Abstract
A generalized Melvin solution for an arbitrary simple finite-dimensional Lie algebra \(\mathcal G\) is considered. The solution contains a metric, n Abelian 2-forms and n scalar fields, where n is the rank of \(\mathcal G\). It is governed by a set of n moduli functions \(H_s(z)\) obeying n ordinary differential equations with certain boundary conditions imposed. It was conjectured earlier that these functions should be polynomials—the so-called fluxbrane polynomials. These polynomials depend upon integration constants \(q_s\), \(s = 1,\dots ,n\). In the case when the conjecture on the polynomial structure for the Lie algebra \(\mathcal G\) is satisfied, it is proved that 2-form flux integrals \(\Phi ^s\) over a proper 2d submanifold are finite and obey the relations \(q_s \Phi ^s = 4 \pi n_s h_s\), where the \(h_s > 0\) are certain constants (related to dilatonic coupling vectors) and the \(n_s\) are powers of the polynomials, which are components of a twice dual Weyl vector in the basis of simple (co-)roots, \(s = 1,\dots ,n\). The main relations of the paper are valid for a solution corresponding to a finite-dimensional semi-simple Lie algebra \(\mathcal G\). Examples of polynomials and fluxes for the Lie algebras \(A_1\), \(A_2\), \(A_3\), \(C_2\), \(G_2\) and \(A_1 + A_1\) are presented.
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1 Introduction
In this paper we start with a generalization of a Melvin solution [1], which was presented earlier in Ref. [2]. It appears in the model which contains a metric, n Abelian 2-forms and \(l \ge n\) scalar fields. This solution is governed by a certain non-degenerate (quasi-Cartan) matrix \((A_{s s'})\), \(s, s' = 1, \dots , n\). It is a special case of the so-called generalized fluxbrane solutions from Ref. [3]. For fluxbrane solutions see Refs. [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28] and the references therein. The appearance of fluxbrane solutions was motivated by superstring/M theory.
The generalized fluxbrane solutions from Ref. [3] are governed by moduli functions, \(H_s(z) > 0\), defined on the interval \((0, +\infty )\), where \(z = \rho ^2\) and \(\rho \) is a radial variable. These functions obey a set of n non-linear differential master equations governed by the matrix \((A_{s s'})\), equivalent to Toda-like equations, with the following boundary conditions imposed: \(H_{s}(+ 0) = 1\), \(s = 1,\ldots ,n\).
In this paper we assume that \((A_{s s'})\) is a Cartan matrix for some simple finite-dimensional Lie algebra \(\mathcal G\) of rank n (\(A_{ss} = 2\) for all s). According to a conjecture suggested in Ref. [3], the solutions to the master equations with the boundary conditions imposed are polynomials:
where the \(P_s^{(k)}\) are constants. Here \(P_s^{(n_s)} \ne 0\) and
where we denote \((A^{s s'}) = (A_{s s'})^{-1}\). The integers \(n_s\) are components of a twice dual Weyl vector in the basis of simple (co-)roots [29].
The set of fluxbrane polynomials \(H_s\) defines a special solution to open Toda chain equations [30, 31] corresponding to a simple finite-dimensional Lie algebra \(\mathcal G\) [32]. In Refs. [2, 33] a program (in Maple) for the calculation of these polynomials for the classical series of Lie algebras (A-, B-, C- and D-series) was suggested. It was pointed out in Ref. [3] that the conjecture on the polynomial structure of \(H_{s}(z)\) is valid for Lie algebras of the A- and C-series. In Ref. [34] the conjecture from Ref. [3] was verified for the Lie algebra \(E_6\) and certain duality relations for six \(E_6\)-polynomials were proved. In Sect. 2 we present the generalized Melvin solution from Ref. [2]. In Sect. 3 we deal with the generalized Melvin solution for an arbitrary simple finite-dimensional Lie algebra \(\mathcal G\). Here we calculate 2-form flux integrals \(\Phi ^s = \int _{M_{*}} F^s\), where \(F^s\) are 2-forms and \(M_{*}\) is a certain 2d submanifold. These integrals (fluxes) are finite when moduli functions are polynomials. In Sect. 3 we consider examples of fluxbrane polynomials and fluxes for the Lie algebras: \(A_1\), \(A_2\), \(A_3\), \(C_2\), \(G_2\) and \(A_1 + A_1\).
2 The solutions
We consider a model governed by the action
where \(g=g_{MN}(x)\mathrm{d}x^M\otimes \mathrm{d}x^N\) is a metric, \(\varphi =(\varphi ^\alpha )\in {\mathbb R}^l\) is a set of scalar fields, \((h_{\alpha \beta })\) is a constant symmetric non-degenerate \(l\times l\) matrix \((l\in {\mathbb N})\), \( F^s = dA^s = \frac{1}{2} F^s_{M N} \mathrm{d}x^{M} \wedge \mathrm{d}x^{N}\) is a 2-form, \(\lambda _s\) is a 1-form on \({\mathbb R}^l\): \(\lambda _s(\varphi )=\lambda _{s \alpha }\varphi ^\alpha \), \(s = 1,\ldots , n\); \(\alpha =1,\dots ,l\). Here \((\lambda _{s \alpha })\), \(s =1,\dots , n\), are dilatonic coupling vectors. In (2.1) we denote \(|g| = |\det (g_{MN})|\), \((F^s)^2 = F^s_{M_1 M_{2}} F^s_{N_1 N_{2}} g^{M_1 N_1} g^{M_{2} N_{2}}\), \(s = 1,\dots , n\).
Here we start with a family of exact solutions to field equations corresponding to the action (2.1) and depending on one variable \(\rho \). The solutions are defined on the manifold
where \(M_1\) is a one-dimensional manifold (say \(S^1\) or \({\mathbb R}\)) and \(M_2\) is a (D-2)-dimensional Ricci-flat manifold. The solution reads [2]
\(s = 1,\dots , n\); \(\alpha = 1,\dots , l\), where \(w = \pm 1\), \(g^1 = \mathrm{d}\phi \otimes \mathrm{d}\phi \) is a metric on \(M_1\) and \(g^2\) is a Ricci-flat metric on \(M_{2}\). Here \(q_s \ne 0\) are integration constants, \(q_s = - Q_s\) in the notations of Ref. [2], \(s = 1,\dots , n\).
The functions \(H_s(z) > 0\), \(z = \rho ^2\), obey the master equations
with the following boundary conditions:
where
\(s = 1,\dots ,n\). The boundary condition (2.7) guarantees the absence of a conic singularity [in the metric (2.3)] for \(\rho = +0\).
The parameters \(h_s\) satisfy the relations
where
\(s, s' = 1,\ldots , n\), with \((h^{\alpha \beta })=(h_{\alpha \beta })^{-1}\). In the relations above we denote \(\lambda _{s}^{\alpha } = h^{\alpha \beta } \lambda _{s \beta }\) and
The latter is the so-called quasi-Cartan matrix.
We note that the constants \(B_{s s'}\) and \(K_s = B_{s s}\) have a certain mathematical sense. They are related to scalar products of certain vectors \(U^s\) (brane vectors, or U-vectors), which belong to a certain linear space (“truncated target space”, for our problem it has dimension \(l+2\)), i.e. \(B_{s s'} =(U^s,U^{s'})\) and \(K_s = (U^s,U^{s})\) [35,36,37]. The scalar products of such a type are of physical significance, since they appear for various solutions with branes, e.g. black branes, S-branes, fluxbranes etc. Several physical parameters in multidimensional models with branes, e.g. the Hawking-like temperatures and the entropies of black holes and branes, PPN parameters, Hubble-like parameters, fluxes etc., contain such scalar products; see [36, 37] and Sect. 3 of this paper. The relation (2.11) defines generalized intersection rules for branes which were suggested in [35]. The constants \(K_s\) are invariants of dimensional reduction. It is well known, see [37] and the references therein, that \(K_s = 2\) for branes in numerous supergravity models, e.g. in dimensions \(D = 10,11\).
It may be shown that if the matrix \((h_{\alpha \beta })\) has an Euclidean signature and \(l \ge n\), and \((A_{ss'})\) is a Cartan matrix for a simple Lie algebra \(\mathcal G\) of rank n, there exists a set of co-vectors \(\lambda _1, \dots , \lambda _n\) obeying (2.11) (for \(l = n\) see Remark 1 in the next section). Thus the solution is valid at least when \(l \ge n\) and the matrix \((h_{\alpha \beta })\) is positive-definite.
The solution under consideration is a special case of the fluxbrane (for \(w = +1\), \(M_1 = S^1\)) and S-brane (\(w = -1\)) solutions from [3] and [25], respectively.
If \(w = +1\) and the (Ricci-flat) metric \(g^2\) has a pseudo-Euclidean signature, we get a multidimensional generalization of Melvin’s solution [1].
In our notations Melvin’s solution (without scalar field) corresponds to \(D = 4\), \(n = 1\), \(l =0\), \(M_1 = S^1\) (\(0< \phi < 2 \pi \)), \(M_2 = {\mathbb R}^2\), \(g^2 = - \mathrm{d}t \otimes \mathrm{d}t + \mathrm{d}x \otimes \mathrm{d}x\) and \(\mathcal{G} = A_1\).
For \(w = -1\) and \(g^2\) of Euclidean signature we obtain a cosmological solution with a horizon (as \(\rho = + 0\)) if \(M_1 = {\mathbb R}\) (\( - \infty< \phi < + \infty \)).
3 Flux integrals for a simple finite-dimensional Lie algebra
Here we deal with the solution which corresponds to a simple finite-dimensional Lie algebra \(\mathcal{G}\), i.e. the matrix \(A = (A_{ss'})\) is coinciding with the Cartan matrix of this Lie algebra. We put also \(n = l\), \(w = + 1\) and \(M_1 = S^1\), \(h_{\alpha \beta } = \delta _{\alpha \beta }\) and denote \((\lambda _{s a }) = (\lambda _{s}^{a}) = \mathbf {\lambda }_{s}\), \(s = 1, \dots , n\).
\(h_s = K_s^{-1}\), and
\(s,l = 1, \dots , n\). [Equation (3.1) is a special case of (3.2)].
It follows from (2.9)–(2.11) that
for any \(i \ne j\) obeying \(A_{ij}, A_{ji} \ne 0\); \(i,j = 1, \dots ,n\). It may be readily shown from (3.3) that the ratios \(\frac{h_i}{h_j} = \frac{K_{j}}{K_{i}}\) are fixed numbers for any given Cartan matrix \(({A_{ij}})\) of a simple (finite-dimensional) Lie algebra \(\mathcal{G}\). (This follows from (3.3) and the connectedness of the Dynkin diagram of a simple Lie algebra.) The ratios (3.3) may be written as follows:
\(i \ne j\), where \(r_i = (\alpha _{i}, \alpha _{i})\) is the length squared of a simple root \(\alpha _{i}\) corresponding to the Lie algebra \(\mathcal{G}\). Here we use the notations \(A_{ij} = 2 (\alpha _{i}, \alpha _{j})/(\alpha _{j}, \alpha _{j})\); \(i,j = 1, \dots ,n\). Equation (3.4) implies
\(i = 1, \dots ,n\), where \(K > 0\). (For simply laced (A, D, E) Lie algebras all \(r_i\) are equal.)
Remark 1
For large enough K in (3.5) there exist vectors \(\mathbf {\lambda }_s\) obeying (3.2) [and hence (3.1)]. Indeed, the matrix \((\Gamma _{sl})\) is positive-definite if \(K > K_{*}\), where \(K_{*}\) is some positive number. Hence there exists a matrix \(\Lambda \), such that \(\Lambda ^{T}\Lambda = \Gamma \). We put \((\Lambda _{as}) = (\lambda _{s}^a)\) and get the set of vectors obeying (3.2).
Now let us consider the oriented 2-dimensional manifold \(M_{*} =(0, + \infty ) \times S^1\). The flux integrals
where
are convergent for all s, if the conjecture for the Lie algebra \(\mathcal{G}\) (on polynomial structure of moduli functions \(H_s\)) is obeyed for the Lie algebra \(\mathcal{G}\) under consideration.
Indeed, due to the polynomial assumption (1.1) we have
as \(\rho \rightarrow + \infty \); \(s =1, \dots , n\). From (3.7), (3.8) and the equality \(\sum _{1}^{n} A_{s l} n_l = 2\), following from (1.2), we get
and hence the integral (3.6) is convergent for any \(s =1, \dots , n\).
By using the master equations (2.6) we obtain
which implies [see (2.8)]
\(s =1, \dots , n \).
Thus, any flux \(\Phi ^s\) depends upon one integration constant \(q_s \ne 0\), while the integrand form \(F^s\) depends upon all constants: \(q_1, \dots , q_n\).
We note that for \(D =4\) and \(g^2 = - \mathrm{d}t \otimes \mathrm{d}t + \mathrm{d}x \otimes d x\), \(q_s\) is coinciding with the value of the x-component of the sth magnetic field on the axis of symmetry.
In the case of the Gibbons–Maeda dilatonic generalization of the Melvin solution, corresponding to \(D = 4\), \(n = l= 1\) and \(\mathcal{G} = A_1\) [5], the flux from (3.11) (\(s=1\)) is in agreement with that considered in Ref. [26]. For Melvin’s case and some higher dimensional extensions (with \(\mathcal{G} = A_1\)) see also Ref. [14].
Due to (3.4) the ratios
are fixed numbers depending upon the Cartan matrix \(({A_{ij}})\) of a simple finite-dimensional Lie algebra \(\mathcal{G}\).
Remark 2
The relation for flux integrals (3.11) is also valid when the matrix \((A_{ss'})\) is a Cartan matrix of a finite-dimensional semi-simple Lie algebra \(\mathcal{G} = \mathcal{G}_1 \oplus \cdots \oplus \mathcal{G}_k\), where \(\mathcal{G}_1, \dots , \mathcal{G}_k\) are simple Lie (sub)algebras. In this case the Cartan matrix \(({A_{ij}})\) has a block-diagonal form, i.e. \(({A_{ij}}) = \mathrm{diag} \left( \left( {A^{(1)}_{i_1 j_1}}\right) , \ldots , \left( {A^{(k)}_{i_k j_k}}\right) \right) \), where \(\left( {A^{(a)}_{i_a j_a}}\right) \) is the Cartan matrix of the Lie algebra \(\mathcal{G}_a\), \(a = 1, \dots , k\). The set of polynomials in this case splits in a direct union of sets of polynomials corresponding to the Lie algebras \(\mathcal{G}_1, \dots , \mathcal{G}_k\). Equations (3.4) and (3.12) are valid, when the indices i, j correspond to one ath block, \(a = 1, \dots , k\). The quantities \(q_i \Phi ^i\) and \(q_j \Phi ^j\) corresponding to different blocks are independent. Equation (3.5) should be replaced by
for any index \(i_a\) corresponding to the ath block; \(a = 1, \dots , k\). The existence of dilatonic coupling vectors \(\mathbf {\lambda }_s\) obeying (3.2) [(and (3.1)] just follows from the arguments of Remark 1, if we put all \(K^{(a)} = K > 0\).
The manifold \(M_{*} =(0, + \infty ) \times S^1\) is isomorphic to the manifold \({\mathbb R}^2_{*} = {\mathbb R}^2 \setminus \{ 0 \}\). The solution (2.3)–(2.5) may be understood (or rewritten by pull-backs) as defined on the manifold \({\mathbb R}^2_{*} \times M_2\), where the coordinates \(\rho \), \(\phi \) are understood as coordinates on \({\mathbb R}^2_{*}\). They are not globally defined. One should consider two charts with coordinates \(\rho \), \(\phi = \phi _1\) and \(\rho \), \(\phi = \phi _2\), where \(\rho > 0\), \(0< \phi _1 < 2 \pi \) and \(- \pi< \phi _2 < \pi \). Here \(\exp (i \phi _1 ) = \exp (i \phi _2)\). In both cases we have \(x = \rho \cos \phi \) and \(y = \rho \sin \phi \), where x, y are standard coordinates of \({\mathbb R}^2\). Using the identity \( \rho \mathrm{d}\rho \wedge \mathrm{d}\phi = \mathrm{d}x \wedge \mathrm{d}y\) we get
\(s =1, \dots , n \). The 2-forms (3.14) are well defined on \({\mathbb R}^2\). Indeed, due to the conjecture from Ref. [3] any polynomial \(H_{s}(z)\) is a smooth function on \({\mathbb R}= (- \infty , + \infty )\) which obeys \(H_{s}(z) > 0\) for \(z \in (- \varepsilon _s, + \infty )\), where \(\varepsilon _s > 0\). This is valid due to the conjecture from Ref. [3] \(H_{s}(z) > 0\) for \(z > 0\) and \(H_{s}(+0) = 1\). Thus, \(\left( \prod _{s' = 1}^{n} \left( H_{s'}\left( x^2 + y^2\right) \right) ^{- A_{s s'}} \right) \) is a smooth function since it is a composition of two well-defined smooth functions \(\left( \prod _{s' = 1}^{n} (H_{s'}(z))^{- A_{s s'}} \right) \) and \(z = x^2 + y^2\).
Now we show that there exist 1-forms \(A^s\) obeying \(F^s = dA^s\) which are globally defined on \({\mathbb R}^2\). We start with the open submanifold \({\mathbb R}^2_{*}\). The 1-forms
are well defined on \({\mathbb R}^2_{*}\) (here \(\mathrm{d}\phi = (x^2 + y^2)^{-1} ( - y \mathrm{d}x + x \mathrm{d}y)\)) and obey \(F^s = dA^s\), \(s = 1, \dots , n \). Using the master equation (2.6) we obtain
\(s = 1, \dots , n \). Here \(H{'}_s = \frac{\mathrm{d}}{\mathrm{d}z} H_s\). Due to the relation \(\rho ^2 \mathrm{d}\phi = - y \mathrm{d}x + x \mathrm{d}y \), we obtain
\(s = 1, \dots , n \). The 1-forms (3.17) are well-defined smooth 1-forms on \({\mathbb R}^2\).
We note that in the case of the Gibbons–Maeda solution [5] corresponding to \(D = 4\), \(n = l= 1\) and \(\mathcal{G} = A_1\) the gauge potential from (3.16) coincides (up to notations) with that considered in Ref. [7].
Now we verify our result (3.11) for flux integrals by using the relations for the 1-forms \(A^s\). Let us consider a 2d oriented manifold (disk) \(D_R = \{ (x,y): x^2 + y^2 \le R^2 \}\) with the boundary \(\partial D_R = C_R = \{ (x,y): x^2 + y^2 = R^2 \}\). \(C_R\) is a circle of radius R. It is an 1d oriented manifold with the orientation (inherited from that of \(D_R\)) obeying the relation \(\int _{C_R} \mathrm{d}\phi = 2 \pi \). Using the Stokes–Cartan theorem we get
\(s = 1, \dots , n \). By using the asymptotic relation (3.8) we find
\(s = 1, \dots , n \), in agreement with (3.11).
Remark 3
We note (for completeness) that the metric and scalar fields for our solution with \(w = +1\) and \(l = n\) can be extended to the manifold \({\mathbb R}^2 \times M_2\). Indeed, in the coordinates x, y the metric (2.3) and scalar fields (2.4) read as follows:
\(a =1, \dots , l \). Here \(H_s = H_s(x^2 + y^2)\), \(s = 1, \dots , n \), and \(f = f(x^2 + y^2)\), where
for \(z \ne 0\) and \(f(0)= \lim _{z \rightarrow 0} f(z)\) (the limit does exist). The function f(z) is smooth in the interval \( (- \varepsilon , + \infty )\) for some \(\varepsilon > 0\). Indeed, it is smooth in the interval \( (0, + \infty )\) and holomorphic in the domain \(\{z | 0< |z| < \varepsilon \}\) for a small enough \(\varepsilon > 0\). Since the limit \(\lim _{z \rightarrow 0} f(z)\) does exist the function f(z) is holomorphic in the disc \(\{z | |z| < \varepsilon \}\) and hence it is smooth in the interval \( (- \varepsilon , + \infty )\). This implies that the metric is smooth on the manifold \({\mathbb R}^2 \times M_2\). (See the text after Eq. (3.14).) The scalar fields are also smooth on \({\mathbb R}^2 \times M_2\).
4 Examples
Here we present fluxbrane polynomials corresponding to the Lie algebras \(A_1\), \(A_2\), \(A_3\), \(C_2\), \(G_2\), \(A_1 + A_1\) and related fluxes. Here as in [32] we use other parameters \(p_s\) instead of \(P_s\):
\(s = 1, \ldots , n\).
\(A_1\) -case. The simplest example occurs in the case of the Lie algebra \(A_1 = sl(2)\). Here \(n_1 = 1\). We get [3]
and
which is also valid for Melvin’s solution with \(D = 4\) and \(h_1 = 2\).
\(A_2\) -case. For the Lie algebra \(A_2 = sl(3)\) with the Cartan matrix
we have [3, 25, 32] \(n_1 = n_2 =2\) and
We get in this case
where \(h_1 = h_2 = h\).
\(A_3\) -case. The polynomials for the \(A_3\)-case read as follows [32, 33]:
Here we have \((n_1, n_2, n_3) = (3,4,3)\) and
with \(h_1 = h_2 = h_3 = h\).
\(C_2\) -case. For the Lie algebra \(C_2 = so(5)\) with the Cartan matrix
we get \(n_1 = 3\) and \(n_2 = 4\). For \(C_2\)-polynomials we obtain [25, 32]
In this case we find
where \(h_1 = 2 h_2\).
\(G_2\) -case. For the Lie algebra \(G_2\) with the Cartan matrix
we get \(n_1 = 6\) and \(n_2 = 10\). In this case the fluxbrane polynomials read [25, 32]
We are led to the relations
where \(h_1 = 3 h_2\).
\((A_1 + A_1)\) -case. For the semi-simple Lie algebra \(A_1 + A_1\) we obtain \(n_1 = n_2 = 1\),
and
where \(h_1\) and \(h_2\) are independent, as well as the quantities \(q_1 \Phi ^1\) and \(q_2 \Phi ^2\).
5 Conclusions
Here we have considered a multidimensional generalization of Melvin’s solution corresponding to a simple finite-dimensional Lie algebra \(\mathcal{G}\). We have assumed that the solution is governed by a set of n fluxbrane polynomials \(H_s(z)\), \(s =1,\dots ,n\). These polynomials define special solutions to open Toda chain equations corresponding to the Lie algebra \(\mathcal{G}\).
The polynomials \(H_s(z)\) depend also upon parameters \(q_s\), which are coinciding for \(D =4\) (up to a sign) with the values of colored magnetic fields on the axis of symmetry.
We have calculated 2d flux integrals \(\Phi ^s = \int F^s\), \(s =1, \dots , n\). Any flux \(\Phi ^s\) depends only upon one parameter \(q_s\), while the integrand \(F^s\) depends upon all parameters \(q_1, \dots , q_n\). The relation for flux integrals (3.11) is also valid when the matrix \((A_{ss'})\) is a Cartan matrix of a finite-dimensional semi-simple Lie algebra \(\mathcal G\).
Here we have considered examples of polynomials and fluxes for the Lie algebras \(A_1\), \(A_2\), \(A_3\), \(C_2\), \(G_2\) and \(A_1 + A_1\). The approach of this paper will be used for a calculation of certain flux integrals for forms \(F^s\) of arbitrary ranks corresponding to certain fluxbrane solutions (of electric type by p-brane notation or magnetic type by fluxbrane classificationFootnote 1) governed by fluxbrane polynomials [38].
An open problem is to find the fluxes for the solutions which are related to infinite-dimensional Lorentzian Kac–Moody algebras, e.g. hyperbolic ones [39, 40]. In this case one should deal with phantom scalar fields in the model (2.1) and non-polynomial solutions to Eqs. (2.6). Another possibility is to study the convergence of flux integrals for non-polynomial solutions for moduli functions corresponding to non-Cartan matrices \((A_{ss'})\) (e.g. for the model with two 2-forms from Ref. [41]).
Notes
We remind the reader that an electric (magnetic) p-brane corresponds to a magnetic (electric) \(F(D-3 -p)\) fluxbrane; see [3] and the references therein.
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Acknowledgements
This work was supported in part by the Russian Foundation for Basic Research Grant No. 16-02-00602 and by the Ministry of Education of the Russian Federation (the Agreement Number 02.a03.21.0008 of 24 June 2016).
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Ivashchuk, V.D. On flux integrals for generalized Melvin solution related to simple finite-dimensional Lie algebra. Eur. Phys. J. C 77, 653 (2017). https://doi.org/10.1140/epjc/s10052-017-5235-5
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DOI: https://doi.org/10.1140/epjc/s10052-017-5235-5