On the perturbative QCD calculation of the R ratio for τ decay

Volume 236, number l

PHYSICS LETTERS B

8 February 1990

O N T H E P E R T U R B A T I V E Q C D C A L C U L A T I O N O F T H E R R A T I O F O R x DECAY C.J, M A X W E L L and J.A. N I C H O L L S Centrefor Particle Theory, Departmentof Physics, Universityof Durham, DurhamDH1 3LE, UK Received 6 November 1989

We show that the perturbative QCD calculation of the R ratio for "~decay cannot be trusted because of the small value of the scheme invariant quantity, Po, for this process. The predictions are strongly dependent on renormalization scheme choice for values of A~s favoured by e+e - annihilation data. This is true to O(c~) and O(a~), and has nothing to do with the large O(a 3) perturbative coefficient.

F o r the decay o f a heavy lepton it is possible to define an analogue o f the R ratio o f e + e - annihilation, R(e+e-) -

a(e+e-~hadrons) a(e+e---+ g+la- )

1)

Hence, for x decay one can define the experimentally measurable ratio

R(x)=-

F ( x - --*v, + hadrons ) F(x--~v~e-ge)

(2)

Assuming that the n u m e r a t o r is a p p r o x i m a t e d by decays into dfi and sO, one obtains the zeroth order estimate [ 1 ] R ( z ) -~Nc= 3, analogous to the zeroth order result N,.~Q 2 for the e + e - ratio. In several recent papers [ 2 - 4 ] it has been argued that one can reliably calculate perturbative corrections to R ( x ) . A fortuitous phase space factor suppresses the time-like contributions to the W self-energy function, and allows a sensible o p e r a t o r product expansion. Electroweak corrections are predicted to give a + 2.4% effect [ 5 ], and non-perturbative Q C D corrections are negative and estimated [3] to be 13%. This means the d o m i n a n t corrections to R (x) -~ 3 should be o f perturbative Q C D origin. Using the calculation o f G o r i s h n y et al. [ 6 ] for the e+e - R ratio, one can deduce [4] the perturbative expansion (for three active quark flavours, N f = 3) R(z)=3(1 +a+5.20a2+

104a 3) ,

where the couplant a=as/n. This expansion is calculated in the MS scheme using the z mass as the scale, /~= m~= 1784 MeV. One can compare eq. ( 3 ) with the N f = 5 expansion for R ( e + e - ) , R ( e + e - ) --'1(1 - 3 +a+i.41a2+64.8a3)

(4)

Again the expansion is for the MS scheme, this time with scale / i = v / s , the CM energy o f the e + e collisions. In this letter we wish to point out that, unfortunately, although a perturbative Q C D expansion can be written down for R ( z ) , the expansion cannot be trusted. F o r values o f A ~ favoured by e+e - annihilation data, the prediction for R (z) is strongly renormalization scheme dependent. F u r t h e r m o r e this statement is true to O ( ~ ) , as well as O ( a 3 ) , and is not due to the worryingly large coefficient o f a 3. Before we can illustrate our claim, we need to briefly review how the perturbative series coefficients and couplants in different renormalization schemes may be obtained. It is convenient to subtract off the zeroth order part o f R (T) and R ( e + e = ) and define the quantities ]~(x) a n d / ~ ( e + e - ), .if(z)-- R ( x ) - 3 3 ~(e+e-)_

(3)

0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )

'

R(e+e-)-3YQ~

3EQf~

(5) 63

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PHYSICS LETTERS B

These then have a perturbation series o f the form /~=a(1 +rla+r2a2+...).

(6)

We can define the Nth order perturbative approximant, R (N) =a(N) ( 1 + rl a(N) + r2a(N)2 +...

+rN--I a(u)N-l) •

(7)

Using the notation of ref. [7], the couplant a (N) in the Nth order is taken to satisfy the truncatedfl-function equation 0a b~r =--ba2(l+ca+c2a2+...+CN_laN-l)

,

(8)

fective charge" (EC) scheme of Grunberg [ 10], in which the higher order coefficients are zero rl=r2 . . . . . rN-l=O, and R(N)=a(N). Thus the couplant is an effective charge equal to the physical quantity. For this scheme one has r=po and c2=P2, where [ 11 ] po=r-rl

CM21S2 =

1

3,9

where (9c) is from ref. [ 8 ]. Eq. (8) can be integrated to obtain a transcendental equation for a (N). For N = 2

1

( ca (2) l +ca(2)j ,

(10)

1

+C In(ca (3)) -- 1C ln l 1 +ca (3)+c2a (3)2 I

+ 2C2--C2[-

~ //2c2a(3)+c"~

--l_arc'ant

d-- (4c2 - c 2) 1/2

.

//ch]

,]-arctanlt~)_ I ' ( 11 )

In integrating eq. (8), boundary conditions corresponding to a Q C D A parameter, (,~), differing by a factor ( 2 c / b ) -c/b from the conventional one, have been used [ 7 ].

clb

,,2) For N = 2 , `i(2) depends only on the choice of r. The N = 3 result depends on r and c2. For given values of r and c2, a (2), a (3) are obtained by solving the transcendental equations ( 10 and ( 11 ). Different choices of r and c2 correspond to different renormalization schemes ( RS' s). One can choose a special scheme, the so-called "ef64

5033Nr+~45 Uf2 -- Y~"

16(11-]Nf)

(13)

(14)

It is clear that perturbation theory can only be trusted if the perturbative approximants do not show a strong dependence on which scheme is chosen, otherwise one result is no more plausible than another. In practice there is typically a substantial flat region surrounding a stationary point at some values r = f, c2 =(2. For N = 2 , ~is defined by d`i(2) (r) ~=~ dr =0,

whilst for N = 3, with 4 c 2 > c 2 one has [9] r3 = ~

2~7

f

2(33-2Nf) ' (9a-c)

r2= a~57 + c l n

, p2=c2+r2-rlc-r~.

These quantities are necessarily RS-invariant, and so Po, P2 may be computed in a calculationally convenient scheme such as MS, and then eq. (13) used to obtain r~ and r2 for any other scheme, (i.e. for other values of r and c2). For the MS calculation r ~ is found by evaluating (9a) at A=/TMS and [ 12]

where

33

8 February 1990

(15)

and for N = 3 , l a n d (2 by 0/~ (3)c(2r') O r

r=z = 0 ,

0/~(3)) 0( r2c' c2 c2=c2=0.

(16)

The prediction for `i(N) should lie in this flat region if the perturbative approximant is to be trustworthy. Stevenson has suggested that one should choose the scheme corresponding to this stationary point, the socalled "principle of minimal sensitivity", or PMS scheme [7]. We are not insisting on the rigid application of PMS, but will show that the flat region in `i(2)(z) and 1~(3) ( r ) moves offtowards R ( r ) = +oo for values of AMS which fit the R ( e + e - ) data, and eventually ceases to exist at all, i.e. eqs. (15), (16) can no longer be satisfied for any values of z and c2. Hence, for instance, the MS prediction for R ( r ) , which is finite and plausible for reasonable values of A ~ ( N f = 5 ), cannot be trusted since, in the vicinity of the MS scheme, the approximants are rapidly varying functions of the scheme. Perturbation theory

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is therefore inapplicable for this process. To keep track of the flat region we shall calculate the PMS approximants. At second order, N = 2 , the condition (15) corresponds to solving the transcendental equation P ° = a1 + c l n

ca

+ ~

.

i

dx a (1 +cx+c>v 2)2 - 1+ ( c + 2 r t ) a '

(20)

for c2 and a. The ri in the equations are implicit functions of a and c2 using eqs. (10), ( 11 ) and ( 13 ). We propose to present results by directly plotting curves o f / ~ ( ~ ) v e r s u s / ~ ( e + e - ). We shall evaluate /~(e+e - ) for x / s = 34 GeV, and it is hence to be calculated in N f = 5 QCD, a n d / ~ ( z ) in N r = 3 QCD. Because of the decoupling theorem [ 13 ], these can be regarded as two versions of Q C D having different A scales, A ~ , and A M . For some given A~s~ we can oblain A M by evolving through flavour thresholds. We have chosen rob=4.2 GeV and mc=1.25 GeV, although the couplant, and hence final results, are insensitive to the precise values chosen. At the b quark threshold, working to third order b, c and c2Ms are

(17)

The solution a = 6 then gives

R b ~ = a( 1 + ½ca) l+cd

8 February 1990

( 18 )

At third order, N = 3, the two conditions of (16) are equivalent to solving the simultaneous transcendental equations (c2 +2rtc+3r2) +a(2rlc2 + 3r2c) +a2(3r2c2) = 0 , 19) 1.0

i

]

q

I

I

I

KEY --MS . . . . . EC --PMS

--

5

.4 /

7

/ / / / / /

0

~

025

l

I

l ~

.030

I

i

i

L~t~

.035

I

k ~

I

.040

i

k

L

I

]

.045

I

I

I

i

I

_~

.050

]

I ~ 1 1 1 .055

.060

~(z) (e + e - ) Fig. 1. *if(x) v e r s u s / ~ ( e + e - ) at second o r d e r for three RS's.

65

cwduarcd using &=5. and cq. ( I I ) IS solved for u. whtrc r is found l’rom !9at using ,?;td, I(=w~,.It. c illId ir' are then rcplaccd with their X,=4 is replawd

cvslumd (9s)

by (I’ =-a+;+’

to find a new r. ,i&

with these new WILES

same prwcdurc

values.

4

an11 thr L.HS cq. ( I I ) ir IS rhcn found from

of 1. h. c. and $“.

The

is used GII thr c quark rhrcshold. TX

II;IW W(S’(r)(RS)

for the aamc KS. Am

is ctimi-

natcd and R(r) c3n be plotted dwccrlp against k[r ‘c- ) for N= 2. 3 and Ihe three KS choices MS. EC. PMS. /;or the 3(s scheme we use Ihc scale II-

,,“o IIU cviduating I?( c+c.. ) andjJT w, for cvalu-

aling liC Y). ( )nc could. m principle. use IHV dlffcrcnc RS’s. using h;x IO tix @.! and PMS ta evalualc a( T). for instance. Howvcr.

only the RS choicr used to

L)* discontinuity a1 the mw lhrcshotds results from the cwcful opphcarion of ihc dccoupkng ihearrm [ 141. hut is numerically uni.?~twrl;rnL. Omitting Ihir

cualuaw &K 1 IW-IISWI lo matter. As shown in ref. 1 I5 J rhc ,i& ckirwml from I?(c+c- ) is rather in-

dwonlinuit) is the only diffcrcncc be~wt~n working in second and third orders Our method was 10 chow,c a value ofR(e*e) and, workmg m a particular RS. find ,?& such thal R(,\‘):e.e-)(RS)=R(c*c..)(.~~2. 3). WC then

for X= 3 in fig. 2. AS &c+c ,,i~.l’hc~~lot.‘c-dalaofrcf.

cvolvc,&&j

66

to.@&.

Clsing this value of,jj&

WC cval-

scnsilivc lo RS. WC display the thrw curves for S= 2 In tie 1. and

RfC+C

t mcreaw.

so dots

[ 16).giwstherr~uh

)(1= lMK.t(iCV~)=3.051~0.007. For both X= 2 and 3’~ 3, Ihe ovrrall brhaviour of Ihe curws IS similar. 411 r.irce schemr% imtiall~ grow

Volume 236, number 1

PHYSICS LETTERS B

.45

I

I

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~

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8 February 1990

[

~

I

~

I

i

I

i

I

i

I

T--

.40

~(@)(e + e - ) .55

= 0,051

~(MS) (7") /

.30

-¢

.25

v

.20

,15

.10

.05

0

t

I .02

L

I .04

I

I .06

t

I .08

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i

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1

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I .22

I .24

i

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k ,30

8.

Fig. 3./~(x) as a function of a at/~(e+e - ) =0.051, the central value from the fit to the e+e - annihilation data of ref. [16]./~Ms (x) is indicated by an asterisk.

slowly, with reasonable values f o r / ~ ( z ) . MS continues to grow but its values are still plausible. PMS and EC however, start to increase rapidly at about the same point ( ~ 0.04 for N = 2 and ~ 0.033 for N = 3), becoming undefined within a very short region. So for values o f / ~ ( e + e - ) favoured by e+e - annihilation data, N = 2 PMS yields unphysically large values o f / ~ ( z ) and eventually becomes undefined, whilst for N = 3 it is not defined at all. This indicates that perturbation theory cannot be trusted for this quantity. Schemes such as MS which yield likely results lie in a region where there is a rapid variation of R ( ~ ) with scheme choice (see figs. 3, 4), whereas the expected flat region corresponds to unreasonably large values, or does not exist at all. The region in which PMS grows rapidly or is un-

defined, corresponds to low or negative values ofpo f o r / ~ ( x ) , defined in ( 13 ). F o r perturbation theory to be valid one requires Po >> 0 [ 7,11 ], and the small value o f Po is what renders perturbation theory untrustworthy for x decay. Decreasing r~ alleviates the situation somewhat, PMS becoming undefined at larger values o f / ~ ( e + e - ) but in third order this is still at the lower end o f the values favoured by e+e annihilation data. The situation for a heavier lepton would also be more favorable as this would raise/t in eq. (9a), and hence increase Po. Whilst the small size of non-perturbative effects and the possibility o f a well-ordered o p e r a t o r product expansion were pleasant surprises in the evaluation o f R ( z ) , the relatively large size o f the perturbative coefficient r~, and the small mass o f the ~, resulting in 67

Volume 236, number 1

PHYSICS LETTERS B

8 February 1990

Rc31(e" e-)= 0051

0.14~

Z~U Fig. 4./~(x) as a function of a and c2 at/~(e+e - ) =0.051. maximum values of/~ (3) (z) in the given range.

L2 m /~MS (.~)

a s m a l l Po, u n f o r t u n a t e l y c o n s p i r e to m a k e p e r t u r b a tive p r e d i c t i o n s u n t r u s t w o r t h y . C . J . M . w o u l d like to t h a n k t h e t h e o r y g r o u p o f L P T H E , Orsay for their hospitality during the period t h i s w o r k was u n d e r t a k e n . We t h a n k Bob O a k e s f o r b r i n g i n g refs. [ 2 , 4 ] to o u r a t t e n t i o n .

References [ 1 ] M.L. Perl, Annu. Rev. Nucl. Part. Sci. 30 (1980) 299. [2] E. Braaten, Phys. Rev. Lett. 60 (1988) 1606. [3] S. Narison and A. Pich, Phys. Lett. B 211 (1988) 183. [4] E. Braaten, Phys. Rev. D 39 (1989) 1458. [5] W.J. Marciano and A. Sirlin, Phys. Rev. Lett. 61 (1988) 1815.

68

-10

is indicated by an asterisk. Zmin and Zmax indicate the minimum and

[6] S.G. Gorishny, A.L. Kataev and S.A. Larin, Phys. Lett. B 212 (1988) 238. [ 7 ] P.M. Stevenson, Phys. Rev. D 23 ( 1981 ) 2916. [8] D.R.T. Jones, Nucl. Phys. B 75 (1974) 531; W. Caswell, Phys. Lett. B 33 (1974) 244. [9] C.J. Maxwell, Phys. Rev. D 28 (1983) 2037. [ 10] G. Grunberg, Phys Lett. B 95 (1980) 70; Phys. Rev. D 29 (1984) 2315. [ 11 ] A. Dhar and V. Gupta, Phys. Rev. D 29 (1984) 2822. [ 12] O.V. Tarasov, A.A. Vladimirov and A. Yu, Zharkov, Phys. Lett. B 93 (1980) 429. [ 13 ] T. Appelquist and J. Carazzone, Phys. Rev. D 11 ( 1975 ) 2856. [ 14] W. Bernreuther and W. Wetzel, Nucl. Phys. B 197 (1983) 228. [15 ] C.J. Maxwell and J.A. Nicholls, Phys. Lett. B 213 (1988) 217. [ 16] R. Marshall, Z. Phys. C 43 (1988) 595.