Phenomenology with F-theory SU(5)

TL;DR

This paper investigates an F-theory SU(5) model's low-energy phenomenology, highlighting new particles, their potential collider signatures, and consistency with gauge unification and proton decay constraints.

Contribution

It presents a detailed phenomenological analysis of an F-theory SU(5) model with novel particles and explores their experimental implications and theoretical consistency.

Findings

Some new particles may be observable at colliders.

The model maintains gauge coupling unification.

Proton decay is suppressed but dimension-six decay is enhanced.

Abstract

We explore the low energy phenomenology of an F-theory based SU(5) model which, in addition to the known quarks and leptons, contains Standard Model singlets, and vector-like color triplets and SU(2) doublets. Depending on their masses and couplings, some of these new particles may be observed at the LHC and future colliders. We discuss the restrictions by CKM constraints on their mixing with the ordinary down quarks of the three chiral familes. The model is consistent with gauge coupling unification at the usual supersymmetric GUT scale, dimension five proton decay is adequately suppressed, while dimension-six decay mediated by the superheavy gauge bosons is enhanced by a factor of 5-7. The third generation charged fermion Yukawa couplings yield the corresponding low-energy masses in reasonable agreement with observations. The hierarchical nature of the masses of lighter generations is accounted for via non-renormalisable interactions, with the perturbative vacuum expectation values of the SM singlet fields playing an essential role.

Tables

Table 1. Table 1: Field content under S ​ U ​ ( 5 ) × U ​ ( 1 ) t i 𝑆 𝑈 5 𝑈 subscript 1 subscript 𝑡 𝑖 SU(5)\times U(1)_{t_{i}} , their homology class and flux restrictions. For convenience, only the properties of 10 , 5 10 5 10,5 are shown. 10 ¯ , 5 ¯ ¯ 10 ¯ 5 \overline{10},\overline{5} are characterised by t i → − t i → subscript 𝑡 𝑖 subscript 𝑡 𝑖 t_{i}\rightarrow-t_{i} . Note that the fluxes satisfy N = N 7 + N 8 + N 9 𝑁 subscript 𝑁 7 subscript 𝑁 8 subscript 𝑁 9 N=N_{7}+N_{8}+N_{9} and ∑ i M 10 i + ∑ j M 5 j = 0 subscript 𝑖 subscript 𝑀 subscript 10 𝑖 subscript 𝑗 subscript 𝑀 subscript 5 𝑗 0 \sum_{i}M_{10_{i}}+\sum_{j}M_{5_{j}}=0 , while χ = χ 7 + χ 8 + χ 9 𝜒 subscript 𝜒 7 subscript 𝜒 8 subscript 𝜒 9 {\chi}=\chi_{7}+\chi_{8}+\chi_{9} .

Curve	Field	$U{\left(1\right)}_i$	defining eq.	homology	$U{\left(1\right)}_Y$-flux	$U\left(1\right)$-flux
${\Sigma }_{{10}^{\left(1\right)}}$:	${10}_3$	$t_1$	$a_1$	$\eta -2c_1-\chi$	$-N=0$	$M_{{10}_1}=1$
${\Sigma }_{{10}^{\left(2\right)}}$:	${10}_1$	$t_3$	$a_4$	$-c_1+{\chi }_7$	$N_7=-1$	$M_{{10}_2}=1$
${\Sigma }_{{10}^{\left(3\right)}}$:	${10}_2$	$t_4$	$a_6$	$-c_1+{\chi }_8$	$N_8=1$	$M_{{10}_3}=1$
${\Sigma }_{{10}^{\left(4\right)}}$:	${10}_2^{\prime }$	$t_5$	$a_8$	$-c_1+{\chi }_9$	$N_9=0$	$M_{{10}_4}=0$
${\Sigma }_{5^{\left(0\right)}}$:	$5_{H_u}$	$-2t_1$	$a_{578}+a_{479}+a_{569}$	$-c_1+\chi$	$N=0$	$M_{5_{H_u}}=1$
${\Sigma }_{5^{\left(1\right)}}$:	${\overline{5}}_2$	$t_1+t_3$	$a_1-c\left(a_{478}+a_{469}\right)$	$\eta -2c_1-\chi$	$-N=0$	$M_{5_1}=-1$
${\Sigma }_{5^{\left(2\right)}}$:	${\overline{5}}_3$	$t_1+t_4$	$a_1-c\left(a_{568}+a_{469}\right)$	$\eta -2c_1-\chi$	$-N=0$	$M_{5_2}=-1$
${\Sigma }_{5^{\left(3\right)}}$:	$5_x$	$-t_1-t_5$	$a_1-c\left(a_{568}+a_{478}\right)$	$\eta -2c_1-\chi$	$-N=0$	$M_{5_3}=n$
${\Sigma }_{5^{\left(4\right)}}$:	${\overline{5}}_1$	$t_3+t_4$	$a_{56}+a_{47}$	$-c_1+\chi -{\chi }_9$	$N-N_9=0$	$M_{5_4}=-1$
${\Sigma }_{5^{\left(5\right)}}$:	${\overline{5}}_{H_d}$	$t_3+t_5$	$a_{58}+a_{49}$	$-c_1+\chi -{\chi }_8$	$N-N_8=-1$	$M_{5_{H_d}}=0$
${\Sigma }_{5^{\left(6\right)}}$:	${\overline{5}}_{\overline{x}}$	$t_4+t_5$	$a_{78}+a_{49}$	$-c_1+\chi -{\chi }_7$	$N-N_7=1$	$-n-1$
	${\theta }_{12}$	$0$	$-$	$-$	$-$	$M_{12}$
${\Sigma }_{5^{\left(6\right)}}$:	${\theta }_{ij}$	$t_i-t_j$	$-$	$-$	$-$	$M_{ij}$
	${\theta }_{\delta }$	$0$	$-$	$-$	$-$	$M_{\delta }$

Table 2. Table 2: Field content under S ​ U ​ ( 5 ) × U ​ ( 1 ) t i 𝑆 𝑈 5 𝑈 subscript 1 subscript 𝑡 𝑖 SU(5)\times U(1)_{t_{i}} . The third column shows the MSSM spectrum and the fourth column displays the predicted exotics. The R-parity assignments appear in the last column. We use assignments 10 t i , 5 − t i − t j subscript 10 subscript 𝑡 𝑖 subscript 5 subscript 𝑡 𝑖 subscript 𝑡 𝑗 {10}_{t_{i}},{5}_{-t_{i}-t_{j}} whith 10 ¯ , 5 ¯ ¯ 10 ¯ 5 \overline{10},\overline{5} characterized by opposite values, t i → − t i → subscript 𝑡 𝑖 subscript 𝑡 𝑖 t_{i}\rightarrow-t_{i} etc. The fluxes eliminated components of the S ​ U ​ ( 5 ) 𝑆 𝑈 5 SU(5) multiplets, giving rise to incomplete representations. There are also n 𝑛 n copies, of 5 + 5 ¯ 5 ¯ 5 5+\bar{5} multiplets.

Irrep	$U{\left(1\right)}_i$	SM spectrum	Exotics	$R$-parity
${\mathrm{\hspace{0.17em}10}}_1$	$t_3$	$Q_1,u_1^c,u_2^c$	$-$	$-$
${\mathrm{\hspace{0.17em}10}}_2$	$t_4$	$Q_2,e_1^c,e_2^c$	$-$	$-$
${\mathrm{\hspace{0.17em}10}}_3$	$t_1$	$Q_3,u_3^c,e_3^c$	$-$	$-$
${\overline{5}}_1$	$t_3+t_4$	$d_1^c,{\ell }_1$	$-$	$-$
${\overline{5}}_2$	$t_1+t_3$	$d_2^c,{\ell }_2$	$-$	$-$
${\overline{5}}_3$	$t_1+t_4$	$d_3^c,{\ell }_3$	$-$	$-$
${\mathrm{\hspace{0.17em}5}}_{H_u}$	$-2t_1$	$H_u$	$D$	$+$
${\overline{5}}_{H_d}$	$t_3+t_5$	$H_d$	$-$	$+$
${\mathrm{\hspace{0.17em}5}}_x$	$-\left(t_1+t_5\right)$	$-$	${(H_{u_i},D_i)}_{i=1,\dots ,n}$	$+$
${\overline{5}}_{\overline{x}}$	$t_4+t_5$	$-$	$D^c+{(H_{d_i},D_i^c)}_{i=1,\dots ,n}$	$+$
${\theta }_{12,21}$	$0$		$S\left(\mathrm{singlet}\right)$	$-$
${\theta }_{14}$	$t_1-t_4$		$⟨{\theta }_{14}⟩=V_1\equiv v_1{M}_{GUT}$	$+$
${\theta }_{15}$	$t_1-t_5$		$⟨{\theta }_{15}⟩=V_2\equiv v_2{M}_{GUT}$	$+$
${\theta }_{43}$	$t_4-t_3$		$⟨{\theta }_{43}⟩=V_3\equiv v_3{M}_{GUT}$	$+$
${\theta }_{ij}^{⟂}$	$t_i-t_j$		$⟨{\theta }_{ij}^{⟂}⟩=0$	$+$

Equations

$$E_8\supset SU(5{)}_{GUT}\times SU(5{)}_{\perp },$$

$$248$$

$$\begin{array}{rl}{\mathcal{C}}_5:& \sum _{k=0}^5{b}_k{s}^{5-k}=0,\end{array}$$

$$\sum _{i=1}^5{t}_i=0.$$

$${\Sigma }_{10_{t_i}}:10{}_{t_i},\overline{10}{}_{-t_i},{\Sigma }_{5_{t_i+t_j}}:{\overline{5}}_{t_i+t_j},5_{-t_i-t_j},{\Sigma }_{1_{t_i-t_j}}:1_{t_i-t_j},$$

$$\mathcal{W}\supset h_{1}10_{t_i}10_{t_j}{5}_{-t_i-t_j}+h_{2}10_{t_i}{\bar{5}}_{t_j+t_k}{\bar{5}}_{t_l+t_m}+h_3{1}_{t_i-t_j}{5}_{-t_i-t_k}{\bar{5}}_{t_j+t_m}+h_4{1}_{t_i-t_j}{1}_{t_j-t_k}{1}_{t_k-t_i},$$

$$Z_2:t_1=t_2.$$

$$\mathcal{W}\supset 10_{t_1}10_{t_2}{5}_{-t_1-t_2}\stackrel{Z_2}{⟶}10_{t_1}10_{t_1}{5}_{-2t_1}$$

$$b_0{s}^5+b_2{s}^3+b_3{s}^2+b_{4}s+b_5=(a_1+a_{2}s+a_3{s}^2)(a_4+a_{5}s)(a_6+a_{7}s)(a_8+a_{9}s),$$

$$N_7={\mathcal{F}}_Y\cdot {\chi }_7,N_8={\mathcal{F}}_Y\cdot {\chi }_8,N_9={\mathcal{F}}_Y\cdot {\chi }_9.$$

$${10}_{t_{i}}=\left\{\begin{array}[]{ll}{\rm Representation}&{\rm flux\,units}\\ n_{{(3,2)}_{1/6}}-n_{{(\bar{3},2)}_{-1/6}}&=\;M_{10}\\ n_{{(\bar{3},1)}_{-2/3}}-n_{{(3,1)}_{2/3}}&=\;M_{10}-N_{y}\\ n_{(1,1)_{+1}}-n_{(1,1)_{-1}}&=\;M_{10}+N_{y}\\ \end{array}\right.\,,$$

$${5}_{t_{i}}=\left\{\begin{array}[]{ll}{\rm Representation}&{\rm flux\,units}\\ n_{(3,1)_{-1/3}}-n_{(\bar{3},1)_{+1/3}}&=\;M_{5}\\ n_{(1,2)_{+1/2}}-n_{(1,2)_{-1/2}}&=\;M_{5}+N_{y}\,\cdot\\ \end{array}\right.$$

$${\theta }_{12}={\theta }_{21}\to S$$

$${\theta }_{13},{\theta }_{14},{\theta }_{15},{\theta }_{34},{\theta }_{35},{\theta }_{45},\mathrm{and}{\theta }_{31},{\theta }_{41},{\theta }_{51},{\theta }_{43},{\theta }_{53},{\theta }_{54}$$

$$⟨{\theta }_{14}⟩\equiv V_1\equiv v_1{M}_{GUT}\ne 0,⟨{\theta }_{15}⟩\equiv V_2\equiv v_2{M}_{GUT}\ne 0,⟨{\theta }_{43}⟩\equiv V_3\equiv v_3{M}_{GUT}\ne 0.$$

$$M_{u}\propto\left(\begin{array}[]{ccc}v_{1}^{2}v_{3}^{2}&v_{1}^{2}v_{3}&v_{1}v_{3}\\ v_{1}^{2}v_{3}&v_{1}^{2}&v_{1}\\ v_{1}v_{3}&v_{1}&1\\ \end{array}\right)\langle H_{u}\rangle,\;M_{d,\ell}=\left(\begin{array}[]{ccc}v_{1}^{2}v_{3}^{2}&v_{1}v_{3}^{2}&v_{1}v_{3}\\ v_{1}^{2}v_{3}&v_{1}v_{3}&v_{1}\\ v_{1}v_{3}&v_{3}&1\\ \end{array}\right)\langle H_{d}\rangle\,,$$

$$5_{H_u}({\bar{5}}_1{\theta }_{14}^2{\theta }_{43}+{\bar{5}}_2{\theta }_{14}{\theta }_{43}+{\bar{5}}_3{\theta }_{14}).$$

$$\frac{{\theta }_{14}{\theta }_{43}{\theta }_{15}}{M_{GUT}^2}{\bar{5}}_{t_3+t_5}{5}_{-2t_1}\to \frac{V_1{V}_2{V}_3}{M_{GUT}^2}{H}_u{H}_d\to \mu H_u{H}_d.$$

$$5_{H_u}{\bar{5}}_{H_d}{\theta }_{14}{\theta }_{43}{\theta }_{15}/M_{GUT}^2+5_{H_u}{\bar{5}}_{\bar{x}}{\theta }_{14}{\theta }_{15}/M_{GUT}+5_x{\bar{5}}_{H_d}{\theta }_{14}{\theta }_{43}/M_{GUT}+5_x{\bar{5}}_{\bar{x}}{\theta }_{14}.$$

$$M_H$$

$${\theta }_{14}{\theta }_{15}{\bar{5}}_{t_4+t_5}{5}_{-2t_1}/M_{GUT}+ϵ{\theta }_{14}{\bar{5}}_{t_4+t_5}{5}_{-t_1-t_5}\to {\theta }_{14}{\theta }_{15}{\bar{5}}_{\bar{x}}{5}_{H_u}/M_{GUT}+ϵ{\theta }_{14}{\bar{5}}_{\bar{x}}{5}_x\cdot$$

$${\cal L}_{D}\supset\tiny{\left(5_{H_{u}},5_{x}\right)M_{D}\left(\begin{array}[]{c}\bar{5}_{H_{d}}\\ \bar{5}_{\bar{x}}\\ \end{array}\right)},$$

$$10_3{\bar{5}}_{\bar{x}}{\bar{5}}_2,10_3{\bar{5}}_{\bar{x}}({\bar{5}}_1{\theta }_{14}+{\bar{5}}_3{\theta }_{34}),10_1{\bar{5}}_{\bar{x}}({\bar{5}}_1{\theta }_{14}{\theta }_{43}+{\bar{5}}_2{\theta }_{43}+{\bar{5}}_3){\theta }_{14},10_2{\bar{5}}_{\bar{x}}({\bar{5}}_1{\theta }_{14}+{\bar{5}}_2+{\bar{5}}_3{\theta }_{34}){\theta }_{14}\cdot$$

$$y=\frac{1}{2}\mathrm{Re}S\int c_1^2({\mathcal{L}}_a),x=-\frac{1}{2}\mathrm{Re}S\int c_1^2({\mathcal{L}}_Y),$$

$$\frac{1}{a_3(M_U)}$$

$$\frac{1}{a_2(M_U)}$$

$$\frac{1}{a_1(M_U)}$$

$$\frac{1}{a_i(M_U)}=\frac{1}{a_i(M_U)}+\frac{b_i^x}{2\pi }\log \frac{M_U}{M_X}+\frac{b_i}{2\pi }\log \frac{M_X}{\mu }\cdot$$

$$\frac{5}{3}\frac{1}{a_1(M_U)}=\frac{1}{a_2(M_U)}+\frac{2}{3}\frac{1}{a_2(M_U)}\cdot$$

$$b_3^x=-3+n_V+n_D,b_2^x=1+n_V,b_1^x=\frac{33}{5}+\frac{2}{5}{n}_D+n_V$$

$$b_3=-3+n_V,b_2=1+n_V,b_1=\frac{33}{5}+n_V\cdot$$

$$\log \frac{M_U}{M_X}=\frac{2\pi }{{\beta }_x}\frac{1}{\mathcal{A}}-\frac{\beta }{{\beta }_x}\log \frac{M_X}{\mu }$$

Full text

Phenomenology with F-theory $SU(5)$

George K. Leontarisa,b 111E-mail: leonta@uoi.gr and Qaisar Shafic 222E-mail: shafi@bartol.udel.edu

a LPTHE, UMR CNRS 7589, Sorbonne Universités,

UPMC Paris 6, 75005 Paris, France

b Physics Department, Theory Division, University of Ioannina,

GR-45110 Ioannina, Greece

c Bartol Research Institute, Department of Physics and Astronomy, University of Delaware,

DE 19716, Newark, USA

We explore the low energy phenomenology of an F-theory based SU(5) model which, in addition to the known quarks and leptons, contains Standard Model (SM) singlets, and vector-like color triplets and SU(2) doublets. Depending on their masses and couplings, some of these new particles may be observed at the LHC and future colliders. We discuss the restrictions by CKM constraints on their mixing with the ordinary down quarks of the three chiral familes. The model is consistent with gauge coupling unification at the usual supersymmetric GUT scale, dimension five proton decay is adequately suppressed, while dimension-six decay mediated by the superheavy gauge bosons is enhanced by a factor of 5-7. The third generation charged fermion Yukawa couplings yield the corresponding low-energy masses in reasonable agreement with observations. The hierarchical nature of the masses of lighter generations is accounted for via non-renormalisable interactions, with the perturbative vacuum expectation values (vevs) of the SM singlet fields playing an essential rôle.

1 Introduction

Models originating from string theory constructions often contain SM singlets and vector-like fields which can mix with the light spectrum and therefore are natural candidates for predicting rare processes that might be discovered in future experiments at the LHC and elsewhere. F-theory models [1], in particular, have the necessary ingredients to describe in a simple and convincing manner a complete picture of such new phenomena. One of the most appealing grand unified theories incorporating these features in an F-theory context, is $SU(5)$ 333For F-theory model building reviews and early references see [2, 3, 4, 5]. For an incomplete list including more recent research papers see [6]-[35].. Indeed, on breaking F-$SU(5)$ to SM symmetry, one ends up with the MSSM spectrum augmented by scalar fields and vector-like states, which are remnants of the underlying GUT representations. In this framework, it is possible to retain gauge coupling unification even in the presence of some additional fields, provided that these form complete multiplets of $SU(5)$. In view of the ongoing experimental searches and possible future signatures, in this work we reconsider some issues regarding the exotic part of these models.

We start with a brief review of the basic features of an $SU(5)$ model [12] derived in an F-theory framework and, in particular, in the context of the spectral cover. We derive an effective theory model by imposing a $Z_{2}$ monodromy and identify the complex surfaces where the chiral matter and Higgs can be accommodated in the quotient theory. We assume a hypercharge flux breaking of the $SU(5)$ symmetry down to the SM one, and proceed with a specific assignment of the MSSM representations on these matter curves and then work out the spectrum and the superpotential. After fixing the necessary free parameters (such as flux units and singlet vevs), we proceed with the investigation of the exotic massless spectrum left over from higher dimensional fields. We then derive their superpotential couplings and analyse the implications for baryon number violating decays as well as other rare processes. We examine the possibility that these states remain massless at low energies being consistent with gauge coupling unification, and discuss the physics implications of the TeV scale exotic states.

2 F-SU(5)

We consider the elliptically fibred case where the highest smooth singularity in Kodaira’s classification is associated with the exceptional group of $E_{8}$ [19, 20]. We assume 7-branes wrapping an $SU(5)$ divisor and interpret this as the GUT symmetry of the effective model. Under these assumptions

[TABLE]

where the first factor is interpreted as the well known $SU(5)_{GUT}$ and the second factor is usually denoted as $SU(5)_{\bot}$.

The MSSM spectrum and possible exotic fields descend from the decomposition of the $E_{8}$ adjoint which, under the assumed breaking pattern (1), decomposes as follows:

[TABLE]

Thus, matter transforms in bi-fundamental representations, with the GUT 10-plets lying in the fundamental of $SU(5)_{\perp}$, and the $\bar{5},5$-plets lying in the antisymmetric representation of $SU(5)_{\perp}$.

We choose to work in the Higgs bundle picture (the spectral cover approach). In this context the properties of the GUT representations with respect to the spectral cover are described by a degree-five polynomial [6]

[TABLE]

where the $b_{k}$ coefficients carry the information of the internal geometry and their homologies, are given by $[b_{n}]=\eta-nc_{1}$, (with $\eta=6c_{1}-t$), where $c_{1}=c_{1}(S)$ is the first Chern class of the tangent bundle and $-t$ that of the normal to the surface $S$. The roots of the equation are identified as the weight vectors $t_{1,...,5}$ satisfying the standard $SU(N)$ constraint ($N=5$ in the present case)

[TABLE]

Under $t_{i}$ the matter curves acquire specific topological and symmetry properties inherited by the fermion families and Higgs fields propagating there. We denote the matter curves accommodating the 10-plets, 5-plets of $SU(5)$ and singlets emerging from $SU(5)_{\perp}$ adjoint decomposition as $\Sigma_{10_{t_{i}}},\Sigma_{5_{t_{i}+t_{j}}},\Sigma_{1_{t_{i}-t_{j}}}.$ Correspondingly, the possible representations residing on these matter curves are denoted by

[TABLE]

where, as far as 5-plets and singlets are concerned, we must have $t_{i}\neq t_{j}$.

Working in the framework of spectral cover, while assuming distinct roots $t_{i}$ of (3), one may further consider the breaking $SU(5)_{\bot}\to U(1)_{\bot}^{4}$. Then, the invariant tree-level superpotential couplings are of the form

[TABLE]

where $h_{1,2,3,4}$ represent the Yukawa strengths. In each of the above terms, the sum of the $t_{i}$ ‘charges’ should add up to zero. Hence, in the second term ${t_{i}}+{t_{j}+t_{k}}+{t_{l}+t_{m}}=0$, which unambiguously implies that all indices in the term proportional to Yukawa coupling $h_{2}$ should differ from each other (due to the fact that ${t_{1}}+{t_{2}+t_{3}}+{t_{4}+t_{5}}=0$).

Returning to the polynomial (3), although its coefficients $b_{n}$ belong to a certain field (holomorphic functions), the roots $t_{i}$ do not necessarily do so. Solutions, in general, imply branch cuts and, as a result, certain roots might be interrelated. The simplest case is if two of them are subject to a $Z_{2}$ monodromy, say, 444For various choices of monodromies, see [7, 8, 10, 11].

[TABLE]

From the point of view of the effective field theory model, the appearance of the monodromy is a welcome result since it implies rank-one mass matrices for the fermions. Indeed, under the $Z_{2}$ monodromy, the coupling

[TABLE]

ensures a top-quark mass at tree-level, while the remaining mass matrix entries are expected to be generated from non-renormalisable terms. After this brief description of the basic features, we proceed in the next section with the analysis of the implications of the hypercharge flux on the symmetry breaking and the massless spectrum of $SU(5)$.

3 Hypercharge Flux breaking of $SU(5)$

The $Z_{2}$ monodromy implies that the spectral cover polynomial factorises as follows:

[TABLE]

where all $a_{i}$ are assumed in the same field as $b_{n}$’s. Thus, while the roots of the three monomials on the right-hand side of (8) are rational functions in this field, it is assumed that the two roots of the binomial $(a_{1}+a_{2}s+a_{3}s^{2})$ cannot be written in terms of functions in the same field.

The $b_{n}(a_{i})$ relations are easily extracted by identifying coefficients of the same powers in $s$ and are of the form $b_{n}=\sum a_{i}a_{j}a_{k}a_{l}$, where the indices satisfy $i+j+k+l+n=24$. Therefore, given the homologies $[b_{n}]$, the corresponding ones for the $a_{i}$ coefficients satisfy $[a_{i}]+[a_{j}]+[a_{k}]+[a_{l}]=[b_{n}]$. Solving the resulting simple linear system of equations, it turns out that these can be determined in terms of the known classes $c_{1},-t$ and three arbitrary ones (dubbed here $\chi_{6,7,8}$), which will be treated as free parameters [10]. Each matter curve is associated with a defining equation involving products of $a_{i}$’s and, as such, it belongs to a specific homological class which subsequently is used to determine the flux restriction on it. If ${\cal F}_{Y}$ represents the hypercharge flux, we will require the vanishing of ${\cal F}_{Y}\cdot c_{1}={\cal F}_{Y}\cdot(-t)=0$, so that all can be expressed in terms of three free (integer parameters) defined by the restrictions

[TABLE]

To construct a specific model, we start by assuming that a suitable $U(1)_{X}$ flux (where the abelian factor $U(1)_{X}$ lies outside $SU(5)$ GUT) generates chirality for the $10$ and $\bar{5}$ representations. Next, the hypercharge flux breaks $SU(5)$ down to the SM and, at the same, time it splits the $10,\overline{10}$ and $5,\bar{5}$’s into different numbers of SM multiplets. If some integers $M_{10},M_{5}$ are associated with the $U(1)_{X}$ flux, and some linear combination $N_{y}$ of $N_{7,8,9}$ represents the corresponding hyperflux piercing a given matter curve, the 10-plets and 5-plets split according to:

[TABLE]

[TABLE]

As already discussed, depending on the restrictions of the flux on the matter curves $\Sigma_{j}$, there are certain conditions on the corresponding hypercharge flux, denoted as $N_{y_{j}}$ (for the specific matter curve $\Sigma_{j}$). These are deduced from the topological properties of the coefficients $a_{i}$ as well as the fluxes.

For a given choice of the flux parameters $M_{i},N_{y_{j}}$, the most general spectrum and its properties under the assumption of a $Z_{2}$ monodromy are exhibited in Table 1. The first column shows the available matter curves and the assumed chiral state propagating on it. The chirality is fixed by the specific choice of $M_{i},N_{y_{j}}$ flux coefficients shown in the last two columns of Table 1. The second column shows the ‘charge’ assignments, $\pm t_{i}$ for the 10-plets, and $\pm(t_{i}+t_{j}),\pm(t_{i}-t_{j})$ for 5-plets and singlets respectively. For this particular arrangement, the structure of the fermion mass matrices exhibits a hierarchical form, consistent with the experimentally measured masses and mixings [12]. In the present work, we will explore other interesting phenomenological implications of this model. The defining equations are shown in the fourth column where, for brevity, the notation $a_{ijk...}=a_{i}a_{j}a_{k}\cdots$ is used. The next column indicates the homologies, the sixth column their associated integers expressing the restrictions of flux on the corresponding matter curves, and the last column lists a choice of $M_{i}$ values consistent with a chiral $SU(5)$ spectrum. Notice that the flux integers are subject to the restrictions [10] $N=N_{7}+N_{8}+N_{9}$ and $\sum_{i}M_{5_{i}}+\sum_{j}M_{10_{j}}=0$. In the minimal case $n=0$ and there are no extra $5+\bar{5}$ pairs. Furthermore, the multiplicities $M_{ij},M_{\delta}$ of singlet fields are not determined in the context of the spectral cover and are left arbitrary.

4 Spectrum of the effective low energy theory

A comprehensive classification of the resulting spectrum is shown in Table 2 where, in the first column, the $SU(5)$ properties are shown. The third column shows the accommodation of the SM representations with their corresponding ‘charges’ given in column 2. Column 4 includes the exotics which, for the specific choice of parameters, involves the triplet pair $D+D^{c}$ and, in principle, $n$ copies of $5+\bar{5}$ representations. In the minimal case we set $n=0$, but perturbativity allows values up to $n\leq 4$. In the modified version of the model we allow for $n\neq 0$ and explore the phenomenological implications. note that restrictions on the number of vector-like 5-plets arise when the model is embedded in an $E_{6}$ framework [25]-[27]. In the last column of the Table, we have also introduced a $Z_{2}$ matter parity to the MSSM field as well as the singlets.

Before proceeding with the main part of our paper we present a few remarks about R-parity in supersymmetric models. A discrete $Z_{2}$ R-parity is often invoked in four dimensional supersymmetric SU(5) models in order to eliminate rapid proton decay mediated by the supesrymmetric partners of the SM quarks and leptons. If left unbroken, this discrete symmetry also yields an attractive candidate for cold dark matter, namely the lightest neutralino. It is perhaps worth noting that this $Z_{2}$ symmetry naturally appears if we employ an $SO(10)$ GUT which is broken down to $SU(3)_{c}\times U(1)_{em}$ by utilizing only tensor representations [36].

The question naturally arises: how do string theory based unified models avoid rapid proton decay? In the ten-dimensional $E_{8}\times E_{8}$ heterotic string framework [37], the compactification process utilizes Calabi-Yau manifolds which typically yields non-abelian discrete symmetries that may contain the desired R-parity ([38] and references therein.)

In F-theory models discrete symmetries including R-parity may arise from a variety of sources. They can emerge from Higgsing $U(1)$ symmetries in F-theory compactifications, or from a non-trivial Mordell-Weil group associated with the rational sections of the elliptic fibration, first invoked in  [28] and further discussed in several works including [29, 30, 31, 32]. More generally, $Z_{n}$ symmetries are associated with Calabi-Yau manifolds whose geometries are associated with the Tate-Shafarevich group [33]. Finally, they may appear as geometric properties of the construction in the spectral cover picture [34]. Based on the existence of such possibilities, in the present model we implement the notion of $R$-parity assuming that it is associated with some symmetry of geometric origin.

4.1 Matter curves and Fermion masses

Returning to the description of the emerging effective model, for further clarification we include a few more details. Initially, in the covering theory there are five matter curves 555Recall from (2), $\Sigma_{10_{i}},i=1,2,\dots,5$ that the 10-plets transform in the fundamental and 5-plets in the antisymmetric representation of $SU(5)_{\bot}$. but due to monodromy $Z_{2}:t_{1}=t_{2}$, two of them are identified and thus they are reduced to four. Similarly, the ten $\Sigma_{5_{t_{i}+t_{j}}}$ reduce to seven matter curves. Furthermore, there are 24 singlets from the decomposition of the adjoint of $SU(5)_{\perp}$ denoted with $\theta_{ij},i,j=1,2\dots,5$, and 20 of them live on matter curves defined by $t_{i}-t_{j}$ while four are ‘chargeless’. However, because of the $Z_{2}$ monodromy among the various identifications, $\theta_{i1}\equiv\theta_{i2}$ and $\theta_{1j}\equiv\theta_{2j}$, the following two singlets:

[TABLE]

are equivalent to one singlet $S$ with zero charge. The remaining singlets with non-zero ‘charges’ are

[TABLE]

The following singlets acquire non-zero vevs which help in realising the desired fermion mass textures:

[TABLE]

All other singlets (designated with $\theta^{\perp}_{ij}$ in Table 2) have zero vevs. Using the SM Higgs and singlet vevs given by (13), we obtain hierarchical quark and charged mass textures

[TABLE]

where, the Yukawa couplings are suppressed for simplicity.

4.1.1 Neutrino sector

The tiny masses accompanied by the relatively large mixings of the neutrinos, as indicated by various experiments, can find a plausible solution in the context of the see-saw mechanism and the existence of family symmetries. In the present F-$SU(5)$ GUT model, the SM singlet fields such as $\theta_{ij}$ form Yukawa terms invariant under the additional family symmetries described above and could be the natural candidates for the right handed neutrinos. Furthermore, observing that the right-handed neutrino mass scale is of the order of the Kaluza-Klein scale in string compactifications, a minimal scenario would be to associate the right handed neutrinos with the KK-modes [7] of these singlet fields, $\theta_{ij}^{KK}\to N_{R}$. An obstruction to this interpretation is that in the covering theory these singlets $\theta_{ij}$ transform in the complex representation, so that $\theta_{ij}^{KK}=N_{R},\theta_{ji}^{KK}=N^{c}_{R}$ and the mass term becomes $M_{KK}N_{R}N_{R}^{c}$, but there are no corresponding Dirac mass terms for both $N_{R},N_{R}^{c}$. However, in the quotient theory under the $Z_{2}$ monodromy $t_{1}=t_{2}$, the KK-modes $\theta_{12}^{KK}\equiv\theta_{21}^{KK}$ transform in the real representation, so that for any KK-level the corresponding modes $N_{R_{k}}=N_{R_{k}}^{c}\to\nu_{k}^{c}$ are identified and a see-saw mechanism is possible. Hence, the non-renormalisable term $5_{-t_{1}-t_{2}}\bar{5}_{t_{1}+t_{4}}\theta_{14}\theta_{21}^{KK}$ under the $Z_{2}$ monodromy is identified with $5_{-2t_{1}}\bar{5}_{t_{1}+t_{4}}\theta_{14}\theta_{21}^{KK}\to 5_{h_{u}}\bar{5}_{3}\theta_{14}\nu^{c}$ and so on. Therefore, under the above assumptions, the KK-modes corresponding to right-handed neutrinos couple to the following combination of the left-handed neutrino components

[TABLE]

The interesting fact is that the right-handed neutrinos are associated with a specific class of wavefuctions [7] such that the emerging mass hierarchy is milder than that of the charged leptons and quarks. It is shown that the mass matrix obtained this way [7] can accommodate the two large mixing angles observed in atmospheric and solar neutrino experiments.

4.2 Mass terms for the doublets and triplets

Returning to the content of Table 2, we observe that there is still freedom to accommodate additional vector-like 5-plets which respect all the required conditions. Hence, aiming to accommodate potential diphoton resonances and other possible experimental signatures of exotic matter beyond the MSSM spectrum, in the present construction we assume the existence of $5+\bar{5}$ pairs and discuss possible implications of the exotic states. As already explained, the $Z_{2}$ monodromy allows a tree-level coupling for the top quark $10_{3}10_{3}5_{H_{u}}$. Furthermore, from the specific accommodation of the fermion generations listed in Table 2, we observe that a tree-level coupling for the bottom quark is also available. A geometric perspective of the Yukawa couplings in the internal manifold is depicted in figure 1. All other mass entries are generated from non-renormalisable terms [12].

Regarding the 5-plets accommodating the MSSM Higgs, we observe that the flux splits the doublet from the triplet in the Higgs sector. As a result, the MSSM $\mu$ term

[TABLE]

does not involve masses for the triplet fields. Fermion mass hierarchies require at least that the singlet vev $V_{1}=\langle\theta_{14}\rangle\gtrsim{\cal O}(10^{-1})M_{GUT}$, so that the MSSM $\mu$ parameter can be kept light for $v_{2}\cdot v_{3}\ll v_{1}$.

In the general case, we need to take into account the extra doublet pairs emerging from the 5-plets remaining in the zero-mode spectrum. As an illustrative example, we take only one additional vector-like pair of 5-plets, that is $n=1$. In this case the available coupling are

[TABLE]

The Higgs mass matrix in the basis {\cal L}\supset{\tiny(H_{d},H_{d}^{\prime})M_{H}\left(\begin{array}[]{c}H_{u}\\ H_{u}^{\prime}\end{array}\right)} is

[TABLE]

where the Yukawa couplings are suppressed to avoid clutter. This implies a light Higgs mass term $\mu\sim V_{1}v_{2}v_{3}$ and a heavy one $M_{H}\sim V_{1}$.

The triplet mass terms emerge from different couplings

[TABLE]

Hence, written in a matrix form

[TABLE]

where the triplet mass matrix is M_{D}={\tiny V_{1}\left(\begin{array}[]{ll}v_{2}&\epsilon\\ \epsilon^{\prime}v_{2}&1\\ \end{array}\right)}, and the parameters $\epsilon\simeq\epsilon^{\prime}\lesssim 1$ stand for corrections when more than one matter multiplets are on the same matter curve. The eigenmasses also depend on the singlet vevs and will be discussed in conjunction with proton decay in the subsequent sections.

In addition to these superpotential couplings, the vector pairs $5+\bar{5}$ generate superpotential terms with the matter fields

[TABLE]

where the non-renormalisable terms are assumed to be scaled by appropriate powers of $M_{GUT}$. In the next sections we will explore possible phenomenological consequences of (21). However, we note that it is feasible to eliminate such couplings from the lagrangian by introducing a different R-parity assignment for the colour triplets.

5 Gauge Coupling Unification

The presence of additional vector-like pairs of colour triplets and higgsinos with masses in the TeV range affect the renormalisation group running of the gauge couplings and the fermion masses. The existence of complete $5+\bar{5}$ $SU(5)$ multiplets at the TeV scale may enhance processes that could be observed in future searches, while they can be consistent with perturbative gauge coupling unification as long as their number is less than four. Threshold corrections from Kaluza-Klein (KK) modes and fluxes play a significant rôle [39] too. Under certain circumstances [40], (for example when the matter fields are localised on genus one surfaces) the KK threshold effects can be universal, resulting to a common shift of the gauge coupling constant at the GUT scale. This has been analysed in some detail in ref [40] and will not be elaborated further. However, in F-theory constructions, there are additional corrections associated with non-trivial line bundles [41, 42]). More precisely, assuming that the $SU(5)$ is generated by D7-branes wrapping a del-Pezzo surface, gauge flux quantization condition [43] implies that D7-branes are associated with a non-trivial line bundle ${\cal L}_{a}$. On the other hand, the breaking of $SU(5)$ occurs with a non-trivial hypercharge flux ${\cal L}_{Y}$ supported on the del Pezzo surface, (but with a trivial restriction on the Calabi-Yau fourfold so that the associated gauge boson remains massless). The flux threshold corrections to the gauge couplings associated with these two line bundles can be computed by dimensionally reducing the Chern-Simons action. If we define

[TABLE]

where, $c_{1}({\cal L})$ denotes the first Chern class of the corresponding line bundle and $S=e^{-\phi}+iC_{0}$ is the axion-dilaton field (and $g_{{}_{IIB}}=e^{\phi}$), the flux corrections to the gauge couplings are expressed as follows

[TABLE]

where $a_{U}$ represents the unified gauge coupling. From  (23-25) we observe that the corrections from the ${\cal L}_{a}$ line bundle are universal and therefore $y$ can be absorbed in a redefinition of $a_{U}$. On the other hand, hypercharge flux thresholds expressed in terms of $x$, are not universal and destroy the gauge coupling unification at the GUT scale $M_{U}$. Notice that in order to eliminate the exotic bulk states $(3,2)_{5}+(\bar{5},2)_{-5}$ emerging from the decomposition of ${\bf 24}$, we need to impose $\int c_{1}^{2}({\cal L}_{Y})=-2$, and therefore we find the simple form $x=e^{-\phi}=\frac{1}{g_{{}_{IIB}}}$. The value of the gauge coupling splitting has important implications on the mass scale of the color triplets discussed in the previous section. In the following we will explore this relation within the matter and Higgs field context of the present model.

We assume that the color triplets $D+D^{c}\in 5_{H}+\bar{5}_{H}$ receive masses at a scale $M_{X}$, while the complete $5+\bar{5}$ extra multiplets obtain masses at a few TeV. The renormalisation group equations take the form

[TABLE]

It can be readily checked that the GUT values of the gauge coupling satisfy

[TABLE]

Assuming $n_{D}$ pairs of $(D+D^{c})$ and $n_{V}$ vector-like 5-plets, the beta functions are

[TABLE]

Using (23,24) and (27) we find

[TABLE]

where we introduced the definitions

[TABLE]

Notice that for the particular spectrum, $\beta_{x},\beta$ are equal, $\beta_{x}=\beta=12$, and independent of the number of multiplets $n_{D}$ and $n_{V}$. Then, from (30) we find that the unification scale is

[TABLE]

i.e., independent of $n_{V},n_{D}$ and the intermediate scale $M_{X}$.

To unravel the relation between the scale $M_{X}$ and the parameter $x$, we proceed as follows. First, we subtract (24) from (23)

[TABLE]

Using (30) and the fact that in our model $n_{D}=1$, we find

[TABLE]

This determines the relation between the parameter $x=e^{-\phi}$ and the scale $M_{X}$ where the Higgs triplets become massive. We can use the expression for $M_{U}$ to express the $M_{X}$ scale as follows

[TABLE]

To determine the value of the GUT coupling $a_{U}$ we use (23,30) and (5) to find

[TABLE]

For the present application, we allow three pairs of 5-plets, $n_{V}=3$, and we obtain the relation

[TABLE]

Substitution of (39) in (37) gives an elegant and very suggestive formula:

[TABLE]

We observe that in order to have $M_{X}\leq M_{U}$, we always need $a_{U}\geq a_{3}\approx\frac{1}{8.5}$. We depict the main results in the figures that follow. In fig.2 we show the variation of the color triplets’ decoupling scale versus the range of values of the dilaton and, in fig.3, we plot the inverse SM gauge couplings taking into account the thresholds of the color-triplets.

5.1 RGEs for Yukawa Couplings

These modifications in the gauge sector and, in particular, the large $g_{U}$ value compared to that of the standard MSSM unification scenario ($g_{U}\sim 1/25$ in MSSM) are expected to have a significant impact on the evolution of the Yukawa couplings. On the other hand, in F-theory constructions the Yukawa coupling strengths at the unification scale are computed analytically and can be expressed in terms of the geometric properties of the internal six-dimensional compact space and the fluxes of the particular construction. For the sake of argument, we assume that all three $5+\bar{5}$ surplus matter fields receive masses in the TeV range, with $\tan\beta$ values $\sim 48-50$ and $M_{GUT}\sim 2\times 10^{16}$ GeV. Then, according to [44], the top mass, in particular, is achieved for Yukawa coupling $h_{t}(M_{GUT})\gtrsim 0.35$ which is significantly lower than the value $\sim 0.6$ obtained in the case of RG running with the beta-functions for the MSSM spectrum.

Turning now to F-theory predictions, as we have seen, the Yukawa couplings are realised at the intersections of three matter curves. The properties of the corresponding matter fields in a given representation $R$ are captured by the wavefunction $\Psi_{R}$ whose profile is obtained by solving the Equations of Motion (EoM) [1]. It is found that the solution exhibits a gaussian profile picked along the matter curve supporting the particular state, $\Psi_{R}\propto f(z_{i})e^{M_{ij}z_{i}\bar{z}_{j}}$. Here $z_{1,2}$ are local complex coordinates, the ‘matrix’ $M_{ij}$ takes into account background fluxes, and $f(z_{j})$ is a holomorphic function. The value of the Yukawa coupling results from integrating over the overlapping wavefunctions. Thus, for the up/down Yukawa couplings,

[TABLE]

The top Yukawa coupling is realised at the intersection where the symmetry is enhanced to ${E}_{6}$, while the bottom and $\tau$ Yukawa couplings are associated with triple intersections of $SO(12)$ enhancements. We note in passing that the corresponding solution of the EoM providing the wavefunction for the up-type quark coupling is rather involved because of the monodromy and must be solved in a non-trivial background where the notion of T-brane is required [45]. Using appropriate background fluxes, we can break ${\cal E}_{6}$ to $SU(5)$, while the latter can break down to the SM gauge group with the hypercharge flux. To estimate the top Yukawa coupling, one has to perform the corresponding integration (41). Varying the various flux parameters involved in the corresponding wavefunctions, it is found that the top quark Yukawa takes values in the interval $h_{t}\sim[0.3-0.5]$, in agreement with previous computations [46, 47], and hence the desired value $h_{t}\sim 0.35$ can be accommodated.

In the present approach, the bottom and $\tau$ Yukawa couplings are formed at a different point of the compact space where the symmetry enhancement is $SO(12)$. Proceeding in analogy with the top Yukawa, one can adjust the flux breaking mechanism to achieve [46, 47]) the successive breaking to $SU(5)$ and $SU(3)\times SU(2)\times U(1)$. Further, for certain regions of the parameter space, one can obtain $h_{b,\tau}$ values in agreement with those predicted by the renormalisation group evolution [48].

6 Decay of Vectorlike Triplets

While analysing the spectrum in section 4, we have seen that the existence of vector-like triplets is a frequently occurring phenomenon. They can be produced in pairs at LHC through their gauge couplings to gluons. However, such exotic particles are not yet observed and must decay through higher dimensional operators through mixing with the MSSM particles.

We start with the minimal model by setting $n=0$, in which case the only states beyond the MSSM spectrum are $D^{c},D$ found in the $\bar{5}_{\bar{x}}$ and $5_{H_{u}}$ respectively. We will consider the case of their mixing with the third family which enhances their decays, due to the large Yukawa coupling compared to the two lighter generations. \textTauhe available Yukawa couplings which mix the down-type triplets are

[TABLE]

where the non-renormalisable terms are scaled by the appropriate powers of the compactification scale or the GUT scale. These terms generate a mixing matrix of the third generation down quark and $D^{c},D$ which can be cast in the form

[TABLE]

where $v_{d}$ stands for the down Higgs vev scaled by the GUT scale. This non-symmetric matrix $M_{D}$ is diagonalised by utilizing the left and right unitary matrices

[TABLE]

implying

[TABLE]

where

[TABLE]

and

[TABLE]

Following standard diagonalisation procedures, in the limit $v_{1}\gg v_{2}\gg v_{d}$, we find that the left mixing angle is

[TABLE]

and for the right-handed mixing we obtain

[TABLE]

From these, we find

[TABLE]

For the assumed hierarchy of vevs we see that the left-mixing prevails. The mixing is restricted by CKM constraints and the contributions of the heavy triplets to the oblique parameters $S,T$ which have been measured with precision in LEP experiments (For detailed computations see [49].). A rough estimate would give the upper bounds $\tan 2\theta_{L}\sim 0.1,\tan 2\theta_{R}\sim 0.3$ which can be easily satisfied for the $v_{1},v_{2}$ values used in this work.

6.1 Proton decay

In this model the dimension-five proton decay R-parity violating tree-level couplings of the form $10_{f}\bar{5}_{f}\bar{5}_{f}$ are absent due to the $t_{i}$ charge assignments of matter fields. However, non-renormalisable terms that could lead to suppressed baryon and lepton number violating processes may still appear. A class of these operators have the general structure

[TABLE]

where $\theta_{pq}^{n}$ represents products of singlet fields required to cancel the non-vanishing combinations of $t_{i,j...}$ charges. Notice, however, that for the particular family assignment in this model none of $t_{i,j,k,l,m}$ in (47) is $t_{5}$ and therefore, to fulfil the condition $\sum_{k=1}^{5}t_{k}=0$ some singlet $\theta_{5s}\equiv 1_{t_{5}-t_{s}}$, with $s=1,2,3,4,$ always must be involved. 666Notice however, that all possible higher order R-parity violating terms

$10_{1}\bar{5}_{1}(\bar{5}_{1}\theta_{14}\theta_{53}+\bar{5}_{2}\theta_{53}+\bar{5}_{3}\theta_{54})\theta_{13}+10_{1}\bar{5}_{2}(\bar{5}_{2}\theta_{43}+\bar{5}_{3})\theta_{53}+10_{2}\bar{5}_{3}\bar{5}_{3}\theta_{54}$

can be eliminated due to the R-parity assignment of the singlets $\theta_{ij}$ shown in Table 2. In the present model no singlet of this kind acquires a non-zero vev, namely $\langle\theta_{5s}\rangle\equiv 0$, and hence dimension four operators are suppressed.

However, as already pointed out, additional Yukawa terms give rise to new tree-level graphs mediated by color triplets. Such graphs induce dimension-5 operators of the form $\frac{1}{M_{{eff}}}QQQ\ell$, $\frac{1}{M_{{eff}}}u^{c}u^{c}d^{c}e^{c}$, where $M_{{eff}}$ is an effective colour triplet mass $M_{eff}\geq M_{GUT}\sim 2.0\times 10^{16}$ GeV[50, 51, 52, 53]. Here, because of the missing triplet mechanism described in the previous section, the $D,D^{c}$ triplets develop masses through mixing with other heavy triplets $D_{i},D_{i}^{\prime}$ emerging from the decomposition of the additional $5+\bar{5}$-pairs. Besides, several couplings are realised as higher order non-renormalisable terms so that, in practice, an effective triplet mass ${M_{eff}}$ is involved which, with suitable conditions on the triplet mixing, could be of the order of the GUT scale. For the case of the Higgsino exchang diagram, for example, with a Higgsino mass identified with the supersymmetry breaking scale $M_{S}$, the proton lifetime is estimated to be [53]

[TABLE]

where the coefficient $C_{R}\geq 0.1$, taking into account the renormalisation group effects on the masses. From (48) we infer that with an effective triplet mass $\gtrsim M_{GUT}$ and a relatively high supersymmetry breaking scale, proton decay can be sufficiently suppressed in accordance with the Super-Kamiokande bound on the proton lifetime.

To estimate the effects of these operators in this model, we consider the triplet mass matrix derived in the previous section

[TABLE]

with $v_{2}=\frac{\langle\theta_{15}\rangle}{M_{GUT}}$ and $v_{1}=\frac{\langle\theta_{14}\rangle}{M_{GUT}}$ as defined in (13). As before, the left and right unitary matrices $V_{L},V_{R}$, as well as the eigenmasses are determined by ${M_{T}^{\delta}}^{2}=V_{L}^{\dagger}M_{T}M_{T}^{\dagger}V_{L}=V_{R}^{\dagger}M_{T}^{\dagger}M_{T}V_{R}$ where, in general, $M_{T}M_{T}^{\dagger}$ and $M_{T}^{\dagger}M_{T}$ are Hermitian but, for simplicity, we will take to be symmetric, M^{2}\sim\left(\begin{array}[]{ll}a&b\\ b&d\\ \end{array}\right)\langle\theta_{14}\rangle^{2}, with real entries and triplet eigenmasses $M^{2}_{1,2}=\frac{1}{2}\left(a+d\pm\sqrt{4b^{2}+(a-d)^{2}}\right)\langle\theta_{14}\rangle^{2}$.

In figure 4 a representative graph is shown mediated by the colour triplets leading to the dominant proton decay mode $p\to K^{+}\bar{\nu}$. The mass insertion (red bullet) in the graph is

[TABLE]

After summing over the eigenstates, one finds that the effective mass involved is

[TABLE]

while there is an additional suppression factor $v_{1}=\langle\theta_{14}\rangle/M$ from the non-renormalisable term (yellow bullet in the graph). Finally $\frac{1}{M_{eff}}\sim\frac{v_{1}}{M^{0}_{eff}}$.

For the $V_{L}$ mixing, assuming reasonable values for the parameters $\epsilon,\epsilon^{\prime}<1$, while taking $v_{1}\sim O(10^{-1})$ and $\lambda\sim 1$ we find

[TABLE]

For the $V_{R}$ case we find

[TABLE]

For a supersymmetry breaking scale $M_{S}$ in the TeV region, we conclude that the lifetime of the proton is consistent with the experimental bounds for an effective mass $M_{eff}$ a few times larger than $M_{GUT}$ which can be satisfied for 777Since the mass insertion $\langle\Phi\rangle 5_{H}\bar{5}_{\bar{x}}\propto v_{2}$ one would expect that for $v_{2}\ll 1$ the contribution of the graph to Proton Decay would be small. However, the element $b$ cancels the effect because it is also proportional to $b\propto v_{2}$. $v_{2}>v_{1}$ and $\epsilon<1$.

There are implications for the $\mu$ term given by $\mu\sim v_{1}v_{2}v_{3}$ (see Eq. 16). Since $v_{1},v_{2}$ cannot be small, in order to sufficiently suppress this term we must have $v_{3}=\langle\theta_{43}\rangle/M_{GUT}\ll 1$. On the other hand, the smallness of $v_{3}$ suppresses also the mass scale of the lighter generations and might lead to inconsistences with the experimental values. We should recall, however, that there are significant contributions to the fermion masses from one-loop gluino exchange diagrams [54] implying masses of the order $m_{u/d}\propto\frac{a_{3}}{4\pi}A_{q}\frac{m_{\tilde{q}}m_{t/b}}{m_{\tilde{g}}^{2}}$ for the up/down quarks, where $A_{q},m_{\tilde{g}},m_{\tilde{q}}$ are respectively the trilinear parameter, the gaugino and squark masses.

We have already stressed that the presence of additional vector-like 5-plets in the model under consideration, is compatible with a smaller value of the unified gauge coupling $g_{U}\sim 10^{-1}$ at the GUT scale. This has significant implications for the proton decay rate which occurs through the exchange of the gauge bosons (dimension six operators). For the well known case $p\to e^{+}\pi^{0}$ the life-time is estimated to be [53, 55, 57]

[TABLE]

The various quantities in the above formula are as follows: $a_{U}^{0}=\frac{1}{25}$ is assumed to be the value of the unified gauge coupling in the minimal $SU(5)$, while $a_{U}$ stands for its value for the present model which is taken to be $a_{U}\approx\frac{1}{8.5}$ (see section 5). The factor $A_{R}$ takes into account the various renomalisation effects, $a_{H}$ is the hadronic matrix element and $M_{V}$ denotes the mass of the gauge boson mediating the process $p\to e^{+}\pi^{0}$. Comparing with the recent experimental limit [56] $\tau(p\to e^{+}\pi^{0})\geq 1.6\times 10^{34}$ years, we find a lower bound on the mass of the gauge boson $M_{V}\geq 1.14\times 10^{16}$ GeV, which is reciting since it just below the GUT scale predicted in this model $M_{U}\approx 2.04\times 10^{16}$ GeV.

6.2 Variation with new physics predictions accessible at LHC

In this section we consider the possibility of predicting new physics phenomena (such as diphoton events) from relatively light ($\sim$ TeV) scalars and triplets. The model discussed so far cannot accommodate a process such as the diphoton event, since there is no direct coupling $D^{c}DS$ with a light singlet $S$. Indeed, the only singlet coupled to $D^{c},D$ is $\theta_{14}$ which acquires a large vev and decouples. To circumvent this we briefly present a modification of the above model by assuming the following non-zero vevs,

[TABLE]

and we maintain the same assignments for the fermion generations listed in Table 2. The mass matrices for the up, down quarks and charged leptons, are given by

[TABLE]

where, as before, we have suppressed the Yukawa couplings expected to be of ${\cal O}(1)$. We observe that the matrices exhibit the expected hierarchical structure. Assuming a natural range of the vevs and Yukawa couplings we estimate that the fermion mass patterns are consistent with the observed mass spectrum.

With this modification, the singlet $\theta_{14}$ is not required to acquire a large vev and it can remain as a light singlet $\theta_{14}=S^{\prime}$. Through its superpotential coupling

[TABLE]

where ${D^{\prime\prime}}^{c}$ stands for the linear combination ${D^{\prime\prime}}^{c}=\cos\phi D^{c}+\sin\phi{\tiny}{D^{\prime}}^{c}$, $S^{\prime}$ could contribute to diphoton emission.

7 Summary

F-theory appears to be a natural and promising framework for constructing unified theories with predictive power. The SU(5) GUT model in particular, appears to be the most economic unified group containing all those necessary ingredients to accommodate vectorlike fermions that might show up in future experiments. Therefore, in the light of possible new physics at the LHC experiments, in this letter, we reconsidered a class of F-theory $SU(5)$ models aiming to concentrate on the specific predictions and low energy implications.

In the F-theory framework, after the $SU(5)$ breaking down to the Standard Model gauge symmetry, we end up with the MSSM chiral mass spectrum, the Higgs doublet fields and usually a number of vector-like exotics as well as neutral singlet fields. We point out that we dispense with the use of large Higgs representations for the $SU(5)$ symmetry breaking since the latter takes place by implementing the mechanism of the hypercharge flux. The corresponding $U(1)_{Y}$ gauge field remains massless by requiring the hypercharge flux to be globally trivial. As a result of these requirements, the spectrum of the effective theory and the additional abelian symmetries accompanying the GUT group, are subject to certain constraints. In addition to the $SU(5)$ GUT group, the model is subject to additional symmetry restrictions emanating from the perpendicular ‘spectral cover’ $SU(5)_{\perp}$ group, which in the effective theory reduces down to abelian factors according to the ‘breaking’ chain

[TABLE]

where $Z_{2}$ is the monodromy action, chosen for this particular class of models under discussion 888For the $SU(5)_{\perp}$ spectral cover symmetry, the possible monodromies fall into a discrete subgroup of the Weyl group $W(SU(5)_{\perp})\sim S_{5}$, with $S_{5}$ being the permutation symmetry of five objects.. A suitable choice of fluxes along these additional abelian factors is responsible for the chirallity of the $SU(5)$ GUT representations and their propagation on the specific matter curves presented in this paper.

In practice, the effects of the remaining spectral cover symmetry in the low energy effective theory are described by a few integers (associated with fluxes) and the ‘charges’-roots $t_{i},i=1,2,\dots,5$ of the spectral cover fifth-degree polynomial where two of them, namely $t_{1,2}$, are identified under the action of the monodromy $Z_{2}:t_{1}\leftrightarrow t_{2}$ applied in this work.

The implementation of the hyperflux symmetry breaking mechanism has additional interesting effects. As is known, chiral matter and Higgs fields reside on the intersections (i.e., Riemann surfaces, dubbed here as matter curves and characterised by the remaining $U(1)$ factors through the ‘charges’ $t_{i}$) of seven branes with those wrapping the $SU(5)$ singularity. In general the various intersections are characterised by distinct geometric properties and as a consequence flux restricts differently on each of them, while implying splittings of the $SU(5)$ representation content in certain cases. As a result, in the present model doublet Higgs fields are accommodated on matter curves which split the $SU(5)$ representations realising an effective doublet-triplet splitting mechanism in a natural manner. More precisely, this ammounts to removing one triplet from the initial Higgs curve with the simultaneous appearance (excess) of another one on a different matter curve. This displacement however is enough to allow a light mass term for the Higgs doublets while heavy triplet-antitriplet mass terms originate from different terms leading to suppression of baryon number violating processes. Chiral fermion generations are chosen to be accommodated on different matter curves, so that a Froggatt-Nielsen type mechanism is implemented to generate the required hierarchy. Furthermore, certain Kaluza-Klein modes are associated with the right-handed neutrino fields implementing the see-saw mechanism through appropriate mass terms with their left-handed counterparts.

The additional spectrum in the present model consists of neutral singlet fields as well as colour triplets and Higgs-like doublets comprising complete $SU(5)$ vector-like pairs in $5+\bar{5}$ multiplets, characterised by non-trivial $t_{i}$-‘charges’. Some singlet fields are allowed to acquire vevs at the TeV scale inducing masses of the same order for the vector-like exotics through the superpotential terms. Such ‘light’ exotics contribute to the formation of resonances producing excess of diphoton events which could be discovered in future LHC experiments. A RGE analysis shows that the resulting spectrum is consistent with gauge coupling unification and the predictions of the third family Yukawa couplings.

Acknowledgements. The authors would like to thank R. Blumenhagen, S.F. King and Tianjun Li for reading the manuscript and helpful comments. G.K.L. would like to thank the Physics and Astronomy Department and Bartol Research Institute of the University of Delaware for kind hospitality. Q.S. is supported in part by the DOE grant “DOE-SC-0013880”.