PERTURBATIVE QUARK MASS CORRECTIONS TO THE TAU ...

arXiv:hep-ph/9804462v1 30 Apr 1998

FTUV/97-47 IFIC/97-63 UG-FT-77/97 hep-ph/9804462 April 1998

PERTURBATIVE QUARK MASS CORRECTIONS TO THE TAU HADRONIC WIDTH

Antonio Picha and Joaquim Pradesb a

b

Departament de F´ısica Te`orica, IFIC, Universitat de Val`encia — CSIC Dr. Moliner 50, E-46100 Burjassot (Val`encia), Spain. Departamento de F´ısica Te´orica y del Cosmos, Universidad de Granada, Campus de Fuente Nueva, E-18002 Granada, Spain.

Abstract The perturbative quark–mass corrections to the τ hadronic width are analysed to O(α3s m2q ), using the presently available theoretical information. The behaviour of the perturbative series is investigated in order to assess the associated uncertainties. The implications for the determination of the strange quark mass from τ decay data are discussed.

1.

Introduction

The inclusive character of the total τ hadronic width renders possible an accurate calculation of the ratio [1–5] Rτ ≡

Γ [τ − → ντ hadrons (γ)] , Γ [τ − → ντ e− ν¯e (γ)]

(1)

using standard field theory methods. The result turns out to be very sensitive to the value of αs (Mτ2 ). Moreover, the uncertainties in the theoretical calculation are quite small and dominated by the perturbative errors. This has been used to perform a very precise determination of the QCD coupling at low energies [5]. Quark masses play a rather minor rˆole in Rτ . Owing to the tiny values of mu and md , their associated corrections are very small [3] (∼ −0.1%). The strange quark contribution to the total τ hadronic width is suppressed by the Cabibbo factor |Vus |2 , which puts the induced ms correction also at the per cent level. However, if one analyses separately the semi-inclusive decay width of the τ into Cabibbo–suppressed modes (i.e. final states with an odd number of kaons), the relatively large value of ms induces an important effect of a size similar to the massless perturbative correction and of opposite sign [3]. The corresponding Rτ,S prediction is then very sensitive to the strange quark mass and could be used to extract information on this important, and nowadays controversial, parameter. A very preliminary value of ms , extracted from the ALEPH τ decay data, has been already presented in recent workshops [6, 7]. The determinations of light quark masses are usually obtained from analyses of the divergences of the vector and axial–vector current two–point function correlators or related observables [8–13]. These correlators are proportional to quark masses and, therefore, are very sensitive to their numerical values. Unfortunately, one needs phenomenological information on the associated scalar and pseudo-scalar spectral functions, which are not well known at present. The obvious advantage of a possible determination of ms analysing quark mass effects in τ decays is that the experimental error can be systematically reduced in foreseen facilities like tau–charm or B factories. There is then some hope to achieve a precise determination of ms from such analyses. Recently the O(αs3) corrections to the J = 0 quark correlators have been calculated [14], and have been found to be rather large. The influence of these O(αs3) corrections on the determination of quark masses and the uncertainties coming from the truncation of the QCD perturbative series depend very much on the observable. One can see for instance that the QCD perturbative series behaves geometrically to O(αs3) for the divergence of pseudo-scalar (scalar) currents if resummed perturbatively in terms of αs (s) [12, 13]. This convergence improves [13] using other resummations like the Principle of Minimal Sensitivity (PMS) [15] or the one advocated in Ref. [4]. 1

The hadronic τ decay width has also a J = 0 contribution, which, as we shall see, behaves rather badly. However, the largest quark–mass correction originates in a piece of the left–handed current correlation function, involving the J = 0 + 1 combination, which shows a much better perturbative convergence. The purpose of this paper is to study the perturbative behaviour of the corrections to Rτ which are proportional to m2q , in order to assess the associated uncertainties. These are the leading theoretical uncertainties in the ms determination. The O(αs m2q ) contributions were already studied in Ref. [3]. In Ref. [16] the contributions of O(αs2m2q ) to the relevant correlators were worked out. More recently, some partial information on O(αs3m2q ) corrections has become available [14,17]. A much detailed analysis of all contributions up to dimension four will be presented elsewhere. 2.

Theoretical Framework

The theoretical analysis of Rτ involves the two–point correlation functions for the vector Vijµ = ψ¯j γ µ ψi and axial–vector Aµij = ψ¯j γ µ γ5 ψi colour–singlet quark currents (i, j = u, d, s): Πµν ij,V (q)

≡i

Πµν ij,A (q) ≡ i

Z

Z

d4 x eiqx h0|T (Vijµ (x)Vijν (0)† )|0i,

(2)

d4 x eiqx h0|T (Aµij (x)Aνij (0)† )|0i.

(3)

They have the Lorentz decompositions (1)

(0)

µν 2 µ ν 2 µ ν 2 Πµν ij,V /A (q) = (−g q + q q ) Πij,V /A (q ) + q q Πij,V /A (q ),

(4)

where the superscript (J) in the transverse and longitudinal components denotes the corresponding angular momentum J = 1 (T) and J = 0 (L) in the hadronic rest frame. (J) The imaginary parts of the two–point functions Πij,V /A (q 2 ) are proportional to the spectral functions for hadrons with the corresponding quantum numbers. The semi-hadronic decay rate of the τ can be written as an integral of these spectral functions over the invariant mass s of the final–state hadrons: Rτ = 12π

Z

0

Mτ2

ds Mτ2

s 1− 2 Mτ

!2 

!

s 1 + 2 2 ImΠ(1) (s) + ImΠ(0) (s) . Mτ 

(5)

The appropriate combinations of correlators are 

(J)

(J)





(J)

(J)



Π(J) (s) ≡ |Vud |2 Πud,V (s) + Πud,A (s) + |Vus |2 Πus,V (s) + Πus,A (s) .

(6)

We can decompose the predictions for Rτ into contributions associated with specific quark currents: Rτ = Rτ,V + Rτ,A + Rτ,S . 2

(7)

Rτ,V and Rτ,A correspond to the contributions from the first two terms in Eq. (6), while Rτ,S contains the remaining Cabibbo–suppressed contributions. Exploiting the analytic properties of the correlators Π(J) (s), Eq. (5) can be expressed as a contour integral in the complex s plane running counter–clockwise around the circle |s| = Mτ2 : Rτ = −πi

I

|s|=Mτ2

ds s

s 1− 2 Mτ

!3 (

!

)

s 3 1 + 2 D L+T (s) + 4 D L (s) . Mτ

(8)

We have used integration by parts to rewrite Rτ in terms of the logarithmic derivative of the relevant correlators, D L+T (s) ≡ −s

d h (0+1) i Π (s) , ds

D L (s) ≡

s d h (0) i s Π (s) , Mτ2 ds

(9)

which satisfy homogeneous renormalization group equations. Using the Operator Product Expansion to organise the perturbative and nonperturbative contributions to the correlators into a systematic expansion [18] in powers of 1/s, the total ratio Rτ can be expressed as an expansion in powers of 1/Mτ2 , with coefficients that depend only logarithmically on Mτ [3]: 

2

Rτ = 3 |Vud| + |Vus |

2



SEW



′ 1+δEW

+δ

(0)

+

X

D=2,4,...



2

cos

(D) θC δud

+ sin

2

(D) θC δus



,

(10) = + is the average of the vector and axial–vector corwhere ′ rections of dimension D, SEW and δEW contain the known [19, 20] electroweak 2 2 2 corrections, and sin θC ≡ |Vus | /(|Vud | + |Vus |2 ). The dimension–zero contribution is the purely perturbative correction neglecting quark masses, which, owing to chiral symmetry, is identical for the vector and axial–vector correlators. It is fully generated by the Adler function D L+T (s), because D L (s) vanishes in the chiral limit. The correction δ (0) has been investigated in great detail in Ref. [4]. We will follow a similar procedure to analyse the perturbative quark–mass corrections of dimension two. (D) δij

3.

(D) (δij,V

(D) δij,A )/2

Dimension–Two Corrections

For the sake of simplicity, let us take here mu = md = 0. The arguments we shall put forward don’t depend on it. In this limit, the vector and axial–vector J J correlators get the same quark–mass corrections, i.e. Dus,A (s) = Dus,V (s) ≡ J Dus (s) (J = L + T, L). The dimension–two contributions can be written in the form: 3 m2s (−ξ 2 s) X ˜L+T L+T dn (ξ) an (−ξ 2 s) , (11) Dus (s)|D=2 = 2 4π s n=0 L Dus (s)|D=2 = −

3 m2s (−ξ 2 s) X ˜L dn (ξ) an (−ξ 2 s) , 8π 2 Mτ2 n=0 3

(12)

where a = αs /π, ξ is an arbitrary scale factor (of order unity) and the coefficients d˜Jn (ξ) are constrained by the homogeneous renormalization group equations satJ isfied by the corresponding functions Dus (s): ξ

n X d ˜J dn (ξ) = [2γk − (n − k)βk ] d˜Jn−k (ξ) , dξ k=1

for n ≥ 1 and

(13)

d ˜J d (ξ) = 0 dξ 0

(14)

i.e. d˜J0 (ξ) = dJ0 , d˜J1 (ξ) = dJ1 + 2γ1dJ0 log ξ , i h d˜J2 (ξ) = dJ2 + 2γ2 dJ0 + (2γ1 − β1 )dJ1 log ξ + γ1 (2γ1 − β1 )dJ0 log2 ξ ,

d˜J3 (ξ) = dJ3 + 2γ3 dJ0 + (2γ2 − β2 )dJ1 + 2(γ1 − β1 )dJ2 log ξ h

h

i

(15)

i

+ (−γ1 β2 + 2γ2 (2γ1 − β1 )) dJ0 + (γ1 − β1 )(2γ1 − β1 )dJ1 log2 ξ 2 + γ1 (γ1 − β1 )(2γ1 − β1 )dJ0 log3 ξ , 3 J J ˜ d4 (ξ) = d4 + · · · The factors βk and γk are the expansion coefficients of the QCD β and γ functions, µ

µ

da = β(a) a , dµ

β(a) =

X

βk ak ,

(16)

k=1

dm = −γ(a) m , dµ

γ(a) =

X

γk ak ,

(17)

k=1

which are known to four loops [21–23]. The coefficients dJn ≡ d˜Jn (1) are only known to order αs2 for J = L + T and αs3 for J = L [8, 14, 17, 24–28]. For three flavours and in the MS scheme, one has: 9 β1 = − ; 2

β2 = −8 ;

β3 = −

3863 ; 192

140599 445 − ζ3 ≈ −94.456 079 ; (18) 2304 16 8885 91 ; γ3 = − 5 ζ3 ≈ 24.840 410 ; γ1 = 2 ; γ2 = 12 288 2977517 9295 135 125 γ4 = − ζ3 + ζ4 − ζ5 ≈ 88.525 817 ; (19) 20736 216 8 6 β4 = −

4

13 21541 323 520 ; d2L+T = + ζ3 − ζ5 ≈ 37.083 047 ; 3 432 54 27 9631 35 17 ; dL2 = − ζ3 ≈ 45.845 949 ; (20) dL0 = 1 ; dL1 = 3 144 2 4748953 91519 5 715 dL3 = − ζ3 − ζ4 + ζ5 ≈ 465.846 304 . 5184 216 2 12 Notice the rather bad perturbative behaviour of the D = 2 corrections to the J correlation functions Dus . Remember that a(Mτ2 ) ≃ 0.11. Inserting the expansions (11) and (12) in Eq. (8), the D = 2 corrections to Rτ,S can be expressed as d0L+T = 1 ;

(2) δus = −8

d1L+T =

m2s (Mτ2 ) ∆[a(Mτ2 )] , Mτ2

∆[a] ≡

where ∆J [a(Mτ2 )] =

X

o 1 n L+T 3∆ [a] + ∆L [a] , 4

(n) d˜Jn (ξ) BJ (aξ ) ,

(21)

(22)

n=0

and the contour integrations are contained in the functions 2 2 dx 3 m(−ξ Mτ x) (1 + x) (1 − x) m(Mτ2 ) |x|=1 x2

(n) BL+T (aξ )

−1 ≡ 4πi

I

(n) BL (aξ )

1 ≡ 2πi

I

|x|=1

dx (1 − x)3 x

m(−ξ 2 Mτ2 x) m(Mτ2 )

!2

!2

an (−ξ 2 Mτ2 x), (23)

an (−ξ 2 Mτ2 x) .

(24)

(n)

Since the quark mass ratio is flavour independent, the integrals BJ (aξ ) regulate also the small corrections proportional to mu and md , which we are neglecting. These functions depend only on βi , γj , aξ ≡ αs (ξ 2 Mτ2 )/π and log ξ. Moreover, they satisfy the following homogeneous renormalization group equations: ξ

4.

X d (n) (n+k) BJ (aξ ) = (nβk − 2γk ) BJ (aξ ) . dξ k=1

(25)

Perturbative aξ Expansion (n)

The usual perturbative approach expands the BJ (aξ ) functions in powers of aξ . This gives, (0)

i

h

BJ (aξ ) = 1 − γ1 2 log ξ + H1J aξ "





− 2γ2 log ξ − γ1 (2γ1 + β1 ) log2 ξ + γ2 − 2γ12 log ξ H1J !

#

γ1 β1 − γ1 − H2J a2ξ 2 2

5

"

2 − 2γ3 log ξ − (2γ2 (2γ1 + β1 ) + γ1 β2 ) log2 ξ + γ1 (γ1 + β1 )(2γ1 + β1 ) log3 ξ 3 



+ γ3 − 4γ1 γ2 log ξ + γ12 (2γ1 + β1 ) log2 ξ H1J !

!

1 β1 1 log ξ H2J β1 γ2 + β2 γ1 − γ1 γ2 + γ12 γ1 − + 2 4 2 # γ1 + (2γ1 − β1 )(γ1 − β1 )H3J a3ξ + · · · 12 (1) BJ (aξ )

!

"

(26)

#

β1 H1J a2ξ = aξ − 2γ1 log ξ + γ1 − 2

"

!

β2 − 2γ2 log(ξ) − γ1 (2γ1 + β1 ) log ξ + γ2 − − γ1 (2γ1 − β1 ) log ξ H1J 2 ! # β1 1 γ1 − (γ1 − β1 ) H2J a3ξ + · · · (27) − 2 2 (2)

2

i

h

BJ (aξ ) = a2ξ − 2γ1 log ξ + (γ1 − β1 ) H1J a3ξ + · · ·

(28)

(3)

BJ (aξ ) = a3ξ − · · ·

(29)

where HnL+T ≡ HnL

−1 4πi

I

dx (1 + x) (1 − x)3 logn (−x) , |x|=1 x2

1 ≡ 2πi

dx (1 − x)3 logn (−x) . |x|=1 x

I

(30) (31)

To O(αs4), the needed integrals are H0L+T = 1 ,

H1L+T =

H2L+T =

25 π 2 − , 18 3

721 25 2 π 4 − π + , 54 9 5 11 85 π 2 H0L = 1 , H1L = − , H2L = − , 6 18 3 575 11 2 3661 85 2 π 4 H3L = − + π , H4L = − π + . 36 6 54 9 5 The perturbative expansions ∆J [a] then take the form H3L+T =

85 π 2 − , 36 6

1 , 6

∆J [a(Mτ2 )] =

H4L+T =

˜ J (ξ) an , d˜Jn (ξ) + h n ξ

Xh

n=0

6

i

(32)

(33)

(34)

˜ J (ξ) depend on d˜J (ξ), βm4 contributions, which are likely to be small. Thus, a more appropriate approach is to directly use the expansions (22), in terms of the original d˜Jn coefficients, and to fully keep the (n) known four–loop information on the functions BJ (aξ ). (n) (n) Tables 1 and 2 show the exact results for BL+T (a) and BL (a) (n = 0, 1, 2, 3) with ξ = 1 obtained at different orders in the β and γ expansions, together with The O(a2 ) correction agrees with the numerical result recently reported in Ref. [29], which is larger than the value originally quoted in Ref. [16]. This larger O(a2 ) correction has been also confirmed by K. Chetyrkin and A. Kwiatkowski [30]. a

8

the final values of ∆J [a], for a = 0.1 (ξ = 1). For comparison the numbers coming from the truncated perturbative expressions at O(a3 ) are also given. (n)

Table 1: Exact results for BL+T (a) (n = 0, 1, 2, 3) obtained at the k–loop (k = 1, 2, 3, 4) approximation (βj>k = γj>k = 0), together with the final value of P (n) ∆L+T [a] = 2n=0 dnL+T BL+T (a), for a = 0.1 and ξ = 1. For comparison the numbers coming from the truncated expressions at O(a3 ) are also given. (0)

Loops BL+T (a) 1 0.890 32 2 0.817 19 3 0.791 43 4 0.782 37 3 O(a ) 0.793 63

(1)

BL+T (a) 0.069 65 0.056 66 0.052 96 0.051 68 0.064 73

(2)

(3)

BL+T (a) BL+T (a) ∆L+T [a] 0.004 52 0.000 186 1.360 0.002 78 −0.000 008 1.166 0.002 36 −0.000 048 1.108 0.002 22 −0.000 060 1.089 0.008 92 0.001 000 1.405

(n)

Table 2: Exact results for BL (a) (n = 0, 1, 2, 3) obtained at the k–loop (k = 1, 2, 3, 4) approximation (βj>k = γj>k = 0), together with the final value of P (n) ∆L [a] = 3n=0 dLn BL (a), for a = 0.1 and ξ = 1. For comparison the numbers coming from the truncated expressions at O(a3 ) are also given. Loops 1 2 3 4 O(a3)

(0)

BL (a) 1.399 08 1.540 13 1.578 53 1.589 10 1.644 46

(1)

(2)

BL (a) 0.184 73 0.202 47 0.206 17 0.207 06 0.218 94

BL (a) 0.022 55 0.024 21 0.024 44 0.024 46 0.021 92

(3)

BL (a) ∆L [a] 0.002 588 4.686 0.002 692 5.052 0.002 690 5.120 0.002 681 5.133 0.001 000 4.356 (n)

These numerical results show a reasonable convergence of the BJ (a) integrals, as higher–order βk and γk contributions are taken into account. Increasing the number of loops one gets a small decrease (increase) of the transverse (longitudinal) contribution. It is also clear that the truncated O(a3 ) expressions overestimate (underestimate) ∆L+T (∆L ). Taking the full four–loop information into account, we get the following perturbative behaviour: ∆L+T [0.1] = 0.7824 + 0.2239 + 0.0823 − 0.0000601 d3L+T + · · ·

(40)

∆L [0.1] = 1.5891 + 1.1733 + 1.1214 + 1.2489 + · · ·

(41)

The L+T series converges very well. Owing to the negative running contributions the ∆L+T [a] series behaves better than the original perturbative expansion of L+T Dus (s)|D=2 . Unfortunately, the longitudinal series is much more problematic. 9

L The bad perturbative behaviour of Dus (s)|D=2 gets reinforced by the running effects, giving rise to a badly defined series. The combined final expansion,





∆[0.1] = 0.9840 + 0.4613 + 0.3421 + 0.3122 − 0.000045 d3L+T + · · ·

(42)

looks acceptable for the firsts terms because ∆L+T is weighted by a larger factor. In fact, this series behaves better than the one in Eq. (39), obtained with the usual perturbative truncation of the contour integrals. Nevertheless, after the third term the series appears to be dominated by the longitudinal contribution, and the bad perturbative behaviour becomes again manifest. Using the full four–loop result we have certainly gained in convergence for the L+T ∆ series [compare the fourth term in the series (37) and (40) for a = 0.1], which is otherwise the one we don’t know the O(a3) coefficient. We can take advantage that the O(a3 ) correction to ∆[a] is almost completely given by the known ∆L contribution. Using d3L+T ∼ d2L+T (d2L+T /d1L+T ) ≈ 317, the fourth term in (42) becomes 0.298, i.e. a 5% reduction only. Taking the size of the O(a3 ) contribution to ∆L as an educated estimate of the perturbative uncertainty, we finally get ∆[0.1] = 2.1 ± 0.3 . (43) 6.

Renormalization–Scale Dependence

The expansion (22) depends order by order on ξ and this dependence cancels out only when we sum the infinite series. In practice, we only know a few first terms of the series (three for ∆L+T [a] and four for ∆L [a]); so we should worry how much the predictions depend on our previous choice ξ = 1. Obviously, ξ should be close to one in order to avoid large logarithms; but variations within a reasonable range, let us say from 0.75 to 2, should not affect too much the final results. Smaller values of ξ would put the QCD coupling in the non-perturbative regime and are therefore not acceptable. Figures 1, 2 and 3 show the sensitivity to the selection of renormalization scale of the final predictions for ∆L+T , ∆L , and ∆, respectively, for a(Mτ2 ) = 0.1. The behaviour of ∆L+T is quite good. The predicted value remains very stable in the whole range ξ ∈ [0.75, 2], showing that the perturbative series is very reliable. Below ξ ∼ 1/2, the perturbative expansion breaks down, as expected, because the coupling aξ is already outside the radius of convergence of the series. The longitudinal series, on the other side, has a quite wild dependence on the renormalization scale. Changing ξ from 1 to 2, amounts to a reduction of ∆L of about 65%. Thus, the theoretical uncertainty is very large in this case. The ξ dependence of the complete expansion ∆, reflects obviously the behaviour of its two components. The larger weight of ∆L+T keeps the result still acceptable, within the range of ξ considered, but the sizeable ∆L contribution 10

spoils the stability and generates a monotonic decrease of the prediction for increasing values of ξ. Taking this variation into account, the theoretical error in Eq. (43) should be increased to about 0.6, i.e. a 30% uncertainty in the final prediction. 6.

Discussion

The bad perturbative behaviour of the longitudinal contribution does not allow to make an accurate determination of the strange quark mass from Rτ,S . Nevertheless, taking ∆[0.1] = 2.1 ± 0.6 , (44) ms (Mτ2 ) could be still obtained with a theoretical uncertainty of about 15%, which is not so bad. Notice that it is the phase–space integration of the original correlation functions the responsible for the different behaviour of the longitudinal and transverse components. Therefore, the perturbative convergence could probably be improved through an appropriate use of weight factors in Eqs. (5) and (8). This requires an accurate measurement of the final hadrons mass distribution in the τ decay, which so far has only been performed for the dominant Cabibbo–allowed modes [31]. The measurement of Rus (s) could be feasible at the forthcoming flavour factories, where a very good kaon identification is foreseen. From the theoretical point of view, the analysis of weighted moments of the final hadrons mass distribution proceeds in a completely analogous way [32]. A detailed study will be presented in a forthcoming publication. Acknowledgements We would like to thank Kostja Chetyrkin for informing us about the existence of a misprint in Ref. [16]. We have benefit from many discussions (and program exchanges) with Matthias Jamin and Arcadi Santamar´ıa, concerning the numerical implementation of running effects at higher orders. Useful discussions with Michel Davier, Andreas H¨ocker and Eduardo de Rafael are also acknowledged. This work has been supported in part by the European Union TMR Network EURODAPHNE —Contract No. ERBFMRX-CT98-0169 (DG 12 – MIHT)— and by CICYT, Spain, under Grants No. AEN-96/1672 and AEN-96/1718.

11

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[22] K.G. Chetyrkin, B.A. Kniehl, and M. Steinhauser, Phys. Rev. Lett. 79 (1997) 2184. [23] K.G. Chetyrkin, Phys. Lett. B404 (1997) 161. [24] L.R. Surguladze, Sov. J. Nucl. Phys. 50 (1989) 372. [25] S.G. Gosrishny et al, Mod. Phys. Lett. A5 (1990) 2703. [26] S.G. Gosrishny et al, Phys. Rev. D43 (1991) 1633. [27] S.G. Gorishny, A.L. Kataev, and S.L. Larin, Nuovo Cimento 92 ( 1986) 119. [28] W. Bernreuther and W. Wetzel, Z. Phys. C11 (1981) 113. [29] K. Maltman, Problems with Extracting ms from Flavor Breaking in Hadronic τ Decays, York preprint YU-PP-K/M-98-1, hep-ph/9804298. [30] K.G. Chetyrkin, private communication. [31] R. Barate et al (ALEPH Coll.), Measurement of the Spectral Functions of Axial-Vector Hadronic τ Decays and Determination of αs (Mτ2 ), preprint CERN-EP-98-012 [32] F. Le Diberder and A. Pich, Phys. Lett. B289 (1992) 165.

13

Figure Captions

• Figure 1.- Variation of ∆L+T [0.1] with the renormalization–scale factor ξ, to four loops. • Figure 2.- Variation of ∆L [0.1] with the renormalization–scale factor ξ, to four loops. • Figure 3.- Variation of ∆[0.1] with the renormalization–scale factor ξ, to four loops.

14

∆ L+T [0.1] 1.5

1.25

1

0.75

0.5

0.25

0 0.5

0.75

1

1.25

ξ

1.5

1.75

2

Figure 1: Variation of ∆L+T [0.1] with the renormalization–scale factor ξ, to four loops.

∆ L [0.1] 10

8

6

4

2

0 0.5

0.75

1

1.25

ξ

1.5

1.75

2

Figure 2: Variation of ∆L [0.1] with the renormalization–scale factor ξ, to four loops.

∆ [0.1] 4

3

2

1 0.5

0.75

1

1.25

ξ

1.5

1.75

2

Figure 3: Variation of ∆[0.1] with the renormalization–scale factor ξ, to four loops.