Particle Interactions with Modified
Dispersion Relations
Yi Ling （凌意）
IHEP,CAS & Nanchang University
2012两岸粒子物理与宇宙学研讨会, 重庆,05/09/2012
Outlines
• The fate of Lorentz symmetry at Planck scale
• Introduction to deformed special relativity
• What is the difference between Lorentz violation
theory and deformed special relativity
• The composition law of particles with modified
dispersion relations
The fate of Lorentz symmetry at Planck scale
•
“Planck length paradox”：
Lorentz contraction in special relativity:
L  1  v 2 / c 2 L0
v  c, L  0
Energy-momentum relation:
E  p c m c
2
2 2
2 4
E' 
E  vp
1  v2
, p' 
p  vE
1  v2
The existence of the minimal length that can be measured
j Aˆ j  8 l p 2 j ( j  1)
x
1
2 p
 l p2 p 
L  lp
2l p
The fate of Lorentz symmetry at Planck scale
•
•
The possibilities of Lorentz symmetry at high energy level
1. Keeping the original form
2. Manifestly broken
3. Deformed
Deformed special relativity (Doubly special relativity) (DSR)
was originally proposed in the context of quantum gravity
phenomenology to reconcile the relativity principle and the
existence of the minimal length scale which is uniform and
invariant to all observers.
Introduction to deformed special relativity
•
The relativity of inertial frames, two universal constants:
1) In the limit E / E pl  0 , the speed of a photon goes to a
universal constant, c .
2) E pl in the above condition is also a universal constant.
E pl
l p 1
As a result, the energy-momentum relation is usually
modified to
f 2  E / E pl ,  E 2  g 2  E / E pl ,  p 2  m 2
Introduction to deformed special relativity
•
Lorentz transformation in standard special relativity (1+1 dim.)
E '   ( E  vp )
p '   ( p  vE )
m2  E 2  p 2  E '2  p '2
1  v  E , v  0
m  0  E  p  E '  p' 
E
1  v  E , v  0
v  1, E '  0,
•
v  1, E '  .
Lorentz transformation in deformed special relativity
e.g.
2
2
E
p
m2 

2
(1  lE ) (1  lE )2
?
E2
p2
E '2
p '2
m 



2
2
2
(1  lE ) (1  lE ) (1  lE ') (1  lE ')2
2
Introduction to deformed special relativity
•
Lorentz transformation in deformed special relativity
 ( E  vp )
1  l[ ( E  vp )  E ]
 ( p  vE )
p' 
1  l[ ( E  vp )  E ]
E' 
1
1
If E  , then E ' 
l
l
m  0, E '  p ' 
 (1  v ) E
1  l[ (1  v )  1]E
If v  c, then E ' 
1
l
Introduction to deformed special relativity
•
Remark: the Lorentz transformation law depends on the form of
modified dispersion relation
m2  E 2  p 2  (lE ) p 2
l
E '   ( E  vp  Evp)
2
l
p '   [ p  vE  (( Ep   ( p  vE )( E  vP))]
2
Two open problems in DSR
•
Usually the standard dispersion relation in special relativity will
be modified with correction terms
m2  E 2  p2  (l p E )n p2
However, such a modification in theory would lead to some severe problems…
Two open problems in DSR
•
A field theory with MDR is still absent. How to define the
position space?
x

p 

x
?
p
F
Lnon
?
x'
F
1
p'
Two open problems in DSR
•
The soccer problem
n  1,
m2  E 2  p 2  ( E / M p ) p 2
M p :1019 Gev 105 g
This modified dispersion relation is not applicable to composite particles
and macroscopic objects. Thus it is not universal but particle number
dependent.
M  nm, E  nE , P  np 
M 2  E 2  P2  (E / nM p )P2
the difference between Lorentz violation
theory and deformed special relativity
• Example: the derivation of the threshold value of the interaction
p  p   D
A. In standard special relativity
p1  p2   p  pD
( p1  p2  )2  ( p  pD )2
Laboratory reference frame:
LHS  ( E1  E2 )2  ( p1  p2 )2 =( E1  E2 )2  p12  2m2p  2mp E1
Center-of-mass reference frame:
RHS  ( E '  E ' D ) 2  ( p '  p ' D ) 2 =( E  ED ) 2  ( m  mD ) 2
E1th 
(m  mD )2  2m2p
2m p
p m

p D mD
Laboratory reference frame:
RHS  ( E  ED )2  ( p  pD )2 =m2  mD2  2( E ED  p  pD )
=m2  mD2  2m mD  (m  mD )2
the difference between Lorentz violation
theory and deformed special relativity
B.
In Lorentz violation theory special relativity
m2  E 2  p 2   ( E , p)  E '2  p '2   '( E ', p ')
Laboratory reference frame:
LHS  ( E1  E2 )2  ( p1  p2 )2 =2m2p  p1  2mp E1
p2  0
RHS  ( E  ED )2  ( p  pD )2 =m2    mD2  D  2( E ED  p  pD )
=(m  mD ) 2  (m  mD )(
E1th

m

D
)
mD


(m  mD ) 2  2m 2p  ( m  mD )(   D )   p
m mD

2m p
Center-of-mass reference frame: No sense
RHS  ( E '  E ' D ) 2  ( p '  p ' D ) 2 =( E '  E ' D ) 2
 (m  mD )2  (m  mD )(
 '
m

 'D
mD
)  (m  mD ) 2
p m

p D mD
the difference between Lorentz violation
theory and deformed special relativity
p   CMB  p  
e.g.
E m  p
2
2
0
E 
th
p
m p m
E
2 E
2
 EGZK
104 ev
E  m  p   ( E , p)
2
2
0
2
mp
 p 
   p
m p m
m
th
Ep 

2 E
4 E
 1017 (ev)2
Ethp 1021 ev  10EGZK
E2
(1020 ev)2
the difference between Lorentz violation
theory and deformed special relativity
C.
In deformed special relativity
I.
m2  E 2  p 2  (lE ) p 2  E '2  p '2  (lE ') p '2
Laboratory reference frame:
p2  0
LHS  ( E1  E2 )2  ( 1  lE1 p1  1  lE2 p2 )2 =2m2p  2m p E1
RHS  ( E  ED )2  ( 1  lE p  1  lED pD ) 2
=m2  mD2  2( E ED  1  lE p  1  lED pD )
=(m  mD )2
E1th 
(m  mD )2  2m2p
2m p
1  lE p
1  lE D pD

m
mD
Center-of-mass reference frame: the same result can be obtained
the difference between Lorentz violation
theory and deformed special relativity
II. m  E  p  (lE ) E  E '  p '  (lE ') E '
2
2
2
E1th DSR 
2
2
E1th
l
1  E1th
2
2
2
The composition law in special relativity
revisited
•
Consider two elementary particles which may have different
masses
m12  E12  p12 ,
•
m2 2  E2 2  p2 2
(1+1 dim.)
We define a composite particle through a process in which the
covariant momentum is conserved
p1  p2   ( p1  p2 ) 
An invariant quantity of the composite particle is
M 2  ( p1  p2 )  ( p1  p2 )   ( p1  p2  )( p1  p2  )
 ( E1  E2 )2  ( p1  p2 ) 2
 m12  m22  2 m12m22  ( p1E2  E1 p2 )2
The composition law in SR revisited
Some remarks:
•
If we define Ec : E1  E2 ,
pc : p1  p2 , we still have
M 2  Ec 2  pc 2
•
Universal
In general
M  m1  m2
•
They are equal if and only if
v1  v2
•
In general, the composite particle could not be elementary.

M
M
The composition law in SR revisited
It is straightforward to extend it to the composition of many
particles:
( p1  p2 )   p3  ( p1  p2  p3 ) 
• If we define E : E  E
t
1
2
 ...,
pt : p1  p2  ... , we still have
M 2  Et 2  pt 2
• In general
M  m1  m2  m3  ...
The composition law in SR revisited
•
The transformation law of the energy and momentum under
the Lorentz boost in 1+1 space time
[ K , E ]  p,
[ K , p]  E
Thus
[ K , E 2  p 2 ]  2 Ep  2 pE  0
It is easy to check that for a composite particle
[ K , Ec 2  pc 2 ]  [ K , 2( E1E2  p1 p2 )]  0
[ K , Et 2  pt 2 ]  0
The composition law in SR revisited
• An interaction involving n incoming particles and m outgoing
particles
p1  p2   ...  pn  p '1  p '2   ...  p 'm
• The conservation law of momentum is preserved under the
Lorentz boost in the sense that
[ K ,( E1  E2  ...  En )  ( E1 ' E2 ' ...  Em ')] 
( p1  p2  ...  pn )  ( p1 ' p2 ' ...  pm ')  0
[ K ,( p1  p2  ...  pn )  ( p1 ' p2 ' ...  pm ')] 
( E1  E2  ...  En )  ( E1 ' E2 ' ...  Em ')  0
The composition law in DSR
•
Consider an elementary particle with a modified dispersion
relations as
m2  E 2  p 2  (lE ) p 2
•
Obviously, it is not an invariant quantity under the standard
Lorentz boost
[ K , E 2  p 2  (lE ) p 2 ]  lp(2 E 2  p 2 )  0
E 2  p 2  (lE ) p 2  E '2  p '2  (lE ') p '2
•
In DSR, a deformed boost generator is proposed so as to
preserve it to be an invariant quantity up to the first order
correction of the Planck length.
l
l 2
[ K , E ]  p  Ep, [ K , p ]  E  ( p  E 2 )
2
2
[ K , E 2  p 2  (lE ) p 2 ]  0  0(l 2 )
The composition law in DSR
•
However, such a choice is not unique. An alternative
deformation
l 2
[ K , E ]  p, [ K , p ]  E  ( p  2 E 2 )
2
When consider the composition law of particles, one need look
for some specific laws of composition of momenta which are
supposed to be compatible with the deformed boosts one has
chosen. And in general, such choices would unavoidably lead
to the relative-locality of the space of momenta.
( p1  p2 )   p1  p2 
( p1  p2 )0  E1  E2
( p1  p2 )1  p1  p2  lE1 p2
[ K ,( p1  p2 )  ( p1  p2 )  ]  0
[ K ,( p1  p2 )  ( p1  p2 )  ]  0
The composition law in DSR
• Our central goal in this talk
We intend to argue that if we input some rules on picking up
one specific form for deformed boost among all the possible
choices, then the relative-locality of the space of momenta may
be avoided.
The composition law in DSR
•
We introduce a notion of effective momentum
peff : 1  lE p
1
(1  lE ) p
2
l
[ K , E ]  p  Ep,
2
[ K , E ]  peff ,
l 2
[ K , p]  E  ( p  E 2 )
2
[ K , peff ]  E
m2  E 2  p 2  (lE ) p 2
m2  E 2  peff 2    p p
p0 : E , p1 : peff
The composition law in DSR
•
We propose a composition law for two elementary particles
( p1  p2 )   p1  p2 
( p1  p2 )0  Ec  E1  E2
( p1  p2 )1  Peff  p1eff  p2 eff
l
p1  p2  ( E1 p1  E2 p2 )
2
M 2  ( p1  p2 )  ( p1  p2 )   ( p1  p2  )( p1  p2  )
 ( E1  E2 ) 2  ( 1  lE1 p1  1  lE2 p2 ) 2
l
( E1  E2 ) 2  [ p1  p2  ( E1 p1  E2 p2 )]2
2
=( E1  E2 ) 2  ( p1  p2 ) 2  l ( p1  p2 )( E1 p1  E1 p1 )
Remark: it is interesting to show that M  m1  m2 if and only if
v1  v2
The composition law in DSR
•
One can easily check the following identities for a composite
particle
[ K ,( p1  p2 )  ( p1  p2 )  ]  0
( E1  E2 )2  ( p1  p2 )2  l ( p1  p2 )( E1 p1  E1 p1 )
=( E1 ' E2 ')2  ( p1 ' p2 ')2  l ( p1 ' p2 ')( E1 ' p1 ' E1 ' p1 ')
[ K , Ec ]  Peff ,
[ K , Peff ]  Ec
[ K , p1  p2   ( p1  p2 )  ]  0
The composition law in DSR
M 2 =( E1  E2 )2  ( p1  p2 )2  l ( p1  p2 )( E1 p1  E2 p2 )
•
It can be further written into a compact form which depends
on the number of particles manifestly given that
1. E1 E2
2. In the relativistic limit, E1
p1 , E2
3. In the non-relativistic limit, E1
p2
p1, E2
p2 , m1
m2
Et 2
M =Et  pt  l
pt
2
2
2
2
Et : E1  E2  ...,
pt : p1  p2  ...
The composition law in DSR
•
One can easily check
Et 2
M =Et  pt  l
pt
2
2
2
2
l
Peff : 1  Et pt
2
[ K , Ec ]  Peff ,
l
[ K , Et ]  p t  Et pt ,
4
l
(1  Et ) pt
4
[ K , Peff ]  Ec
l
[ K , pt ]  Et  ( pt 2  Et 2 )
4
The composition law in DSR
•
Extension to arbitrary composite particle or macroscopic object
which is composed of n elementary particles
M 2 =( E1  E2 ...  En )2  ( p1  p2 ...  pn )2
 l ( p1  p2 ...  pn )( E1 p1  E2 p2 ...  En pn )
 E t 2  Peff 2
E t :=E1  E2 ...  En
l
Peff : ( p1  p2 ...  pn )  ( E1 p1  E2 p2 ...  En pn )
2
[ K , E t ]  Peff ,
[ K , Peff ]  E t
The composition law in DSR
•
For a macroscopic object with n particles in thermal equilibrium,
it is reasonable to assume that
E1
then
E2
.....En
kT
M 2 =E t 2  Pt 2  l
l
Peff : 1  Et Pt
n
l
[ K , E t ]  P t  E t Pt ,
2n
Et 2
Pt
n
l
(1  Et )Pt
2n
l
[ K , Pt ]  E t 
(Pt 2  Et 2 )
2n
The composition law in DSR
•
A general interaction
P( n )   P( m)   (P( n )  P( m))   P( n ')   P( m ') 
E(n)  E(m)  E( n ')  E( m ')
P(n)eff +P(m)eff  P(n ') eff +P(m ') eff
E t (n )  E t (m)  E t (n ')  E t ( m ')
lE t (n )
lE t (m)
lE t (n ')
lE t (m ')
1
Pt (n )+ 1 
Pt (m)  1 
Pt ( n ')+ 1 
Pt ( m ')
n
m
n'
m'
The composition law for many sorts of
elementary particles
•
Two elementary particles with different dispersion relations
m2  E 2  p 2  (lE ) p 2 (I)
•
 2  k0 2  k12  (lk )k0 2
We introduce a notion of effective energy
Eeff : 1  lk1 k0
1
(1  lk1 )k0
2
l 2
l
2
[ K , k0 ]  k1  (k0  k1 ), [ K , k1 ]  k0  k1k0
2
2
[ K , k1 ]  Eeff ,
[ K , Eeff ]  k1
(II)
The composition law for many sorts of
elementary particles
p  k   ( p  k ) 
t
( p1  p2 )0  Eeff
 E  1  lk1 k0
( p1  p2 )1  Pefft  1  lE p  k1
•
l
E  k0  k1k0
2
l
p  k1  Ep
2
One can easily check that
t
t
[ K , Pefft ]  Eeff
, [ K , Eeff
]  Pefft
[ K ,( p  k  )  ( p  k )  ]  0
•
The invariant quantity for the composite particle
M 2 =( E  k0 )2  ( p  k1 )2  l[k0k1 ( E  k0 )  Ep( p  k1 )]
The composition law for many sorts of
elementary particles
•
It can be further written into a compact form which depends
on the number of particles manifestly if
1. In the relativistic limit E
2. In an equilibrium state E
p, k0
k0 , p
k1
k1
l
M =Et  pt  Et pt ( Et  pt )
4
2
2
2
Et : E  k0  ...,
pt : p  k1  ...
l
[ K , Et ]  p t  ( Et 2  Et pt  pt 2 ),
8
l
[ K , pt ]  Et  ( Et 2  Et pt  pt 2 )
8
The composition law for many sorts of
elementary particles
•
A composite particle which contains n elementary particles
with dispersion relation (I) and m elementary particles with
dispersion relation (II)
l
Et pt
M =Et  pt  Et pt (  )
4
m n
2
2
2
l Et 2  pt 2 Et pt
[ K , Et ]  p t  (

),
8
m
n
l Et 2  pt 2 Et pt
[ K , pt ]  Et  (

)
8
n
m
Et : E  k0  ...,
pt : p  k1  ...
the difference between Lorentz violation
theory and deformed special relativity
•
General modified dispersion relations
m2  f 2 ( E , p ) E 2  g 2 ( E , p ) p 2
Eeff  f ( E , p ) E , peff  g ( E , p ) p
•
[ K , Peff ]  Eeff , [ K , Eeff ]  Peff
A point of view from rainbow spacetime
ds 2  
1
1
2
2
dt

dx
f 2 ( E , p)
g 2 ( E , p)


m   p p  g ( E ) p p
2
p0  E ,
p1  p
p0  Eeff ,
p1  Peff
p1  p2  p3  p4
p1  p2   ( p1  p2 )   p3  p4 
Summary
•
•
We propose a composition law of momenta for a multiparticle system in deformed special relativity. The form of
modified dispersion relation for a composite particle or
macroscopic object is not universal but dependent on the
number of elementary particles it consists of.
We introduce a notion of effective energy and momentum for
particles such that a specific deformed Lorentz boost
generator can be constructed. The benefits of such deformed
Lorentz boosts are twofold.
i) A composition law of momenta compatible with the deformed Lorentz
boost can be defined without introducing a notion of relative-locality of
the space of momenta.
ii) We provide a specific law of composition of momenta for interactions
involving non-universal dispersion relations such that the invariance
of the conservation law under the deformed Lorentz boost can be
easily achieved.