Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://qfthep.sinp.msu.ru/talks2013/St._PetersburgStech_.pdf Äàòà èçìåíåíèÿ: Sat Jun 22 11:35:08 2013 Äàòà èíäåêñèðîâàíèÿ: Thu Feb 27 20:29:33 2014 Êîäèðîâêà:

general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

Phenomenology of SU (3)L â SU (3)R â SU (3)C â SO(3)F and the Higgs Boson
Ber thold Stech University of Heidelberg

St. Petersburg, June 2013

Ber thold Stech Phenomenology of trinification

University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

Trinification
A little discussed extension of the Standard Model is to go from SU (2)L â U (1) â SU (3)C to

SU (3)L â SU (3)R â SU (3)C E
6

which is a maximal subgroup of

some references to trinification: Y. Achiman, B.S. (1978-1979), A. de Rujula, H. Georgi, S.L. Glashow (1984), K.S. Babu, X.G. He, S.Pakvasa (1986), B.S. Phys. Rev. D 86, 055003 (2012)

Ber thold Stech Phenomenology of trinification

University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

3 gauge couplings can unify Trinification is a GUT ! combine GUT with Flavor (Generation) Symmetry : GUT â Flavor with the consequence: all fermions, quarks and leptons, are in the same flavor representation. This requirement leaves only very few models discussed in the literature. We take

Trinification â SO(3)F all
Ber thold Stech Phenomenology of trinification

fermions are

3 vectors

with respect to this flavor group.
University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

Aim:
Construction of a relatively simple GUTâ Flavor model in which all Higgs fields are flavor singlets and all flavon fields are GUT singlets . Few parameters. This appears difficult if not impossible for SO(10) GUT's but possible for the gauge groups

E6

â flavor and Trinification â flavor !

Phys. Rev. D77, 076009 (2008) Z. Tavar tkiladze, B.S., For tschr. Phys. 58, No 7-9 (2010) 692 B.S., arXiv hep-ph 1012.6028 .B.S.

non supersymmetric model favoured by simplicity
Ber thold Stech Phenomenology of trinification University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

There are several excellent features of SUSY but susy seems not relevant for TeV scale physics (LHC ! ) non supersymmetric models regain impor tance: i) the quadratic divergencies causing the hierarchy problem affect vacuum expectation values and only indirectly par ticle masses. Vev's are not understood (cosmological constant !) and may have there origin at a very high scale. ii) Vev's are due to tadpole diagrams which are momentum independent, can be subtracted and have then no influence on par ticle proper ties. iii) the neutrino masses likely require a two step unification process which in non supersymmetric theories occur naturally by electroweak unification at MI 2 · 1013 GeV.
Ber thold Stech Phenomenology of trinification University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

Concorde
60 50 40 30 20 10 2
Ber thold Stech Phenomenology of trinification

1/

4

6

8

Log10[µ/GeV]

10

12

14

16

18

University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

Single generation for fermions (27)

E6 SU (3)L â SU (3)R â SU (3)C
L

27 = QL (x)+L(x)+QR (x)
QL QR

(Q

a L )i

ua = da , Da





i Lk

1 L1 + = E e+



E- 2 L2 ^

e- , 3 L3

^ ^ (QR )k = ^a , da , D u a

a

,

3 QL (x) = 3, 1, ¯ ,

3 L(x) = ¯, 3, 1 ,

3 QR (x) = 1, ¯, 3

mixing: d D
Ber thold Stech Phenomenology of trinification

UL -spin,

^ ^ d D UR -spin
University of Heidelberg

B. S., Z. Tavar tkiladze, Phys. Rev. D 77 (2008) 076009


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

~ Higgs fields H , H
2 Higgs fields (3 the standard model v1 H = 0 0 , 3, 1) break the trinification group down to : 00 v2 0 0 ~ 0 b2 b3 b 0 H 0 MR M3 0 MI
C

MI = 0 gives SU (2)L â SU (2)R â U (1) , MI = 0 and MR = 0 leads to SU (2)L â U (1)Y â SU (3) With v1 = 0, v2 = 0 only U (1)e remains. mt = gt v1 , mb = gb b, MD = MI , parity like quantum number
Ber thold Stech Phenomenology of trinification

v2 + v 1

2 2

(174 GeV)2

~ H is diagonal, H fields not directly coupled to fermions.

University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

Potentials

ol' man river

~ The Higgs fields H , H are two (3 , 3R ) matrix fields L formed from 2 â 18 = 36 scalar fields
?? Can one construct a Potential for 36 scalar fields leading to the hierarchy MI , MR mt , mb ?? This potential should provide for spontaneous symmetry breaking giving 36 - 15 = 21 massive Higgs par ticles and 15 Goldstone bosons eaten up by W + , W - , Z and 12 heavy vector bosons! A strict treatment is complicate. We use a phenomenological ansatz. Only vevs, no explicit mass terms.

Ber thold Stech Phenomenology of trinification

University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

we star t with the simplest 4 invariants V0 = c1 J1 + c2 J2 + c3 J3 + c4 J4
J1 = (Tr[H · H ])2 , J2 = Tr[H · H · H · H ], ~~ ~ ~~ ~ J3 = (Tr[H · H ])2 , J4 = Tr[H · H · H · H ]

vevs:

J1 = (M 2 + v2 + b2 )2 , 1

J2 = (M 4 + v4 + b4 ) 1
4 3

J3 = (M 2 + v2 + b2 )2 , 2 3

J4 = M 4 + v4 + b 2

for MI = MR = M and M3 = b2 = 0 (most symmetric case)

no linear combination of the 4 invariants produces these vevs The potential needs a logarithmic dependence on invariants (Coleman- Weinberg type). We take 1 V= V0 3J k + log[ J1 J1 J2 JJ3 4 J4 ] J2
Ber thold Stech Phenomenology of trinification University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

requirement : derivatives with respect to all 36 scalar fields have to vanish at the proposed minimum: 1 2 3 ~1 H1 = v1 , H2 = b, H3 = M , H1 = v2 etc 2 result: k = 4 and for c2 , c3 , c
2

4
2 2

c

1

b2 + M 2 + v2 b2 + M 2 + v2 1 1 , c1 4 + M 4 + v4 2 + v2 + M 2 b b3 1 2

, c1

b2 + M 2 + v2 1 . b4 + v4 + M 4 3 2

2

Obviously, one has c2 = c3 = c4 = c1 in the large M limit. 1 b2 + b2 + v2 + v2 = (174 GeV)2 = (246 GeV) 3 1 2 2 for v1 = v2 and b, b
Ber thold Stech Phenomenology of trinification

2

3

v1 one finds v1

123 GeV.
University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

The potential V is fully invariant and provides for the spontaneous symmetry breaking to U (1) â U (1)e . The 36 â 36 matrix of second derivatives
2 Mab =

1 2V 2 ha h

b

has still a high symmetry because the invariants used so far do ~ not connect H with H . There are 4 massive states, 15 Goldstone bosons and 17 still massless states. To second order in v1 and v2 the masses obtained are m2 = c1 (v2 + b2 ), m2 = c1 (v2 + b2 ) light 1 1 2 2 3 m2 = c1 (4M 2 + 5v2 + 5b2 ), m2 = c1 (4M 2 + 5v2 + 5b2 ) heavy 3 1 4 2 3 .
Ber thold Stech Phenomenology of trinification University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

The Higgs
for v1 = v2 one obtains v1 = v2 m
c1
2 Higgs 1 2

174 = 123 GeV
1

= m2 = m2 = c1 v 1 2
1 2

2 1

(125 GeV)2 for c

1.04

1 (or c1

for v2 = 0) appears natural

(predicted 2010 and 2012)

Input - Eigenvalues 0 for M

Is the Higgs a

Twin

??

2 degenerate states: a combination of normal and fermiophobic Higgs fields ? seems a possibility, but the necessary additional invariants and states not yet considered can remove the degeneracy.
arXiv hep-ph 1303.6931 B.S.
Ber thold Stech Phenomenology of trinification University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

Inclusion of more invariants for V : J1 , .......J9 , ~~ J5 = Tr[H · H · H · H ] etc

The 36 first derivatives at the chosen minimum fixes the coefficients in terms of 3 parameters result: Eigenvalues of the 36 â 36 mass matrix : m2 i m2 gs , Hig 15 would be Goldstone par ticles 00 All new masses can be taken to be the first higher ones are charged (±). The degeneracy of the Higgs (125 GeV) can be kept or removed. ~ it depends on the vev's of H

Ber thold Stech Phenomenology of trinification

University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

V

eff

with shifted fields
1 1 3 3 H1 v + H1 , H3 M + H3 etc

power expansion in M

neglecting inverse powers of M

V becomes polynomial with field configurations up to 4th 3 powers only (h3 = Re[H3 ], ......) 3 Veff 4(h3 )2 M 2 + 4 h1 h3 M + 2(h1 )2 v2 + ..... 1 1 13 3 ~ ~ + O(H 3 ) + O(H 4 ) + O(H 3 ) + O(H 4 ) replaces the standard model in presence of a huge hierarchy symmetr y breaking proper ties remain unchanged valid at low scales, renormalizable
Ber thold Stech Phenomenology of trinification University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

Trinification âFlavor Flavor = SO(3)F



: scalar flavon fields (GUT singlets) , = 1, 2, 3

M = coupling matrix

L

eff Y

=

1 M



( T H ) + .......

effective interaction to be understood on a deeper level
Ber thold Stech Phenomenology of trinification University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

Flavor SO(3)
The scalar flavon fields are represented by the hermitian matrix field (x):

(x) =





(symmetric) +i (antisymmetric)
"1" + "5" "3"



The 3 â 3 coupling matrices in front of the Higgs fields are then obtained from the VEV's of L
eff Y

=

( M

T

H ) + i

( M

T

HA )

Ber thold Stech Phenomenology of trinification

University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

Phenomenology The coupling matrix G =
MI

determines the mass hierarchy

4 mu 0 00 1 G= = 0 mc 0 = 0 2 0 M mt 0 0 mt 0 0 1 at = 0.050 correct up quark mass ratios
The coupling matrix A = i M describes par ticle mixings. It is antisymmetric and hermitian, 1 real parameter: 0 - 0 1/2 A=i = i - M -1/2 0
Ber thold Stech Phenomenology of trinification

µ=MI

University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

The generation matrices G and A in the effective Yukawa interaction



appear

L

eff Y

= G ( T H ) + A ( T HA ) (G2 ) ~ ( T H )1 (H )1 + h.c. + MN

the 3rd term gives masses to the heavy leptons hierarchy of masses and the mixings of all fermions are now fully determined
¨ The A term is CP odd unique ¨ CP
Ber thold Stech Phenomenology of trinification University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

The Masses and Mixings of all 3 â 27 fermions are obtained from the mass matrix
ij

M

=G



H ij +A



H

A ij

~ +(G2 ) H




i

1~ H MN



j

v1 0 0 H = 0 b 0 0 0 MI 000 2 2 = 0 f2 f3 3 3 0 f2 f3



v2 0 0 ~ H = 0 b2 b3 0 MR M3

HA

quarks

HA

leptons

= similar to quarks

mD = mL = MI
Ber thold Stech Phenomenology of trinification University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

Mass Matrices
Up quark matrix Down quark matrix = d D m0 b ^ ^ d D 2A , f 2A G + f2 3 3 f2 A , MI G Mu = G mt

Md

,D

Md = m0 G + f22 A - f0 AG-1 A b
H
A

(¯, 3, 1), 3

fji = H

i Aj

:

3 f2 = 3 xg MI ,

23 f0 = f3 f2 / MI

This gives a ver y good fit for the CKM unitarity triangle
Ber thold Stech Phenomenology of trinification University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

Results
mu = G mt md m
2 G mb + A f2 - f0 A · G-1 · A v2 1 MI

¨ CP small mixing and ¨ ¨ CP large mixing and ¨



1+

3 f2 MI

(G-1 · A - A · G-1 )

obser vables: quarks 10, charged leptons 3, light neutrinos 9, heavy fermions 15 22 + 15 = 37 9 parameters:
(besides the gauge couplings)
2 3 mt , mb , m , f2 , f0 , f2 , , , 1/2

heavy neutrino masses: ver y large hierarchy

Ber thold Stech Phenomenology of trinification

University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

Neutrinos
with xg from HA the 3 â 3 neutrino matrix is m
2 v1 MI

i xg i xg -i xg i xg 2
2 xg 2 2 + 2 x g + 2 2 xg 2 2 + 2 xg - 2 4 v 2 12 MI

-i xg i xg 2 v4 1 , 2 MI v4 1 , 2 MI

Eigenvalues of m m : (m2 )2 (m1 )2 (m3 )2
Ber thold Stech Phenomenology of trinification

University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

i) Inver ted hierarchy ii) degeneracy in the no mixing limit xg 0 iii) R = (m2 - m2 )/(m2 - m2 ) / 2 0.035 2 1 2 3 iv) xg fixed from xg v2 1 MI m2 tm a 1 2
3

exp: 0.032

m2 a xg . 0.034

tm

0.034 eV ,

xg 0.04

=m

v) MI together with fixes masses of all heavy fermions vi) tiny increase of (m )33 element by neutrino mixing pattern.
Ber thold Stech Phenomenology of trinification

1.01 gives correct

University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

Neutrino proper ties
example : setting m3 = 0.07 and fixing m3,3 = 1.007 m3 one finds masses: m2 = 0.08542, mixing angles: 12
¨ CP ¨:

m1 = 0.08487, 50 ,

m3 = 0.07025 eV 3.7 not good (-26 , -96 )

33 ,


23 70

13

sensitive to charged lepton matrix ( "Cabibbo" angle too small) Majorana phases:

Neutrinoless double decay parameter: |m | = 0.07 eV
Ber thold Stech Phenomenology of trinification

|m | scales with m

3

University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

Summary E6 â Flavor Trinification â SO(3)F
The effective Yukawa interaction at the weak scale has a simple form: only flavor singlet Higgs fields and GUT singlet flavon fields. MI 2 · 1013 GeV, the meeting point of g1 and g2 , fixes the mass scales of light and heavy neutrinos and new physics. phenom. Higgs potential desired spont. symmetry breaking, 15 w. b. Goldstones, 1 or 2 Higgs of 125 GeV and heavier (par tly degenerate) scalars. Some scalars are fermiophob , some have tiny decay widths.
Ber thold Stech Phenomenology of trinification University of Heidelberg


general remarks

trinification

Potentials

the Higgs

trinification â flavor

Results

Neutrinos

Summary

¨ mixing of generations and ¨ is combined with the mixing CP of standard model states with heavy par ticles.

The known quark and charged lepton proper ties determine to some extent the neutrino proper ties. ¨ CP Predictions for hierarchy, ¨ , Majorana phases as well as the mass parameter for O decays. few symmetr y breaking parameters allow for a fit of all fermion masses and mixings !

it's fun to work on non fashionable models!

Ber thold Stech Phenomenology of trinification

University of Heidelberg