fulltext

A new contribution to the flavour-changing lepton-photon vertex (*)

D. PALLE

Department of Theoretical Physics, Rugjer Bosˇkovic´ Institute P.O. Box 1016, Zagreb, Croatia

(ricevuto il 30 Ottobre 1996; approvato il 23 Dicembre 1996)

Summary. — We show that the correct perturbation theory for mixed fermion states leads to nonvanishing contributions of the dimension-four vertex operators for flavour-changing transitions. Their contributions to the amplitude are of the same order of magnitude as the dimension-five vertex operators. Considerations are valid irrespective of the electroweak model.

PACS 11.10.Gh – Renormalization.

PACS 12.15.Ff – Quark and lepton masses and mixing. PACS 13.40.Hq – Electromagnetic decays.

It seems that current measurements in astro- and particle physics (solar and atmospheric neutrinos, COBE data, LSND data, observed ionization of the Universe, etc.) strongly suggest that neutrinos should be massive particles. In this paper we show that the correct treatment of the flavour mixing of massive leptons in perturbation theory results in new terms in the decay amplitude due to the dimension-four operators.

Contributions to flavour-changing radiative decays of leptons in the perturbative calculations of the majority of electroweak models appear through quantum loops owing to the existence of flavour-changing charged weak currents. We can write the general form of the f1K f21 g amplitude (vertex) with the following Lorentz

structures [1]: Mm

(

f1(p1) Kf2(p2) 1g(q)

)

4 2ı u(p2) Gm ; f1f2u(p1) 4 (1) 4 2ı u(p2)[gm

(

F1L(q2) PL1 F1R(q2) PR) 1ısmnqn

(

F2L(q2) PL1 F2R(q2) PR

)

1 1qm

(

F3L(q2) PL1 F3R(q2) PR

)

] u(p1) , where PL , Rf 1 2( 1 Z g5) .

(*) The author of this paper has agreed to not receive the proofs for correction.

Previous calculations [1] of the decay amplitude were focused on the F2L , R form

factors because the F3L , R form factors gave zero contribution to f1K f21 g. However,

the F1L , Rform factors were claimed to give contributions that should vanish because of

the conservation of the electromagnetic current. We show that the latter claim is incorrect and that the F1L , R form factors give contributions to the amplitude

comparable with that of the F2L , R[1].

The most natural choice for the renormalization scheme in electroweak theory is the on-shell renormalization scheme [2]. In this scheme, we repeat the most important ingredients concerning mixed fermion states. The on-shell renormalization conditions for mixed fermions (propagators) are

Sijren[ pole ] 4

dij

mi2 pO

, where SijfFourier transform ıa 0 NTci(x) cj(y) N0b . (2)

These renormalization conditions ensure a correct input for fermion masses and a correct form of renormalized propagators. These conditions can be written in a more transparent form by introducing on-shell spinors [2]:

.

`

`

/

`

`

´

Kijrenu(mj) 40 , u(mi) Kijren4 0 ,

{

1 mi2 pO

Kiiren

}

u(mi) 4u(mi) ,

u(mi)

{

Kiiren 1

mi2 pO

}

4 u(mi) ,

definition : Kijren(p) f ]Sren(p)(ij21, i , j 4flavour indices . (3)

For Majorana fields, one obtains the same formulae for on-shell renormalization conditions, but with a different number of conditions in comparison with Dirac fermions [2].

Furthermore, any correctly quantized electroweak theory preserves the BRST symmetry [2]. Thus, the generalized Ward-Takahashi identity for the flavour-changing lepton-photon vertex can be written as

qm_G m ; ij ren (p 1q, pNq) 42eSren ij (p) 1eSrenij (p 1q) . (4)

From the general Lorentz structure of the lepton-photon amplitude one can see that only the F1L , R form factors are related to flavour off-diagonal self-energies

through Ward-Takahashi identities [2, 3]. These identities are valid for renormalized Green functions, and the on-shell renormalization conditions (3) should be used to uniquely fix finite terms of self-energies:

. / ´ Srenij (p) u(p , mj) 40 , u(p , mi) Srenij (p) 40 . (5)

145

It is now evident that the electromagnetic current remains conserved because of the on-shell conditions:

qm_{u(p , m}

i) Grenm ; ij(p 1q, pNq) u(p1q, mj) 40 .

In addition, we can evaluate the F1L , R(q2) form factors from the on-shell conditions

at q2

4 0. Let us write the most general form of the renormalized self-energy (flavour indices suppressed):

Sren_{(p) f}

₍

_s

1(p2) 1dZ1

)

p_{O P}L1 (s2(p2) 1dZ2) p_{O P}R1

(6)

1

(

s3(p2) 1dZ3

)

PL1

(

s4(p2) 1dZ4

)

PR.

Inserting the above into the Ward-Takahashi identity and setting pm4 0 and q24 0, we obtain the following expressions for the form factors:

. / ´ F1L( 0 )f1cf24 e

(

s1( 0 ) 1dZ1

)

, F1R( 0 )f1cf24 e

(

s2( 0 ) 1dZ2

)

. (7)

The four renormalization constants are defined by the on-shell renormalization conditions (5) ( fi fermion has a mass mi):

.

`

`

`

`

/

`

`

`

`

´

dZ14 1 m122 m22 [2m1 2 s1(m1 2 ) 1m2 2 s1(m2 2 ) 1m1m2

(

s2(m1 2 ) 2s2(m2 2 )

)

2 2m2

(

s3(m12) 2s3(m22)

)

1 m1

(

s4(m12) 2s4(m22)

)

] , dZ24 1 m122 m22 [m1m2

(

s1(m12) 2s1(m22)

)

2 m12s2(m12) 1m22s2(m22) 1 1m1

(

s3(m12) 2s3(m22)

)

2 m2

(

s4(m12) 2s4(m22)

)

] , dZ34 1 m122 m22 [m2m12

(

s1(m12) 2s1(m22)

)

2 m1m22

(

s2(m12) 2s2(m22)

)

1 1m22s3(m12) 2m12s3(m22) 2m1m2

(

s4(m12) 2s4(m22)

)

] , dZ44 1 m122 m22 [2m1m22

(

s1(m12) 2s1(m22)

)

1 m2m12

(

s2(m12) 2s2(m22)

)

2 2m1m2

(

s3(m12) 2s3(m22)

)

1 m22s4(m12) 2m12s4(m22) ] . (8)

For definiteness, let us write the interaction Lagrangian with flavour-changing lepton charged currents with Dirac neutrinos, in a form that is valid irrespective of the symmetry-breaking mechanism: LI4 g k2

!

ijW m niUijgmPLlj1 g k2 MW

!

_ijf1_n i[mljUijPR2 mniUijPL] lj1h.c. ,

n 4neutrino,

l

4 charged lepton , f14 Nambu-Goldstone scalar . Then we have to find renormalized neutrino self-energies with the above interaction Lagrangian and the renormalization conditions (5).

In the ’t Hooft-Feynman gauge, one can easily verify that (9) s1 ; 2 ; 3 ; 4(p2) 42ı g2 32 p2

!

l UilU *ml3

{

g

22 2 ml2 MW2

h

B1(p2; ml2, MW2) ; 2 mnimnm MW2 B1(p2; ml 2_{, M} W2); ml2 MW2 mniB0(p 2_{; m} l 2_{, M} W2); ml2 MW2 mnmB0(p 2_{; m} l 2_{, M} W2)

}

,

]m , i , l( are flavours of ]n1, n2, charged lepton

l

( .

The scalar functions B0(p2) and B1(p2)[3] have to be evaluated for p2b(M12 M2)2, so

it would be useful to make an expansion in the vicinity of p2

4 0: ı 16 p2] 1 ; pm( B0 ; 1(p 2_{; M} 1, M2) 4



d4k ( 2 p)4 ] 1 ; km( (k2 2 M121 ıe)

(

(k 1p)22 M221 ıe

)

, B0(p2; M1, M2) 4u(M1, M2) 1b2p21 b4p41 O(p6) , B1(p2; M1, M2) 4h(M1, M2) 1 1 2

g

b4 M222 M12 2 2 b2

h

p 2 1 O(p4) , b24 1 2 M121 M22 (M122 M22)2 1 2 M1 2 M22 (M122 M22)3 ln M2 M1 , b44 M141 10 M12M221 M24 6(M122 M22)4 2 2 M1 2 M22(M121 M22) (M122 M22)5 ln M2 M1 ,

u , h functions contain ultraviolet infinity .

From the above we can evaluate the leading terms of the F1L , R( 0 )f1cf2form factors

(m2bm1 and mlbMW)

.

`

/

`

´

F1L( 0 )i c mC 2ı eGF 4k2p2m1 2

!

l UilUml*

u

2 1 8 m_l2 MW2 1 3 m_l2 MW2 lnMW ml

v

, F1 R ( 0 )i c mC 2ı eGF 4k2p2m1m2

!

_l UilUml*

u

5 8 ml2 MW2 2 3 ml2 MW2 lnMW ml

v

, (10)

and at the same time (see eq. (10.27) in the book of ref. [1])

.

`

/

`

´

F2L( 0 )i c mC 2ı eGF 4k2 p2m2

!

_l UilUml*

g

3 4 ml 2 MW2

h

, F2R( 0 )i c mC 2ı eGF 4k2 p2m1

!

_l UilUml*

g

3 4 ml2 MW2

h

. (11)

147

It is straightforward to evaluate the rate

G

(

f1(m1) Kf2(m2) 1g

)

4 m122 m22 4 pm12 [ 2 p2 (p 1

k

p2 1 m22)(NF2VN21 NF2AN2) 1 1NF1VN2(

k

p21 m222 m2) 1NF1AN2(

k

p21 m221 m2) 2m2p(F1VF2V* 1F1V* F2V2 2F1AF2A* 2F1A* F2A) 1p(p1

k

p21 m22)(F1 V F2 V * 1F1 V * F2 V 1 F1 A F2 A * 1F1 A * F2 A ) ] , where p 4 m1 2 2 m22 2 m1 , FiV , A4 FiR6 FiL 2 .

Thus, the contributions from the F1L , R(n1cn2) and F2L , R(n1cn2) form factors to

the rate of n1K n21 g (or m K e 1 g) are of the same order of magnitude [1] (however

the neutrino flavour diagonal charges vanish FL , R

1 ( 0 )ni4 0).

To conclude, one can say that, if perturbation theory is correctly applied to mixed fermion states, one has to calculate all F1 , 2L , R( f1cf2) form factors to evaluate decay

amplitudes in any electroweak model with lepton mixing. In our calculation we respect Lorentz, gauge and BRST symmetries, as well as the renormalization conditions for mixed fermion states. A correct evaluation of the neutrino lifetime for certain electroweak models could be of great importance in the theoretical cosmology and astrophysics [1, 4], with Sciama’s decaying neutrino hypothesis as an example [5].

R E F E R E N C E S

[1] LEEB. W. and SHROCKR. E., Phys. Rev. D, 16 (1977) 1444; MOHAPATRAR. N. and PALP. B.,

Massive Neutrinos in Physics and Astrophysics (World Scientific, Singapore) 1991, and

references therein.

[2] AOKIK. et al., Prog. Theor. Phys., Suppl., 73 (1982) 1.

[3] BO¨HMM., SPIESBERGERH. and HOLLIKW., Fortschr. Phys., 34 (1986) 687.

[4] KOLBE. W. and TURNERM. S., The Early Universe (Addison-Wesley Pub. Co., California, USA) 1990.

[5] SCIAMA D. W., Nature, 346 (1990) 40; Modern Cosmology and the Dark Matter Problem (Cambridge University Press, Cambridge, UK) 1995.