particle interactions with modified dispersion relations yi ling （凌意） ihep,cas &...

Particle Interactions with Modified Dispersion Relations

Yi Ling （凌意）

IHEP,CAS & Nanchang University2012两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:

Energy-momentum relation:

The existence of the minimal length that can be measured

2 201 /L v c L , 0v c L

2 2 2 2 4E p c m c

pL l2ˆ 8 ( 1)pj A j l j j

212

2 p px l p lp

2 2' , '

1 1

E vp p vEE p

v v

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.

• The relativity of inertial frames, two universal constants:

1) In the limit , the speed of a photon goes to a

universal constant, .

2) in the above condition is also a universal constant.

/ 0plE E

plE

c

2 2 2 2 2/ , / ,pl plf E E E g E E p m

As a result, the energy-momentum relation is usually modified to

1pl pE l

Introduction to deformed special relativity

Introduction to deformed special relativity• Lorentz transformation in standard special relativity (1+1 dim.)

• Lorentz transformation in deformed special relativity

' ( )

' ( )

E E vp

p p vE

2 2 2 2 2' 'm E p E p

, 010 ' '

, 01

E vvm E p E p E

E vv

1, ' 0, 1, ' .v E v E

2 22

2 2(1 ) (1 )

E pm

lE lE

2 2 2 2?2

2 2 2 2

' '

(1 ) (1 ) (1 ') (1 ')

E p E pm

lE lE lE lE

e.g.

Introduction to deformed special relativity• Lorentz transformation in deformed special relativity

(1 )0, ' '

1 [ (1 ) 1]

v Em E p

l v E

( )'

1 [ ( ) ]

( )'

1 [ ( ) ]

E vpE

l E vp E

p vEp

l E vp E

1 1If , then 'E E

l l

1If , then 'v c E

l

Introduction to deformed special relativity• Remark: the Lorentz transformation law depends on the form of

modified dispersion relation

' ( )2

' [ (( ( )( ))]2

lE E vp Evp

lp p vE Ep p vE E vP

2 2 2 2( )m E p lE p

Two open problems in DSR

• Usually the standard dispersion relation in special relativity will be modified with correction terms

However, such a modification in theory would lead to some severe problems…

2 2 2 2( )npm E p l E p


• A field theory with MDR is still absent. How to define the position space?

F

nonL

'p

px

1F'x

??px


• The soccer problem

2 2 2 21, ( / )pn m E p E M p

This modified dispersion relation is not applicable to composite particles and macroscopic objects. Thus it is not universal but particle number dependent.

2 2 2 2

, E , P

E P (E / )Pp

M nm nE np

M nM

19 5:10 10pM Gev g

the difference between Lorentz violation theory and deformed special relativity• Example: the derivation of the threshold value of the interaction

A. In standard special relativity

p p D

1 2 Dp p p p 2 21 2( ) ( )Dp p p p

Center-of-mass reference frame:

Laboratory reference frame:

2 2 2 2 21 2 1 2 1 2 1 1LHS ( ) ( ) =( ) 2 2p pE E p p E E p m m E

2 2 2 2D RHS ( ' ' ) ( ' ' ) =( ) ( )D D DE E p p E E m m

2 2

1

( ) 2

2D p

thp

m m mE

m

Laboratory reference frame:2 2 2 2

D

2 2 2

RHS ( ) ( ) = 2( )

= 2 ( )

D D D D

D D D

E E p p m m E E p p

m m m m m m

D D

p m

p m

the difference between Lorentz violation theory and deformed special relativity

B. In Lorentz violation theory special relativity

Center-of-mass reference frame: No sense


2 2 21 2 1 2 1 1LHS ( ) ( ) =2 2p p pE E p p m m E

2 2 2D

2 2

RHS ( ' ' ) ( ' ' ) =( ' ' )

' ' ( ) ( )( ) ( )

D D

DD D D

D

E E p p E E

m m m m m mm m

2 2

1

( ) 2 ( )( )

2

DD p D p

Dth

p

m m m m mm m

Em

2 2 2 2D

2

RHS ( ) ( ) = 2( )

=( ) ( )( )

D D D D D

DD D

D

E E p p m m E E p p

m m m mm m

D D

p m

p m

2 2 2 2 2( , ) ' ' ( ', ')'m E p E p E p E p

2 0p

CMBp p

2pth

p GZK

m mE E

E

2 2 20E m p 2 2 2

0 ( , )E m p E p

42p

pp

thp

pm mE

E

m

m

E

410E ev 17 2 2 20 210 ( ) (10 )ev E ev

2110 10thp GZKE ev E


e.g.


C. In deformed special relativity

Center-of-mass reference frame: the same result can be obtained


2 2 21 2 1 1 2 2 1LHS ( ) ( 1 1 ) =2 2p pE E lE p lE p m m E

2 2

1

( ) 2

2D p

thp

m m mE

m

2 2D

2 2

2

RHS ( ) ( 1 1 )

= 2( 1 1 )

=( )

D D

D D D D

D

E E lE p lE p

m m E E lE p lE p

m m

1

1 DD D

lE p m

mlE p

2 2 2 2 2 2 2( ) ' ' ( ') 'm E p lE p E p lE p 2 0p

I.


11

112

DSR thth

th

EE

lE

2 2 2 2 2 2 2( ) ' ' ( ') 'm E p lE E E p lE E II.

The composition law in special relativity revisited

• Consider two elementary particles which may have different masses

• We define a composite particle through a process in which the covariant momentum is conserved

2 2 2 2 2 21 1 1 2 2 2, m E p m E p

1 2 1 2( )p p p p

An invariant quantity of the composite particle is2

1 2 1 2 1 2 1 2

2 21 2 1 2

2 2 2 2 21 2 1 2 1 2 1 2

( ) ( ) ( )( )

( ) ( )

2 ( )

M p p p p p p p p

E E p p

m m m m p E E p

(1+1 dim.)

The composition law in SR revisited

Some remarks:

• If we define , we still have

• In general

• They are equal if and only if

• In general, the composite particle could not be elementary.

1 2M m m

1 2 1 2: , :c cE E E p p p 2 2 2

c cM E p

1 2v v

Universal

M

M


It is straightforward to extend it to the composition of many particles:

• If we define , we still have

• In general

1 2 3 ...M m m m

1 2 1 2: ..., : ...t tE E E p p p

2 2 2t tM E p

1 2 3 1 2 3( ) ( )p p p p p p


• The transformation law of the energy and momentum under

the Lorentz boost in 1+1 space time

Thus

It is easy to check that for a composite particle

[ , ] , [ , ]K E p K p E

2 2[ , ] 2 2 0K E p Ep pE

2 21 2 1 2[ , ] [ ,2( )] 0c cK E p K E E p p

2 2[ , ] 0t tK E p


• An interaction involving n incoming particles and m outgoing particles

•The conservation law of momentum is preserved under the Lorentz boost in the sense that

1 2 1 2

1 2 1 2

[ , ( ... ) ( ' ' ... ')]

( ... ) ( ' ' ... ') 0n m

n m

K E E E E E E

p p p p p p

1 2 1 2... ' ' ... 'n mp p p p p p

1 2 1 2

1 2 1 2

[ , ( ... ) ( ' ' ... ')]

( ... ) ( ' ' ... ') 0n m

n m

K p p p p p p

E E E E E E

The composition law in DSR

• Consider an elementary particle with a modified dispersion relations as

• Obviously, it is not an invariant quantity under the standard Lorentz boost

• 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.

2 2 2 2( )m E p lE p

2 2 2 2 2[ , ( ) ] (2 ) 0K E p lE p lp E p

2 2[ , ] , [ , ] ( )2 2

l lK E p Ep K p E p E

2 2 2 2[ , ( ) ] 0 0( )K E p lE p l

2 2 2 2 2 2( ) ' ' ( ') 'E p lE p E p lE p


• However, such a choice is not unique. An alternative deformation

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.

2 2[ , ] , [ , ] ( 2 )2

lK E p K p E p E

1 2 1 2( )p p p p

1 2 1 2[ , ( ) ( ) ] 0K p p p p

1 2 0 1 2

1 2 1 1 1 22

( )

( )

p p E E

p lE pp p p

1 2 1 2[ , ( ) ( ) ] 0K p p p p


• 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 a

voided.


• We introduce a notion of effective momentum

1: 1 (1 )

2effp lE p lE p

2 2[ , ] , [ , ] ( )2 2

l lK E p Ep K p E p E

[ , ] , [ , ]eff effK E p K p E

2 2 2 2( )m E p lE p 2 2 2effm E p p p

0 1: , : effp E p p


• We propose a composition law for two elementary particles

1 2 1 2( )p p p p

1 2 0 1 2

1 2 1 1 2 1 2 1 1 2 2

( )

( ) ( )2

c

eff eff eff

p p E E E

lp p P p p p p E p E p

21 2 1 2 1 2 1 2

2 21 2 1 1 2 2

2 21 2 1 2 1 1 2 2

2 21 2 1 2 1 2 1 1 1 1

( ) ( ) ( )( )

( ) ( 1 1 )

( ) [ ( )]2

=( ) ( ) ( )( )

M p p p p p p p p

E E lE p lE p

lE E p p E p E p

E E p p l p p E p E p

Remark: it is interesting to show that if and only if 1 2M m m

1 2v v


• One can easily check the following identities for a composite particle

1 2 1 2[ , ( ) ( ) ] 0K p p p p

[ , ] , [ , ]c eff eff cK E P K P E

1 2 1 2[ , ( ) ] 0K p p p p

2 21 2 1 2 1 2 1 1 1 1

2 21 2 1 2 1 2 1 1 1 1

( ) ( ) ( )( )

=( ' ') ( ' ') ( ' ')( ' ' ' ')




• It can be further written into a compact form which depends on the number of particles manifestly given that

1.

2. In the relativistic limit,

3. In the non-relativistic limit,

2 2 21 2 1 2 1 2 1 1 2 2=( ) ( ) ( )( )M E E p p l p p E p E p

1 2E E1 1 2 2,E p E p

1 1 2 2 1 2, ,E p E p m m

2 2 2 2

2= t

t t t

EM E p l p

1 2 1 2: ..., : ...t tE E E p p p


• One can easily check

[ , ] , [ , ]c eff eff cK E P K P E

2: 1 (1 )

4eff t t t t

l lP E p E p

2 2 2 2

2= t

t t t

EM E p l p

2 2[ , ] , [ , ]4

( )4t t t t t t t t

l lK E p E p K p E p E


• Extension to arbitrary composite particle or macroscopic object which is composed of n elementary particles

2 2 21 2 1 2

1 2 1 1 2 2

2 2t eff

=( ... ) ( ... )

( ... )( ... )

E P

n n

n n n

M E E E p p p

l p p p E p E p E p

t 1 2

eff 1 2 1 1 2 2

E := ...

P : ( ... ) ( ... )2

n

n n n

E E E

lp p p E p E p E p

[ , E ] P , [ , P ] Et eff eff tK K


• For a macroscopic object with n particles in thermal equilibrium, it is reasonable to assume that

then

: 1 E P (1 E )P2eff t t t tn n

l lP

2 2 2 2E=E P Ptt t tM l

n

2 2[ , E ] P E P , [ , P ] E2

(P E )2t t t t t t t tn n

l lK K

1 2 ..... nE E E kT


• A general interaction

P( ) P( ) (P( ) P( )) P( ') P( ')n m n m n m

E( ) E( ) E( ') E( ')

P( ) +P( ) P( ') +P( ')eff eff eff eff

n m n m

n m n m

t t t t

t t t tt t t t

E ( ) E ( ) E ( ') E ( ')

E ( ) E ( ) E ( ') E ( ')1 P ( )+ 1 P ( ) 1 P ( ')+ 1 P ( ')

' '

n m n m

l n l m l n l mn m n m

n m n m

The composition law for many sorts of elementary particles• Two elementary particles with different dispersion relations

• We introduce a notion of effective energy

2 2 2 2( )m E p lE p 2 2 2 20 1 0( )k k lk k

2 20 1 0 1 1 0 1 0[ , ] ( ), [ , ]

2 2

l lK k k k k K k k k k

1 0 1 0

1: 1 (1 )

2effE lk k lk k

1 1[ , ] , [ , ]eff effK k E K E k

(I) (II)

The composition law for many sorts of elementary particles

• One can easily check that

• The invariant quantity for the composite particle

[ , ] , [ , ]t t t teff eff eff effK P E K E P

2 2 20 1 0 1 0 1=( ) ( ) [ ( ) ( )]M E k p k l k k E k Ep p k

( )p k p k

1 2 0 1 0 0 1 0

1 2 1 1 1

( ) 12

( ) 12

teff

teff

lp p E E lk k E k k k

lp p P lE p k p k Ep

[ , ( ) ( ) ] 0K p k p k

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

2. In an equilibrium state

0 1, E p k k

2 2 2

4= ( )t t t t t t

lM E p E p E p

0 1: ..., : ...t tE E k p p k

0 1, E k p k

2 2

2 2

[ , ] ( ), 8

, (8

[ ] )

t t t t t t

t t t t t t

lK E p E E p p

lK p E E E p p

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)

2 2 2

4= ( )t t

t t t t

l E pM E p E

m np

0 1: ..., : ...t tE E k p p k

2 2

2 2

[ , ] ( ), 8

8

[ , ] ( )

t t t tt t

t t t tt t

l E p E pK E p

l E p E pK p E

m n

n m

the difference between Lorentz violation theory and deformed special relativity• General modified dispersion relations

• A point of view from rainbow spacetime

2 2 2 2 2( , ) ( , )m f E p E g E p p

[ , ] , [ , ]eff eff eff effK P E K E P

( , ) , ( , )eff effE f E p E p g E p p

2 2 22 2

1 1

( , ) ( , )ds dt dx

f E p g E p

2 ( )m p p g E p p

0 1, p E p p

0 1, eff effp E p P

1 2 1 2 3 4( )p p p p p p

1 2 3 4p p p p

Summary

• We propose a composition law of momenta for a multi-particle 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.

particle interactions with modified dispersion relations yi ling （凌意） ihep,cas &...

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