PoS(ICHEP2020)241 https://pos.sissa.it/ groups fail to shed light on this issue, since they also contain ) 10 ( SO and ) 5 ( SU [email protected] three copies of fermion representations ofbe an enlarged so: gauge group. the Standard However, it Model doesunified families not model, might need in be to which distributed case over distinctand the representations explain gauge (at of symmetry least a itself partially) grand the mighton flavor discriminate puzzle. embedding the all The various SM most fermions families ambitious in version a of single this irreducible idea representation consists of the gauge group. Institute of Particle and Nuclear Physics Faculty of Mathematics and Physics, CharlesV University, Holešovičkách 2, 18000 Prague 8, Czech Republic E-mail: We do not know why thereexplain the are observed pattern three of fermion fermion masses families and mixing inbased angles. the on Standard Standard grand the unified Model theories (SM), nor can we Copyright owned by the author(s) under the terms of the Creative Commons 40th International Conference on High EnergyJuly physics 28 - - ICHEP2020 August 6, 2020 Prague, Czech Republic (virtual meeting) © Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). Renato Fonseca Explaining the SM flavor structure withtheories grand unified PoS(ICHEP2020)241 , ) ) + 2 )× 5 (1) / SM c ( 2 1 ( d G which − SU + , and the of SU 2 c , c ··· u X 1 )× d larger than ( 3 . Assigning + + ( = 0 G Q R ( L SU 3 , + flavor  3 R / 1 or a , matrices which account 1 follow directly from the , 3 3 ∗  × H SM 3 j c = . G e c i G d L generation , is a simple group — such as ij  ) 3 , a G E / Y 2 ( . In particular, the hypercharge matrix ) − , + 5 being directly related to the way in which model [9, 10] where fermions are assigned ( 1 family ∗ , ) 3 3 H /  SU 11 c j 5 ( d = p i [5–8]. However, it can also happen with simple of c 2 ) SU Q 5 u 1 ( ij , ) ) U 6 D / Y ( 1 ) × : family replication is nowhere to be seen at a fundamental , 3 + 2 ( , 3 H 330 ( j c SU + u = i , with the factor s Q ) × Q g 165 3 ij ( ) + = U 3 SU 55 Y g ( ). Each copy is often called a + c e and 11 g and are somewhat curious; for example, only the fundamental and anti-fundamental = L ) 2 , 1 g c , [1–4] — it then becomes possible to relate the three Standard Model gauge coupling , with the fermions distributed over 15 irreducible representations: , 1 d 6 0 , , g E 1 c ( SM 3 u / G or , = . The quantum numbers 5 ) ) Q c p c , which is spontaneously broken at very high energies. If This constitutes a strong motivation for studying viable ways of embedding the Standard Model The Standard Model of particle physics is a gauge theory built around the group We also do not know why these matrices have the values that they do, but perhaps once we have Given that grand unified theories can account for the quantum numbers of the Standard Model e e 10 ) ≡ = ( = 1 fermions are contained in the representation + SM ( 1 X L L SO g fermions in GUT representations beyond the usualthe scenarios. way SM There fermions are are embedded other can reasons: have aof rather for curiously example, the and dramatic three effect on gauge the couplings. unification the value To of appreciate these it,lines let couplings in us to), figure1 consider which higher are first energies, known the not assuming standard to only extrapolation unify. of the Importantly, Standard one usually Model compares fields the (full values of level, only emerging at low energies due to spontaneous symmetry breaking. to the representations do not have a trifold repetition.on This the is in semi-simple fact group what happensgroups. with A several particularly (but interesting not example all) is models the based fermions are presumed to beknown embedded. at low In energies; it other also words, incorporates figure1 anis assumption. not just The a assumption product is of that the what is an explanation for the existence of three familiesStandard that Model might family become replication clear. is One an possibility accidentgauge is in group, that the the fermions following might sense: be under assigned a more to fundamental a combination of representations Explaining the SM flavor structure with grand unified theories 1. Flavor and grand unification It is worth noting thatrepresentations in also standard GUTs have — flavor based indicesstill on and be the therefore seen groups as at mentioned matrices high earlier in energies — flavor space. the the fermion Yukawa couplings can for the measured fermion masses and mixing parameters: constants. Furthermore, the quantum numbers of fermions under a flavor index to each fermion, Yukawa interactions are controlled by representations of the special unitary groups arerational used, numbers. plus Grand it unified is possible theories to (GUTs)numbers. provide write an all elegant The hypercharges as explanation idea for these is quantum that the true gauge symmetry of Nature is given by a group fermions in an appealing way, weModel, are namely left the with existence of what three seems copies to of be every a irreducible deeper fermion representation mystery of the Standard and transformation properties of these fields under the enlarged group ( G U PoS(ICHEP2020)241 . ) 7 ( ab (2) δ 1 is the . The SU ∝ 2 i α g  / b control the π T c 4 a T T ≡ (where  . generators, so 1 0 c is the diagonal abc model utilizing − i ) g into the 3 chiral α ) ) ic α 5 3 b µ must be a completely 2 7 ( . / A ( = / . But other scenarios 5 ) SM 1  5 SU p abc b − ( G bca SU : for example an c , T ) c , = s 2 a SU − g / 1 , T 1 a µ  g g , ( A 0 (dashed line in1). figure , , 0 → and g , ) 2 0 3 y a µ ( g 5 A √ , / 2 = 3 g real representation ( diag 2 p + g . For example, an = ) = s c g ) e 2 . ( = + ab must decompose under norm. 3 3 SU δ y L g T ∝ + ···  3 b c and + GUTs, but different arrangements could conceivably T d 0 g a 6 R + T E ) which can be reducible:  = c + would imply that 2 — where u , therefore appearing in the commutator relation ) g SM R 5 and  0 + must have the same norm as all other / ) G , ) 2 3 ) 2 ( abc Q , and it is even possible that 2 c ( p 10 1 / ( 3 0 SU ( 3 1 g = T ) 3 + − SO 2 n / , ( L 5 2 / 3 p SU + 1 · · · → T c  = − + , d 1 0 3 g / R 1 → , + 7 3 / with Tr R 1 , potentially leading to a situation where the three gauge couplings unify with no ) , 5 3 / / YY 3 ( 1 generator — yields ( p Bottom-up evolution of the Standard Model gauge couplings at one loop, with ) 2 , ( diag n n It might not sound like much, but this simple constraint on the fermion field content of grand The only fermions which can have a pure electroweak mass are most likely the Standard Model This factor is the same for SU One might ask why is it so important to normalize all generators of the GUT group such that Tr 1 = unified theories turns out to be quite stringent. In reference [10] a search was made over (a) 2. Viable fermion GUT representations ones, so if there arebroken. more These fermions extra they fermions must —true, have if the a they fermion mass exist GUT at representations even all before — electroweak must symmetry be is vector-like. In order for this to be embedding might yield the dashed line. families plus a real representation (of equating Tr The reason is this: the structure constants Figure 1: extra fields lighter than the unification scale.is This whether is or certainly possible not mathematically; those thecan question conceivably scenarios even spoil are the associated relations with viable models. In fact, mathematically one transformation of gauge bosons undertransformation infinitesimal parameter). global And transformations: in orderantisymmetric for tensor, this which to in correspond turn to requires a that Tr unitary transformation, Explaining the SM flavor structure with grand unified theories SM might spoil the relation yield the branching rule usual picture (full lines) makes the assumption that fermions are embedded as in Y PoS(ICHEP2020)241 . ) s is be g (4) (3) 2 27 and . In E = E × Y model Y 3 and (c) 3 ) g ( 6 G 10 and 10830 and at very low ( E and c D D ) or g Y SO Y 3 ( , . , contains several = ) and U U ij E , ... of 2 under the Standard Y Φ Y SU 0 e 3876 g H 10 ) R , , . 2 0 × , / ij 5 D g 1 R 3 e 3 H ≥ − / , representations inside the + , , or even bigger special unitary 5 -invariant term above into , ) 2 N ) 5 , i p DE c ij 45 1 SM ( e × 19 H ( N = ( , , G 3 i SU 1 , e ≡ ( ’s for , g N ) ) SU c i e H 5 e gauge group, the fermion-fermion- H N ( , . ( , ) 2 L Φ and SU 19 , ij SU · QN ) ( 1 for each scalar irreducible representation 2 H L subgroup. Fortunately, only a few of them / , , y SU 1 c 5 171 N , · S L 2 there are various ways of embeddings the SM SM , , , c ) and the i 1 G ( 6 L i at the electroweak scale one needs to calculate the EL 6 4 : it could either transform as . This possibility was also mentioned in [11]. 171 , of the S ) E y ≡ 2 E ≥ Φ , , d Y 19 ) = , H ( N 1 , 171 ( 10 d ) i QDL ( , and 0 SU S c 5 , , SU because, as we shall see below, in order to compute the D d 1 SO Yukawa , , (there might be more than one). Under some reasonable Y c 1 i , i UD ( L d S U G are , , model exploring this last idea were considered in [12], which Y ≡ u D , ) G in S S c i glimpse , u 19 , ( in equation (3) merely acts as an overall normalization factor for these c SM DL y , (b) different combinations of representations S Q SU G , , i G L representation of S Q 3 3 171 ). There is no viable alternative to the relations 171 15 3876 ones: ). On the other hand, for ≥ ) 16 19 N ( ( × 3 SU SU The Standard Model group can only be embedded in one way in all these groups, except for fermions, and crucially family replication is not a requirement. The only non-trivial case where allby using fermions the can be embedded in a single representation The viable simple groups GUTs is unique. Furthermore,( for all practical purposes so is the one in an Apart from trivial variations, the fermion content in With a single fermion representation In order to go further, we need to break down the single • • • • This can also trivially be achieved with the fundamental representation of 2 , energies, the scan showed that The main features ofprovides an a glimpse ofreplication. how flavor I might used ariseStandard Model the from Yukawa matrices word a fundamentalratio of theory several which vacuum expectations has values (VEVs), no which is family a daunting task3. still to be addressed. A model for flavorgenesis reference [12] considered the first possibility. Butcomponents which how transform as might the entries of assumptions, such as the nonexistence of confining interactions besides those of different embeddings of generated from just a single number? The answer is the following. The scalar groups if we were to consider extra vector-like fermions. Explaining the SM flavor structure with grand unified theories different simple groups scalar interactions are controlled byΦ a singlet number There are two possible quantum numbers for Model gauge group, and it is the ratio of their VEVs that control the entries of three matrices. fact, the coupling constant several pieces which are only invariant under the larger are relevant at low energies. Among others, we find the following PoS(ICHEP2020)241 T is ) (5) (7) (9) (6) (8) 0 ,  0 QN , c c 1 2 Λ 1 L L ,  -invariant 0 must be a ) ( is a singlet L are massless c 19 = M 4 ( SM , where Q

T is a quadruplet 3 , , H  i 1 c jk SU N i 5 which ensures that ··· Q S Q L N L . QDL c N  + S X Y ! ijkl

+  H T  2 3  i EL 2 1 i 5 S DL d d QN ij r L L 2 S   Λ 1 4 √ D + = L fields: = M transforms as the antisymmetric . S c i N e 3 T H e c i QDL 2 ij ª ® ® ¬ . For example,  √ M E ij QN S N c c then the fermions flavor group and therefore it can be i 5 Y , − − H d d can be found in [12]. Recall that the 0 SM QN QN ’s, and so on. All in all, there is always 14 34 and T . Furthermore, the right-hand side of

T F  ) 4 d )  , X i G Λ Λ EL 0 3 H 3 4 i 5 , S + , Y 1 ( L c 0 2 √ L 2 3 , c u kl  QN 34 0 0 QN 13 SU = indicate that a field transforms under an extra N , r Λ + j Λ 1 matrix would be given by the expression N L − ( ’s and 2 ! = over its vector-like partner M M c scalars: indeed we find that the U c c j 5 = E c 5 Y ij , d X d d and S . In this way, flavor might be generated effectively QN QN 23 24 i M N

QN 12 ! i ,β Λ Λ , ; 5 Λ D + L c − − UD is an anti-quadruplet (upper index) and D Y ν i e UD S © ­ ­ « i S Q i h S UD E ,α 2 y Q S 2 i EL L M 3 2 √ 2 S — a sextet. ν c √ i -invariant pieces + 3 and − e √ j c F is a sextet of the ) αβ T u + = SM ) ) = 4 DL U ’s and 1 ν 0 ( i QDL u G fields, so we can write that S U , U Y S Q U m i ij Y QN 0 − ( e M , − which are the vector-like partners of the Standard Model fermion H SU M Q H 2 1 , and which commutes with

, c i ) , e / u ⊃ 0 2 1 ( 3 19 and + √ as an example, the SM ( and formulas for the remaining i = QDL c and H T S = )

Q SU c 1 3 0 D Yukawa Q ij QN , L 0 M = M L H , , i QDL 1 3 0 d Q S − , / Q

y , 1 2 M ( u ⊃ − , = c =

Q found inside U Yukawa Y i UD F S ) L We then see that there are 4 Similar calculations can be done for the remaining Standard Model Yukawa matrices, as well

4 1 ( − y SU an excess of 3 copies of a fermionat representation low energies only the Standardlight Model fermions fermions is are controlled observed. by the The VEVs precise of composition the of these product of two quadruplets of term in equation (3) contains the as for the neutrino masses (no index). These indices must be antisymmetrized, therefore where For example, if at low energies from a fundamental theory which is flavorless. Still, in order to confront this model Explaining the SM flavor structure with grand unified theories The quantum numbers associated to the symbolsthose appearing here of have been mentioned before except representations. On the other hand the indices of this group (lower index), the scalar until the electroweak symmetry is broken, and so are equation (3) also contains the following interactions with where two upper indices mean that written as an anti-symmetriccombination 4 of by all 4 the matrix.some anti-symmetric The matrix Standard of Model coefficients. Higgs boson With this notation and taking and PoS(ICHEP2020)241 897 93 ]. (1974) Phys. 32 , 6 E (1992) 2889. Phys. Rev. D , 69 Annals Phys. Nucl. Phys. B , , 1606.01109 . (2018) 091B01 (1979) 126. (1975) 575. 2018 156 Phys. Rev. Lett. , 23 (2016) 003[ (1980) 738. PTEP Phys. Rev. Lett. 2009.03909 , , , 08 22 Nucl. Phys. B JHEP , , AIP Conf. Proc. , Flavorgenesis Phys. Rev. D , 6 model for electroweak interactions ) 1 ) ( 1 ( U A Universal gauge theory model based on U Canonical neutral current predictions from the weak ) × ]. 3 ) × ( 3 ( Unified interactions of leptons and hadrons A flipped 331 model SU SU Unity of all elementary particle forces ]. An hep-ph/9206242 On the chirality of the SM and the fermion content of GUTs ]. Chiral dilepton model and the flavor question Family unification in special grand unification 1504.03695 The state of the art: gauge theories Towards a grand unified theory of flavor (1976) 177. 60 (1992) 410[ 1807.10855 (2015) 757[ [ 46 electromagnetic gauge group 438. Lett. B (1975) 193. Grand unified theories, which have been proposed and studied for more than four decades, I am grateful to Andreas Ekstedt and Michal Malinský for their collaboration in the paper [12], [9] H. Georgi, [8] R.M. Fonseca and M. Hirsch, [7] P. Frampton, [6] F. Pisano and V. Pleitez, [3] H. Fritzsch and P. Minkowski, [5] M. Singer, J. Valle and J. Schechter, [2] H. Georgi, [1] H. Georgi and S. Glashow, [4] F. Gürsey, P. Ramond and P. Sikivie, [10] R.M. Fonseca, [12] A. Ekstedt, R.M. Fonseca and M. Malinský, [11] N. Yamatsu, provide a potential explanation forof the the three Standard gauge Model couplings.family quantum replication. However, numbers, standard In GUTs as this do work well not I as explain discussed the the how non-standard phenomena values GUTs of might fermion doAcknowledgments so. on which the present workAgency is of partially the based. Czech Republic I (GAČR)University through acknowledge Research contract the Center number financial UNCE/SCI/013. 20-17490S support and from from the the Grant Charles References Explaining the SM flavor structure with grand unified theories with the observed fermion masses and mixingwhich data, minimize it would the be scalar necessary potential. to also examine the VEVs 4. Conclusions