) 2 2 1 χ = 1 χ ⎛ ⎞ i 1 Nbi B.2. Bi-charged vector-like fermion ⎝ ⎠ gY gH χ 35 49 2 + 2 qHi + + Nbi g 1 − χ 2 6 6 2 i=1 In the case with Nbi pairs of bi-charged vector-like fermions, ⎤ the two-loop RGEs are given as follows. Here, we assume that L,i ¯ = − 2 3 2⎦ (D¯ ,i ) is 2 of SU(2)L (3 of SU(3)C ); (Q L,i, qH L,i) ( 1/2, qH,i) and + (12) g − y = 3 2 t (Q D¯ ,i, qH D¯ ,i) (1/3, qH,i). 3 3 dg g 41 10 dg3 g 2 Y = Y + N = 3 −7 + N 2 bi 2 bi dt 16π 6⎡ 9 dt 16π ⎡ 3 3 2 3 2 gY ⎣ 199 35 gY g3 ⎣ 11 2 gY + + Nbi + + Nbi (16π 2)2 18 54 1 − χ 2 (16π 2)2 6 9 1 − χ 2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ Nbi 2 Nbi Nbi 2 Nbi 10 g 10 g gH χ g 4 g gH χ + ⎝ q2 ⎠ H + ⎝ q ⎠ Y + ⎝2 q2 ⎠ H + ⎝− q ⎠ Y Hi − 2 Hi − 2 Hi − 2 Hi − 2 3 = 1 χ 9 = 1 χ = 1 χ 3 = 1 χ i 1 i 1 ⎤ i 1 i 1 ⎤ 9 3 2 44 16 2 17 2⎦ 9 2 38 2 2⎦ + + Nbi g + + Nbi g − y + g + −26 + Nbi g − 2y 2 2 2 3 9 3 6 t 2 2 3 3 t R. Daido et al. / Physics Letters B 768 (2017) 30–37 37 2 dyt yt 17 g 9 9 [21] A. Kamada, K. Kohri, T. Takahashi, N. Yoshida, arXiv:1604.07926 [astro-ph.CO]. = − Y − g2 − 8g2 + y2 dt 16π 2 12 1 − χ2 4 2 3 2 t [22] J.M. Cline, Z. Liu, W. Xue, Phys. Rev. D 85 (2012) 101302, arXiv:1201.4858 [hep- ph]. [23] R. Foot, S. Vagnozzi, Phys. Rev. D 91 (2015) 023512, arXiv:1409.7174 [hep-ph]. dλ 1 g 2 = 2 − Y − 2 + 2 [24] R. Foot, S. Vagnozzi, Phys. Lett. B 748 (2015) 61, arXiv:1412.0762 [hep-ph]. 24λ 3λ 9λg2 12λyt dt 16π 2 1 − χ 2 [25] R. Foot, S. Vagnozzi, J. Cosmol. Astropart. Phys. 1607 (07) (2016) 013, arXiv: 1602.02467 [astro-ph.CO]. 4 2 [26] V. Cardoso, C.F.B. Macedo, P. Pani, V. Ferrari, J. Cosmol. Astropart. Phys. 3 gY 3 gY 2 9 4 4 + + g + g − 6y 1605 (05) (2016) 054, arXiv:1604.07845 [hep-ph]. − 2 2 − 2 2 2 t 8 (1 χ ) 4 1 χ 8 [27] R. Foot, Int. J. Mod. Phys. A 29 (2014) 1430013, arXiv:1401.3965 [astro-ph.CO]. [28] B. Holdom, Phys. Lett. B 166 (1986) 196. References [29] S.L. Glashow, Phys. Lett. B 167 (1986) 35. [30] E.D. Carlson, S.L. Glashow, Phys. Lett. B 193 (1987) 168. [1] P. Langacker, Precision tests of the standard model, in: Proceedings of the PAS- [31] K.S. Babu, C.F. Kolda, J. March-Russell, Phys. Rev. D 54 (1996) 4635, arXiv:hep- COS90 Symposium, World Scientific, 1990. ph/9603212. [2] J.R. Ellis, S. Kelley, D.V. Nanopoulos, Phys. Lett. B 260 (1991) 131. [32] F. Lyonnet, I. Schienbein, F. Staub, A. Wingerter, Comput. Phys. Commun. 185 [3] U. Amaldi, W. de Boer, H. Furstenau, Phys. Lett. B 260 (1991) 447. (2014) 1130, arXiv:1309.7030 [hep-ph]. [4] P. Langacker, M.x. Luo, Phys. Rev. D 44 (1991) 817. [33] F. Lyonnet, I. Schienbein, arXiv:1608.07274 [hep-ph]. [5] C. Giunti, C.W. Kim, U.W. Lee, Mod. Phys. Lett. A 6 (1991) 1745. [34] E. Del Nobile, M. Nardecchia, P. Panci, J. Cosmol. Astropart. Phys. 1604 (04) [6] H. Murayama, T. Yanagida, Mod. Phys. Lett. A 7 (1992) 147. (2016) 048, arXiv:1512.05353 [hep-ph]. [7] C. Bachas, C. Fabre, T. Yanagida, Phys. Lett. B 370 (1996) 49, arXiv:hep-th/ [35] L. Chuzhoy, E.W. Kolb, J. Cosmol. Astropart. Phys. 0907 (2009) 014, arXiv:0809. 9510094. 0436 [astro-ph]. [8] J. Redondo, arXiv:0805.3112 [hep-ph]. [36] H. Davoudiasl, R. Kitano, T. Li, H. Murayama, Phys. Lett. B 609 (2005) 117, [9] F. Takahashi, M. Yamada, N. Yokozaki, Phys. Lett. B 760 (2016) 486, arXiv:1604. arXiv:hep-ph/0405097. 07145 [hep-ph]. [37] B. Patt, F. Wilczek, arXiv:hep-ph/0605188. [10] M. Marinelli, G. Morpurgo, Phys. Rep. 85 (1982) 161. G.[38] Y. Kim, K.Y. Lee, Phys. Rev. D 75 (2007) 115012, arXiv:hep-ph/0611069. [11] S. Davidson, M.E. Peskin, Phys. Rev. D 49 (1994) 2114, arXiv:hep-ph/9310288. [39] V. Barger, P. Langacker, M. McCaskey, M.J. Ramsey-Musolf, G. Shaughnessy, [12] S. Davidson, S. Hannestad, G. Raffelt, J. High Energy Phys. 0005 (2000) 003, Phys. Rev. D 77 (2008) 035005, arXiv:0706.4311 [hep-ph]. arXiv:hep-ph/0001179. G.[40] Y. Kim, K.Y. Lee, S. Shin, J. High Energy Phys. 0805 (2008) 100, arXiv:0803. [13]. I.T Lee, et al., Phys. Rev. D 66 (2002) 012002, arXiv:hep-ex/0204003. 2932 [hep-ph]. [14] S.L. Dubovsky, D.S. Gorbunov, G.I. Rubtsov, JETP Lett. 79 (2004) 1, Pisma Zh. [41] A. Beniwal, F. Rajec, C. Savage, P. Scott, C. Weniger, M. White, A.G. Williams, Eksp. Teor. Fiz. 79 (2004) 3, arXiv:hep-ph/0311189. Phys. Rev. D 93 (11) (2016) 115016, arXiv:1512.06458 [hep-ph]. C.[15] P. Kim, E.R. Lee, I.T. Lee, M.L. Perl, V. Halyo, D. Loomba, Phys. Rev. Lett. 99 [42] D.S. Akerib, et al., arXiv:1608.07648 [astro-ph.CO]. (2007) 161804. [43] Y. Nomura, J. Thaler, Phys. Rev. D 79 (2009) 075008, arXiv:0810.5397 [hep-ph]. [16] S.D. McDermott, H.B. Yu, K.M. Zurek, Phys. Rev. D 83 (2011) 063509, arXiv: 1011.2907 [hep-ph]. [44] R.D. Peccei, H.R. Quinn, Phys. Rev. Lett. 38 (1977) 1440. [17] R. Essig, et al., Working group report: new light weakly coupled particles, [45] S. Weinberg, Phys. Rev. Lett. 40 (1978) 223. arXiv:1311.0029 [hep-ph]. [46] F. Wilczek, Phys. Rev. Lett. 40 (1978) 279. [18] A.D. Dolgov, S.L. Dubovsky, G.I. Rubtsov, I.I. Tkachev, Phys. Rev. D 88 (11) (2013) [47] J.E. Kim, Phys. Rev. Lett. 43 (1979) 103. 117701, arXiv:1310.2376 [hep-ph]. [48]. M.A Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B 166 (1980) 493. [19] H. Vogel, J. Redondo, J. Cosmol. Astropart. Phys. 1402 (2014) 029, arXiv:1311. [49] N. Arkani-Hamed, L. Motl, A. Nicolis, C. Vafa, J. High Energy Phys. 0706 (2007) 2600 [hep-ph]. 060, arXiv:hep-th/0601001. [20] D.C. Moore, A.D. Rider, G. Gratta, Phys. Rev. Lett. 113 (25) (2014) 251801, arXiv: [50] M.x. Luo, Y. Xiao, Phys. Lett. B 555 (2003) 279, arXiv:hep-ph/0212152. 1408.4396 [hep-ex].