

 20 Dec 1994

The Two Loop Long Range Effect on the

Proton Decay Effective Lagrangian

Takeshi Nihei and Jiro Arafune

Institute for Cosmic Ray Research, University of Tokyo,

Midori-cho, Tanashi-shi, Tokyo 188 JAPAN

Abstract We calculate the two loop long range effect on the proton decay effective Lagrangian. Numerical calculation for suppression factor gives A

L

(2-loop) =

0:321 for the value of the strong coupling constant ff

s

(m

Z

) = 0:116. Two loop

effect to more general effective Lagrangian is also given.

1 Introduction

Proton decay is one of the most important prediction of grand unified theory (GUT). The main decay mode p ! K

+

+ * in the minimal supersymmetric (SUSY)

SU(5) GUT has been searched for with the underground experiments, and the lifetime bounds of this mode have been given by Kamiokande and IMB as o/ ? 1:0 \Theta 10

32

years and o/ ? 0:62 \Theta 10

32

years, respectively.[1] These bounds are rather close to the

theoretical prediction[2][3][4] in case of the SUSY breaking parameters of less than 1 TeV and the Superkamiokande is expected to improve these bounds drastically in the near future. It is now desirable to make a more careful estimation of the theoretical bounds.

In the following we estimate the two loop QCD correction to the effective proton decay Lagrangian. This correction is estimated by solving the renormalization group equation from the weak energy scale to the low energy scale, say 1 GeV.

1

In order to formulate the renormalization group equation we need to estimate the two loop renormalization constant of the effective Lagrangian, which is not calculated so far though the two loop effect of the quark wavefunction, the quark masses and fi-function are calculated in the literature[5]; this is why reliable calculation with the two loop approximation is not made yet. In this paper, the renormalization constant is calculated for rather general effective Lagrangian for proton decay in the two loop approximation and the renormalization group equation of the coefficient of the proton decay effective operator is given. The numerical estimation of the effect is also made.

2 The renormalization constant of the effective

operator

The dominant proton decay effective Lagrangian at the weak scale is written as[6]

L

eff

= C

ijkl

ffl

fffifl

(q

ff

i

; q

fi

j

)(q

fl

k

; `

l

) + h:c:; (1)

where q and ` are left-handed Weyl spinors of quark and lepton field respectively. Here ff; fi; fl are color indices, i; j; k; l are flavor indices and the bracket ( , ) means the inner product for Weyl spinors.

We shall also consider the proton decay Lagrangian with mixed quark chiralities, like

L

0

eff

= D

ijkl

ffl

fffifl

(q

c

ff

i

; oe

_

q

fi

j

)(q

c

fl

k

; oe

_

`

l

) + h:c:; (2)

or

L

00

eff

= E

ijkl

ffl

fffifl

(q

c

ff

i

; q

c

fi

j

)(q

fl

k

; `

l

) + h:c:; (3)

to discuss the two loop corrections to different four fermion vertices. Here oe

_

denotes

the Pauli matrices and q

c

is the Weyl spinor for antiquark field.

The long range effect of renormalization on the coefficient C

ijkl

in Eq.(1) is to be

evaluated by the ratio of its value at 1GeV to that at the weak scale m

Z

' 90GeV,

2

that is,

R

L

=

C

ijkl

(1GeV)

C

ijkl

(m

Z

)

: (4)

In literatures, however, the following quantity is more frequently used:

A

L

=

"

C

ijkl

m

u

m

d

#

_ = 1GeV "

C

ijkl

m

u

m

d

#

_ = m

Z

; (5)

where _ represents the renormalization point and m

u

or m

d

means the up-quark or

the down-quark mass respectively.

The one loop correction has already been given as[2]

A

L

(1 loop) =



ff

s

(m

b

)

ff

s

(m

Z

)

!

\Gamma

18

23



ff

s

(m

c

)

ff

s

(m

b

)

!

\Gamma

18

25



ff

s

(1GeV)

ff

s

(m

c

)

!

\Gamma

18

27

= 0:45: (6) Here the strong coupling constant at the renormalization point _ = m

Z

(m

Z

' 90GeV)

is taken as ff

s

(m

Z

) = 0:116.

We take account of only QCD correction since the strong coupling constant is much larger than the weak or electromagnetic coupling constants. Up to the two loop level, there are six different types of Feynman diagrams for the QCD correction as given in the Fig.'s 1(a), 1(b), \Delta \Delta \Delta , 1(f).

Each type of diagram in Fig.1 is calculated by straightforward calcutations and the result is given in Table 1. In the table, MS scheme is adopted[7],

1

ffl

=

1

ffl

\Gamma fl + ln(4ss); (7)

where n = 4 \Gamma 2ffl ( n-dimensional space-time ) and fl is an Euler number.

In Table 1, we take A = g

2

s

=(4ss)

2

= ff

s

=(4ss) ( g

s

is a strong coupling constant )

and j as the gauge parameter in the gluon propagator in the following form:

D

_*

F fffi

(k

2

) =

\Gamma iffi

fffi

k

2



g

_*

\Gamma j

k

_

k

*

k

2

!

: (8)

3

From Table 1, we obtain the renormalization constant Z

C

as

Z

C

= 1 \Gamma 4

A

ffl

+

A

2

ffl

`

\Gamma

74

3

+

8N

F

9

+

29

9

j \Gamma

3

4

j

2

+ \Delta

'

+

A

2

ffl

2

`

25 \Gamma

2N

F

3

\Gamma

31

2

j +

7

2

j

2

'

; (9)

including the one loop effect. Here \Delta = 0 for the left-handed ( or right-handed ) quarks and lepton as in Eq.(1), while \Delta = \Gamma 10=3 for the mixed fermion chiralities as in Eq.(2), Eq.(3).

3 The renormalization group equation of C

ijkl

The quark wavefunction renormalization constant Z

q

, QCD fi-function fi, quark

mass renormalization factor fl

m

are calculated in the literature[5] as followes;

Z

q

= 1 \Gamma

A

ffl

ffT

2

\Gamma

A

2

ffl

T

2

(

ff

2

8

+ ff +

25

8

!

C

2

\Gamma N

F

t \Gamma

3

4

T

2

)

+

A

2

ffl

2

T

2

(

ff

2

4

+

3ff

4

!

C

2

+

ff

2

2

T

2

)

; (10)

fi(g

s

) = \Gamma Ag

s

`

11

3

C

2

\Gamma

4tN

F

3

'

\Gamma A

2

g

s

`

34

3

C

2

2

\Gamma 4N

F

tT

2

\Gamma

20

3

N

F

tC

2

'

; (11)

fl

m

(g

s

) = 8A + 8A

2

`

101

6

\Gamma

5N

F

9

'

: (12)

with N

F

being the number of quark flavors, ff = 1 \Gamma j, C

2

= 3, T

2

=

4

3

and t =

1

2

.

The running of g

s

and the quark mass m

q

are given by

_

d

d_

g

s

= fi(g

s

);

_

d

d_

m

q

= \Gamma fl

m

(g

s

)m

q

: (13)

Combining Eq.(9) and Eq.(10), we get the renormalization group equation of the coefficient C

ijkl

as

_

d

d_

C

ijkl

= \Gamma

(

_

d

d_

log(Z

C

Z

\Gamma

3

2 q

)

)

C

ijkl

4

= \Gamma

ae

4A + A

2

`

14

3

+

4N

F

9

+ \Delta

'

+ O(A

3

)

oe

C

ijkl

: (14)

In the following we take \Delta = 0 unless otherwise stated, since the dominant decay mode in the minimal SUSY SU(5) GUT is derived from the Lagrangian in Eq.(1) with left-handed quarks and lepton.

Solving Eq.(14), one can get the two loop long range effect on the proton decay effective Lagrangian between weak scale and 1GeV:

A

L

(2 loop) =



ff

s

(m

b

)

ff

s

(m

Z

)

!

\Gamma

18

23



ff

s

(m

c

)

ff

s

(m

b

)

!

\Gamma

18

25



ff

s

(1GeV)

ff

s

(m

c

)

!

\Gamma

18

27

\Theta



ff

s

(m

b

) +

46ss

58

ff

s

(m

Z

) +

46ss

58

!

18

23

\Gamma

327

116

\Theta



ff

s

(m

c

) +

50ss

77

ff

s

(m

b

) +

50ss

77

!

18

25

\Gamma

341

154

\Theta



ff

s

(1GeV) +

54ss

96

ff

s

(m

c

) +

54ss

96

!

18

27

\Gamma

355

192

: (15)

The result of numerical calculation is given in Table 2. Two loop correction changes A

L

, but R

L

does not change very much. Uncertainty of the strong coupling constant

has a great influence on these corrections. Here we give some comments on the Lagrangians with mixed chiralities. We find the one loop correction is common to all types of the local four fermion proton decay Lagrangian with three quarks and one lepton, while we find that the two loop corrections to them are common except for the diagram of the type Fig.1(e) as seen in the Table 1.

4 Conclusion

We have obtained the two loop correction for the long range effect on the proton decay effective Lagrangian and got the result R

L

(2-loop) = 1:33 and A

L

(2-loop) =

0:32 for ff

s

(m

Z

) = 0:116. We have found that the two loop correction of A

L

is

appreciably large though the two loop four fermion vertex correction itself has a small contribution compared with the quark mass and gauge coupling renormalization

5

correction. We note that the four fermion vertex correction, though found to be small, is necessary to obtain a consistent evaluation of the two loop renormalization effect. We have also found that the two loop correction to the effective proton decay Lagrangian depends on the combination of the chiralities of the four fermion fields while the one loop correction does not.

Acknowledgement

We would like to thank T.Goto for useful discussions.

References

[1] K.S.Hirata et al., Phys. Lett. B220 (1989) 308 ;

R.Becker-Szendy et al., "Proceedings of 23rd International Cosmic Ray Confer- ence, Calgary 1993 Vol.4 589.

[2] J.Ellis, D.V.Nanopoulos and S.Rudaz, Nucl. Phys. B202 (1982) 43. [3] P.Nath, A.H.Chamseddine and R.Arnowitt, Phys. Rev. D32 (1985) 2348 ;

R.Arnowitt and P.Nath, Phys. Rev. D49 (1994) 1479, and references therein.

[4] M.Matsumoto, J.Arafune, H.Tanaka and K.Shiraishi, Phys. Rev. D46 (1992)

3966 ; T.Goto, T.Nihei and J.Arafune, ICRR-Report-317-94-12 (1994).

[5] D.Gross and F.Wilczek, Phys. Rev. Lett. 30 (1973) 1343 ;

H.D.Polizer, Phys. Rev. Lett. 30 (1973) 1346 ; D.R.T.Jones, Nucl. Phys. B75 (1974) 531 ; D.V.Nanopoulos and D.A.Ross, Nucl. Phys. B157 (1979) 273 ;

' E.Sh.Egoryan and O.V.Tarasov, Teoreticheskaya i Matematicheskaya Fizika

41(1979) Vol.1, 26.

6

[6] S.Weinberg, Phys. Rev. D26 (1982) 287 ;

N.Sakai and T.Yanagida, Nucl. Phys. B197 (1982) 533.

[7] W.A.Bardeen, A.J.Buras, D.W.Duke and T.Muta, Phys. Rev. D18 (1978) 3998.

7 Figure caption

Figure 1 : The types of diagrams in the two loop level. The bold line in the center represents that we take the inner product of q

i

and q

j

and that of q

k

and `

l

. Fig.1(a)

and Fig.1(b) represent a typical two loop diagram with a quark and a gluon self- energy correction subgraph, respectively. Fig.1(c) is that with a gluon-quark vertex correction subgraph. Fig.1(d) contains a four fermion vertex correction subgraph. Fig.1(e) has gluon crossing propagators. In Fig.1(f) gluons are emitted from three quark lines and meet at the same point.

Table captions

Table 1 : Contribution from each type of diagrams in Fig.1 to Z

C

. Definitions of

A(= ff

s

=(4ss)), ffl, j ( gauge parameter ), N

F

and \Delta are given in the text. ( \Delta = 0 for

the left-handed ( or right-handed ) quarks and lepton as in Eq.(1), while \Delta = \Gamma 10=3 for the mixed fermion chiralities as in Eq.(2), Eq.(3). )

Table 2 : Numerical results for R

L

and A

L

. The value of the strong coupling

constant is ff

s

(m

Z

) = 0.116, 0.111 and 0.121 for (a), (b) and (c), respectively.

8

q

i

q

j

q

k

`

l

(a)

q

i

q

j

q

k

`

l

(d)

q

i

q

j

q

k

`

l

(b)

q

i

q

j

q

k

`

l

(e)

q

i

q

j

q

k

`

l

(c)

q

i

q

j

q

k

`

l

(f)

Figure 1: The types of diagram in the two loop level. The bold line in the center represents that we take the inner product of q

i

and q

j

and that of q

k

and `

l

. Fig.1(a)

and Fig.1(b) represent a typical two loop diagram with a quark and a gluon self- energy correction subgraph, respectively. Fig.1(c) is that with a gluon-quark vertex correction subgraph. Fig.1(d) contains a four fermion vertex correction subgraph. Fig.1(e) has gluon crossing propagators. In Fig.1(f) gluons are emitted from three quark lines and meet at the same point.

9

type number of diagrams coefficient of A

2

=ffl

2

coefficient of A

2

=ffl

a 6 \Gamma

8

3

(1 \Gamma j)(2 \Gamma j)

16

9

(1 \Gamma j)

b 3 5 \Gamma

2N

F

3

+

3

2

j \Gamma

26

3

+

8N

F

9

+

7

2

j \Gamma

3

4

j

2

c 6

52

3

\Gamma 17j +

25

6

j

2

\Gamma

112

9

\Gamma

2

9

j

d 9 2(2 \Gamma j)

2

\Gamma

16

3

+

8

3

j

e 3 0

10

3

j + \Delta

f 1 0 0 total 28 25 \Gamma

2N

F

3

\Gamma

31

2

j +

7

2

j

2

\Gamma

74

3

+

8N

F

9

+

29

9

j \Gamma

3

4

j

2

+ \Delta

Table 1: Contribution from each type of diagrams in Fig.1 to Z

C

. Definitions of

A(= ff

s

=(4ss)), ffl, j ( gauge parameter ), N

F

and \Delta are given in the text. ( \Delta = 0 for

the left-handed ( or right-handed ) quarks and lepton as in Eq.(1), while \Delta = \Gamma 10=3 for the mixed fermion chiralities as in Eq.(2), Eq.(3). )

10

(a) ff

s

(m

Z

) = 0:116

R

L

A

L

1-loop 1.31 0.449

2-loop 1.34 0.321

(b) ff

s

(m

Z

) = 0:111

R

L

A

L

1-loop 1.28 0.476

2-loop 1.31 0.363

(c) ff

s

(m

Z

) = 0:121

R

L

A

L

1-loop 1.33 0.421

2-loop 1.38 0.277

Table 2: Numerical results for R

L

and A

L

. The value of the strong coupling constant

is ff

s

(m

Z

) = 0.116, 0.111 and 0.121 for (a), (b) and (c), respectively.

11

