Advertisement

Gauge field and brane-localized kinetic terms on the chiral square

  • Ricardo G. LandimEmail author
Open Access
Regular Article - Theoretical Physics

Abstract

Extra dimensions have been used as attempts to explain several phenomena in particle physics. In this paper we investigate the role of brane-localized kinetic terms (BLKT) on thin and thick branes with two flat extra dimensions (ED) compactified on the chiral square, and an abelian gauge field in the bulk. The results for a thin brane have resemblance with the 5-D case, leading to a tower of massive KK particles whose masses depend upon the compactification radius and the BLKT parameter. On the other hand, for the thick brane scenario, there is no solution that satisfy the boundary conditions. Because of this, the mechanism of suppressed couplings due to ED (Landim and Rizzo, in JHEP 06:112, 2019) cannot be extended to 6-D.

1 Introduction

Extra dimensions (ED) have been considered over the decades as tools to address a wide range of issues in particle physics, such as the hierarchy  [2, 3, 4, 5, 6, 7] and flavor problems  [8, 9, 10]. Quantum field theory with two ED, for instance, may provide explanations for proton stability [11], origin of electroweak symmetry breaking  [12, 13, 14, 15], breaking of grand unified gauge groups  [16, 17, 18, 19] and the number of fermion generations  [20, 21, 22, 23, 24, 25]. The Standard Model (SM) itself might be extended by employing ED, in the so-called Universal Extra Dimension model (UED). In this context, the whole SM content is promoted to fields which propagate in compact ED, having Kaluza–Klein (KK) excitations, in either one  [26] or two ED  [27, 28, 29, 30]. The zero-mode of each KK tower of states in 4-D is thus identified with the correspondent SM particle and a lowest KK state can be a dark matter (DM) candidate. Current results from LHC  [31, 32] impose bounds on the UED compactification radius R for one (\(R^{-1}>1.4-1.5\) TeV)1  [33, 34, 35] or two ED (\(R^{-1}>900\) GeV) [36].

On the other hand, the 4-D gravity might be an emergent phenomenon from ED, as in the DGP model [37], where the brane-induced term was initially obtained for a massless spin-2 field. Such a mechanism is possible for a spin-1 field as well, in which a brane-localized kinetic term (BLKT) is generated on the brane by radiative corrections due to the interaction of localized matter fields on the brane with the gauge field in the bulk  [38], and it holds for infinite-volume, warped and compact ED. The same mechanism also works for two ED  [39, 40, 41] and the role of such a term has been investigated in several different scenarios  [42, 43, 44, 45, 46, 47, 48, 49]. The localization of matter/gauge fields in branes was studied in other contexts, for thin  [5, 50, 51, 52, 53, 54, 55, 56, 57] and thick branes [58, 59].

ED can also be employed in order to elucidate the nature of the DM and its possible interaction with the SM. Usually, a DM candidate may couple with the SM through a scalar mediator (or directly through Higgs if DM is a scalar field), via the so-called Higgs portal  [60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83], or through a vector mediator, which is introduced by a kinetic-mixing term  [84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97]. In both cases, much of the parameter space has been excluded by a diverse set of experiments and observations  [93, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134]. The small value of both couplings constants may be explained if we consider a single, flat ED and a thick brane with BLKT spread inside it [1], where inside the ‘fat’ brane the SM fields behaves as in the UED model with one ED.

An obvious generalization of this previous work then would be investigate the possibility of suppressed couplings, along with the presence of BLKT on thin branes, for higher dimensional spacetimes. In this paper we investigate this possibility in 6-D, which is a natural extension since UED has been built for two ED as well. Alongside with this aim, we consider BLKT on thin branes, leading to results that can be compared with the 5-D case  [42]. We assume the same compactification of the UED model in 6-D, where the so-called chiral square was chosen because it is the simplest compactification that leads to chiral quarks and leptons in 4-D [27]. For simplicity we will only consider an abelian gauge field in the bulk, although for other fields the results are analogue. The presence of BLKT on thin branes has a similar result as in the 5-D case, where the masses of the 4-D KK tower of states are determined by a transcendental equation. A thick brane with a BLKT, on the other hand, is not allowed by the boundary conditions (BC) at the intersection between the regions thick brane/bulk. Therefore, the mechanism in 5-D can be consistently extended for 6-D only for thin branes.

This paper is organized as follows. Section 2 reviews the 6-D gauge field without any BLKT on the chiral square. In Sect. 3 we introduce thin branes with BLKT and analyze the spectrum of masses while in Sect. 4 a fat brane is considered. Sect. 5 is reserved for conclusions.

2 Gauge field in the bulk

We will consider two flat and transverse ED (\(x^4\) and \(x^5\)) compactified on the chiral square. The square has size \(\pi R\) and the adjacent sides are identified \((0,y)\sim (y,0)\) and \((\pi R,y)\sim (y,\pi R)\), with \(y\in [0,\pi R]\), which means the Lagrangians at those points have the same values for any field configuration: \({\mathscr {L}}(x^\mu , 0,y)={\mathscr {L}}(x^\mu ,y,0)\) and \({\mathscr {L}}(x^\mu , \pi R,y)={\mathscr {L}}(x^\mu ,y,\pi R)\).

There is only an abelian gauge field \(V^A, ~A=0-3, 4, 5\) in the bulk and the action is similar to the one of UED with two ED  [27, 28], given by
$$\begin{aligned} S=\int d^4x\int _0^{\pi R} dx^4\int _0^{\pi R}dx^5 \left( -\frac{1}{4}V_{AB}V^{AB}+{\mathscr {L}}_{GF} \right) \,, \end{aligned}$$
(1)
where A is the 6-D index and the gauge fixing term has the following form to cancel the mixing between \(V_4\) and \(V_5\) with \(V_\mu \) [28]
$$\begin{aligned} {\mathscr {L}}_{GF}=-\frac{1}{2\xi }\Big [\partial _\mu V^\mu -\xi (\partial _4V_4+\partial _5V_5)\Big ]^2\,, \end{aligned}$$
(2)
where \(\xi \) is the gauge fixing parameter and we will work in the Feynman gauge (\(\xi =1\)). After integrating by parts the action (1) is written as
$$\begin{aligned} S= & {} \int d^4x\int _0^{\pi R} dx^4\int _0^{\pi R}dx^5 \bigg \{ -\frac{1}{4}V_{\mu \nu }V^{\mu \nu }-\frac{1}{2\xi }(\partial _\mu V^\mu )^2\nonumber \\&+\,\frac{1}{2}\Big [(\partial _4 V_\mu )^2+(\partial _5 V_\mu )^2\Big ]+\frac{1}{2}\Big [(\partial _\mu V_4)^2+(\partial _\mu V_5)^2\nonumber \\&-\,\xi (\partial _4V_4+\partial _5V_5^2)-(\partial _4V_5-\partial _5V_4)^2\Big ]\nonumber \\&+\,\text {surface terms}\bigg \}. \end{aligned}$$
(3)
In the Feynman gauge, the equations of motion for the components of \(V^A\) are
$$\begin{aligned} (\Box -\partial _4^2-\partial _5^2)V_A=0\,, \end{aligned}$$
(4)
where \(\Box \equiv \partial _\mu \partial ^\mu \). Furthermore, it is required that the surface terms vanish on the boundary, in order not to have flow of energy or momentum across it, i.e.,
$$\begin{aligned}&\int d^4 x \biggl \{\int dx^4 \Big [V_{5\mu }\delta V^\mu +V_{45} \delta V_4+(\partial _\mu V^\mu -\partial _4V_4\nonumber \\&\quad -\,\partial _5V_5)\delta V_5 \Big ]\biggr |_{x^5=0}^{x^5=\pi R} \nonumber \\&\quad +\, \int dx^5 \Big [V_{4\mu }\delta V^\mu -V_{45} \delta V_5+(\partial _\mu V^\mu -\partial _4V_4\nonumber \\&\quad -\,\partial _5V_5)\delta V_4 \Big ]\biggr |_{x^4=0}^{x^4=\pi R}\biggr \}=0. \end{aligned}$$
(5)
Vanishing the surface terms lead to the following BC for \(V_\mu \)
$$\begin{aligned} V_\mu (y,0)= & {} V_\mu (0,y)\nonumber \,,\\ \partial _4 V_\mu |_{(x^4,x^5)=(y,0)}= & {} \partial _5 V_\mu |_{(x^4,x^5)=(0,y)}\,,\nonumber \\ \partial _5 V_\mu |_{(x^4,x^5)=(y,0)}= & {} - \partial _4 V_\mu |_{(x^4,x^5)=(0,y)}\,, \end{aligned}$$
(6)
and for \(V_4\) and \(V_5\)
$$\begin{aligned} V_4(y,0)= & {} V_5(0,y)\nonumber \,,\\ \partial _4 V_4 |_{(x^4,x^5)=(y,0)}= & {} \partial _5 V_5 |_{(x^4,x^5)=(0,y)}\,,\nonumber \\ \partial _5 V_4 |_{(x^4,x^5)=(y,0)}= & {} - \partial _4 V_5 |_{(x^4,x^5)=(0,y)}\,, \end{aligned}$$
(7)
$$\begin{aligned} V_5(y,0)= & {} -V_4(0,y)\nonumber \,,\\ \partial _4 V_5 |_{(x^4,x^5)=(y,0)}= & {} - \partial _5 V_4 |_{(x^4,x^5)=(0,y)}\,,\nonumber \\ \partial _5 V_5 |_{(x^4,x^5)=(y,0)}= & {} \partial _4 V_4 |_{(x^4,x^5)=(0,y)}\,, \end{aligned}$$
(8)
for any \(0\le y \le \pi R\). The same relations exist for the fields at \((y, \pi R)\) and \((\pi R, y)\). From the above relations it is possible to see the transformation law \((V_4,V_5) \rightarrow (V_5,-V_4)\) satisfied by the fields under \((x^4,x^5)\rightarrow (-x^5,x^4)\) [28].
We expand the components of the 6-D gauge field in KK towers of states
$$\begin{aligned} V_\mu (x^\nu , x^4,x^5)= & {} \sum _j\sum _k v_0^{(j,k)}(x^4,x^5) V_\mu ^{(j,k)}(x^\nu )\,, \end{aligned}$$
(9)
$$\begin{aligned} V_4(x^\nu , x^4,x^5)= & {} \sum _j\sum _k v_4^{(j,k)}(x^4,x^5) V_4^{(j,k)}(x^\nu )\,, \end{aligned}$$
(10)
$$\begin{aligned} V_5(x^\nu , x^4,x^5)= & {} \sum _j\sum _k v_5^{(j,k)}(x^4,x^5) V_5^{(j,k)}(x^\nu )\,, \end{aligned}$$
(11)
which yields the equation of motion for \(v_i^{(j,k)}(x^4,x^5)\), with \(i=0, 4\) or 5,
$$\begin{aligned} \left[ \partial _4^2+\partial _5^2+(M_i^{(j,k)})^2\right] v_i^{(j,k)}(x^4,x^5)=0\,, \end{aligned}$$
(12)
where \(M_i^{(j,k)}\) are the physical masses of the gauge field \( V_\mu \) and the scalar fields \(V_4\) and \(V_5\), respectively. The solutions for the equation of motion, which satisfy the BC above, and are normalized through the relation
$$\begin{aligned} \int _0^{\pi R} dx^4\int _0^{\pi R} dx^5\, v_i^{(j,k)}(x^4,x^5)v_i^{(j',k')}(x^4,x^5)= & {} \delta _{j,j'}\delta _{k,k'}\,,\nonumber \\ \end{aligned}$$
(13)
are given by
$$\begin{aligned} v_0^{(j,k)}(x^4,x^5)= & {} \frac{1}{\pi R}\left[ \cos (m_j x^4+m_k x^5)\pm \cos (m_k x^4-m_j x^5)\right] \,,\nonumber \\ \end{aligned}$$
(14)
$$\begin{aligned} v_4^{(j,k)}(x^4,x^5)= & {} \frac{\sqrt{2}}{\pi R}\sin \Big (\frac{j x^4+k x^5}{R}\Big )\,, \end{aligned}$$
(15)
$$\begin{aligned} v_5^{(j,k)}(x^4,x^5)= & {} -\frac{\sqrt{2}}{\pi R}\sin \Big (\frac{k x^4-j x^5}{R}\Big )\,, \end{aligned}$$
(16)
where j and k are integers and the parameters \(m_j\) and \(m_k\) are \(m_j=j/R\) and \(m_k=k/R\), for the \(+\) sign in Eq. (14) or \(m_j=(j+1/2)/R\) and \(m_k=(k+1/2)/R\) for the − sign. The physical masses of the scalar fields are \( (M_{4,5}^{(j,k)})^2=(j^2+k^2)/R^2\) while for the tower of states of the 4-D vector field they are given by
$$\begin{aligned} (M_0^{(j,k)})^2=m_j^2+m_k^2\,, \end{aligned}$$
(17)
Unlike the 5-D case, where the new scalar field, which is the extra component of the vector field, can be gauged away, in 6-D there is an additional degree of freedom that remains. This fact is explicitly seen if one works in the unitary gauge, where only one of the two linear combinations of the the scalar fields \(V_4\) and \(V_5\) is eaten by the vector boson \(V_\mu ^{(j,k)}\)  [28].

3 BLKT on thin branes

Applying the same ideas of the last section, we will now consider the effect of BLKT on branes localized at the points (0, 0), \((\pi R,\pi R)\) and \((\pi R,0)\sim (0,\pi R)\). We should recall that preserving KK parity implies that operators at (0, 0) and \((\pi R,\pi R)\) are identical.

3.1 BLKT at (0, 0)

We will first analyze the change in the wave-function due to the presence of a BLKT term on a brane localized at (0, 0).2 The localized kinetic term is four-dimensional for distances shorter than R, and it is given by [38, 40]
$$\begin{aligned} {\mathscr {L}}_{BLKT}=\left[ -\frac{1}{4}V_{\mu \nu }V^{\mu \nu }-\frac{1}{2\xi }(\partial _\mu V^\mu )^2\right] \cdot \delta _A R^2\,\delta (x^4,x^5)\,, \end{aligned}$$
(18)
where we conveniently added a gauge-fixing term. After expanding the 6-D gauge field into a tower of KK states, the equation of motion for the wave-function \( v_0^{(j,k)}(x^4,x^5)\) has the same structure of the 5-D case
$$\begin{aligned} \Big [\partial _4^2+\partial _5^2+M_{j,k}^2+ M_{j,k}^2\delta _AR^2\delta (x_4,x_5)\Big ]v_0^{(j,k)}(x^4,x^5)=0\,, \end{aligned}$$
(19)
where we relabeled \(M_0^{(j,k)}\equiv M_{j,k}\).
The 4-D Lagrangian is found integrating the wave-function over the ED. The resulting Lagrangian has diagonal terms
$$\begin{aligned} {\mathscr {L}}_4=\sum _{j,k} \biggr [-\frac{1}{4}Z_{(j,k)}V_{\mu \nu }^{(j,k)}V^{\mu \nu }_{(j,k)} +Z_{(j,k)}M_{j,k}^2V_\mu ^{(j,k)}V^{\mu }_{(j,k)} \biggr ]\,, \end{aligned}$$
(20)
where \(Z_{(j,k)}\) is a normalization factor, if the wave-function satisfies the relations
$$\begin{aligned}&\int _0^{\pi R} dx^4\int _0^{\pi R} dx^5\,\Big [1+\delta _AR^2\delta (x^4,x^5)\Big ] v_0^{(j,k)}v_0^{(j',k')}\nonumber \\&=Z_{(j,k)}\delta _{j,j'}\delta _{k,k'}\,,\nonumber \\&\quad \int _0^{\pi R} dx^4\int _0^{\pi R} dx^5\,\Big [\partial _4v_0^{(j,k)}\partial _4v_0^{(j',k')}\nonumber \\&\quad +\,\partial _5v_0^{(j,k)}\partial _5v_0^{(j',k')}\Big ]=Z_{(j,k)}M_{j,k}^2\delta _{j,j'}\delta _{k,k'} \,. \end{aligned}$$
(21)
The normalization factor for a delta-function at the origin is
$$\begin{aligned} Z_{(j,k)}=1+\delta _A R^2 v_0^{(j,k)}(0,0)\,, \end{aligned}$$
(22)
and the gauge field in 4-D becomes canonically normalized after dividing it by \(Z_{(j,k)}^{-1/2}\).
Due to the presence of a BLKT the surface terms are no longer zero. The non-trivial solution (\(\delta _A\ne 0\)) for the Eq. (19) is
$$\begin{aligned} v_0^{(j,k)}(x^4,x^5)= & {} A_{j,k}\Big [\cos (m_j x^4)\cos (m_k x^5)\nonumber \\&+\,\cos (m_k x^4)\cos (m_j x^5)\Big ]\nonumber \\&+\,B_{j,k}\Big [\sin (m_j x^4)\sin (m_k x^5)\nonumber \\&+\,\sin (m_k x^4)\sin (m_j x^5)\Big ]\,. \end{aligned}$$
(23)
The solution above no longer satisfy the last BC in Eq. (6). Equation (14) is recovered if \(A_{j,k}=-B_{j,k}\) and \(\sin (m_k x_4)\sin (m_j x_5)\) is replaced by \(-\sin (m_k x_4)\sin (m_j x_5)\). The coefficients \(A_{j,k}\) and \(B_{j,k}\) are found requiring the familiar conditions of continuity of the function and discontinuity of its derivative at (0, 0). Similar to the case of a delta-function in 1-D, we integrate the equation of motion (19) over \(x_4\) and \(x_5\), from \((0^-,0^-)\) to \((0^+,0^+)\). Performing a replacement of dummy variables we get
$$\begin{aligned}&\int _{0^-}^{0^+} \, dy\Big [\partial _4 v_0^{(j,k)}(x^4,y)|_{x^4=0^-}^{x^4=0^+}+\partial _5v_0^{(j,k)}(y,x^5)|_{x^5=0^-}^{x^5=0^+}\nonumber \\&\quad -\, \partial _4 {\overline{v}}_0^{(j,k)}(x^4,y)|_{x^4=0^-}^{x^4=0^+}-\partial _5{\overline{v}}_0^{(j,k)}(y,x^5)|_{x^5=0^-}^{x^5=0^+}\Big ]=\nonumber \\&\quad -\, M_{j,k}^2\delta _AR^2v_0^{(j,k)}(0,0)\,, \end{aligned}$$
(24)
where \(v_0^{(j,k)}\) is the wave-function for \(x^4,x^5>0\) and \({\overline{v}}_0^{(j,k)}\) is the wave-function for \(x^4,x^5<0\). Terms with crossed coordinates such as \(\sim v_0^{(j,k)}(0^+,0^-)\) are zero. Using Eqs. (24) and (17) we get the wave-function due to a two-dimensional delta-function source
$$\begin{aligned} v_0^{(j,k)}(x^4,x^5)= & {} N_{j,k}\Big [\cos (m_j x^4)\cos (m_k x^5)\nonumber \\&+\,\cos (m_k x^4)\cos (m_j x^5)\nonumber \\&-\,\frac{\delta _A}{2}x_j x_k\Big (\sin (m_j x^4)\sin (m_k x^5)\nonumber \\&+\,\sin (m_k x^4)\sin (m_j x^5)\Big )\Big ]\,, \end{aligned}$$
(25)
where \(m_j=x_j/R\), \(m_k=x_k/R\) and \(N_{j,k}\) is the normalization constant defined through Eq. (13), which gives
$$\begin{aligned} N_{j,k}^{-2}= & {} \frac{\pi ^2R^2}{2}\biggr \{1 +\frac{\delta _A}{4\pi ^2} \cos ^2(\pi x_j)\Big [1+\cos ^2(\pi x_k)\Big ]\nonumber \\&+\,\frac{ \sin (2 \pi x_k)}{2 \pi x_k}+\frac{1}{4} \delta _A^2 x_j^2 x_k^2\nonumber \\&-\,\frac{ \delta _A}{2\pi }\Big [ x_k \cos ^2(\pi x_j) \cot (\pi x_k)+ x_j \cot (\pi x_j) \cos ^2(\pi x_k)\Big ]\nonumber \\&-\,\frac{x_j x_k \sin (2 \pi x_j) \sin (2 \pi x_k)}{ \pi ^2(x_j^2-x_k^2){}^2}\nonumber \\&+\,\frac{4 x_k^2 \cos ^2(\pi x_j) \csc ^2(\pi x_k)}{\pi ^2(x_j^2-x_k^2){}^2}+\frac{4 x_j^2 \csc ^2(\pi x_j) \cos ^2(\pi x_k)}{\pi ^2(x_j^2-x_k^2){}^2}\biggr \}\,.\nonumber \\ \end{aligned}$$
(26)
As in the 5-D case [48], the transcendental equation that determines the roots \(x_j\) and \(x_k\) is found requiring the Dirichlet BC \( v_0^{(j,k)}(\pi R,\pi R)=0\), whose solutions depend only upon \(\delta _A\)
$$\begin{aligned} \cot (\pi x_j)\cot (\pi x_k)=\frac{\delta _A}{2}x_j x_k\,. \end{aligned}$$
(27)
Equation (27) has an evident resemblance to the root equation in 5-D (\(\cot (\pi x_n)=\delta _Ax_n/2\)) [48]. Since only one Eq. (27) determines both roots \(x_j\) and \(x_k\), it is expected the existence of a continuous set of values \(x_j\) and \(x_k\) that satisfies Eq. (27). The solutions of Eq. (27) are shown in Fig. 1, for \(\delta _A=1\), while different values of \(\delta _A\) are plotted in Fig. 2.
From Fig. 1 we see that there are \((2n+1)\) quantized masses for each curve n, where we labeled n as being each one of the dashed lines. Each mode is described by the segments in the dashed lines, i.e., at \(n=0\) (first dashed line) there is one mode \(M_{0,0}\), a massive zero-mode, the second dashed line (\(n=1\)) has three quantized masses \(M_{0,1}\), \(M_{1,0}\) and \(M_{1,1}\) (being the first two degenerate), and so on. The segments in the middle of each dashed line are the levels corresponding to \(M_{j,j}\) and since the curves are symmetric under reflection over the line \(x_j=x_k\), the masses \(M_{j,k}\) and \(M_{k,j}\) are degenerate. These features are the usual behavior of quantum systems in two dimensions. Although there is a continuous set of values \((x_j, x_k)\) in each segment, the whole set represent only one (mass) state, being narrow the range of each state. In Table 1 it is presented the masses \(M_{j,k}\) for the first three curves of Fig. 1. The masses correspondent to each KK level are either increased or decreased in an alternated pattern, when the parameter \(\delta _A\) is increased, as seen in Fig. 2.
Fig. 1

Solutions of the transcendental Eq. (27) until \(x_j\sim x_k\sim 3\) for \(\delta _A=1\), for the thin-brane model

Table 1

Mass range \( M_{j,k}R=\sqrt{x_j^2+x_k^2}\) for the first three curves plotted in Fig. 1

(jk)

\( M_{j,k}R\,\)

(0, 0)

\(0.5-0.6\)

(1, 1)

\(0.9-1.1\)

(0, 1)

\(1.1-1.5\)

(1, 0)

\(1.1-1.5\)

(2, 2)

1.8

(0, 2)

\(1.8-2.1\)

(1, 2)

\(2.1-2.5\)

(2, 0)

\(1.8-2.1\)

(2, 1)

\(2.1-2.5\)

Fig. 2

Solutions of the transcendental Eq. (27) for different values of \(\delta _A\), for the thin-brane model

Fig. 3

Solutions of the transcendental Eq. (30) for different values of \(\delta _A\) and two values of \(\delta _B\), for the thin-brane model

3.2 BLKT at (0, 0) and (\(\pi R\), \(\pi R\))

We consider now branes localized at (0, 0) and \((\pi R, \pi R)\) with BLKT on them. For the sake of completeness we add the following term in the Lagrangian
$$\begin{aligned} {\mathscr {L}}_{BLKT}= & {} -\frac{1}{4}V_{\mu \nu }V^{\mu \nu }\cdot \Big [ \delta _A R^2\,\delta (x^4,x^5)\nonumber \\&+\,\delta _B R^2\, \delta (x^4-\pi R, x^5-\pi R)\Big ]\,, \end{aligned}$$
(28)
where \(\delta _A\) is not necessarily equal to \(\delta _B\). The equation of motion (19) is modified by an extra term proportional to \(\delta _B\). The normalization factor has now the following terms
$$\begin{aligned} Z_{(j,k)}=1+\delta _A R^2 v_0^{(j,k)}(0,0)+\delta _B R^2 v_0^{(j,k)}(\pi R,\pi R)\,. \end{aligned}$$
(29)
The wave-function is equal to Eq. (25) for \(x^4,x^5\le \pi R\), and the transcendental equation is found through the non-continuity of the derivative of the wave-function, whose expression is similar to (24). The quantized masses are therefore found through the transcendental equation
$$\begin{aligned} \left( 1+\frac{\delta _A \delta _B}{4}x_j^2x_k^2\right) \cot (x_j \pi )\cot (x_k \pi )=\frac{x_j x_k}{2}(\delta _A+\delta _B)\,, \end{aligned}$$
(30)
which is reduced to Eq. (27) when \(\delta _B=0\). This root equation is also similar to the one in the 5-D case  [42]. The solutions of Eq. (30) are depicted in Fig. 3. For lower (higher) values of \(\delta _B\) the larger (smaller) roots start having the same value, roughly independent of \(\delta _A\). The case \(\delta _A=\delta _B\) preserves KK-parity and the roots are presented in Fig. 4. Their values are similar to the case \(\delta _B=0\).

3.3 BLKT at \((0,\pi R)\)

Since the points \((0,\pi R)\) and \((\pi R,0)\) are identified it is sufficient to consider only one case. We will consider now the BLKT inside a brane localized at \((0,\pi R)\), whose Lagrangian is
$$\begin{aligned} {\mathscr {L}}_{BLKT}=-\frac{1}{4}V_{\mu \nu }V^{\mu \nu }\cdot \delta _A R^2\,\delta (x^4,x^5-\pi R)\,. \end{aligned}$$
(31)
Similar to the previous cases, the normalization constant becomes
$$\begin{aligned} Z_{(j,k)}=1+\delta _A R^2 v_0^{(j,k)}(0,\pi R)\,. \end{aligned}$$
(32)
The solution for the equation of motion with a delta function source at \((0,\pi R)\), satisfying similar BC as Eq. (24), is
$$\begin{aligned} v_0^{(j,k)}(x^4,x^5)= & {} N_{j,k}\Big [\cos (m_j x^4+m_k x^5)+\cos (m_k x^4-m_j x^5)\nonumber \\&+\,\sin (m_j x^4+m_k x^5)\nonumber \\&+\,\sin (m_k x^4-m_j x^5)\Big ]\,, \end{aligned}$$
(33)
where
$$\begin{aligned} N_{j,k}^{-2}= & {} 2\pi ^2 R^2\biggr \{1+\frac{1}{\pi ^2(x_j^2-x_k^2)}+\frac{\sin ^2(\pi x_j) \sin (2 \pi x_k)}{2\pi ^2 x_j x_k}\nonumber \\&+\frac{ \cos (2 \pi x_k)- \sin (2 \pi x_j)}{\pi ^2(x_j^2-x_k^2)}\nonumber \\&+\frac{2 \cos (\pi x_k)}{\pi ^2(x_j^2-x_k^2)}\Big [\sin (\pi x_j) - \cos (\pi x_j)\Big ]\biggr \} \,. \end{aligned}$$
(34)
The dependence of \(\delta _A\) appears in the transcendental equation, which is identical to Eq. (27), thus having the same solutions for the pair of roots \((x_j,x_k)\).
Fig. 4

Solutions of the transcendental equation (30) for different values of \(\delta _A\), when \(\delta _A=\delta _B\), for the thin-brane model

4 BLKT on a thick brane

We consider now the effect of a BLKT on the thick brane, lying between \( \pi r < x^4,x^5 \le \pi R\), with a width \(\pi (R-r)\equiv \pi L\). The thin brane is obtained in the limit \(L\rightarrow 0\). In 5-D, the difference between thin and thick branes leads to the suppressed coupling mechanism [1], existing for branes with a finite thickness. In 6-D, thin branes carry localized operators on the conical singularities at the corners of the square. On the other hand, BLKT are spread inside the thick branes, thus for thin branes the surface terms (5) are non-zero, while for a fat brane the operators are not localized at the conical singularities and there are two regions in the two-dimensional (ED) space, each one having (in principle) vanishing surface terms.

The BLKT with gauge fixing term is [38, 40]
$$\begin{aligned} {\mathscr {L}}_{TB}=\left[ -\frac{1}{4}V_{\mu \nu }V^{\mu \nu }-\frac{1}{2\xi }(\partial _\mu V^\mu )^2\right] \cdot \delta _A R^2\,\theta (x^4,x^5)\,, \end{aligned}$$
(35)
where the step function is non-zero only inside the thick brane, i.e.
$$\begin{aligned} \theta (x^4,x^5)= & {} \alpha ^2 \quad \text {for } \pi r< x^4,x^5 \le \pi R, \qquad \nonumber \\&\theta (x^4,x^5)=0\quad \text {for } x^4,x^5<\pi r. \end{aligned}$$
(36)
The equation of motion for the wave-function inside the thick brane is now
$$\begin{aligned} \Big [\partial _4^2+\partial _5^2+( M_0^{(j,k)})^2+( M_0^{(j,k)})^2\delta _A\alpha ^2R^2\Big ]{\overline{v}}_0^{(j,k)}(x^4,x^5)=0\,. \end{aligned}$$
(37)
Similar to the 5-D case [1], we may define an effective mass as \({\overline{M}}_0^{(j,k)}\equiv M_0^{(j,k)}\sqrt{1+ \delta _A\alpha ^2 R^2 }\), thus the presence of the step-function changes the mass term in the equation of motion for \(V_\mu ^{(j,k)}\) inside the thick brane. It has the same structure of Eq. (12), but with the replacement \(M_0^{(j,k)}\rightarrow {\overline{M}}_0^{(j,k)}\) [1]. Defining the effective mass parameters as
$$\begin{aligned} {\overline{m}}_j\equiv m_j\sqrt{1+ \delta _A\alpha ^2 R^2 }\,, \quad {\overline{m}}_k\equiv m_k\sqrt{1+ \delta _A\alpha ^2 R^2 }\,, \end{aligned}$$
(38)
the wave-function inside the thick brane \({\overline{v}}_0^{(j,k)}(x^4,x^5)\) has also the same structure of Eq. (14). The wave-function outside the thick brane is
$$\begin{aligned} v_0^{(j,k)}(x^4,x^5)=A_1^{(j,k)}\left[ \cos (m_j x^4+m_k x^5)\pm \cos (m_k x^4-m_j x^5)\right] \,, \end{aligned}$$
(39)
while inside the thick brane the wave-function is
$$\begin{aligned} {\overline{v}}_0^{(j,k)}(x^4,x^5)=A_2^{(j,k)}\left[ \cos ({\overline{m}}_j x^4+{\overline{m}}_k x^5)\pm \cos ({\overline{m}}_k x^4-{\overline{m}}_j x^5)\right] \,, \end{aligned}$$
(40)
where \(A_1^{(j,k)}\) and \(A_2^{(j,k)}\) are coefficients to be determined.

Both wave-functions have this form in order to satisfy the BC (6). The mass parameters, however, should be either \({\overline{m}}_i=m_i=i/R \) (\(+\)) or \({\overline{m}}_i=m_i=(i+1/2)/R\) (−), for \(i=j,k\), in order to satisfy the same BC. This is only possible if \(\delta _A\alpha =0\). Even if we assume that the fields no longer need to satisfy all previous BC, the situation remains the same by the following reason. The wave-function should be continuous at \((y,\pi r)\) and \((\pi r, y)\), as well as its derivative with respect to both ED coordinates \(x^4\) and \(x^5\). Thus \({\overline{v}}_0^{(j,k)}(\pi r,y)=v_0^{(j,k)}(\pi r,y)\) and \(\partial _4 {\overline{v}}_0^{(j,k)}|_{(x^4,x^5)=(\pi r,y)}=\partial _4v_0^{(j,k)}|_{(x^4,x^5)=(\pi r,y)}\) (the continuity conditions at \((y, \pi r)\) give exactly the same expressions). These conditions can be satisfied at a point \(y_c\) on the boundary, but it is not possible to match the functions all along the boundary, being the only possibility \({\overline{m}}_i=m_i\). Therefore the only viable solution is \(\delta _A\alpha =0\), which leads to a thin brane.

5 Conclusions

In this paper we have investigated the implications of BLKT on thin and thick branes, for a model of two ED compactified on the chiral square, when a vector field is present in the bulk. For thin branes the presence of BLKT gives mass to all modes of the KK tower of states, being the masses dependent upon the compactification radius and the BLKT parameter. The roots are roughly the same for branes at different positions, i.e., they have similar values for branes localized at (0, 0), \((0,\pi R)\sim (\pi R,0)\) and \((\pi R,\pi R)\). The transcendental equations and other relations resemble the 5-D case. The model presents the usual behavior of quantum systems in two dimensions, i.e., there are \((2n+1)\) quantized masses for each curve n, and each mode is described by the segments in the dashed lines: one massive zero-mode \(M_{0,0}\) at \(n=0\), three quantized masses \(M_{0,1}\), \(M_{1,0}\) and \(M_{1,1}\) at \(n=1\) (being \(M_{0,1}\) and \(M_{1,0}\) degenerate), etc.

The BLKT on thick branes, on the other hand, does not provide a non-trivial result \(\delta _A\alpha \ne 0\) due to the BC. This scenario is similar to the refraction of a wave-function by a two-dimensional step function. Suppose an incident wave-function \(e^{i(k_x x+ k_y y) }\), a refracted wave \(e^{i(q_x x+ q_y y) }\) and a step-function different from zero for \(x>0\). The BC implies that \(k_y=q_y\) and the analogue situation in our problem is therefore \(\bar{m}_i=m_i\). Hence the mechanism of suppressed coupling in 5-D  [1] cannot be applied in 6-D.

The results presented here works for different fields in the bulk and can be used in several further proposals, as for instance, in a model of ED with the dark photon as mediator. This model was done in 5-D [48, 49], but an extension might be able with two ED as well, or even its inclusion in the UED model. In both cases, the BLKT breaks the extra \(U(1)_D\) gauge symmetry via BC without adding an extra Higgs-like field, avoiding, in turn, constraints on the Higgs-portal coupling. Potential signatures for such massive spin-1 KK particles depend upon the specific model considered but it usually includes missing energy searches, which might constrain the two parameters in this model.

Footnotes

  1. 1.

    For \(\Lambda R \sim 5-35\), where \(\Lambda \) is the cutoff scale.

  2. 2.

    As explained in [40] the propagator of the 6-D gauge field is found after a regularization procedure.

Notes

Acknowledgements

The author would like to thank Gia Dvali for clarifications and Thomas Rizzo for comments. This work was supported by by CAPES under the process 88881.162206/2017-01 and Alexander von Humboldt Foundation.

References

  1. 1.
    Ricardo G. Landim, Thomas G. Rizzo, Thick Branes in Extra Dimensions and Suppressed Dark Couplings. JHEP 06, 112 (2019)MathSciNetGoogle Scholar
  2. 2.
    Ignatios Antoniadis, A Possible new dimension at a few TeV. Phys. Lett. B 246, 377–384 (1990)ADSGoogle Scholar
  3. 3.
    R. Keith, Dienes, Emilian Dudas, Tony Gherghetta, Extra space-time dimensions and unification. Phys. Lett. B 436, 55–65 (1998)MathSciNetGoogle Scholar
  4. 4.
    Ignatios Antoniadis, Nima Arkani-Hamed, Savas Dimopoulos, G.R. Dvali, New dimensions at a millimeter to a Fermi and superstrings at a TeV. Phys. Lett. B 436, 257–263 (1998)Google Scholar
  5. 5.
    Nima Arkani-Hamed, Savas Dimopoulos, G.R. Dvali, The Hierarchy problem and new dimensions at a millimeter. Phys. Lett. B 429, 263–272 (1998)Google Scholar
  6. 6.
    Lisa Randall, Raman Sundrum, A Large mass hierarchy from a small extra dimension. Phys. Rev. Lett. 83, 3370–3373 (1999)ADSMathSciNetzbMATHGoogle Scholar
  7. 7.
    Nima Arkani-Hamed, Timothy Cohen, Raffaele Tito D’Agnolo, Anson Hook, Hyung Do Kim, David Pinner, Solving the Hierarchy Problem at Reheating with a Large Number of Degrees of Freedom. Phys. Rev. Lett. 117(25), 251801 (2016)ADSGoogle Scholar
  8. 8.
    Kaustubh Agashe, Gilad Perez, Amarjit Soni, Flavor structure of warped extra dimension models. Phys. Rev. D 71, 016002 (2005)ADSGoogle Scholar
  9. 9.
    J. Stephan, Huber, Flavor violation and warped geometry. Nucl. Phys. B 666, 269–288 (2003)Google Scholar
  10. 10.
    A. Liam Fitzpatrick, Gilad Perez, Lisa Randall, Flavor anarchy in a Randall-Sundrum model with 5D minimal flavor violation and a low Kaluza-Klein scale. Phys. Rev. Lett. 100, 171604 (2008)Google Scholar
  11. 11.
    Thomas Appelquist, Bogdan A. Dobrescu, Eduardo Ponton, Ho-Ung Yee, Proton stability in six-dimensions. Phys. Rev. Lett. 87, 181802 (2001)ADSGoogle Scholar
  12. 12.
    Nima Arkani-Hamed, Hsin-Chia Cheng, Bogdan A. Dobrescu, Lawrence J. Hall, Selfbreaking of the standard model gauge symmetry. Phys. Rev. D 62, 096006 (2000)ADSGoogle Scholar
  13. 13.
    Michio Hashimoto, Masaharu Tanabashi, Koichi Yamawaki, Top mode standard model with extra dimensions. Phys. Rev. D 64, 056003 (2001)ADSGoogle Scholar
  14. 14.
    Csaba Csaki, Christophe Grojean, Hitoshi Murayama, Standard model Higgs from higher dimensional gauge fields. Phys. Rev. D 67, 085012 (2003)ADSGoogle Scholar
  15. 15.
    C.A. Scrucca, M. Serone, L. Silvestrini, A. Wulzer, Gauge Higgs unification in orbifold models. JHEP 02, 049 (2004)ADSMathSciNetGoogle Scholar
  16. 16.
    Arthur Hebecker, John March-Russell, The structure of GUT breaking by orbifolding. Nucl. Phys. B 625, 128–150 (2002)ADSMathSciNetzbMATHGoogle Scholar
  17. 17.
    J. Lawrence, Hall, Yasunori Nomura, Takemichi Okui, David Tucker-Smith, SO(10) unified theories in six-dimensions. Phys. Rev. D 65, 035008 (2002)Google Scholar
  18. 18.
    T. Asaka, W. Buchmuller, L. Covi, Bulk and brane anomalies in six-dimensions. Nucl. Phys. B 648, 231–253 (2003)ADSMathSciNetzbMATHGoogle Scholar
  19. 19.
    T. Asaka, W. Buchmuller, L. Covi, Quarks and leptons between branes and bulk. Phys. Lett. B 563, 209–216 (2003)ADSzbMATHGoogle Scholar
  20. 20.
    A. Bogdan, Dobrescu, Erich Poppitz, Number of fermion generations derived from anomaly cancellation. Phys. Rev. Lett. 87, 031801 (2001)MathSciNetGoogle Scholar
  21. 21.
    M. Fabbrichesi, M. Piai, G. Tasinato, Axion and neutrino physics from anomaly cancellation. Phys. Rev. D 64, 116006 (2001)ADSGoogle Scholar
  22. 22.
    Nicolas Borghini, Yves Gouverneur, Michel H.G. Tytgat, Anomalies and fermion content of grand unified theories in extra dimensions. Phys. Rev. D 65, 025017 (2002)Google Scholar
  23. 23.
    M. Fabbrichesi, R. Percacci, M. Piai, M. Serone, Cancellation of global anomalies in spontaneously broken gauge theories. Phys. Rev. D 66, 105028 (2002)ADSMathSciNetGoogle Scholar
  24. 24.
    J.M. Frere, M.V. Libanov, Sergey V. Troitsky, Neutrino masses with a single generation in the bulk. JHEP 11, 025 (2001)Google Scholar
  25. 25.
    T. Watari, T. Yanagida, Higher dimensional supersymmetry as an origin of the three families for quarks and leptons. Phys. Lett. B 532, 252–258 (2002)ADSzbMATHGoogle Scholar
  26. 26.
    Thomas Appelquist, Hsin-Chia Cheng, Bogdan A. Dobrescu, Bounds on universal extra dimensions. Phys. Rev. D 64, 035002 (2001)ADSGoogle Scholar
  27. 27.
    Bogdan A. Dobrescu, Eduardo Ponton, Chiral compactification on a square. JHEP 03, 071 (2004)Google Scholar
  28. 28.
    Gustavo Burdman, Bogdan A. Dobrescu, Eduardo Ponton, Six-dimensional gauge theory on the chiral square. JHEP 02, 033 (2006)ADSMathSciNetGoogle Scholar
  29. 29.
    Eduardo Ponton, Lin Wang, Radiative effects on the chiral square. JHEP 11, 018 (2006)ADSMathSciNetGoogle Scholar
  30. 30.
    Gustavo Burdman, Bogdan A. Dobrescu, Eduardo Ponton, Resonances from two universal extra dimensions. Phys. Rev. D 74, 075008 (2006)ADSGoogle Scholar
  31. 31.
    Georges Aad et al., Search for squarks and gluinos in events with isolated leptons, jets and missing transverse momentum at \(\sqrt{s}=8\) TeV with the ATLAS detector. JHEP 04, 116 (2015)ADSGoogle Scholar
  32. 32.
    The ATLAS collaboration. Search for squarks and gluinos in events with isolated leptons, jets and missing transverse momentum at \(\sqrt{s}=8\) TeV with the ATLAS detector. 2013Google Scholar
  33. 33.
    Nicolas Deutschmann, Thomas Flacke, and Jong Soo Kim. Current LHC Constraints on Minimal Universal Extra Dimensions. Phys. Lett., B771:515–520, 2017Google Scholar
  34. 34.
    Jyotiranjan Beuria, AseshKrishna Datta, Dipsikha Debnath, Konstantin T. Matchev, LHC Collider Phenomenology of Minimal Universal Extra Dimensions. Comput. Phys. Commun. 226, 187–205 (2018)ADSGoogle Scholar
  35. 35.
    M. Tanabashi et al., Review of Particle Physics. Phys. Rev. D 98(3), 030001 (2018)ADSGoogle Scholar
  36. 36.
    G. Burdman, O.J.P. Eboli, D. Spehler, Signals of Two Universal Extra Dimensions at the LHC. Phys. Rev. D 94(9), 095004 (2016)ADSGoogle Scholar
  37. 37.
    G.R. Dvali, Gregory Gabadadze, Massimo Porrati, 4-D gravity on a brane in 5-D Minkowski space. Phys. Lett. B 485, 208–214 (2000)ADSMathSciNetGoogle Scholar
  38. 38.
    G.R. Dvali, Gregory Gabadadze, Mikhail A. Shifman, (Quasi)localized gauge field on a brane: Dissipating cosmic radiation to extra dimensions? Phys. Lett. B 497, 271–280 (2001)ADSzbMATHGoogle Scholar
  39. 39.
    G.R. Dvali, Gregory Gabadadze, Gravity on a brane in infinite volume extra space. Phys. Rev. D 63, 065007 (2001)ADSMathSciNetGoogle Scholar
  40. 40.
    Gia Dvali, Gregory Gabadadze, Xin-rui Hou, Emiliano Sefusatti, Seesaw modification of gravity. Phys. Rev. D 67, 044019 (2003)ADSMathSciNetGoogle Scholar
  41. 41.
    Gia Dvali, Gregory Gabadadze, M. Shifman, Diluting cosmological constant in infinite volume extra dimensions. Phys. Rev. D 67, 044020 (2003)Google Scholar
  42. 42.
    Marcela Carena, Timothy M.P. Tait, C.E.M. Wagner, Branes and orbifolds are opaque. Acta Phys. Polon. B 33, 2355 (2002)Google Scholar
  43. 43.
    Marcela Carena, Eduardo Ponton, Timothy M.P. Tait, C.E.M. Wagner, Opaque Branes in Warped Backgrounds. Phys. Rev. D 67, 096006 (2003)Google Scholar
  44. 44.
    F. del Aguila, M. Perez-Victoria, Jose Santiago, Physics of brane kinetic terms. Acta Phys. Polon. B 34, 5511–5522 (2003)ADSGoogle Scholar
  45. 45.
    F. del Aguila, M. Perez-Victoria, Jose Santiago, Bulk fields with general brane kinetic terms. JHEP 02, 051 (2003)Google Scholar
  46. 46.
    H. Davoudiasl, J.L. Hewett, T.G. Rizzo, Brane localized kinetic terms in the Randall-Sundrum model. Phys. Rev. D 68, 045002 (2003)ADSMathSciNetzbMATHGoogle Scholar
  47. 47.
    H. Davoudiasl, J.L. Hewett, T.G. Rizzo, Brane localized curvature for warped gravitons. JHEP 08, 034 (2003)ADSMathSciNetGoogle Scholar
  48. 48.
    G. Thomas, Rizzo, Kinetic mixing, dark photons and an extra dimension. Part I. JHEP 07, 118 (2018)Google Scholar
  49. 49.
    G. Thomas, Rizzo, Kinetic mixing, dark photons and extra dimensions. Part II: fermionic dark matter. JHEP 10, 069 (2018)Google Scholar
  50. 50.
    G.R. Dvali, S.H. Henry Tye, Brane inflation. Phys. Lett. B 450, 72–82 (1999)ADSMathSciNetzbMATHGoogle Scholar
  51. 51.
    G. Alencar, R.R. Landim, M.O. Tahim, R.N. Costa Filho, Gauge Field Localization on the Brane Through Geometrical Coupling. Phys. Lett. B 739, 125–127 (2014)ADSMathSciNetzbMATHGoogle Scholar
  52. 52.
    G. Alencar, Hidden conformal symmetry in Randall-Sundrum 2 model: Universal fermion localization by torsion. Phys. Lett. B 773, 601–603 (2017)ADSzbMATHGoogle Scholar
  53. 53.
    G. Alencar, R.R. Landim, C.R. Muniz, R.N. Costa Filho, Nonminimal couplings in Randall-Sundrum scenarios. Phys. Rev. D 92(6), 066006 (2015)Google Scholar
  54. 54.
    G. Alencar, C.R. Muniz, R.R. Landim, I.C. Jardim, R.N. Costa Filho, Photon mass as a probe to extra dimensions. Phys. Lett. B 759, 138–140 (2016)ADSzbMATHGoogle Scholar
  55. 55.
    G. Alencar, I.C. Jardim, R.R. Landim, C.R. Muniz, R.N. Costa Filho, Generalized nonminimal couplings in Randall-Sundrum scenarios. Phys. Rev. D 93(12), 124064 (2016)Google Scholar
  56. 56.
    G. Alencar, I.C. Jardim, R.R. Landim, \(p\)-Forms non-minimally coupled to gravity in Randall-Sundrum scenarios. Eur. Phys. J. C 78(5), 367 (2018)ADSGoogle Scholar
  57. 57.
    Luiz F. Freitas, G. Alencar, and R. R. Landim. Universal Aspects of \(U(1)\) Gauge Field Localization on Branes in \(D\)-dimensions. JHEP, 02:035, 2019Google Scholar
  58. 58.
    A. De Rujula, A. Donini, M.B. Gavela, S. Rigolin, Fat brane phenomena. Phys. Lett. B482, 195–204 (2000)ADSGoogle Scholar
  59. 59.
    Howard Georgi, Aaron K. Grant, Girma Hailu, Chiral fermions, orbifolds, scalars and fat branes. Phys. Rev. D 63, 064027 (2001)ADSMathSciNetGoogle Scholar
  60. 60.
    V. Silveira, A. Zee, Scalar Phantoms. Phys. Lett. 161B, 136–140 (1985)ADSGoogle Scholar
  61. 61.
    J. McDonald, Gauge singlet scalars as cold dark matter. Phys. Rev. D 50, 3637–3649 (1994)ADSGoogle Scholar
  62. 62.
    C.P. Burgess, M. Pospelov, T. ter Veldhuis, The Minimal model of nonbaryonic dark matter: A Singlet scalar. Nucl. Phys. B 619, 709–728 (2001)ADSGoogle Scholar
  63. 63.
    M.C. Bento, O. Bertolami, R. Rosenfeld, L. Teodoro, Selfinteracting dark matter and invisibly decaying Higgs. Phys. Rev. D 62, 041302 (2000)ADSGoogle Scholar
  64. 64.
    O. Bertolami, R. Rosenfeld, The Higgs portal and an unified model for dark energy and dark matter. Int. J. Mod. Phys. A 23, 4817–4827 (2008)ADSzbMATHGoogle Scholar
  65. 65.
    M.C. Bento, O. Bertolami, R. Rosenfeld, Cosmological constraints on an invisibly decaying Higgs boson. Phys. Lett. B 518, 276–281 (2001)ADSGoogle Scholar
  66. 66.
    J. March-Russell, S.M. West, D. Cumberbatch, D. Hooper, Heavy Dark Matter Through the Higgs Portal. JHEP 07, 058 (2008)ADSGoogle Scholar
  67. 67.
    A. Biswas, D. Majumdar, The Real Gauge Singlet Scalar Extension of Standard Model: A Possible Candidate of Cold Dark Matter. Pramana 80, 539–557 (2013)ADSGoogle Scholar
  68. 68.
    R. Costa, A.P. Morais, M.O.P. Sampaio, R. Santos, Two-loop stability of a complex singlet extended Standard Model. Phys. Rev. D 92, 025024 (2015)ADSGoogle Scholar
  69. 69.
    A. Eichhorn, M.M. Scherer, Planck scale, Higgs mass, and scalar dark matter. Phys. Rev. D 90(2), 025023 (2014)ADSGoogle Scholar
  70. 70.
    N. Khan, S. Rakshit, Study of electroweak vacuum metastability with a singlet scalar dark matter. Phys. Rev. D 90(11), 113008 (2014)ADSGoogle Scholar
  71. 71.
    F.S. Queiroz, K. Sinha, The Poker Face of the Majoron Dark Matter Model: LUX to keV Line. Phys. Lett. B 735, 69–74 (2014)ADSGoogle Scholar
  72. 72.
    C. Kouvaris, I.M. Shoemaker, K. Tuominen, Self-Interacting Dark Matter through the Higgs Portal. Phys. Rev. D 91(4), 043519 (2015)ADSGoogle Scholar
  73. 73.
    S. Bhattacharya, S. Jana, S. Nandi, Neutrino Masses and Scalar Singlet Dark Matter. Phys. Rev. D 95(5), 055003 (2017)ADSGoogle Scholar
  74. 74.
    O. Bertolami, C. Cosme, J.G. Rosa, Scalar field dark matter and the Higgs field. Phys. Lett. B 759, 1–8 (2016)ADSGoogle Scholar
  75. 75.
    R. Campbell, S. Godfrey, H.E. Logan, A. Poulin, Real singlet scalar dark matter extension of the Georgi-Machacek model. Phys. Rev. D 95(1), 016005 (2017)ADSGoogle Scholar
  76. 76.
    M. Heikinheimo, T. Tenkanen, K. Tuominen, V Vaskonen. Observational Constraints on Decoupled Hidden Sectors. Phys. Rev., D94(6):063506, (2016). [Erratum: Phys. Rev.D96,no.10,109902(2017)]Google Scholar
  77. 77.
    K. Kainulainen, S. Nurmi, T. Tenkanen, K. Tuominen, V. Vaskonen, Isocurvature Constraints on Portal Couplings. JCAP 1606(06), 022 (2016)Google Scholar
  78. 78.
    S. Nurmi, T. Tenkanen, K. Tuominen, Inflationary Imprints on Dark Matter. JCAP 1511(11), 001 (2015)Google Scholar
  79. 79.
    T. Tenkanen, Feebly Interacting Dark Matter Particle as the Inflaton. JHEP 09, 049 (2016)ADSGoogle Scholar
  80. 80.
    J.A. Casas, D.G. Cerdeño, J.M. Moreno, J. Quilis, Reopening the Higgs portal for single scalar dark matter. JHEP 05, 036 (2017)ADSzbMATHGoogle Scholar
  81. 81.
    Catarina Cosme, João G. Rosa, O. Bertolami, Scalar field dark matter with spontaneous symmetry breaking and the \(3.5\) keV line. Phys. Lett. B 781, 639–644 (2018)Google Scholar
  82. 82.
    M. Heikinheimo, T. Tenkanen, K. Tuominen, WIMP miracle of the second kind. Phys. Rev. D 96(2), 023001 (2017)ADSGoogle Scholar
  83. 83.
    G. Ricardo, Landim, Dark energy, scalar singlet dark matter and the Higgs portal. Mod. Phys. Lett. A 33(15), 1850087 (2018)Google Scholar
  84. 84.
    Bob Holdom, Two U(1)’s and Epsilon Charge Shifts. Phys. Lett. 166B, 196–198 (1986)ADSGoogle Scholar
  85. 85.
    Bob Holdom, Searching for \(\epsilon \) Charges and a New U(1). Phys. Lett. B 178, 65–70 (1986)ADSGoogle Scholar
  86. 86.
    Keith R. Dienes, Christopher F. Kolda, John March-Russell, Kinetic mixing and the supersymmetric gauge hierarchy. Nucl. Phys. B 492, 104–118 (1997)ADSGoogle Scholar
  87. 87.
    F. Del Aguila, The Physics of z-prime bosons. Acta Phys. Polon. B 25, 1317–1336 (1994)Google Scholar
  88. 88.
    K.S. Babu, Christopher F. Kolda, John March-Russell, Leptophobic U(1) \(s\) and the R(\(b\)) - R(\(c\)) crisis. Phys. Rev. D 54, 4635–4647 (1996)ADSGoogle Scholar
  89. 89.
    G. Thomas, Rizzo, Gauge kinetic mixing and leptophobic \(Z^\prime \) in E(6) and SO(10). Phys. Rev. D 59, 015020 (1998)Google Scholar
  90. 90.
    Daniel Feldman, Boris Kors, Pran Nath, Extra-weakly Interacting Dark Matter, Phys. Rev. D 75, 023503 (2007)ADSGoogle Scholar
  91. 91.
    Daniel Feldman, Zuowei Liu, Pran Nath, The Stueckelberg Z-prime Extension with Kinetic Mixing and Milli-Charged Dark Matter From the Hidden Sector. Phys. Rev. D 75, 115001 (2007)ADSGoogle Scholar
  92. 92.
    Maxim Pospelov, Adam Ritz, Mikhail B. Voloshin, Secluded WIMP Dark Matter. Phys. Lett. B662, 53–61 (2008)ADSGoogle Scholar
  93. 93.
    Maxim Pospelov, Secluded U(1) below the weak scale. Phys. Rev. D 80, 095002 (2009)ADSGoogle Scholar
  94. 94.
    Hooman Davoudiasl, Hye-Sung Lee, William J. Marciano, Muon Anomaly and Dark Parity Violation. Phys. Rev. Lett. 109, 031802 (2012)ADSGoogle Scholar
  95. 95.
    Hooman Davoudiasl, Hye-Sung Lee, William J. Marciano, ’Dark’ Z implications for Parity Violation, Rare Meson Decays, and Higgs Physics. Phys. Rev. D 85, 115019 (2012)ADSGoogle Scholar
  96. 96.
    Essig, Rouven et al: Working Group Report: New Light Weakly Coupled Particles. In: Proceedings, 2013 Community Summer Study on the Future of U.S. Particle Physics: Snowmass on the Mississippi (CSS2013): Minneapolis, MN, USA, July 29-August 6, 2013, (2013)Google Scholar
  97. 97.
    Eder Izaguirre, Gordan Krnjaic, Philip Schuster, Natalia Toro, Analyzing the Discovery Potential for Light Dark Matter. Phys. Rev. Lett. 115(25), 251301 (2015)ADSGoogle Scholar
  98. 98.
    Michael Duerr, Pavel Fileviez Pérez, Juri Smirnov, Scalar Dark Matter: Direct vs. Indirect Detection. JHEP 06, 152 (2016)Google Scholar
  99. 99.
    P. Athron et al., Status of the scalar singlet dark matter model. Eur. Phys. J. C 77(8), 568 (2017)ADSGoogle Scholar
  100. 100.
    A. Djouadi, O. Lebedev, Y. Mambrini, J. Quevillon, Implications of LHC searches for Higgs-portal dark matter. Phys. Lett. B 709, 65–69 (2012)ADSGoogle Scholar
  101. 101.
    K. Cheung, Y.-L.S. Tsai, P.-Y. Tseng, T.-C. Yuan, A. Zee, Global Study of the Simplest Scalar Phantom Dark Matter Model. JCAP 1210, 042 (2012)ADSGoogle Scholar
  102. 102.
    A. Djouadi, A. Falkowski, Y. Mambrini, J. Quevillon, Direct Detection of Higgs-Portal Dark Matter at the LHC. Eur. Phys. J. C 73(6), 2455 (2013)ADSGoogle Scholar
  103. 103.
    Cline, J.M., Kainulainen, K., Scott, P., Weniger, C.: Update on scalar singlet dark matter. Phys. Rev., D88:055025, (2013). [Erratum: Phys. Rev.D92,no.3,039906(2015)]Google Scholar
  104. 104.
    M. Endo, Y. Takaesu, Heavy WIMP through Higgs portal at the LHC. Phys. Lett. B 743, 228–234 (2015)ADSGoogle Scholar
  105. 105.
    A. Goudelis, Y. Mambrini, C. Yaguna, Antimatter signals of singlet scalar dark matter. JCAP 0912, 008 (2009)ADSGoogle Scholar
  106. 106.
    A. Urbano, W. Xue, Constraining the Higgs portal with antiprotons. JHEP 03, 133 (2015)Google Scholar
  107. 107.
    D.S. Akerib et al., Improved Limits on Scattering of Weakly Interacting Massive Particles from Reanalysis of 2013 LUX Data. Phys. Rev. Lett. 116(16), 161301 (2016)ADSGoogle Scholar
  108. 108.
    X.-G. He, J. Tandean, New LUX and PandaX-II Results Illuminating the Simplest Higgs-Portal Dark Matter Models. JHEP 12, 074 (2016)ADSGoogle Scholar
  109. 109.
    M. Escudero, A. Berlin, D. Hooper, M.-X. Lin, Toward (Finally!) Ruling Out Z and Higgs Mediated Dark Matter Models. JCAP 1612, 029 (2016)ADSGoogle Scholar
  110. 110.
    P.A.R. Ade et al., Planck 2015 results. XIII. Cosmological parameters. Astron. Astrophys. 594, A13 (2016)Google Scholar
  111. 111.
    Cline, J.M., Scott, P.: Dark Matter CMB Constraints and Likelihoods for Poor Particle Physicists. JCAP, 1303:044, (2013). [Erratum: JCAP1305,E01(2013)]Google Scholar
  112. 112.
    T.R. Slatyer, Indirect dark matter signatures in the cosmic dark ages. I. Generalizing the bound on s-wave dark matter annihilation from Planck results. Phys. Rev. D 93(2), 023527 (2016)Google Scholar
  113. 113.
    M. Ackermann et al., Searching for Dark Matter Annihilation from Milky Way Dwarf Spheroidal Galaxies with Six Years of Fermi Large Area Telescope Data. Phys. Rev. Lett. 115(23), 231301 (2015)ADSGoogle Scholar
  114. 114.
    D.S. Akerib et al., Results from a search for dark matter in the complete LUX exposure. Phys. Rev. Lett. 118(2), 021303 (2017)ADSGoogle Scholar
  115. 115.
    A. Tan et al., Dark Matter Results from First 98.7 Days of Data from the PandaX-II Experiment. Phys. Rev. Lett. 117(12), 121303 (2016)Google Scholar
  116. 116.
    R. Agnese et al., Search for Low-Mass Weakly Interacting Massive Particles with SuperCDMS. Phys. Rev. Lett. 112(24), 241302 (2014)ADSGoogle Scholar
  117. 117.
    E. Aprile et al., Dark Matter Results from 225 Live Days of XENON100 Data. Phys. Rev. Lett. 109, 181301 (2012)ADSGoogle Scholar
  118. 118.
    M.G. Aartsen et al., Search for dark matter annihilations in the Sun with the 79-string IceCube detector. Phys. Rev. Lett. 110(13), 131302 (2013)ADSGoogle Scholar
  119. 119.
    M.G. Aartsen et al., Improved limits on dark matter annihilation in the Sun with the 79-string IceCube detector and implications for supersymmetry. JCAP 1604(04), 022 (2016)ADSGoogle Scholar
  120. 120.
    Battaglieri, Marco et al.: US Cosmic Visions: New Ideas in Dark Matter 2017: Community Report. In: U.S. Cosmic Visions: New Ideas in Dark Matter College Park, MD, USA, March 23-25, 2017, (2017)Google Scholar
  121. 121.
    E.M. Riordan et al., A Search for Short Lived Axions in an Electron Beam Dump Experiment. Phys. Rev. Lett. 59, 755 (1987)ADSGoogle Scholar
  122. 122.
    J.D. Bjorken, S. Ecklund, W.R. Nelson, A. Abashian, C. Church, B. Lu, L.W. Mo, T.A. Nunamaker, P. Rassmann, Search for Neutral Metastable Penetrating Particles Produced in the SLAC Beam Dump. Phys. Rev. D 38, 3375 (1988)ADSGoogle Scholar
  123. 123.
    A. Bross, M. Crisler, Stephen H. Pordes, J. Volk, S. Errede, J. Wrbanek, A Search for Shortlived Particles Produced in an Electron Beam Dump. Phys. Rev. Lett. 67, 2942–2945 (1991)ADSGoogle Scholar
  124. 124.
    D. James, Bjorken, Rouven Essig, Philip Schuster, Natalia Toro, New Fixed-Target Experiments to Search for Dark Gauge Forces. Phys. Rev. D 80, 075018 (2009)Google Scholar
  125. 125.
    Hooman Davoudiasl, Hye-Sung Lee, William J. Marciano, Dark Side of Higgs Diphoton Decays and Muon g-2. Phys. Rev. D 86, 095009 (2012)ADSGoogle Scholar
  126. 126.
    Motoi Endo, Koichi Hamaguchi, Go Mishima, Constraints on Hidden Photon Models from Electron g-2 and Hydrogen Spectroscopy. Phys. Rev. D 86, 095029 (2012)ADSGoogle Scholar
  127. 127.
    D. Babusci et al., Limit on the production of a light vector gauge boson in phi meson decays with the KLOE detector. Phys. Lett. B 720, 111–115 (2013)ADSGoogle Scholar
  128. 128.
    F. Archilli et al., Search for a vector gauge boson in \(\phi \) meson decays with the KLOE detector. Phys. Lett. B 706, 251–255 (2012)ADSGoogle Scholar
  129. 129.
    P. Adlarson et al., Search for a dark photon in the \(\pi ^0 \rightarrow e^+e^-\gamma \) decay. Phys. Lett. B 726, 187–193 (2013)ADSGoogle Scholar
  130. 130.
    S. Abrahamyan et al., Search for a New Gauge Boson in Electron-Nucleus Fixed-Target Scattering by the APEX Experiment. Phys. Rev. Lett. 107, 191804 (2011)ADSGoogle Scholar
  131. 131.
    H. Merkel et al., Search for Light Gauge Bosons of the Dark Sector at the Mainz Microtron. Phys. Rev. Lett. 106, 251802 (2011)ADSGoogle Scholar
  132. 132.
    Matthew Reece, Lian-Tao Wang, Searching for the light dark gauge boson in GeV-scale experiments. JHEP 07, 051 (2009)ADSGoogle Scholar
  133. 133.
    Bernard Aubert et al., Search for Dimuon Decays of a Light Scalar Boson in Radiative Transitions Upsilon \(\rightarrow \) gamma A0. Phys. Rev. Lett. 103, 081803 (2009)ADSGoogle Scholar
  134. 134.
    K. Herbert, Dreiner, Jean-François Fortin, Christoph Hanhart, Lorenzo Ubaldi, Supernova constraints on MeV dark sectors from \(e^+e^-\) annihilations. Phys. Rev. D 89(10), 105015 (2014)Google Scholar

Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP3

Authors and Affiliations

  1. 1.Physik Department T70Technische Universität MünchenGarching bei MünchenGermany

Personalised recommendations