Transcript MMM-Primorsko_2010 - Indico

A MAXIMAL MASS MODEL
Matey Mateev, University of Sofia
Primorsko, 24.06.2010
Matey Mateev
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V. G. Kadyshevsky, V. N. Rodionov and A. S. Sorin
(M. V. Chizhov, P. Danev)
Almost all detailes may be found in
Towards a Maximal Mass Model
hep-ph/30.08.2007
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We investigate consequences
of introduction
a new principle in local QFT:
Principle of existence of
a maximal mass M:
m<M
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On this ground we generalize the Standard Model to
a Maximal Mass Model
The three main components of the Standard Model
are:
1. Local QFT
2. Local gauge SU(3)xSU(2)xU(1) invariance
3. Higgs mechanism for generation of masses
SM describes well the existing experimental data.
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The fundamental physical constants
с — velocity of light
ħ — Planck constant
G — Newton gravitational constant
allow simple geometrical or group theoretical
interpretation.
For instance in special relativity the 3-dimensional
velocity space is a Lobachevsky space with curvature
-1/с² .


v


 1

c
u 
,
2
2 
v 
 1  v
1 2 
2
c
c 

u02  u 2  1
“Lobachevsky geometry”,
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(□  m ) ( x)  0
2
 ( x) 
1
(2 )
3
2
e
ip x 
 ( p)d p
4

( p x  p x  p.x )

0 0
2
(m  p ) ( p)  0, p  p  p
2
2
2
2
0
2
m p p
2
2
0
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m ≤ M
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We use Anti-De Sitter
geometry of momentum space:
2
2
2
p  p  p5  M
2
0
2
2
p p m
2
0
Here we have:
m ≤ M
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2

2
2
( p  p  p5  M ) ( p0 , p, p5 )  0

2

2
2
2 ~
 ( p0 , p, p5 )   ( p0  p  p5  M ) ( p0 , p, p5 )
2
0
  ( p, p5 )   1 ( p) 

2
2
~
  

 ( p0 , p, p5 )   ( p, p5 )  
,
p

M

p
.
5

  ( p)

(
p
,

p
)
5   2


The sign of
p5
is a new degree of freedom
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2

(m  p  p ) ( p0 , p, p5 )  0
2
2
0

m 
m  p  M  p  M 1  2 
 M 
2
2
2
5
2
2
5
2
2
m
cos   1  2 ,
M
( p5  M cos  )( p5  M cos  ) ( p, p5 )  0
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Klein-Gordon equation in Anti de Sitter momentum space
2M ( p5  M cos  ) ( p, p5 )  0
2M ( p5  M cos  )1 ( p)  0,
2M ( p5  M cos  ) 2 ( p)  0,
~
1 ( p)   ( p  m )1 ( p)
2
2
 2 ( p)  0
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“Flat” limit is defined as transition from Anti De Sitter
to Minkowski momentum space and it corresponds to:
M  p
2
2
and
p5  M
In the flat limit:
p2
p2
m2
m2
p5  M 1  2  M 
, cos   1  2  1 
2
M
2M
M
2M
and
2 M ( p5  M cos  )  p  m
2
2
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The action may be written in 5-dimensional form:
Euclidean formulation
De Sitter – O(4, 1)
We may integrate over p5
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 ( p) 
1 ( p)   2 ( p)
1 ( p ) 
p5
M ,  ( p )  1 ( p )   2 ( p )
 ( p) p5  M ( p)
2M
,  2 ( p) 
 ( p ) p5  M ( p)
2M
S0 ( M )  M 

  d 4 p   ( p)( p 2  m 2 ) ( p)  (  ( p)  M cos  ( p)) 2

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Fourier transform and configuration
space:
2M
3
e
ipK x K
 ( pL p L  M 2 ) ( p, p5 )d 5 p   ( x, x5 )
(2 ) 2
K , L  1,2,3,4,5.
2
2
( 2  □  M ) ( x, x5 )  0
x5
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initial data are given at
x5  0 :
Why Euclidean formulation?
Higgs potential:
2 2
1


2
U ( ,  ; M )  ( M 
)( ( x)   ( x)) 2 
2
2
2  2 ( x)   2 ( x) 2 2
 (
 ) .
4
2
mo2
m  m0 1 
4M 2
,
m0  2v.
2

m 
m
 0mM
1  2  1 
2 
M
 2M 
2
2
0
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Electromagnetic field
Fermion fields
Fermion fields
I.
m 2  p 2  (m  pn n )( m  pn n ), n  1,2,3.4
2
m
m 2  p 2  2M ( p5  M cos  ), cos   1  2
M
2M ( p5  M cos  )  2M sin   pn n  ( p5  M ) 5  2M sin   pn n  ( p5  M ) 5 
2
2

 

I.
II.
D( p, M )  pn n  ( p5  M ) 5  2M sin 
2
2M ( p5  M cos  ) 
  pn n   5 ( p5  M )  2M cos    pn n   5 ( p 5  M )  2M cos  
2  
2 

Dexotic ( p, M )  pn  n  5 ( p5  M )  2M cos  2
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2 M ( p5  M cos  ) 
 


n
5
n
5
 pn  ( p5  M )  2 M sin  pn  ( p5  M )  2 M sin 
2 
2

Once more the Dirac’s trick (De Sitter space) :
In the flat limit
|p|<<M :
Chiral fermion fields
1  5
1  5
r 
 ; l 

2
2
Weyl spinors
In our case:
( p 2  p52  M 2 ) ( p, p5 )  0
( pK Г K  M )( pN Г N  M ) ( p, p5 )  0
K , N  0,1,2,3,4,5
1
 L ( p, p5 ) 
( M  pK Г K ) ( p, p5 )
2M
1
 R ( p, p5 ) 
( M  pK Г K ) ( p, p5 )
2M
Chirality depends on energy momentum!
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In 1956 Lee ,Yang and Wu discovered parity
violation in weak interactions i.e. violation of
mirror symmetry. From our point of view this
effect is a direct consequence of de Sitter
geometry of 4-momentum space.
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Now we have all the bricks to construct a
generalization of the SM based on De Sitter
momentum space geometry and SU c (3)  SU L (2) UY (1)
gauge invariance.
It is local and naturally incorporates the new
physical principle – existence of a maximal
mass M of the objects described by quantum
fields.
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For instance the term:
becomes:
New P-odd effects
The new free chirality fields may be
represented as:
i n
( R , L ) ( x)  r ,l ( x) 
 ( x)
2M
n
and one predicts corrections to all weak
interaction processes. A global fit of all SM
LEP data gives M ~ 1 TeV
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New interactions
New Higgs decay mode
H  2l
( )

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
MMM
1
 2
M
a  a  a  292(63)(58) 10
exp
SM
11
 M  300GeV
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• Specific relations between the
Youkawa coupling constants and M:
M 
2
 v
4 4
f1
2 2
f1
8( v  m )
2
f1

 v
4
f2
2
4
8( v  m )
2
f2
2
f2
 .....
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Exotic fields
Exotic fields are good
candidates for dark matter:
- they are completely different
from ordinary fermions.
- in the flat limit they do not
have an ordinary analogue.
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Dark matter production
q
DM
mq
mDM
M
M

q
H
DM
Exotic matter is connected to ordinary matter
through the same mass generation Higgs
mechanism.
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The new “dark matter” world
may happen to be connected
with us only through gravitation
and Higgs exchange!
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Conclusions:
1. On the basis of a new physical principle
– the existence of a maximal mass M and
purely geometrical considerations - a
local QFT MMM is constructed.
2. Chirality (parity violation) has clear
geometrical origin.
New P-odd effects are predicted.
3. New interactions are appearing in the
Higgs sector.
4. M is predicted to be in the TeV region.
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5. We predict “exotic” fermions
(candidates for dark matter) coupled
to the ordinary matter through the
Higgs field.
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Yuri Manin
«GEOMETRY IS A
SPECIFIC
PRESERVATIVE FOR
QUICKLY ROTTENING
PHYSICS»
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2
«EXPERIMENT =
GEOMETRY +
PHYSICS»
A. EINSTEIN
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LHC = Geometry + Physics
CERN
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59
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THANK YOU!
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