Algebraic Structure of Lepton and Quark Flavor Invariants Elizabeth - PowerPoint PPT Presentation

Algebraic Structure of Lepton and Quark Flavor Invariants Elizabeth Jenkins Department of Physics University of California, San Diego GGI, Firenze, July 5, 2012 E. Jenkins Algebraic Structure of Lepton and Quark Flavor Invariants

Introduction Observation of neutrino oscillations ⇒ neutrino flavor eigenstates ν e , ν µ , ν τ are not mass eigenstates ν 1 , ν 2 , ν 3 . Constitutes first evidence for new physics beyond the Standard Model. Leading theory for massive light neutrinos is non-renormalizable theory = SM + d = 5 Weinberg operator + . . . in which weak-doublet neutrinos acquire Majorana masses upon EWSB from unique d = 5 operator which respects gauge symmetry. This theory is the low-energy EFT obtained from the renormalizable seesaw theory by integrating out the gauge-singlet neutrinos with Majorana masses M ≫ M W , M Z , m t , m H ! Flavor structure of these theories is of interest. Useful to discuss flavor structure in terms of flavor invariants, which are basis independent.

Introduction There is extensive literature on flavor invariants, both quark invariants Jarlskog, Greenberg, Kusenko & Shrock, · · · and lepton invariants Branco & Rebelo, Branco, Rebelo & Silva-Marcos, Kusenko & Shrock, Dreiner, Kim, Lebedev & Thormeier, · · · It is interesting to address the classification of flavor invariants using invariant theory. Mathematics of invariant theory describes the algebraic structure of invariants. The number of invariants of a given degree in the flavor-symmetry breaking mass matrices is encoded in Hilbert series. Flavor invariants with usual operations of addition and multiplication form a ring, which is finitely generated. It is interesting to determine the generators of the ring and the non-trivial relations (syzygies) among invariants.

Introduction This talk is based on the references E. E. Jenkins and A. V. Manohar, “Algebraic Structure of Lepton and Quark Flavor Invariants and CP Violation,” JHEP10 (2009) 094. A. Hanany, E. E. Jenkins, A. V. Manohar and G. Torri, “Hilbert Series for Flavor Invariants of the Standard Model," JHEP03 (2011) 096. E. Jenkins and A. V. Manohar, “Rephasing Invariants of Quark and Lepton Mixing Matrices," Nucl. Phys. B792 (2008) 187.

Lepton and Quark Flavor Invariants Flavor structure of our leading theories (SM + d = 5 operator, seesaw) is encoded by flavor invariants constructed from the quark and lepton mass matrices. There are a finite number of basic invariants, and a general invariant can be written as a polynomial in the basic invariants. Number of basic invariant generators is equal to number of independent physical parameters: quark and lepton masses, mixing angles and phases The basic invariants and all non-trivial relations (syzygies) between these invariants determines the flavor structure of a given theory.

Primer: Invariant Theory Before addressing the physical problem of interest, it is useful to consider some simple examples which illustrate the mathematics of invariant theory in a very simple context.

Model I Two complex couplings m 1 and m 2 which transform under G = U ( 1 ) × U ( 1 ) m 1 → e i φ 1 m 1 , m 2 → e i φ 2 m 2 . (1) Ring of invariant polynomials generated by two basic invariants I 1 = m 1 m ∗ 1 and I 2 = m 2 m ∗ 2 with no non-trivial relations (syzygies) between I 1 and I 2 General invariant is of the form 1 ) r 1 ( m 2 m ∗ 2 ) r 2 ( m 1 m ∗ (2) Hilbert series Definition ∞ ∞ c r q r = 1 + � � c r q r H ( q ) = (3) r = 0 r = 1 c r = the number of invariants of degree r , c r ≥ 0

Model I General invariant is of the form 1 ) r 1 ( m 2 m ∗ 2 ) r 2 ( m 1 m ∗ Hilbert series of Model I 1 + 2 q 2 + 3 q 4 + 4 q 6 + 5 q 8 + . . . H ( q ) = ∞ � ( n + 1 ) q 2 n = n = 0 1 = ( 1 − q 2 ) 2 � 2 � 1 + q 2 + q 4 + q 6 + . . . = (5)

Model I Hilbert series � 2 1 � 1 + q 2 + q 4 + q 6 + . . . H ( q ) = = (6) ( 1 − q 2 ) 2 Theorem H ( q ) = N ( q ) D ( q ) N ( q ) = 1 + c 1 q + c 2 q 2 + · · · + c d N − 2 q d N − 2 + c d N − 1 q d N − 1 + q d N c r ≥ 0 , c r = c d N − r p � 1 − q d r � � D ( q ) = r = 1 � d D = d r r

Model I 2 ] U ( 1 ) × U ( 1 ) of all polynomials which Ring C [ m 1 , m ∗ 1 , m 2 , m ∗ are invariant under G = U ( 1 ) × U ( 1 ) p = dim V − dim G , dim V = 4 , dim G = 2 Knop’s Theorem Theorem dim V ≥ d D − d N ≥ p dim V = 4 , d D = 4 , d N = 0 , p = 2 4 ≥ 4 ≥ 2

Model II Two couplings m 1 and m 2 which transform under G = U ( 1 ) m 1 → e i φ m 1 , m 2 → e 2 i φ m 2 . Invariants generated by four basic invariants I 1 = m 1 m ∗ 1 , I 2 = m 2 m ∗ 2 , I 3 = m 2 m ∗ 2 1 and I 4 = m ∗ 2 m 2 1 , but the four basic invariants are not all independent since I 3 I 4 = I 2 1 I 2 Hilbert series 1 + 2 q 2 + 2 q 3 + 3 q 4 + 6 q 6 + . . . H ( q ) = 1 + q 3 = (7) ( 1 − q 2 ) 2 ( 1 − q 3 ) I 1 , I 2 , I 3 , I 4 not all independent is encoded in Hilbert series 1 H ( q ) � = ( 1 − q 2 ) 2 ( 1 − q 3 ) 2 1 + 2 q 2 + 2 q 3 + 3 q 4 + 7 q 6 + . . . = (8)

Model II ( I 3 − I 4 ) cannot be written in terms of I 1 , I 2 , ( I 3 + I 4 ) Syzygy I 3 I 4 = I 2 1 I 2 ( I 3 − I 4 ) 2 = ( I 3 + I 4 ) 2 − 4 I 3 I 4 = ( I 3 + I 4 ) 2 − 4 I 12 I 2 General polynomial in basic invariants P 1 ( I 1 , I 2 , I 3 + I 4 ) + ( I 3 − I 4 ) P 2 ( I 1 , I 2 , I 3 + I 4 )

Model II 2 ] U ( 1 ) of all polynomials which are Ring C [ m 1 , m ∗ 1 , m 2 , m ∗ invariant under G = U ( 1 ) p = dim V − dim G , dim V = 4 , dim G = 1 Knop’s Theorem Theorem dim V ≥ d D − d N ≥ p dim V = 4 , d D = 7 , d N = 3 , p = 3 4 ≥ 4 ≥ 3

Model III Three couplings m 1 , m 2 and m 3 which transform under G = U ( 1 ) m 1 → e i φ m 1 , m 2 → e 2 i φ m 2 , m 3 → e 3 i φ m 3 . 13 basic invariants m 1 m ∗ m 3 2 m ∗ 2 I 1 = 1 , I 8 = 3 , m 2 m ∗ m ∗ 3 2 m 2 I 2 = 2 , I 9 = 3 , m 3 m ∗ m 1 m 2 m ∗ I 3 = 3 , I 10 = 3 , m 2 1 m ∗ I 4 = m ∗ 1 m ∗ 2 , I 11 = 2 m 3 , m ∗ 2 m 1 m 3 m ∗ 2 I 5 = 1 m 2 , I 12 = 2 , m 3 1 m ∗ I 6 = m ∗ 1 m ∗ 3 m 2 3 , I 13 = 2 . m ∗ 3 I 7 = 1 m 3 ,

Model III 35 relations between products I i I j , but now there are relations among relations (syzygies) Example I 4 I 5 I 6 I 7 = I 4 1 I 10 I 11 obtained by multiplying relations I 4 I 7 = I 2 1 I 11 and I 5 I 6 = I 2 1 I 10 OR by multiplying I 4 I 5 = I 2 1 I 2 , I 6 I 7 = I 3 1 I 3 and using I 10 I 11 = I 1 I 2 I 3 , so I 4 I 5 I 6 I 7 = I 5 1 I 2 I 3 = I 4 1 I 10 I 11 Hilbert series 1 + q 2 + 3 q 3 + 4 q 4 + 4 q 5 + 4 q 6 + 3 q 7 + q 8 + q 10 H ( q ) = . ( 1 − q 2 ) 2 ( 1 − q 3 )( 1 − q 4 )( 1 − q 5 )

Model III 3 ] U ( 1 ) of all polynomials which Ring C [ m 1 , m ∗ 1 , m 2 , m ∗ 2 , m 3 , m ∗ are invariant under G = U ( 1 ) p = dim V − dim G , dim V = 6 , dim G = 1 Knop’s Theorem Theorem dim V ≥ d D − d N ≥ p dim V = 6 , d D = 16 , d N = 10 , p = 5 6 ≥ 6 ≥ 5

Lepton and Quark Flavor Invariants Use invariant theory to solve classification of quark and lepton mass matrix invariants in the (i) seesaw model and (ii) SM + dim-5 operator (giving Majorana masses to weakly interacting neutrinos) Invariant structure in lepton sector is highly non-trivial with many non-linear relations (syzygies) among the basic invariants. Invariant structure depends on number of generations n g of SM quarks and leptons and n ′ g of neutrino singlets Able to solve problem for low-energy EFT with n g = 2 , 3 and for high-energy seesaw theory with n g = n ′ g = 2 , 3. Hilbert series obtained in cases of physical interest. Number of independent invariants and syzygy structure encoded by Hilbert series. Algebraic structure of lepton invariants is much more complicated than for quark invariants.

News About ν s What is new? for What is ν ? Hilbert series of flavor invariants for Lagrangians (i) SM+ d = 5 operator and (ii) seesaw model determined. Syzygy relations follow from Hilbert series. Solution dependent on number of families. Cases of physical interest: n g = 3 families of SM fermions and n ′ g = 2 , 3 right-handed neutrinos now solved. Algebraic structure of lepton invariants is very complicated.

For the purposes of this talk: Definition Standard Model ≡ nonrenormalizable EFT containing only SM fields with gauge symmetry SU ( 3 ) × SU ( 2 ) × U ( 1 ) truncated after unique d = 5 operator (higher dimensional operators d = 6, · · · neglected) Definition Seesaw Model ≡ renormalizable SU ( 3 ) × SU ( 2 ) × U ( 1 ) theory with additional gauge-singlet neutrinos N

Flavor Matrices High-Energy Seesaw Model i ( Y D ) ij Q j H † − E c − U c i ( Y U ) ij Q j H − D c i ( Y E ) ij L j H † L = I ( Y ν ) Ij L j H − 1 − N c 2 N c I M IJ N c J + h.c. Mass matrices: m U , m D , m E , m ν , M Low-Energy Effective Theory ≡ SM + d = 5 operator i ( Y D ) ij Q j H † − E c L EFT − U c i ( Y U ) ij Q j H − D c i ( Y E ) ij L j H † = + 1 2 ( L i H ) ( C 5 ) ij ( L j H ) + h.c. Mass matrices: m U , m D , m E , m 5

Quark Flavor Invariants � SU ( n g ) Q × SU ( n g ) Uc × SU ( n g ) Dc × U ( 1 ) 2 � m U , m † U , m D , m † C D U U c T m U U Q m U → U D c T m D U Q m D → m † X U ≡ U m U m D † m D X D ≡ U † X U , D → Q X U , D U Q