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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 16.5 equations at two-loop level and find that the GUT unification scale is around 10 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 3 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . 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 g kinetic mixing, but it is rather insensitive to the size of the hidden Y 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, gY is the gauge couplings of U(1)Y , and gH remains un- the hidden matter fields carry a non-zero electric charge, and they 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 i now acquires a frac- sufficiently small. Thus the minicharged dark matter is a natural 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- 2 smaller by 1 − χ compared to gY . Thus, the kinetic mixing with tion we explain how the gauge coupling unification is improved by unbroken U(1)H modifies the normalization of the hypercharge adding U(1)H , and show the results of solving RG equations at two- 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 μ On the other hand, the beta-functions of the gauge couplings and iγμ(g Q i A + gH qH A )i, (3) Y i H the kinetic mixing parameter in the original basis take a surpris- ingly simple form.
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