IIT-BHU

A new solution of the fermionic mass hierarchy ofthe standard model

Gauhar Abbasa1

a Department of Physics, Indian Institute of Technology (BHU), Varanasi 221005, India

Abstract

We present one of the most minimal realizations of the Froggatt-Nielson mech-anism through the abelian discrete symmetries Z2 and Z5 within the frameworkof the standard model. In this realization, masses of individual fermions fromthe third to first family appear in terms of the ascending power of the expansion

parameter〈 k〉

Λ.

1email: gauhar.phy@iitbhu.ac.in

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1 Introduction

One of the most challenging puzzles of the standard model (SM) of Glashow, Weinberg andSalam is the observed mass pattern of the charged fermions [1, 2]. In the SM, masses ofall fermions and quark-mixing angles are arbitrary parameters, and their masses and mixingpatterns are remarkably fascinating.

For instance, the masses of the third family fermions are much larger than that of thesecond family, and the masses of the second family fermions are much larger than that of thefirst family, i.e. mτ >> mµ >> me, mb >> ms >> md and mt >> mc >> mu. This is thefermionic mass hierarchy among the three fermionic families of the SM.

There is second interesting and challenging aspect of the mass pattern of the chargedfermions which is the mass hierarchy within the each family. This mass hierarchy is reallybizarre in the sense that masses of up type quarks of the second and third families are muchlarge than that of down type quarks of same families, on the other side, mass of the downtype quark of the first family is greater than the twice of the mass of the up type quark of thesame family. This can be written as md > mu, mc >> ms, mt >> mb.

We should also note that there is a third side of the quark mass hierarchy among the threequark families. This is the observed mixing among the three generations of the quarks.There is again interesting and remarkable peculiarity in the mixing pattern of the threegenerations of quarks in the form of the hierarchy among the quark-mixing angles, i.e.sin θ12 >> sin θ23 >> sin θ13 where θ12 is the Cabibbo angle, the mixing angle between thefirst and second quark families, θ23 is the mixing angle between the second and third quarkfamilies, and θ13 is the mixing angle between the first and third quark families.

Explaining the origin of the fermionic mass hierarchy among and within the fermionicfamilies along with the quark-mixing pattern is a formidable problem. There are seriousefforts addressing this problem in literature [1–49]. Recently, an incongruous solution of thisproblem is discussed in Ref. [1] where real singlet scalar fields and vector-like fermions areused to achieve this goal.

One of the most elegant and beautiful explanations of the fermionic mass hierarchy andquark-mixing pattern of the SM can come from the celebrated Froggatt-Nielson mechanism[3]. In this mechanism, an abelian flavour symmetry U(1)F is added to the SM in such a waythat only top quark acquires its mass through renormalized operator. This abelian flavoursymmetry U(1)F is weakly broken and is capable to distinguish fermions among differentfamilies. For instance, if there exists a flavon field k1 which has charge −1 under the abelianflavour symmetry U(1)F , and charges of the fermions ψci and ψj under U(1)F symmetry are θiand θj, respectively then the Yukawa operator of the type ψiϕψj, where ϕ represents the SMHiggs field, is forbidden by the U(1)F symmetry. However, an effective operator of the typeψiϕψj( k/Λ)(θi+θj) is still allowed where Λ is the scale at which new physics reveals itself.

Thus masses of fermions are recovered through higher order effective operators having

1Consonant letter “ k"(k@) is taken from the Devanagari script. It is pronounced as “Ka" in Kashmir [1].

2

the following structure :

O = y(kΛ

)(θi+θj)ψϕψ, (1)

where y is the coupling constant. The flavon field acquires a vacuum-expectation value (VEV)〈 k〉 which breaks the flavour U(1)F symmetry spontaneously.

The new physics scale Λ can be anywhere between the weak and the Planck scale. The

only essential condition is the ratio〈 k〉

Λshould be much smaller than unity. The effect of

flavon field k will be observably very small in the limit where the scale of new physics Λ islarger than the weak scale. However, the scenario where the symmetry breaking scale is nearthe weak scale is promisingly interesting from the phenomenological point of view keepingin mind that Large Hadron Collider is about to enter in its high luminosity phase. Hence, thecrucial question is how low this scale could be given the present bounds on flavour-changingand CP-violating processes.

This interesting question depends on the underlying unknown dynamics, for instancewhether abelian flavour symmetry U(1)F is local or global. For instance a gauged abelianflavour symmetry U(1)F can affect low energy phenomenology via exchange of the corre-sponding gauge boson. If it is global and spontaneously broken the there must exists amassless Goldestone boson.

Although a very large numbers of models have been inspired from the Froggatt-Nielsonmechanism based on an abelian flavour symmetry U(1)F , there are less efforts to implementthe Froggatt-Nielson mechanism strictly through simple minimal abelian discrete symme-tries within the minimal framework of the SM. Such a scenario is interesting from the the-oretical point of view due to an extensive use of abelian discrete symmetries such as Z2 inmodel building, for instance two-Higgs-doublet model and minimal supersymmetric standardmodel. Furthermore, as discussed earlier, low energy phenomenology is expected to take ashift since there is no local or global abelian flavour symmetry U(1)F to affect it.

Hence, in this work, we propose one of the most minimal realizations of the Froggatt-Nielson mechanism within the framework of the SM to explain origin of the observed masspattern of fermions among and within the three fermionic families along the quark-mixingpattern where one does not need to impose a continuous U(1)F symmetry. Instead of acontinuous abelian U(1)F symmetry, we use two simple discrete symmetries Z2 and Z5 inthe framework of the SM. It should be noted that this is the most minimal discrete symmetryrequirement.

We comment that solution of the fermionic mass hierarchy discussed in this work is one ofthe most complete realizations of the Froggatt-Nielson mechanism. This is because it providesa complete solution of the fermionic mass hierarchy of the SM by explaining the origin of themass hierarchy among and within the three fermionic families along with the quark-mixingpattern.

There are a few models already present in literature where Abelian discrete symmetriesare partially or fully employed to achieve the Froggatt-Nielson mechanism. For instance, inRef. [7], the Abelian discrete symmetry is a very large symmetry ZN ⊂ U(1) where U(1)

is again a continuous flavour symmetry. Hence, ZN is effectively a continuous symmetry.

3

Ref. [8] only explores neutrino masses through the Froggatt-Nielson mechanism such thatthe discrete symmetries are of the form ZM × ZN . Here, again ZN is required to be largeenough so that the symmetry is effectively ZM × U(1). Ref. [9] is in fact using a largersymmetry Z2×Z ′2×U(1). Similarly, the symmetry of Ref. [10] is also non-minimal Z2×U(1).Thus, we observe in these papers that the Froggatt-Nielson mechanism either is achieved byextending the U(1) symmetry by a discrete symmetry or by using a very large discrete ZNsymmetry so that it approximates to a continuous U(1) symmetry. Hence, these models basedon discrete as well as continuous symmetries cannot be considered minimal models. Thusthe model presented in this work is an attempt to achieve the Froggatt-Nielson mechanismwith the minimal Abelian discrete symmetry.

The organization of this paper is as follows: in section 2 we present our model. The scalarpotential is discussed in section4. An ultra-violet completion is presented in section 3. Weconclude in section 5.

2 Froggatt-Nielson mechanism with Z2 × Z5 discrete sym-metries

For achieving the Froggatt-Nielson mechanism with the minimal discrete symmetry, we em-ploy a gauge singlet scalar field k which behaves in the following way under SU(3)c ×SU(2)L × U(1)Y symmetry of the SM

k : (1, 1, 0). (2)

We then write masses of the three fermionic families in terms of the expansion parameter〈 k〉/Λ where Λ is the scale of new physics which renormalizes our model.

We begin by introducing two discrete symmetries Z2 and Z5 which are being added to thesymmetry of the SM, and impose them on the singlet right handed fermions of the SM and acomplex gauge singlet scalar field k. The only renormalized scalar-fermion coupling is theYukawa coupling of the top quark.

We note that Yukawa couplings of other fermions are completely forbidden by the discretesymmetries Z2 and Z5

2 now. Masses of fermions other than top quark, are now recovered byhigher dimension operators which appears in ascending power of the expansion parameterkΛ

.

The mass Lagrangian for the model reads,

Lmass = yttRϕ†ψ3,q

L +kΛ

(yb bRϕ

†ψ3,qL + yτ τRϕψ

3,lL

)+

(kΛ

)2

yc cRϕ†ψ2,q

L (3)

+

(kΛ

)3 (ys sRϕ

†ψ2,qL + yµµRϕ

†ψ2,lL

)+

(kΛ

)4

yuuRϕ†ψ1,q

L

2More detail is given in appendix.

4

Fields Z2 Z5

uR, cR, tR + ω3

dR, eR, sR, µR, bR, τR - ω4

ψ1L + ω4

ψ2L + ω

ψ3L + ω3

k - ω

Table 1: The charges of right-handed fermions of three families of the SM and singlet scalarfield under Z2 and Z5 symmetries where ω is the fifth root of unity.

+

(kΛ

)5 (yddRϕ

†ψ1,qL + yeeRϕ

†ψ1,lL

)+ H.c..

where superscripts denote family number, quark and leptonic doublets.It is essential to check if the model explains quark-mixing. For this purpose we write down

quark mass matrices. We define the expansion parameter as〈 k〉

Λ= ε. In terms of expansion

parameter ε, the up-type quark mass matrix is3

MU = v

yu11ε4 yu12ε2 yu13

yu21ε4 yu22ε

2 yu23yu31ε

4 yu32ε2 yu33

. (5)

Now we write a Hermitian matrix

Ru =M†UMU = v2

Y u11ε

8 yu12ε6 yu13ε

4

Y u21ε

6 yu22ε4 yu23ε

2

Y u31ε

4 yu32ε2 Y u

33

. (6)

The matrix Ru can be diagonalized via unitary transformation given by

V †URuVU = diag(m2u,m

2c ,m

2t ). (7)

The form of matrix VU can be parameterized through the three mixing angles [51, 52].These angles can be read from the Eq.(16) of Ref. [51], and given as,

tan θu12 ≈R12u

R22u

= Cu12ε

2, tan θu23 ≈R23u

R33u

= Cu23ε

2, tan θu13 ≈R13u

R33u

= Cu13ε

4. (8)

3This type of mass matrix was discussed in Ref. [50]

M =

a1 b1 c1a2 b2 c2a3 b3 c3

, (4)

where ci >> bi >> ai.

5

Similarly, the mass matrix of down type quarks can be written as,

MD = v

yd11ε5 yd12ε3 yd13ε

yd21ε5 yd22ε

3 yd23ε

yd31ε5 yd32ε

3 yd33ε

. (9)

We further define a Hermitian matrix

Rd =M†DMD = v2

Y d11ε

10 yd12ε8 yd13ε

6

Y d21ε

8 yd22ε6 yd23ε

4

Y d31ε

6 yd32ε4 Y d

33ε2

. (10)

The matrix Rd can be diagonalized via unitary transformation given by

V †DRdVD = diag(m2d,m

2s,m

2b). (11)

The form of matrix VD can be parameterized through the following three mixing angles[51]:

tan θd12 ≈R12d

R22d

= Cd12ε

2, tan θd23 ≈R23d

R33d

= Cd23ε

2, tan θd13 ≈R13d

R33d

= Cd13ε

4. (12)

The Cabibbo-Kobayashi-Maskawa matrix is given by VCKM = V †UVD, and the three mixingangles in the standard parameterization are,

sin θ12 ≈ ε2 C12, sin θ23 ≈ ε2 C23, sin θ13 ≈ ε4 C13. (13)

From the above results, we observe that sin θ13 is much suppressed relative to sin θ12 andsin θ23. The sin θ12 is found to be of the same order as sin θ23. However, similar conclusion isalso reported in Ref. [12].

The masses of quarks at leading order are given by [51]

mu ≈ yuε4, mc ≈ ycε

2, mt ≈ ytv, (14)

md ≈ ydε5, ms ≈ ysε

3, mb ≈ ybε.

Now we observe the power of the discrete symmetries Z2 and Z5. The mass of the topquark yields from the usual Yukawa coupling. However, the masses of fermions of the thirdto first family are recovered through higher dimension operators in the ascending power of

the expansion parameterkΛ

. It is self evident that masses of the first family fermions are

much suppressed with respect to the second family fermions due to the descending powers

of the expansion parameterkΛ

. Similarly, the masses of the second family fermions are

further much suppressed relative to the masses of the third family fermions again due to the

suppression caused by the expansion parameterkΛ

. This explains naturally the origin of the

fermionic mass hierarchy among the three fermionic families.

6

The mass hierarchy within the families is also a natural outcome for the second and third

generations of fermions due to the suppression caused by the expansion of the parameterkΛ

.

The masses of down type quarks of the second and third families are suppressed by the one

power of the expansion parameterkΛ

with respect to that of the masses of up type quarks

of those families. Similarly, the mass of the d quark is suppressed by the one power of the

expansion parameterkΛ

with respect to that of the u quark. Here, one needs a small amount

of fine-tuning since the mass of the d quark is slightly bigger than that of the u quark.

3 Ultra-violet completion

It is desired to discuss a UV completion of the model. This can be done by introducing thefollowing vector-like fermions:

Q = U1L,R : (3, 1,

4

3)), U2

L,R : (3, 1,4

3)), (15)

D1L,R : (3, 1,−2

3)), D2

L,R : (3, 1,−2

3)), D3

L,R : (3, 1,−2

3)),

L = E1L,R : (1, 1,−2)), E2

L,R : (1, 1,−2)), E3L,R : (1, 1,−2)),

and three gauge singlet scalar fields k′, k′′ and k′′′. The behaviour of the vector-likefermions and the singlet scalar fields k′, k′′ and k′′′ under Z2 and Z5 symmetries is givenin table 2.

Fields Z2 Z5

U1L - ω2

U1R + ω4

U2L - ω2

U2R + ω

D1L, E

1L + ω3

D1R, E

1R + ω4

D2L, E

2L + ω3

D2R, E

2R + ω

D3L, E

3L + ω3

D3R, E

3R + ω3

k′ - ω3

k′′ + ω4

k′′′ + ω2

Table 2: The charges of vector-like fermions and singlet scalar field under Z2 and Z5 discretesymmetries.

7

(a) (b) (c)

(d) (e)

Figure 1: Masses of quarks: (a) diagram contributing to the mass of u quark, (b) diagramcontributing to the mass of d quark, (c) diagram contributing to the mass of c quark, (d)diagram contributing to the mass of s quark, (e) diagram contributing to the mass of b quark.We denote the particles corresponding to singlet scalar fields k, k′ and k′′ by S, S ′ and S ′′,respectively.

The mass interactions of, for instance, vector-like quarks are given by,

LV = M1U U

1LU

1R k

′ +M2U U

2LU

2R k+M1

DD1LD

1R k

′′ (16)

+ M2DD

2LD

2R k

′′′ +M3DD

3LD

3R + H.c.

The interactions of the SM fermionic fields with vector-like quarks can be written as,

L = g1q1LϕU

1R + g2q

2LϕU

2R + g3q

1LϕD

1R + g4q

2LϕD

2R + g5q

3LϕD

3R + H.c. (17)

The interactions of scalar field with vector-like fermions are the following:

L = C1uRU1L k+ C2cRU

2L k+ C3dRDL k+ C4sRDL k+ C4bRDL k (18)

A similar Lagrangian can be written for leptons.

8

4 The scalar potential

We can now discuss the pure scalar sector of the model. The terms of the scalar potentialwhich are relevant to our present discussion can be written in the following form:

V = µϕ†ϕ+ λ1(ϕ†ϕ)2 + (µ1 k† k+H.c) + (µ2 k′† k′ +H.c) + (µ3 k′′† k′′ +H.c) (19)

+ (µ4 k′′† k′′ +H.c) + ρ1 k2 k′ + ρ2 k2 k′′ + ρ3 k2 k′′′ + ρ4 k3

+ (other allowed terms),

where the seventh and tenth terms break the Z2 symmetry, and eight and ninyh terms breakZ5 symmetry softly. A potential having cubic terms similar to this is already studied in litera-ture in the context of strong eletroweak phase transition [53].

Phenomenological investigation of the flavon field assuming it a complex singlet scalar ispresented in Refs. [54,55]. The parameter space of flavon field is explored for defined bench-mark values in Ref. [54] by imposing constraints from flavour physics data. It is observed thatthe quartic self-coupling of the flavon field plays an important role in constraining the param-eter space of the flavon field. The phenomenological investigation of the model presented inthis work is beyond the scope of this paper, and will be discussed elsewhere [56].

It is noted that within the renormalized model, the mass of the u to s-quarks appear atone-loop level. The masse of the b-quark originates from a tree-level contribution. This isshown in figure 1.

As we just discussed, the renormalization of our model is provided by vector-like fermions.The large hadron collider has searched for these fermions and currently excludes them ap-proximately below 1 TeV in simplified scenarios [57].

5 Conclusion

We conclude now by observing that the standard way to achieve the Froggatt-Nielson mech-anism is by employing a continuous flavour abelian U(1)F symmetry and a flavon field.

In this paper, we have shown that the basic and central idea of the Froggatt-Nielsonmechanism can be achieved without employing a continuous abelian U(1)F flavour symmetryby imposing the minimal discrete symmetries. This can be done by adding simple and theminimal Abelian discrete symmetries Z2 and Z5 to the SM.

We must note that the Froggatt-Neilson mechanism for charged leptons and quarks isidentical so far, and has been achieved through the gauge singlet scalar field k by imposingdiscrete symmetries Z2 and Z5. In principle, charged leptons could have their own Froggat-Nielson mechanism. This can be implemented by adding one more gauge singlet scalar fieldkl and a set of discrete abelian symmetry Z ′2 and Z ′5 under which the charged leptons and

gauge singlet scalar field kl have charges identical to table 1 and quarks behave trivially.It should be noted that the UV completion of the model discussed in this work make

the existence of the vector-like fermions inevitable, thus providing an interesting and richphenomenology which will be explored in future.

9

Appendix

There are five fifth roots of unity which are:

1, ω, ω2, ω3, ω4.

Their main properties can be written as,

1 + ω + ω2 + ω3 + ω4 = 0, ω5 = 1, ω∗ = ω4, ω2∗ = ω3. (20)

Acknowledgement

I am extremely grateful to Prof. Anjan S. Joshipura for crucial comments and advice on thiswork.

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