Muon electric dipole moment induced by unparticles.

Muon electric dipole moment induced by unparticlesMuon electric dipole moment induced by unparticles

VIII Latin American Symposium on High Energy Physics, Valparaíso, Chile, 6-12 December 2010

1. Unparticle Physics (UP) overview2. Moun g-23. Unparticle contributions to af

4. Constraints on UP parameters from muon g-25. Fermion electric dipole moment (EDM)6. Unparticle contributions to EDM7. Mathematical structure of UP parameters8. Conclusions

SU(2)L

U(1)Y

BZG Λ<v

Λ=v

Λ=Λu

Λ=Λplanck

The standard model does not have the property of scale invariance… But there could be a sector of the theory, as yet unseem, that is exactly scale invariant and very weakly interacting with the rest of the standard model… unparticles.

Unparticle Physics★

★H. Georgi, Phys. Rev. Lett. 98, 221601 (2007)

E.W.

4

SM

SM

d d

SMd dC

Λ

u

U BZM O UL O OMu

u

u

LM u

:O

SMOBZ

M u

dSM dBZ 4

ΛMU

Scheme

ΛΛu

renormalizable couplingsdimensional transmutation

nonrenormalizable couplings


The standard model does not have the property of scale invariance… But there could be a sector of the theory, as yet unseem, that is exactly scale invariant and very weakly interacting with the rest of the standard model… unparticles.

Unparticle Physics★Scalar-pseudoscalar

Vector-Axial Tensor

F( p2 )

Ad

2sin(d )( p2 )d 2

A

d

16 5/ 2

(2 )2d

(d 12)

(d 1)(2d)

F ( p2 )

F( p2 ) F

, ( p2 )F

( p2 )T ,

g p p / p2 ,

T , 1

2[ 2

3 ]

where

ijk COU

Λu

dBZ

M u

dSM dBZ 4

LU s

ijs

ΛUd 1

fif

jOU i

ijp

ΛUd 1

fi5 f

jOU

LU s

ijv

ΛUd 1

fi f

jOU

ija

ΛUd 1

fi

5 fjOU

LU s

ijT

4ΛUd

i(D D )

jOU

Kingman Cheung, Wai-Yee Keung and Tzu-Chiang Yuan, Phys. Rev. D.76 055003 (2007)


L

spin-1/2

eaf

4mf

( f f )F

f

f

A (q2 )

(q2)F1(q

2) iF2(q2) q2mf

Λ

a a

exp aSM 255(63)(49)10 11

The E821 Muon (g-2), Brookhaven National Laboratory, Phys. Rev. Lett. 92 (2004) 161802

The relevant interaction Lagrangian of the fermion Magnetic Dipole Moment al is given by

The anomalous magnetic moment of a fermion l is related to the photon-lepton vertex function Γμ as follows:

It is well known that between the experimental value and the SM prediction there is a small discrepancy


F s,p(d ,r

fi

12 ) dxx1 d (1 x)d (1r

fi

12 x)(1 r

fix)d 2

0

1

F v,a (d ,r

fi

12 ) dx d yy1 d (1 x y)d 1(1 r

fi(x y))d 3

0

1 x

0

1

3(1 3x 3y)

r

fi

32 (x y) 1 x y 4(x y)2 d(x y 1)(1 2x 2y)( )

r

fi

12 x y 3 d(x y 1) 8(x y)2( )

r

fi2 x y 3(x y)2 d(x y 1)(1 3x 3y)( )

a

f

U k | fik |2

mi2d 2

ΛU2d 2

ie, ,

Adr

fi

12

16 2 sin(d )F k (d ,r

fi

12 )

L

spin-1/2

eaf

4mf

( f f )F

ud( p k) A

(q2 )

f ( p ')

f ( p)

fi(k q)

fi(k)

New contributions to af from unparticles are generated at the one-loop level. The results are

Scalar-pseudoscalar

Vector-Axial

Andi Hektor, Yuji Kajiyama and Kristjan Kannike, Phys. Rev. D78 053008 (2008)

where rfi=mf2/mi

2 and the functions F are given by:

from the chirality structure, η=+1(-1) for scalar and vector (pseudo scalar and axial) unparticles.


1| i

s |2

(is )2

| ip |2

(ip )2

| iv |2

(iv )2

| ia |2

(ia )2

ie, ,

(ik )2

ΛU2d 2

mi2d 2

16 2 sin(d )(a 207.512)

Adr

fi

12 F k (d ,r

fi

12 )

From the total contributions to af from unparticles

we obtain a constraint on the λμik parameters

given by a quadric equation of twelve dimensions

207.512 stands for the systematic and experimental errors of Δaμ with 2 standard deviations. Numerical evaluation shows that αμe

k is undefined in the interval where dU guarantees unitarity, Also;

(

s,v,a )2 ,(s,v,a )2 0

(

p )2 ,(p )2 0

cosh2 |

s |2

|s |2

| s |2

|s |2

| v |2

|v |2

| v |2

|v |2

| a |2

|a |2

| a |2

|a |2

sinh2 |

p |2

|p |2

| p |2

|p |2

We can however project the quadric surface by requiring that λμe

k=0, thus the resulting constraint equation represents an eight dimensional hyperboloid hipervolume with parametric equations (Ω6 coshϑ, Ω2 sinhϑ), where:

(0,0, z1)

(x1, y

1,0)

(x, y, z)

Ω6 and Ω2 represent a six and two dimensional ellipses.


a

fUP a

f

U s af

U p af

U v af

U a

a

fUP a

f

U s af

U p af

U v af

U a

1| i

s |2

(is )2

| ip |2

(ip )2

| iv |2

(iv )2

| ia |2

(ia )2

ie, ,

From the total contributions to af from unparticles

1 TeV

10 TeV

100 TeV

1000 TeV

✜Kingman Cheung, Wai-Yee Keung and Tzu-Chiang Yuan, Phys. Rev. D.76 055003 (2007)

du Λu(TeV)

2.0 1.35

1.8 4

1.6 23

1.4 660

i

s,v,a | cosh || is,v,a |i

s,v,a | cosh |

|i

p sinh || ip ||i

p sin |

Limit on Λu with λμμv=1

from monophoton production data of σ(e+e- γ + X)≈0.2 pb at LEP2(95% C.L.)✜

a 207.512


αμμs αμτ

s αμμv αμτ

v αμμa αμμ

a

i

s,v,a | cosh || is,v,a |i

s,v,a | cosh |


|i

p sinh || ip ||i

p sin |

but, what is the allowed interval for the hyperbolic angle ?


ud( p k) A

(q2 )

f ( p ')

f ( p)

fi(k q)

fi(k)

(q2 )F1(q2 ) iF

2(q2 ) q2m

f F3(q

2) q2mf

5 Λ

The presence of a non-zero df would mean that there are both parity and time-reversal violation. In a world with perfect CPT symmetry, a search for df would also be a

direct test of CP symmetry

J. Bailey et al., J. Phys. G 4, 345 (1978) | d

exp | 3.73.410 19

The dipole magnetic moment of a fermion l is related to the T violating form factor F3(q2) term of the photon-lepton vertex function Γμ as follows:

The experimental limit on the electric dipole moments [e-cm] with 95% C.L. and the theoretical standard model values are:

| d

SM |10 35

| dnexp |; 6.3×10−26

| deexp |: 1.6×10−27 | d

eSM |10 38

| dnSM |10 31

L

spin-1/2

eaf

4mf

( f f )F

i

2d

f( f 5

f )F


ud( p k) A

(q2 )

f ( p ')

f ( p)

fi(k q)

fi(k)

The presence of a non-zero df would therefore signify the existence of both parity and time-reversal violation. In a world with perfect CPT symmetry, a search for df would

also be a direct test of CP symmetry


exp | 3.73.410 19

| d

SM |10 35

L

spin-1/2

eaf

4mf

( f f )F

i

2d

f( f 5

f )F

dfk k '

Ad

16 2 sin(dU )[(

fik )*

fik '

fik (

fik ' )*]r

fi

12

mi

2dU 2

ΛU

2dU 2Gk k '(dU ,r

fi

12 )

ie, ,

Gs-p(dU ,r

fi

12 ) i dxx1 dU (1 r

fix)dU 2(1 x)dU

0

1

Gv-a (dU ,r

fi

12 ) dx y1 dU (1 3x 3y)(1 x y)dU 1[1 r

fi(x y)]dU 3{3 r

fi[2 x y dU (x y 1)]}

0

1 x

0

1

Unparticle contribution to EDM is induced by interference of scalar (vector) and pseudo scalar (axial) operators:

where

(

fik )*

fik '

fik (

fik ' )* 2 |

fik ||

fik ' | cos(

fikk ' )

fikk '

fik

fik '



exp | 3.73.410 19

| d

SM |10 35

(

sp ,sp )( 1

4 , 1

4 )

(

va ,va )( 1

4 , 1

4 )

i 1

16

(

va ,va )( 1

6 , 1

3 )

(

sp ,sp )( 1

3 , 1

6 )

(

va ,va )( 1

3 , 1

6 )

we used the parameterization of the hyperboloid (Ω6cosh ϑ, Ω2sinh ϑ) to correctly use the phenomenological bound of effective parameters discussed above, and we found:

dfk k '

Ad

16 2 sin(dU )[(

fik )*

fik '

fik (

fik ' )*]r

fi

12

mi

2dU 2

ΛU

2dU 2Gk k '(dU ,r

fi

12 )

ie, ,

fikk '

fik

fik '

our prediction is well above the experimental bound!

(

sp ,sp )( 1

6 , 1

3 )

i 1

16


(

sp ,sp )( 1

4 , 1

4 )

(

va ,va )( 1

4 , 1

4 )

(

va ,va )( 1

6 , 1

3 )

(

sp ,sp )( 1

3 , 1

6 )

(

va ,va )( 1

3 , 1

6 )

i 1

16

i 1

16

(

sp ,sp )( 1

6 , 1

3 )

dfk k '

Ad

16 2 sin(dU )[(

fik )*

fik '

fik (

fik ' )*]r

fi

12

mi

2dU 2

ΛU

2dU 2Gk k '(dU ,r

fi

12 )

ie, ,

fikk '

fik

fik '

i

kk ' 2

i

sp isp 0

i

s,p,v,a 0


exp | 3.73.410 19

| d

SM |10 35

d

fk k ' 0

| d

k k ' || dexp |

| d

k k ' || dexp | ✗

✗

✔

we will study another particular case

If ? i

kk ' 2

all effective parameters are real numbers


dfk k '

Ad

16 2 sin(dU )[(

fik )*

fik '

fik (

fik ' )*]r

fi

12

mi

2dU 2

ΛU

2dU 2Gk k '(dU ,r

fi

12 )

ie, ,

fikk '

fik

fik '

i 1

16

i 1

16

φ

φ 89.9999973º

φ

v φv 89.9999999º

| d

exp | 3.73.410 19

| d

exp | 3.73.410 19

There are other possibilities for the hyperbolic angle:

| |1

i

va 2

and λμip small and vanishing vector-axial contribution to EDM

i

va 2

| |1 and λμip small and nonvanishing vector-axial contribution to

EDM sinh : 1

i

kk' 2

and All effective parameters about the same order of magnitude


VIII Latin American Symposium on Higth Energy Physics, Valparaíso, Chile, 6-12 December 2010

1. We find phenomenological bounds for some effective parameters (λμik)

on framework of Unparticles Physics Model.2. The muon electric dipole moment(dμ) induced by unparticles operators

show a complex structure of the coupling parameters. Therefore it reveals an new phases of CP violation.

3. Since the muon electric dipole moment depends very much on the different phases of the effective parameters, it is more sensitive than the standard model approach.

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