Bounds on the W-boson Electric Dipole Moment  Revisited




P. Saxena, Prachi Parashar, N.K. Sharma, Ashok K. Nagawat and Sardar Singh


Department of Physics, University of Rajasthan, Jaipur (India) 302 004.





Abstract

Keeping in view the suggestion [7] that the cut-off procedure involves a good deal of


uncertainity in the prediction of E.D.M. of W  boson, we have re-examined the earlier


calculation by Marciano and Queijeiro [2] by replacing the Cut-off regularization


process by BPHZ regularization [8]. This works in a clean and unambiguous manner


without involving any approximation. We also examine apparently inapplicable


approximations like using m = m in [2]. The bounds on and d are significantly
f f W W



changed in all cases as compared with those reported in [2]. The necessary caution that


is to be exercised while using approximations is explicitly pointed out.

arXiv: 2 Apr 2002 PACS number (s): 13.40.Em





pranav_saxena19@hotmail.com
prachi_parashar@hotmail.com
nksharma_thep@hotmail.com
nagawat@hotmail.com
singhjaipur@hotmail.com





1


I. INTRODUCTION


In the past few years, there have been some efforts to evaluate Electric Dipole


Moment (EDM) of W- boson [1-3]. In 1986, Marciano and Queijeiro [2] obtained limit


on and d of W- boson updating an earlier suggestion by Salzman and Salzman [1].
W W



They made use of the concept of induced fermion electric dipole moment, which is


induced in the presence of W- boson or vice-versa. In order to overcome the ultraviolet


divergence, they made use of the cut-off procedure, first used by Paulli-Villars [4]. This


work has been extensively been used by the later authors in this field in their calculations


[3,5,6]. As rightly pointed out by Barr and Marciano [7], the bounds derived from the


cut-off dependent loop effects can sometimes be misleading. Unforeseen cancellations


could reduce the value of EDM estimate particularly if the scale of new physics defined


through the cut-off dependent parameter is equal to m . Apart from this, the authors
W



of ref. [2] made use of some approximations, which might have eventually influenced


the outcome. In particular, the assumption making m = m , is not physically correct. In
f f



order to check whether the observation made in [7] plus the approximations used in [2]


have some impact on the results we have re-performed this calculation. In order to


overcome the uncertainty in the use of cut-off procedure, we make use of the versatile


BPHZ regularization [8] procedure without making any approximation whatsoever. The

calculation for

D , and d go on very smoothly till the end. However, the limits
W W



obtained on and d are drastically different from those given in ref. [2].
W W



Next to check the effects of the approximation made in [2], we have re-

performed complete calculation for

D , and d using Pauli-Villars cut-off method
W W





2


[4] without making any approximation. Here again the limits obtained on and d are
W W



significantly changed as compared with those reported in ref. [2]. Thus this outcome


apart from substantiating the observation made by Barr and Marciano [7], also points


towards being careful about making approximations while carrying out renormalization


program. This aspect will be further elucidated below.




II. CALCULATIONS USING BPHZ REGULARIZATION PROCEDURE


Beginning with the CP- violating amplitude reported in ref. [2], namely, Eq. (5) of


[2] corresponding to fig. (1), we notice that the integral is logarithmically divergent.


Therefore BPHZ regularization scheme [8] can be justifiably applied [9]. Using this


scheme and after a lengthy and tedious algebra we obtain for the CP- violating


amplitude, the expression:


2
eg -
D W
 = - u ( p q p
/  - p
 u p
/ ; (1a)
2 2 2 ( 2 1 ) 1
) 5 ( )
1
128 m 2
W


which leads to

eG m
F f W
d = ,
f
8 2
2



= ( .
4 126 10-21 ecm)m f W , (1b)
G
2 eV

where


3
R - R2R - R 2
R - 2
R R + 3
R - R2 - R - 2
R - R +
= 1
2
R 2
R - 2RR - 2
R + 2
R - 2R + 1



(
R - R + )
1 + 2
R - 2
R R - 2
R + 2
R - 2R + 1 2
R + 2
R -
 ln - 1 ln(
R ) +
2
(
R - R + )
1 - 2
R - 2RR - 2
R + 2
R - 2R + 1 R




3


2 2
R R + 6
R R - 3
R + 2
10R + 6R . (1c)
2
3R

2 2
m m
with f
R = , f
R= .
2 2
m m
W W

In arriving at Eq. (1), the following identities have also been used:
1
 =  +
 =  , ( ) ; (2a)
5
2 5
2


p p =0. (2b)
1 1 = p p
2 2 =
   q q q


It may be emphasized that we have used m m and R, R' 0 in this calculation. We
f f



2
notice that the factor ln + O )
1
( of ref. [2] Eq. (8) is replaced by in our
2
mW


formulation. We have evaluated and bounds on and d by using the experimental
W W



bounds [10] on d , d , d and d . The results are shown in Table 1. For comparison
e  n



we have shown the corresponding values arising from the calculations of ref. [2], Eq. (9)


in Table 2 where the aforesaid approximations have been used.




III. CALCULATIONS USING CUT-OFF PROCEDURE ( m m , R, R 0 )
f f



Again beginning with the Eq. (5) of ref. [2] we obtained after a very lengthy


algebraic manipulation the following form of expression for D :

- 2
 = eg
D u( p )q p   p u p (3)
2 2 W 2 [ 2 5 + 5 1 ] ( )
1 
64 mW


d = eGF m , (4)
f W f
4 2
2





4


where


1
= - 2R(D - )
1 (D - 5 - 2R'+2R) + 6D - 3D + D - 6R'+3R' 3
- DR' +2R'
2 [ { 2 3 2 2 3
12(D - )
1 R
- 1
(
6 + R')R + 3 2
R (D -1 + 2R') - 2 3
R } ln (D) - (D - )
1 3 ln(R') -

1 {-2R'4+R'3 1(-8R)+R'2 9(+R-12 2R)+3D R'3-R'2 1(+R)+(R- )12(R+ )1
(1- R'+R)2 - 4R
- R'(R + )
1 2 - (R - )
1 2 5
( R + 2 2
R - )
1 + R' 8
( 3
R + 2
R + 10R - 7) }


1 + R'2 -2RR -
' 2R + 2
R + (R -
' R + )
1 1
( - R +
' R)2 - 4R
ln -
2R'


1 {4 2R -3+D(-3+2D) -(D-R+'R)2 6+ 2D -3R+'D(-3+R-'R)-
(R -
' R + D)2 - 4RD

- (
2 R - 2
' R) + 3R - 2R (
6 D - )
1 D - (
6 D + 2
)
1 (D - R +
' R) + 5
( D - )
3 (D - R + 2
' R) }
D 2 + R 2 + R'2 -2RR -
' 2RD + (D + R -
' R) (D - R +
' R)2 4RD
ln . (5)
2R' D


It may be pointed out that the following Feynman parameterization and integrations have


been used in the aforesaid calculation:


1 x x x
1 1 2 3 1
= )
5
( dx dx dx dx (6)
1 2 3 4
abcde 0 0 0 0 ([R'-R) 2
x + Rx + x - x D - x + 1
4 4 ( 2 4) ]5
2



where x , x ...... x are the Feynman parameters and
1 2 4



a = ( 2
k - k. 2
q - m ) , b = ( 2
k + k. 2
q - m ) ;
W W



c = ( 2
k - k. 2
q - ) , d = ( 2
k + k. 2
q - ) ; (7)


2
e = ( 2 2 2
k + k.( p + p ) + m - m and D = .
1 2 f f ' ) 2
mW





5


Now if we introduce the approximation m = m but not R, R'= 0 in Eq. (5), the above
f f



expression reduces to


1
= = - R - DR + D R + D - D + D - R D - D - R
R R' 2 { 5
(
2 6 2 ) (6 3 2 3 12 ) ln ( ) ( )
1 3 ln ( )
1 (
2 D - )
1 R


1 2 2 1 - 2R + 1 -
- 1 + 4
3D -14R - 6DR + 28R - R
12DR ln -
1 - 4R 2R


1 2 3 4 2 3 2 2 2 2
- 6D + 3D - D + 24DR - 6D R + 2D R -12R -12DR + 8D R
2
D - 4RD

D - 2R + D2 - 4RD
ln . (8)
2R


We notice that the approximations R, R' 0 cannot be used in Eq. (5) and Eq. (8)


because both expressions diverge.


The authors of ref. [2] could use these approximations as they have done so


before performing Feynman parametric integrations, whereas we are trying to use these


after performing the integration over parametric space. To our knowledge, the use of


approximations before doing integration in parametric space as is done in ref. [2] need


some caution as elucidated below [11]:


The justification of the approximations R, R'= 0 used in ref. [2], lies in the fact


that these are negligible as compared with the Cut-off parameter under the assumption

2
>>> m or . An application of this approximation in the denominator of our
W



2

Eq. (6) allows only the term (x - x ) to survive. This term on being integrated over
2 4 2
mW

x and x diverges when . As such the use of this approximation before
2 4



integration over parametric space is not justified [11] as is done in ref. [2]. On the other



6


hand there is absolutely no justification to use the approximation m = m anywhere in
f f



the calculation. As such no approximation of the type used in [2] is justifiably applied in


the Cut-off procedure calculation.


As the expression (4,5) is not a cozy, as Eq. (8) of [2], we can extract


information about and d from it by plotting d against as shown in Fig. (2)
W W W



using experimental limits on d (f=e,, and n) from [10]. We notice from Fig. (2) that
f



at about = 3 TeV, the values of various d are relatively stabilized. These are given in
W



Table 3. The corresponding variation of d against , using Eq. (9) of [2], are shown in
W



Fig. (3). For completeness, we also show the variations of d against corresponding
W



to our Eq. (4,8) in Fig. (4). The values of d corresponding to = 3 TeV using Fig. (3)
W



and (4) are shown in Table 3 and 4 respectively.


As a corollary, we take the limit on d corresponding to most stringent limit on
W



neutron EDM ( d ) as obtained in our BPHZ regularization procedure as a reference
n



value and obtain there from the limits on d , d and d . These are given in Table 5. For
e 


another similar exercise, we take d 30
10-
ecm and re-calculate d , d and d and
W e 


d . We have preferred this value since identical value occurs in the calculation by Booth
n



[5] i.e. d -30
810 ecm , who make use of QCD radiative correction for its evaluation
W



as also in our calculations, namely, d 3
- 0
4.22810 ecm corresponding the d limit
W e



(exp.) using BPHZ method (Table 1). The results are shown in Table 6 These values are





7


very close to the corresponding experimental limits. This may lead us to conclude that


limits on d may lie in the vicinity of 30
10- ecm , if these methods are to be believed.
W





IV. CONCLUSION


In the absence of any experimental limit on d it is not possible to make any
W



definite statement about the outcome of the aforesaid calculations and limits. In reality


this is not the objective of this note  we have attempted some clarification about the use


of Cut-off procedure and in applicable approximations in the calculations of ref. [2]. An


unambiguous statement about the use of BPHZ regularization procedure is obvious. On


the other hand, uses of approximations, however, obvious need caution before use as they


sometimes may lead to altogether wrong conclusions.




V. ACKNOWLEDGEMENT


The authors acknowledge the Financial Assistance from DST (Department of


Science & Technology), New Delhi (India) for carrying out this work. P. Parashar wishes


to thank CSIR (Council of Scientific & Industrial Research), New Delhi (India) for


granting her the Financial Assistance in the form Senior Research Fellow (SRF).





8


REFERENCES


[1] F. Salzman and G. Salzman, Phys. Lett. 15, 91 91965); Nuovo Cimento 41A, 443

(1966).

[2] W.J. Marciano and A. Queijeiro, Phys. Rev. D33, 3449 (1986); F. Salzman and

G. Salzman, Phys. Lett. 15, 91 (1965); Nuvo-Cimento 41A, 443 (1966).

[3] I.B. Khriplovich and M. Pospelov, Sov. J. Nucl. Phys. 53, 638 (1991).

[4] W. Pauli and F. Villars, Rev. Mod. Phys. 21, 433 (1949).

[5] Michal J. Booth, EFI-93-01, .

[6] F. Hoogeveen, Nucl. Phys. B341, 322 (1990).

[7] S.M. Barr and William J.Marciano, "CP- violation" (Advance series on

directions in High Energy Physics, Vol. 3); ed. C. Jarlskog, World Scientific,

1989, pp. 455-499.

[8] N.N. Bogluibov and O. Parasuik, Act. Math. 97, 27 (1957); K. Hepp, Comm.

Math. Phys. 2, 301 (1966); W. Zimmerman, Comm. Math. Phys. 15, 208 (1969).

[9] T.P. Cheng and L.F. li, "Gauge theory of elementary particle", Clarendon,

Oxford, 1984.

[10] D.E. Groom et al. (Particle Data Group), European Physics Journal C15,

1(2000).

[11] S. Weinberg, "Quantum Theory of Fields", Vol.- I, Cambridge University Press,


(1995), pp. 495-496.





9


Table 1: Theoretical Bounds on and d using BPHZ regularization procedure
W W


with m m .
f f



d Experimental limits d
f Square bracket term (ecm)
W W
on d (ecm)
f


d -27
1.810 10
= 4.95310 -14
3.447 10 -30
4.22810
e



d -19 6 -1 -17
 3.7 10 = 1.15910 1.464 10 1.79610

- 3 1 -15

d 16
3.110 = 4.096 10 2.065 10 2.56310

d 2
- 6
6.310 2 1
- 1
1.77110 -27
2.17210
n md 8
R = = 1
.
5 73 10
(Bare quarks) mW

2
mu 8
R = = 3.59310
mW





Table 2: Theoretical Bounds on and d using the expression (4) & (5) for = 3
W W



TeV.


d Experimental limits on d (ecm) d
f f (ecm)
W W


d -27
1.810 -32
7.517 10 -48
9.22210
e


d -19 -16 -32
 3.7 10 2.28010 2.797 10

- -9 -25

d 16
3.110 1.24110 1.52210

d 2
- 6
6.310 -20
5.82510 -36
7.147 10
n





10


Table 3: Theoretical Bounds on and d using the expression (9) of ref. [2].
W W


d Experimental limits on d (ecm) d (ecm)
f f W W



d -27
1.810 4
1.19810 1
- 9
1.470 10
e


d -19 3 -13
 3.7 10 1.19110 1.46110

- 4 -12

d 16
3.110 5.93310 7.279 10

d 2
- 6
6.310 -7
2.05210 2
- 3
2.51810
n




Table 4: Theoretical Bounds on and d using the expression (4) & (8) for = 3
W W



TeV


d Experimental limits on d (ecm) d (ecm)
f f W W



d -27
1.810 -31
1.38510 -47
1.70010
e


d -19 1
- 6 3
- 2
 3.7 10 3.82810 4.697 10

- -9 -25

d 16
3.110 2.10610 2.58410

d 2
- 6
6.310 -20
5.827 10 -36
7.14810
n





Table 5: Theoretical EDM Limits on d , d , using d -27
2.172 10 ecm .
e  d W

Theor. Bounds on d (ecm) Theor. Bounds on d Exp. limits on d
d f f f
f

using Eq. (9) of ref. [2] (ecm) using Eq. (1) (ecm)

3
- 5
 2
- 5
 -27

d 8
.
1 65 10 2
.
9 36 10 1.8 10
e


3
- 3
 3
- 1
 -19

d 3.855 10 4
.
4 68 10 3.7 10


-32
6.484 10 -35
-
.
2 65610 16
3.110

d





11


Table 6: Theoretical EDM Limits on d , d , d , d using d 30
10-
ecm .
e  n W


Theor. Bounds on d (ecm) Theor. Bounds on d Exp. limits on
d f f
f
using Eq. (9) of ref. [2] (ecm) using Eq. (1) d (ecm)
f


3
- 9
 -28
 -27

d 5
.
8 85 10 .
4 252 10 1.8 10
e

-36
 3
- 0
 -19

d 2.533 10 2
.
2 99 10 3.7 10


-35
4.25910 3
- 1
-
2
.
7 24 10 16
3.110

d

d -32
2.25110 -29
-
7
.
5 2210 26
6.310
n(bar q
e uark )
s


d -36
 -28
 2
- 6

n(constitue t
n s quarks) 5.263 10 2
.
8 70 10 6.3 10





f f' f



W W













Fig. 1: Diagram for the fermion induced electric dipole moment of W-boson.





12


Fig. 2: Graph between d and using our expression (4) & (5).
W





13


Fig. 3: Graph between d and using expression (9) of ref. [2].
W





14


Fig. 4: Graph between d and using our expression (4) & (8).
W





15



