electric dipole moment of the τ-lepton at polarized e+e− colliders

Physics Letters B 305 (1993) 306-311 North-Holland PHYSICS LETTERS B

Electric dipole moment of the z-lepton at polarized e + e - colliders

Gll les C o u t u r e 1

Physws Department, Concordta Umverstty, 1455 de Matsonneuve Ouest, Montreal, Canada H3B 1M8

Received 2 January 1993, revxsed manuscript received 2 March 1993

We consider the process e +e--, z +z- and calculate the sensitivity of the az,muthal asymmetry in the •- decay channel to F3 At x/s = 6 GeV, this asymmetry could lead to a bound of 3 × 10-3 on Fs, given 10 7 z +z- pairs and transversely polarized initial beams We also consider partially polarized beams

1. Introduction

Electric dipole moments (edm) are interesting in their own rights since they constitute another window on the poorly understood phenomenon of CP violation [ 1 ] Although one can expect all (non-self conjugate) par- tlcles with spin to have an electric dipole moment arising from weak Interaction, the predictions from the minimal standard model are well beyond levels that can be reached In the near future [ 2 ] Therefore, the observa- tion of any edm within a few orders of magnitude of current levels would be a striking signature of physics beyond the SM

In this paper, we want to consider the Try vertex with an edm [ 3 ] We will also include the magnetic moment term as a potential background The vertex we consider is then [4 ]

--Ie( FI(Q2) 'u+ F2(Q2)2m~ Q~((Y~'P/I)+ F3(Q2)2m~ QP((Yu~/i)?5) ' ( l )

where Q is the photon momentum We will set Fl to be 1 since this defines the electric charge of the fermlon F2 and F3 are the magnetic and electric dipole moment terms, respectively Note, F2 ¢ 0 in the minimal SM this term includes all the higher order corrections and is typically of order c~/Tr The current direct, experimental bound on this term is F2 ~< 0 02 [ 5 ] We will not include here the anapole moment [6 ] term although work is currently in progress on this term [ 7 ], current literature is inconsistent regarding ~ts gauge invarlance and deft- ni t ion [ 8,4] Furthermore, since its CP properties are different from both F~ and F2, we will assume that appro- priate cuts will allow us to isolate it

2. Calculation

The first step is to calculate the process e ÷ (S~, p~ ) e - ($2 , P2)~ ' r +(83, P3)z-(84, P4) using the vertex as defined in eq ( 1 ) We use the complete vertex only at the ~zz vertex the bounds on the yee vertex are far too tight for us to take them into account here [ 9 ] We follow Tsai 's procedure [ 10] We define the polarization vectors in a shghtly different way in order to take into account the polarization of the initial state

Address after June 1, 1993 Physics Department, UQAM, Unlverslte du Quebec a Montreal, CP 8888, succ A, Montreal, PQ, Canada H3C 3P8

306 0370-2693/93/$ 06 oo © 1993 Elsevier Science Publishers B V All rights reserved

Volume 305, number 3 PHYSICS LETTERS B 13 May 1993

83 = (/~)'S~, S~ cos(0) cos(0) - S ~ s in(0) +)'S~ cos(0) sin(0),

S~ cos(0) s in(0) +S~ cos(0) +~,S~ s in(0) sin(0), 7S~ cos(0) - S ~ sin(0) ) (2a)

and

$4 = ( -flyS~, S~ cos(0) cos(0) - S ~ s in(0) +yS~ cos(0) sin(0),

S~ cos(0) s in(0) +S~ cos(0) +yS~, s in(0) sm (0), yS~ cos(0) - S ~ sin(0) ) , (2b)

where/~= x/1 - 4 m 2/s and 7 = x/~s/2m, The upper (spatial) indices refer to the r rest frame and the angles refer to the angles of the z in the laboratory 0 is defined with respect to the direction of the e - and 0 is defined with respect to the alignment of its spin Using these, after some calculation, we obtain a very lengthy expression Space does not allow us to write it completely here We will g~ve only the relevant terms

d o o/2]~ d-Q~ - s (au.pol+ ( S l ) (S2)o 'po l ) , (3a)

where

O'unpo I ~--- 1 "~ COS2(0) + sin2 (0)/72+S~S~4[ 1 + cos2 (0) - s i n2 (0 ) /72 ] +S~S~ sin2 (0) ( 1 + 1/~ 2 ) +S~S{fl 2 sin2(0)

+ (S~SI+S]S~) sln(20)/y+F3B[2(S~ -S~) sin2(0) +~ sin(20) (S~-S~) ] (3b)

and

O.po I =f12 sln2 (0) cos(20) + {S~S~[ 1 + cos2(0) - -s in2(0) /y] --S~S~[ 1 + cos2 (0) + sin2(0)/):1

+S~S~, sIn2(0) ( 1 + 1/72) + (S~$5~ +S~S~) s in(20) /y} cos(20)

- 2 [ ( S~S~ + S~S~) cos(0) + ( S~S~ + S~S~) sm (0)/~,] s in(20)

-F3~fl{ [ ( S ~ - S ~ ) 2 sin2(0)/~,+ (S~-S~) sin(20) ] c o s ( 2 ¢ ) - 2 ( S ~ - S ~ ) sin(0) sin(20)} (3c)

In the proper hmlts, the full equation agrees with Tsai [ 10], Barr and Marcmno [ 2], Sflverman and Shaw [5 ], and P1 and Sanda [ 11 ] It is important to note that most terms are either spin-independent or depend on the product of the two spins This product implies that its effect will require the measurement of correlation products between the two z's On the other hand, the terms proportional to F3 depend on a single spin at a time This means that by considering only one T at a Ume, the effect will survive while it will be washed out in the other case

We see here that the presence of an edm in the 7rr vertex polarizes slightly the produced r's This has been shown previously [ 12 ] The effect of lmtial polarization is to enhance further th~s effect in some region of the azimuthal d~stribution (the ¢~ angle) In order to see how measurable these effects are, we let the r decay to n v~ Being a two-body decay, th~s decay mode is the most sensiuve to the polarization of the z In complete analogy w~th the spin precession experiments done on the muon, we would expect the effect to be m i m m u m when we integrate over the whole energy range of the p~ons As we concentrate on a given pxon energy range, we are sensitive to a particular orientation of the spin of the r in its rest frame At th~s point, one has two options

(a) Method I one searches for an energy asymmetry In the p~on spectrum In some parts of the detector, th~s asymmetry will be enhanced by the polarization o f the mmal beams This is very s~milar to the method described prewously [ 12 ]

(b) Method II one concentrates on a g~ven pion energy range and measure the angular asymmetry (in ¢~) within th~s energy range

Since we are deahng with a multi-body final state and we want to implement several binning procedures, we performed the calculation by Monte Carlo methods Since the opt imum CM energy for the measurement ofF3 through an energy asymmetry ~s 6 GeV [ 12 ], we will concentrate on this energy We have shown that working

307


at C M energy o f 10 G e V does not degrade the bounds very m u c h [ 12 ], by app rox ima te ly a factor 1 5

3. Results

In what follows, we will a ssume that F2 is 0 We take the approach that this t e rm can be ca lcula ted wi th very

high prec is ion in the SM and will s imply add a wel l -known t e rm to the d i f ferent d l s t r i buhons We wdl also

neglect the in te r fe rence t e r m FEF3 N o t e that this t e rm is po tent ia l ly ha rmfu l it has exact ly the same symmet r i e s

as the t e rm we want to measu re We feel jus t i f i ed to d rop it because we ver i f ied that the fac tor in f ront o f it is

numer ica l ly small since F2 << l, we can d rop F2F3 m front o f F 3

In fig l a we present the results f r o m m e t h o d I As expected , we see e n h a n c e m e n t s and reduc t ions m the energy

a s y m m e t r y m some parts o f the de tec to r We de f ined the fo l lowing sect ions o f the de tec to r Assume that the spin

o f the e - is a long the x axis and the e - is m o v i n g along the z axis The U part o f the detector represents ~ ~z < ~ < 3 ~z

and ~ < ~ < 7~z and the L part o f the de tec to r represents the o ther areas Fig l a represents the ra t io

d a / d E ~ _ Ipol ~-3=0

da/dE,~- [ unpol F3=0

The bumps are for the U region whi le the dtps are for the L region The bumps with large a m p l i t u d e are for

( S I ) = ( $ 2 ) = 1 while the smal le r " b u m p s " are for ( S ~ ) = ( $ 2 ) = 0 5 We see here only the effect o f ini t ial

po la r i za t ion on the energy spec t rum o f the n - In fig l b we show the d i s to r t ion induced in this spec t rum by F3 we now plot

de/dEn- I F3=O Ol de/dE,~_ I~3=o

for a po la r i za t ion o f 1 The curve that crosses the axis at h igher energies represents the U sect ion o f the de tec tor

115

1 1

I 05

' ~ ~ o 9 5

o 9

0 85

0 4 0 8 12 16 2 24 2 6

I O r2

I 008

~ 1 o II

, 0 0 4

_ o 0 0 0 . ,

i I i i r l l r l l l l l l l l l l l , , I . . . . I , t ~ 2 0 4 0 8 12 15 2 21 8

.E n E ~

Fig 1 (a) (energy distribution with polarized beams)/(energy distribution with unpolanzed beams) The curves with large amphtude correspond to ( S~ ) = 1 = ($2) while those with small amplitude correspond to (S~) = 0 5 = ($2) The bumps are for the U region and the dtps are for the L region (b) (energy distribution with F3=0 01 )/(energy distribution with F3=0) for a fully polarized beam The curve that crosses the axis at higher energies represents the U section while the one that crosses the axis at lower energies represents the L part of the detector The case of unpolarlzed initial beams is represented by the middle curve

308


while the curve that crosses the axis at lower energies represents the L part of the detector For comparison, we also plot the same ratio for unpolarized initial beams, this gives the middle curve Unfortunately, using this method, we have not been able to improve substantially the bounds on F3 obtained from an unpolarlzed incom- ing beam Therefore, it appears that method I with polarized initial beams has a marginal advantage over an energy asymmetry in the pion spectrum obtained with unpolarized initial beams

We now turn to method II As expected, we still see a t iny effect in the ~ angular distr ibution of the pxons when we integrate over the whole range of energy of the pions this is due to the azimuthal asymmetry that we have to start with Then, we divide the energy range into two (0 3-1 5 and 1 5-2 7 GeV), three (0 3-1 0, 1 0-2 0, 2 0 - 2 7 GeV) and five (0 3-1 0, 1 0-1 5, 1 5-2 0, 2 0-2 5, 2 5-2 7 GeV) energy bins It turned out that the two energy bins gave us the more sensitive results The narrower, higher energy bins (in the five bin cut) are more sensitive to F 3 but the statistics is degraded too much to take advantage of the greater sensitivity

In fig 2a we show the departure from uniform angular (~) distr ibution of the pIons one would expect, for fully polarized beams The ratio we present is the angular distr ibution one would get with polarized beams divided by what one would get for unpolarized beams, F3 = 0 here In fig 2b, we show the distortion induced in this distr ibution by the presence of an electric dipole moment F3 = 0 01 here The ratio is now what one would expect from fully polarized beams with an edm divided by what one would expect from fully polarized beams without an edm In both cases, we have split the energy spectrum into two bins 0 3~<E.~< 1 5 GeV and 1 5 ~< E~< 2 7 GeV Figs 3a, 3b are similar to these ones but for ( S ] ) = 0 5 = ($2 ) instead of 1 It is clear from these figures that one should try to measure the distortion induced in the high energy part of the spectrum It is also easy to unders tand that the range ¼7r ~< 0. ~ 3 rr and ~- rr ~< 0. ~< 7 g should lead to better results since the distortion is maximal in this part of the distr ibution So, with polarized beams, we should expect to measure slightly less events than with unpolarized beams and with an edm, we should even measure a little bit less When we integrate over this angular range, we get an average value of 0 867 for the polarized beams with F 3 = 0 and 0 860 for the polarized beams with F3 = 0 01 Clearly, this is a difficult measurement The statistical error that comes with the production of 1 mil l ion z + z - pairs would allow a l - a b o u n d of 0 008 or so on F3, we have assumed a branching ratm of 10%

In table 1 we compare three different methods to set bounds on F3 we consider method II discussed here with

ii

, ] , , , , I , , , 1 1 ¢ , t z q i i , ~ i r t l ~ I 2 3 4 5 S

qE

LI

1 oc5

1 Dc]25

1

o 9975

o 995

0 9925

0 9 9

Q g875 ' ' , ~ . . . . ~ . . . . I . . . . ~ . . . . ] . . . .

1 2 3 4 S

Fig 2 (a) (angular distribution with polarized beams) / (angular distribution with unpolanzed beams) for a fully polarized beam The large amphtude corresponds to the high energy b m ( 1 5 ~<E~< 2 7 GeV) while the small amphtude corresponds to the low energy bm (0 3 ~< E~ ~< 1 5 GeV ), (b) (angular distribution with F3 = 0 01 ) / (angular distribution with F3 = 0) for a fully polarized beam The upper curve corresponds to the low energy bm (0 3 ~< E~< 1 5 GeV) while the lower curve corresponds to the high energy b m ( 1 5 ~< E~ ~< 2 7 GeV )

309


T

q

~ 1 oo2

7

l n , , n l t l l l f n n , , ~ . . . . I . . . . I . . . . I J

I 2 $ ,~ 5 6 . . . . r . . . . , . . . . , . . . . , . . . . , ,

I 2 3 4 5 6

Fxg 3 (a) Sameas f ig 2 a b u t f o r ( S l ) = 0 5 = ( S 2 ) , ( b ) s a m e a s f i g 2 b b u t f o r ( S l ) = 0 5 = ( S 2 )

Table 1 Different bounds obtained on F3 from the azimuthal asymmetry &scussed m this paper (Ao.), from the pmn energy asymmetry (AE.) &scussed prewously and from the total pair production cross-section (tz.+T-) All hmlts are at the l-or level

Bound from r + z - pairs Polarization

1X 10 6 3 × 10 6 1 0 X 10 6

A#. 0 0084 0 0049 0 0027 ( S j ) = 1 = ($2) 0 016 00093 0 0051 ( $ 1 ) = 0 5 = < $ 2 )

AE. 0 013 0 0076 0 0041 ($1) = 0 = (5'2)

a~+~- 0 026 0 020 0 015 (S~) = 0 = ($2)

polar ized beams, the energy asymmetry discussed previously with unpolanzed beams and finally, the bound one could obta in from a measurement of the total cross-sect~on Recall that the hmlts on F3 from an energy asymmetry and total cross-section measurements are given by

2 + ( l / R ) X / 2 2 x / I _ ( 1 / R ) 2 x / ~ - and )~ ~x/~+~_,

respectively, where R m Q2 /4 m ~ It lS clear that the method discussed here is superior but requires rather highly polar ized beams Polar izat ions of at least 50% are required m order to reach bounds comparable to those obta ined from an energy asymmetry measurement Both methods are superior to a measurement of the total product ion cross-section, the total cross-sect~on is sensmve only to F~ and a very h~gh precision ts reqmred m order to put an interesting bound on F3

Although a bound of 0 008 on F3 would be an enormous improvement over current bounds, we would still be very far from probing a composl teness scale of a TeV or so [2] Probing such a composlteness scale would reqmre, typical ly #'3 ~< 3 X 10 -6 Clearly, this would reqmre an enormous statistics However, in the Welnberg model of CP violation, the edm of a fermlon has an m f3 dependence and a relatively loose bound on Fs on the lepton, might m fact be as constraining m this model as the very good bound on F3 of the e - It appears that this enhancement is greatly reduced at the two-loop level but some crltlcal cancellat]ons might be impor tan t [ 13 ]

310


4. Conclusions

We have shown that po l a r i za t ion o f the mi tml b e a m s in the process e + e - - , z + r - can be useful m set t ing

b o u n d s on the electr ic & p o l e m o m e n t o f the z, the F3 f o r m fac tor Cons ide r ing the n - decay mode , concent ra t - 5 ~ 7 lng on the energy range l 5 ~< E . ~< 2 7 GeV, and record ing the even t s wi th in ¼ n ~< ~. ~< 3 n and a n ~< q~...~ z n, com-

pletely po la r i zed b e a m s could lead to a b o u n d F3~< 8 × l0 -3, whi le ha l f -po la r ized b e a m s lead to a b o u n d

F3 ~< 1 6 × 10 -2, g iven one m d h o n tau pairs Th is is to be con t ras ted wtth the b o u n d s f rom an energy a s y m m e t r y

in the p lon spec t rum and the to ta l p r o d u c t i o n cross-sect ion (bo th for u n p o l a n z e d b e a m s ) which are 0 013 and

0 026, respec t ive ly It appears that a highly po la r i zed b e a m ts r equ t red m o rde r to surpass the b o u n d reached by

an energy a s y m m e t r y using u n p o l a n z e d beams , at least for the m e t h o d o u t l m e d in this paper H o w e v e r , the

advan tage o f the m e t h o d p resen ted here is to reduce the sys temat tc errors the angular m e a s u r e m e n t (q~.) re-

q m r e d here ~s m u c h m o r e rehable than the energy m e a s u r e m e n t (E~) r equ i r ed for an energy a s y m m e t r y Fur-

t he rmore , &vxdlng the energy s p e c t r u m o f the p ion m t o m o r e than two b ins does not seem to i m p r o v e the

b o u n d s An energy a s y m m e t r y m e a s u r e m e n t xs cer ta in ly possible w~th po la r tzed beams, the b o u n d s do no t ap-

pear to ~mprove ove r wha t has been p resen ted here

Acknowledgement

I want to thank the SSC Labora to ry , I S E M and the physics d e p a r t m e n t at S M U , and the physics d e p a r t m e n t

at U Q A M for the use o f the t r c o m p u t i n g faci l i t tes I also want to thank Rysza rd S t roynowskt for s t tmula t lng

d iscuss ions

References

[ 1 ] For a review on electric &pole moments, see N F Ramsey, Annu Rev Nucl Part Sc~ 32 ( 1982 ) 211, and references thereto [2] See, e g W Mancano and S Barr, m CP-vlolatlon, ed C Jarlskog (World Scientific, Singapore, 1989) [ 3 ] For a review on r physics, see B C Bansb and R Stroynowskl, Plays Rep 157 (1988) 1 [4] M J Musolfand B R Holstein, Phys Rev D 43 (1990) 2956 [5] D J Sflverman and G L Shaw, Phys Rev D 27 (1983) 1196 [ 6 ] See e g F Boudjema and C Hamzaoul, Phys Rev D 43 ( 1991 ) 3748 [7 ] G Couture, C Hamzaom, E Bondy and G Jacklmow, work m progress [ 8 ] A Gonora and R G Stuart, preprmt CERN-th-6348-91 [9] W Bernreuther and M Suzuki, Rev Mod Plays 63 ( 1991 ) 313

[10]Y-S Tsal, Phys Rev D4 (1971) 2821 [11 ] S-Y P1 and A I Sanda, Phys Rev D 14 (1976) 1772 [ 12] G Couture, Pbys Lett B 272 ( 1991 ) 404 [ 13] S M Barr and A Zee, Pbys Rev Lett 65 (1990) 2l,

J Gumon and R Vega, Phys Lett B 251 (1990) 157

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electric dipole moment of the τ-lepton at polarized e+e− colliders

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