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On the Higgs potential in the Minimal S 3 -Invariant Extension of the Standard Model U. J. Salda na Salazar A. Mondrag on M. Mondrag on XIII Mexican Workshop on Particles and Fields Le on, Guanajuato. 25 October, 2011. U. J.

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SLIDE 1

On the Higgs potential in the Minimal S3-Invariant Extension of the Standard Model

  • U. J. Salda˜

na Salazar

  • A. Mondrag´
  • n
  • M. Mondrag´
  • n

XIII Mexican Workshop on Particles and Fields Le´

  • n, Guanajuato.

25 October, 2011.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 1 / 61

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SLIDE 2

1 Stage I: A quick overview...

Massless massive fundamental particles The Higgs Mechanism

Spontaneous Symmetry Breaking Yukawa interactions

The flavour problem

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 2 / 61

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SLIDE 3

1 Stage I: A quick overview...

Massless massive fundamental particles The Higgs Mechanism

Spontaneous Symmetry Breaking Yukawa interactions

The flavour problem

2 Stage II: A quick glance into the MS3IESM

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 2 / 61

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SLIDE 4

1 Stage I: A quick overview...

Massless massive fundamental particles The Higgs Mechanism

Spontaneous Symmetry Breaking Yukawa interactions

The flavour problem

2 Stage II: A quick glance into the MS3IESM 3 Stage III: Flavouring the Higgs potential

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 2 / 61

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SLIDE 5

Motivation

So, why to derive it again?

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 3 / 61

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SLIDE 6

Motivation

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 4 / 61

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SLIDE 7

Motivation

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 5 / 61

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SLIDE 8

Motivation

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 6 / 61

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SLIDE 9

Motivation

In what sense am I asking: Which is the most general S3-invariant Higgs potential?

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 7 / 61

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SLIDE 10

Motivation

In what sense am I asking: Which is the most general S3-invariant Higgs potential? It has the highest level of flavour symmetry.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 7 / 61

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SLIDE 11

Motivation

In what sense am I asking: Which is the most general S3-invariant Higgs potential? It has the highest level of flavour symmetry. It has the highest arbitrariness without breaking the flavour symmetry.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 7 / 61

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SLIDE 12

Stage I: A quick overview...

Stage I: A quick overview...

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 8 / 61

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SLIDE 13

Stage I: A quick overview...

GSM = SU(3)C ⊗ SU(2)L ⊗ U(1)Y

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 9 / 61

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SLIDE 14

Stage I: A quick overview...

GSM = SU(3)C ⊗ SU(2)L ⊗ U(1)Y ⊗U(1)b−l

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 9 / 61

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SLIDE 15

Stage I: A quick overview... Massless massive fundamental particles

Massless massive fundamental particles

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 10 / 61

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SLIDE 16

Stage I: A quick overview... Massless massive fundamental particles

Gauge invariance kills any possibility of adding mass terms −mψψψ to the fermions as well as 1 2m2

BBµBµ to the gauge bosons.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 11 / 61

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SLIDE 17

Stage I: A quick overview... Massless massive fundamental particles

Gauge invariance kills any possibility of adding mass terms −mψψψ to the fermions as well as 1 2m2

BBµBµ to the gauge bosons.

In order to preserve the renormalizability of the theory, mass terms putted by hand for the massive vector bosons, are forbidden. (we can not break by hand the gauge symmetry)

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 11 / 61

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SLIDE 18

Stage I: A quick overview... Massless massive fundamental particles

Gauge invariance kills any possibility of adding mass terms −mψψψ to the fermions as well as 1 2m2

BBµBµ to the gauge bosons.

In order to preserve the renormalizability of the theory, mass terms putted by hand for the massive vector bosons, are forbidden. (we can not break by hand the gauge symmetry) Therefore all particles within the SM (without the Higgs mechanism) appear massless.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 11 / 61

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SLIDE 19

Stage I: A quick overview... Massless massive fundamental particles

But then,

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 12 / 61

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SLIDE 20

Stage I: A quick overview... Massless massive fundamental particles

But then, the theory itself demands the introduction of a scalar particle that help us to remove some residual divergences in order to guarantee renormalizability through diagrams of the type:

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 12 / 61

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SLIDE 21

Stage I: A quick overview... Massless massive fundamental particles

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 13 / 61

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SLIDE 22

Stage I: A quick overview... Massless massive fundamental particles

Summarizing the problems of the SM without a Higgs mechanism:

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 14 / 61

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SLIDE 23

Stage I: A quick overview... Massless massive fundamental particles

Summarizing the problems of the SM without a Higgs mechanism: Theoretically massive gauge bosons are massless

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 14 / 61

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SLIDE 24

Stage I: A quick overview... Massless massive fundamental particles

Summarizing the problems of the SM without a Higgs mechanism: Theoretically massive gauge bosons are massless ... massive fermions are massless

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 14 / 61

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SLIDE 25

Stage I: A quick overview... Massless massive fundamental particles

Summarizing the problems of the SM without a Higgs mechanism: Theoretically massive gauge bosons are massless ... massive fermions are massless Introduction of a scalar particle is essential for renormalizability...

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 14 / 61

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SLIDE 26

Stage I: A quick overview... Massless massive fundamental particles

Summarizing the problems of the SM without a Higgs mechanism: Theoretically massive gauge bosons are massless ... massive fermions are massless Introduction of a scalar particle is essential for renormalizability... All these issues are related.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 14 / 61

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SLIDE 27

Stage I: A quick overview... The Higgs Mechanism

The Englert-Brout-Higgs-Guralnik-Hagens-Kibble Mechanism

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 15 / 61

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SLIDE 28

Stage I: A quick overview... The Higgs Mechanism

The Englert-Brout-Higgs-Guralnik-Hagens-Kibble Mechanism The Higgs Mechanism

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 15 / 61

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SLIDE 29

Stage I: A quick overview... The Higgs Mechanism

Spontaneous Symmetry Breaking → massive gauge bosons

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 16 / 61

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SLIDE 30

Stage I: A quick overview... The Higgs Mechanism

Spontaneous Symmetry Breaking → massive gauge bosons Yukawa interactions → massive fermions

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 16 / 61

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SLIDE 31

Stage I: A quick overview... The Higgs Mechanism

Spontaneous Symmetry Breaking → massive gauge bosons Yukawa interactions → massive fermions The needed scalar particle

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 16 / 61

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SLIDE 32

Stage I: A quick overview... The Higgs Mechanism

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 17 / 61

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SLIDE 33

Stage I: A quick overview... The Higgs Mechanism

Spontaneous Symmetry Breaking

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 18 / 61

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SLIDE 34

Stage I: A quick overview... The Higgs Mechanism

It is known that:

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 19 / 61

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SLIDE 35

Stage I: A quick overview... The Higgs Mechanism

It is known that: Only those local gauge symmetries which are broken will imply their generators to acquire mass.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 19 / 61

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SLIDE 36

Stage I: A quick overview... The Higgs Mechanism

It is known that: Only those local gauge symmetries which are broken will imply their generators to acquire mass. W +, W −, and Z are the only massive gauge bosons...

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 19 / 61

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SLIDE 37

Stage I: A quick overview... The Higgs Mechanism

It is known that: Only those local gauge symmetries which are broken will imply their generators to acquire mass. W +, W −, and Z are the only massive gauge bosons... Gluons and photons are massless...

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 19 / 61

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SLIDE 38

Stage I: A quick overview... The Higgs Mechanism

It is known that: Only those local gauge symmetries which are broken will imply their generators to acquire mass. W +, W −, and Z are the only massive gauge bosons... Gluons and photons are massless... An isospin weak doublet is introduced, and with it the Higgs lagrangian is constructed as a GSM invariant.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 19 / 61

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SLIDE 39

Stage I: A quick overview... The Higgs Mechanism

LH = (Dµφ)†(Dµφ) − µ2φ†φ − λ(φ†φ)2

www.nature.com/nphys/journal/v7/n1/fig tab/nphys1874 F1.html

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 20 / 61

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SLIDE 40

Stage I: A quick overview... The Higgs Mechanism

GSM = SU(3)C ⊗ SU(2)L ⊗ U(1)Y ⊗U(1)b−l (before spontaneous symmetry breaking)

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 21 / 61

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SLIDE 41

Stage I: A quick overview... The Higgs Mechanism

GSM = SU(3)C ⊗ SU(2)L ⊗ U(1)Y ⊗U(1)b−l (before spontaneous symmetry breaking) GPhys.World = SU(3)C ⊗ U(1)EM⊗U(1)b−l (after spontaneous symmetry breaking)

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 21 / 61

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SLIDE 42

Stage I: A quick overview... The Higgs Mechanism www.quantumdiaries.org/2011/10/10/who − ate − the − higgs/

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 22 / 61

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SLIDE 43

Stage I: A quick overview... The Higgs Mechanism

Yukawa interactions

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 23 / 61

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SLIDE 44

Stage I: A quick overview... The Higgs Mechanism

Y f LφfR + h.c.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 24 / 61

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SLIDE 45

Stage I: A quick overview... The Higgs Mechanism

Y f LφfR + h.c. vY f LfR + h.c.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 24 / 61

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SLIDE 46

Stage I: A quick overview... The flavour problem

Houston, we have a flavour problem...

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 25 / 61

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SLIDE 47

Stage I: A quick overview... The flavour problem

Matter in the universe is found to be constituted by a single family of fermions (νe,e,u,d), nevertheless,

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 26 / 61

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SLIDE 48

Stage I: A quick overview... The flavour problem

Matter in the universe is found to be constituted by a single family of fermions (νe,e,u,d), nevertheless, in the “quantum world” there are additionally two more replicas.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 26 / 61

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SLIDE 49

Stage I: A quick overview... The flavour problem

Matter in the universe is found to be constituted by a single family of fermions (νe,e,u,d), nevertheless, in the “quantum world” there are additionally two more replicas. Why 3 (at least) generations of matter?

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 26 / 61

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SLIDE 50

Stage I: A quick overview... The flavour problem

Let’s take a quick deeper look into this mystery.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 27 / 61

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SLIDE 51

Stage I: A quick overview... The flavour problem

Some aspects of the flavour problem: Quark weak mixing angles (PDG 2010):

θ12 ≈ 13.0o θ23 ≈ 2.4o θ13 ≈ 0.2o

Lepton weak mixing angles (Shwetz, Tortola, & Valle 2011):

Θ12 ≈ 33.9o Θ23 ≈ 46.1o Θ13 ≈ 6.5o(IH7.2o)

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 28 / 61

slide-52
SLIDE 52

Stage I: A quick overview... The flavour problem

Some aspects of the flavour problem: Quark weak mixing angles (PDG 2010):

θ12 ≈ 13.0o θ23 ≈ 2.4o θ13 ≈ 0.2o

Lepton weak mixing angles (Shwetz, Tortola, & Valle 2011):

Θ12 ≈ 33.9o Θ23 ≈ 46.1o Θ13 ≈ 6.5o(IH7.2o)

CP-violation just occurs in the weak sector.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 28 / 61

slide-53
SLIDE 53

Stage I: A quick overview... The flavour problem

Some aspects of the flavour problem: Quark weak mixing angles (PDG 2010):

θ12 ≈ 13.0o θ23 ≈ 2.4o θ13 ≈ 0.2o

Lepton weak mixing angles (Shwetz, Tortola, & Valle 2011):

Θ12 ≈ 33.9o Θ23 ≈ 46.1o Θ13 ≈ 6.5o(IH7.2o)

CP-violation just occurs in the weak sector. The mass hierarchy.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 28 / 61

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SLIDE 54

Stage I: A quick overview... The flavour problem

Mass-Hierarchy Plot

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 29 / 61

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SLIDE 55

Stage II: A quick glance into the MS3IESM

Stage II: A quick glance into the MS3IESM

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 30 / 61

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SLIDE 56

Stage II: A quick glance into the MS3IESM

Some references of works with an S3 symmetry...

  • S. Pakvasa et al, Phys. Lett. 73B, 61 (1978)
  • E. Derman, Phys. Rev. D19, 317 (1979)
  • D. Wyler, Phys. Rev. D19, 330 (1979)
  • R. Yahalom, Phys. Rev. D29, 536 (1984)
  • A. Mondrag´
  • n et al, Phys. Rev. D59, 093009, (1999)
  • J. Kubo et al, Prog. Theor. Phys. 109, 795 (2003)
  • J. Kubo et al, Phys. Rev. D70, 036007 (2004)
  • A. Mondragon et al, Phys. Rev. D76, 076003, (2007)
  • D. Meloni et al, Nucl. Part. Phys. 38 015003, (2011)
  • T. Teshima et al, arXiv:1103.6127 (2011)
  • G. Bhattacharyya et al, Phys. Rev. D83, 011701 (2011)

And many more... I apologize for those references I don’t include.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 31 / 61

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SLIDE 57

Stage II: A quick glance into the MS3IESM

Why S3 as the flavour symmetry?

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 32 / 61

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SLIDE 58

Stage II: A quick glance into the MS3IESM

Why S3 as the flavour symmetry? Before the introduction of the Yukawa interactions,

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 32 / 61

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SLIDE 59

Stage II: A quick glance into the MS3IESM

Why S3 as the flavour symmetry? Before the introduction of the Yukawa interactions, 3 families are indistinguishable.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 32 / 61

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SLIDE 60

Stage II: A quick glance into the MS3IESM

Why S3 as the flavour symmetry? Before the introduction of the Yukawa interactions, 3 families are indistinguishable. A group theory study reveals:

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 33 / 61

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SLIDE 61

Stage II: A quick glance into the MS3IESM

Why S3 as the flavour symmetry? Before the introduction of the Yukawa interactions, 3 families are indistinguishable. A group theory study reveals:

S3 irreps: 1S, 1A, and 2

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 33 / 61

slide-62
SLIDE 62

Stage II: A quick glance into the MS3IESM

Why S3 as the flavour symmetry? Before the introduction of the Yukawa interactions, 3 families are indistinguishable. A group theory study reveals:

S3 irreps: 1S, 1A, and 2 In a 3d-real representation: 1S and 2

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 33 / 61

slide-63
SLIDE 63

Stage II: A quick glance into the MS3IESM

Why S3 as the flavour symmetry? Before the introduction of the Yukawa interactions, 3 families are indistinguishable. A group theory study reveals:

S3 irreps: 1S, 1A, and 2 In a 3d-real representation: 1S and 2

After fermions gain mass,

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 33 / 61

slide-64
SLIDE 64

Stage II: A quick glance into the MS3IESM

families become distinguishable...

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 34 / 61

slide-65
SLIDE 65

Stage II: A quick glance into the MS3IESM

Why S3 as the flavour symmetry? Before the introduction of the Yukawa interactions, 3 families are indistinguishable. A group theory study reveals:

S3 irreps: 1S, 1A, and 2 In a 3d-real representation: 1S and 2

After fermions gain mass, families become distinguishable.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 35 / 61

slide-66
SLIDE 66

Stage II: A quick glance into the MS3IESM

Why S3 as the flavour symmetry? Before the introduction of the Yukawa interactions, 3 families are indistinguishable. A group theory study reveals:

S3 irreps: 1S, 1A, and 2 In a 3d-real representation: 1S and 2

After fermions gain mass, families become distinguishable. Was the mass distributed following the irreps of S3 in the 3d-real representation?

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 35 / 61

slide-67
SLIDE 67

Stage II: A quick glance into the MS3IESM

The Log-Mass Plot

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 36 / 61

slide-68
SLIDE 68

Stage II: A quick glance into the MS3IESM

The Log-Mass Plot

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 37 / 61

slide-69
SLIDE 69

Stage II: A quick glance into the MS3IESM

Smile!

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 38 / 61

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SLIDE 70

Stage II: A quick glance into the MS3IESM

Three generations (flavours): Grandparents Parents Children are distributed in different families (Sectors: neutrinos, charged leptons, u quarks, d quarks).

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 39 / 61

slide-71
SLIDE 71

Stage II: A quick glance into the MS3IESM

Three generations (flavours): Grandparents Parents Children are distributed in different families (Sectors: neutrinos, charged leptons, u quarks, d quarks). But any age (mass) comparison that we want to do

  • n this global picture will be among all the families.
  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 39 / 61

slide-72
SLIDE 72

Stage II: A quick glance into the MS3IESM

Three generations (flavours): Grandparents Parents Children are distributed in different families (Sectors: neutrinos, charged leptons, u quarks, d quarks). But any age (mass) comparison that we want to do

  • n this global picture will be among all the families. Nevertheless,

families are independent from each other, they don’t share both last names (different quantum numbers).

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 39 / 61

slide-73
SLIDE 73

Stage II: A quick glance into the MS3IESM

Three generations (flavours): Grandparents Parents Children are distributed in different families (Sectors: neutrinos, charged leptons, u quarks, d quarks). But any age (mass) comparison that we want to do

  • n this global picture will be among all the families. Nevertheless,

families are independent from each other, they don’t share both last names (different quantum numbers).

Any conclusion about the mass hierarchy (flavour structure) can not be done from this global picture.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 39 / 61

slide-74
SLIDE 74

Stage II: A quick glance into the MS3IESM

To see the flavour structure we need to interrelate families in each independent sector.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 40 / 61

slide-75
SLIDE 75

Stage II: A quick glance into the MS3IESM

Say Tequila!

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 41 / 61

slide-76
SLIDE 76

Stage II: A quick glance into the MS3IESM

Mass-Percent Plot

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 42 / 61

slide-77
SLIDE 77

Stage II: A quick glance into the MS3IESM

The Flavour Spectrum

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 43 / 61

slide-78
SLIDE 78

Stage II: A quick glance into the MS3IESM

The Flavour Spectrum

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 44 / 61

slide-79
SLIDE 79

Stage II: A quick glance into the MS3IESM

The Flavour Spectrum S3 it’s conserved.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 44 / 61

slide-80
SLIDE 80

Stage II: A quick glance into the MS3IESM

The Flavour Spectrum S3 it’s conserved. A single Higgs weak doublet: Mf =   µ1 µ1 µ3  

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 44 / 61

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SLIDE 81

Stage II: A quick glance into the MS3IESM

The Flavour Spectrum S3 it’s conserved. We need at least to introduce additionally two Higgs weak-doublets more to the SM.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 45 / 61

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Stage II: A quick glance into the MS3IESM

The Flavour Spectrum S3 it’s conserved. We need at least to introduce additionally two Higgs weak-doublets more to the SM. Mf =   µ1 + µ2 µ4 µ5 µ4 µ1 − µ2 µ6 µ7 µ8 µ3  

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 45 / 61

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Stage II: A quick glance into the MS3IESM

A special feature arises from the theory:

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 46 / 61

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Stage II: A quick glance into the MS3IESM

A special feature arises from the theory: The concepts of flavours and generations are taken to a more fundamental level.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 46 / 61

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Stage III: Flavouring the Higgs potential

Stage III: Flavouring the Higgs potential

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 47 / 61

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SLIDE 86

Stage III: Flavouring the Higgs potential

Some references of works with an S3 invariant Higgs potential...

  • S. Pakvasa and H. Sugawara, Phys. Lett. 73B, 61 (1978)
  • E. Derman, Phys. Rev. D19, 317 (1979)
  • D. Wyler, Phys. Rev. D19, 330 (1979)
  • R. Yahalom, Phys. Rev. D29, 536 (1984)
  • Y. Koide, Phys. Rev. D60, 077301 (1999)
  • J. Kubo et al, Phys. Rev. D70, 036007 (2004)
  • S. Chen et al, Phys. Rev. D70, 073008 (2004)
  • O. F´

elix-Beltr´ an et al, J. Phys.: Conf. Ser. 171 012028 (2009)

  • D. Meloni et al, Nucl. Part. Phys. 38 015003, (2011)
  • G. Bhattacharyya et al, Phys. Rev. D83, 011701 (2011)

Again, there are many more, I apologize for those not included.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 48 / 61

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Stage III: Flavouring the Higgs potential

Cooking recipe for a Higgs Potential with S3 flavour

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 49 / 61

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Stage III: Flavouring the Higgs potential

The essential ingredients to work it out were:

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 50 / 61

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Stage III: Flavouring the Higgs potential

The essential ingredients to work it out were: The tensorial products between irreps: 1S ⊗ 1S = 1S 1A ⊗ 1A = 1S 1A ⊗ 1S = 1A 1S ⊗ 2 = 2 1A ⊗ 2 = 2 2 ⊗ 2 = 1S + 1A + 2

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 50 / 61

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SLIDE 90

Stage III: Flavouring the Higgs potential

The essential ingredients to work it out were: The tensorial products between irreps: 1S ⊗ 1S = 1S 1A ⊗ 1A = 1S 1A ⊗ 1S = 1A 1S ⊗ 2 = 2 1A ⊗ 2 = 2 2 ⊗ 2 = 1S + 1A + 2 To carefully carry the weak (SU(2)L) index.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 50 / 61

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Stage III: Flavouring the Higgs potential

  • 1. Find out all the l.i. S3-invariant terms for 2 and 4 scalar fields.
  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 51 / 61

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Stage III: Flavouring the Higgs potential

  • 1. Find out all the l.i. S3-invariant terms for 2 and 4 scalar fields.

n = 2: 1S ⊗ 1S [2 ⊗ 2]S

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 51 / 61

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SLIDE 93

Stage III: Flavouring the Higgs potential

  • 1. Find out all the l.i. S3-invariant terms for 2 and 4 scalar fields.

n = 2: 1S ⊗ 1S [2 ⊗ 2]S n = 4: 1S ⊗ 1S ⊗ 1S ⊗ 1S [(1S ⊗ 2) ⊗ (1S ⊗ 2)]S [(1S ⊗ 2) ⊗ (2 ⊗ 2)2]S (2 ⊗ 2)A ⊗ (2 ⊗ 2)A (2 ⊗ 2)S ⊗ (2 ⊗ 2)S [(2 ⊗ 2)2 ⊗ (2 ⊗ 2)2]S

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 51 / 61

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Stage III: Flavouring the Higgs potential

  • 2. Take an explicit convention for the whole theory (Yukawa lagrangian

and Higgs potential) of where to place the symmetric and antisymmetric doublet components. HD = HDA HDS

  • (fDA, fDS)T ⊗ (gDA, gDS)T =

1 √ 2 (fDAgDA + fDSgDS)1S ⊕ 1 √ 2 (fDAgDS − fDSgDA)1A ⊕ 1 √ 2 fDAgDS + fDSgDA fDAgDA − fDSgDS

  • 2
  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 52 / 61

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Stage III: Flavouring the Higgs potential

  • 3. For each S3-invariant term make all the different independent weak

indexes contractions. For example:

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 53 / 61

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SLIDE 96

Stage III: Flavouring the Higgs potential

  • 3. For each S3-invariant term make all the different independent weak

indexes contractions. For example: (2 ⊗ 2)S ⊗ (2 ⊗ 2)S = 1S3

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 53 / 61

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SLIDE 97

Stage III: Flavouring the Higgs potential

  • 3. For each S3-invariant term make all the different independent weak

indexes contractions. For example: (2 ⊗ 2)S ⊗ (2 ⊗ 2)S = 1S3

(2w ⊗ 2w)S ⊗ (2w′ ⊗ 2w′)S = 1S3⊗GSM (2w ⊗ 2w′)S ⊗ (2w ⊗ 2w′)S = 1S3⊗GSM (2w ⊗ 2w′)S ⊗ (2w′ ⊗ 2w)S = 1S3⊗GSM

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 53 / 61

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SLIDE 98

Stage III: Flavouring the Higgs potential

  • 3. For each S3-invariant term make all the different independent weak

indexes contractions. For example: (2 ⊗ 2)S ⊗ (2 ⊗ 2)S = 1S3

(2w ⊗ 2w)S ⊗ (2w′ ⊗ 2w′)S = 1S3⊗GSM (2w ⊗ 2w′)S ⊗ (2w ⊗ 2w′)S = 1S3⊗GSM (2w ⊗ 2w′)S ⊗ (2w′ ⊗ 2w)S = 1S3⊗GSM

In terms of the Higgs fields:

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 53 / 61

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SLIDE 99

Stage III: Flavouring the Higgs potential

  • 3. For each S3-invariant term make all the different independent weak

indexes contractions. For example: (2 ⊗ 2)S ⊗ (2 ⊗ 2)S = 1S3

(2w ⊗ 2w)S ⊗ (2w′ ⊗ 2w′)S = 1S3⊗GSM (2w ⊗ 2w′)S ⊗ (2w ⊗ 2w′)S = 1S3⊗GSM (2w ⊗ 2w′)S ⊗ (2w′ ⊗ 2w)S = 1S3⊗GSM

In terms of the Higgs fields:

1 2(H† 1wH1w + H† 2wH2w)2 1 2[(H† 1wH1w)2 + (H† 2wH2w)2 + (H† 1wH2w)2 + (H† 2wH1w)2] 1 2[(H† 1wH1w)2 + (H† 2wH2w)2 + (H† 1wH2w)2 + (H† 2wH1w)2]

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 53 / 61

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Stage III: Flavouring the Higgs potential

  • 4. Assign the same self-coupling parameter for each different contraction
  • f the same S3-invariant term.
  • U. J. Salda˜

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Stage III: Flavouring the Higgs potential

  • 4. Assign the same self-coupling parameter for each different contraction
  • f the same S3-invariant term.
  • 5. Group similar terms and relate their couplings by new parameters.
  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 54 / 61

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Stage III: Flavouring the Higgs potential

The most general and renormalizable S3-invariant Higgs potential is: V = V2H + V4H V2H = −µ2

sxs − µ2 D(x1 + x2)

V4H = ax2

s +b[(y2 S1+y2 1S +y2 S2+y2 2S)+xs(x1+x2)+yS1y1S +yS2y2S]

+cfijk(ySiyjk + h.c.) + d(x1 + x2)2 +e(y12 − y21)2 + f[(x1 − x2)2 + (y12 + y21)2] where xs = H†

s Hs, xi = H† i Hi (i = 1, 2), yij = H† i Hj (i, j = 1, 2, s), and

f112 = f121 = f211 = −f222 = 1.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 55 / 61

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Stage III: Flavouring the Higgs potential

a ≡ 3λ1 b ≡ λ2 √ 2 c ≡ 3λ3 2 d ≡ λ4 + √ 2λ6 2 e ≡ λ4 + λ5 − √ 2λ6 2 f ≡ λ6 + √ 2λ4 2 √ 2

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 56 / 61

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SLIDE 104

Summary and Conclusions

Summary and Conclusions

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 57 / 61

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SLIDE 105

Summary and Conclusions

Summary

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 58 / 61

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SLIDE 106

Summary and Conclusions

Summary Recall of the original motivation for the Higgs mechanism.

Mass for gauge bosons and fermions in a gauge invariant way. The need for a scalar particle...

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 58 / 61

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SLIDE 107

Summary and Conclusions

Summary Recall of the original motivation for the Higgs mechanism.

Mass for gauge bosons and fermions in a gauge invariant way. The need for a scalar particle...

The flavour problem.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 58 / 61

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SLIDE 108

Summary and Conclusions

Summary Recall of the original motivation for the Higgs mechanism.

Mass for gauge bosons and fermions in a gauge invariant way. The need for a scalar particle...

The flavour problem. A quick glance into the MS3IESM.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 58 / 61

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SLIDE 109

Summary and Conclusions

Summary Recall of the original motivation for the Higgs mechanism.

Mass for gauge bosons and fermions in a gauge invariant way. The need for a scalar particle...

The flavour problem. A quick glance into the MS3IESM. Recipe for constructing the most general S3-invariant Higgs potential.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 58 / 61

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SLIDE 110

Summary and Conclusions

Conclusions

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 59 / 61

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SLIDE 111

Summary and Conclusions

Conclusions The MS3IESM provides a flavour framework that explains both theoretical and experimental facts about the massive nature of fermions:

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 59 / 61

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SLIDE 112

Summary and Conclusions

Conclusions The MS3IESM provides a flavour framework that explains both theoretical and experimental facts about the massive nature of fermions:

Three indistinguishable families. (theoretical)

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 59 / 61

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SLIDE 113

Summary and Conclusions

Conclusions The MS3IESM provides a flavour framework that explains both theoretical and experimental facts about the massive nature of fermions:

Three indistinguishable families. (theoretical) The flavour spectrum for known fermion masses. (experimental)

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 59 / 61

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SLIDE 114

Summary and Conclusions

Conclusions The MS3IESM provides a flavour framework that explains both theoretical and experimental facts about the massive nature of fermions:

Three indistinguishable families. (theoretical) The flavour spectrum for known fermion masses. (experimental)

The recipe can be extended to the construction of other flavour-invariant Higgs potentials.

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 59 / 61

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SLIDE 115

Summary and Conclusions

Conclusions The MS3IESM provides a flavour framework that explains both theoretical and experimental facts about the massive nature of fermions:

Three indistinguishable families. (theoretical) The flavour spectrum for known fermion masses. (experimental)

The recipe can be extended to the construction of other flavour-invariant Higgs potentials. Just six self-couplings parameters are needed. (more predictive power)

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 59 / 61

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SLIDE 116

Thanks for your attention. Any Questions?

Thanks for your attention.

  • U. J. Salda˜

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SLIDE 117

Thanks for your attention. Any Questions?

Professor Alfonso Mondrag´

  • n Ballesteros

(Scientific Research Prize 2011 from the Sociedad Mexicana de F´ ısica)

  • U. J. Salda˜

na Salazar (IF-UNAM) On the Higgs potential in the MS3IESM 25 October, 2011. 61 / 61