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Pole and running lepton masses in QED and "maximal transcendentality" hypothesis Andrei L. Kataev (INR, Moscow)

Abstract Three discovered charged leptons have pole masses me =0.510998910±13в10-9 MeV ( CODATA-06), mµ =105.6583668±38в10-9 MeV (CODATA-06) and m =1776.69±0.19±15 MeV from e+ e- Novosibirsk data for threshold e+ e- + - ( KEDR- collab (07)). We study questions: 1. Definitions of running lepton masses in the MS-schemeespecially important for -lepton mass due to interest in (H + - )- how many order O()-corrections important? 2. QED structure of RG-function m ()= - , comparison Ї with structure of QED (). May it give argument PRO correctness of Baikov-Chetyrkin-Kuhn (08) results ? 3. Conclusions.
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Why running lepton masses in QED ? a) Not often considered. Interesting to use say for -lepton. b) H + - - important decay mode: H + - l + j ets in W -boson fusion may be detectable at LHC if MH = 115-135 GeV and L=60 f b-1 (CMS-collab note referred by N.Krasnikov (07)at 13th Lomonosov Conf. talk). Up to MH 150GeV B r(H + - ) > B r(H cc) (see e.g. Kataev,Kim (96):)

We used H - + = = 2/8 MH m2 (m -pole) like in some other codes H cc = 3 2/8 MH m2 (MH )CH (s ) with running quark mass. c
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3 At present CH (s ) is known up to O(s )-terms (Baikov, Chetyrkin, Kuhn (06). As was shown by Kataev, Kim (07)13 Lomonosov Conf, Kataev, Kim (08)- ACAT08- Erice and Bakulev, Mikhaiklov, Stefanis (08-09-work in progress) Tevatron and LHC experimental precision do not need at present 4 the inclusion of O(s )- they may be important in case of finding Higgs-boson.

Therefore it is appropriate to study the similar approximation for H - + . In the MS-scheme its O(3 )-approximation takes the following form
(M H - + = 0 ( mm H ) )2 [1 + a(MH )s1 + a(MH )2 s2 + a(MH )3 s3 + a(MH )2 as (MH ) QC D ] ,

a(MH ) =



MS

(M H )

, m (MH ) - are QED parameters (they are

MS s

related to the on-shell and m - see below), as (MH ) = the QCD parameter.
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(MH )

is


Evaluation of running -lepton mass: m (MH ) m (m ) m :
a(ml )

ml (MH ) = ml (ml ) exp - a(MH ) = m l (m l ) a(ml )

a(MH ) 20 /

QE m D (x) dx QE D (x)
0

AD(a(MH )) AD(a(ml ))

(1)

where AD(a) are defined up to O(3 )-corrections, which depend from first 4 terms of the RG-functions QE D 2 3 4 5 6 2 = M S (a) = 0 a + 1 a + 2 a + 3 a + 4 a Ї ln µ
ln ml Ї ln µ2

=

QE D m Ї

(a) = -0 a - 1 a2 - 2 a3 - 3 a4 - 4 a

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3 known from Gorishny et al (91); 1-fermion loop contribution to 4 from Baikov, Chetyrkin and Kuhn (07); QED 3 Chetyrkin (97) agrees with QED limit of Vermaseren, Larin, van Ritb ergen (97) QCD 3 ; 4 , 4 are added for theoretical reasons. Next- relation m (m ) m
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m (m ) = m [1 - ( ) + ( )2 (-0.51 + (NL = 3 2) 1.56 - 0.15) + ( ) (2.06 + (NL = 2) 0.84 + 2.49 - (NL = 2) - 0.07 - 0.19 - (NL = 2)2 1.96)] Obtained from analytical result of Melnikov, van Ritb ergen (OO), which agrees with semi-analytical result of Chetyrkin, Steinhauser. m (m ) = m [1 - ( ) + ( )2 2.46 - ( )3 1.94)] Conclusion: a) for -lepton sign-alternating structure of QED PT series seems to manifest itself.electron (NL = 0)- unclear - ( -, -, +). b) It is enough to take into account one term in PT (others are rather small). In the coefficient function- the same:

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Indeed, in the M S -scheme QED coefficients of the coefficient function read s1 = d E = 1 - [ 65 - 3 ](NL + 1 + 3 F Q2 ) (at QF = 0 dE = 2 F 16 E d2 = -.49 (in case of ) 2.37 (in case of µ)
QC D 17 4 691 - 9 3 64 4

4.25

s2 = dE - 0 (0 + 20 ) 2 /3 2

s3 = dE - [d1 (0 + 0 )(0 + 20 ) + 1 0 + 21 (0 + 20 )] 2 /3 3 dE = 3 = E -
QE D -QC D 0 2 1

/3
F

23443 - 239 3 + 45 5 ) - [ 88 - 65 3 - 3 4 + 55 ](NL + 1 + 3 768 16 8 3 4 4 +[ 15511 - 3 ](NL + 1 + 3 F Q2 )2 (dE = -2.03 f or ) 3 F 3888 E = [ 15511 - 4 3 ]4 F Q2 12.95 (In Euclidean region F 2916 3 2 (/ ) s /pi-coefficient is larger- but overall contribution is

Q4 ) F

smaller

than 1-loop term. These results are obtained from Gorishny et al (90-91) and Chetyrkin (97)
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Structure of analytically known QED terms of RG-functions (it is possible to separate the dependence from the number of leptons N = NL + 1 and quark charges QF ): 0 = 1 (N + 3 3 2 = 3 =
616 243 1 32 1 128 F

Q2 ) 1 = 1 (N + F 4
F

F

Q2 ) F
F

- (N + 3

Q6 ) + F
F

22 9

(N + 3

Q2 ) F

2

- 23(N + 3
F

Q8 ) - (N + 3 F
F

F 352 9

Q4 ) F -

2

676 27

-

352 9 3

+

(N + 3

Q 2 ) 3 + (N + 3 F

Q4 ) F

2

256 3 3

(lig ht - by -

lig ht - scattering term)
[1] 4

=

1 1024

4157 6

+ 128

3

(N + 3

F

Q10 ) Notice the appearance F

of 3 -terms in 4 - scheme-independent part of 4 - did not appear 5 in similar lower terms (in QCD this term has group weight CF )rather special feature- interesting the check/study (Kataev (08)).
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In to 1 2

the case of anomalous mass dimension at present it us possible get similar result up to 3-loops 0 = 3 4 13 10 2 = 16 2 - 3 (N + 3 F QF ) 1 = 64 129 + (-46 + 483 )(N + 3 F Q4 ) - 140 (N + 3 F Q2 )2 F F 2 27 not possible to get similar expressions from final second structure is composed from (N + 3 F Q6 ) F Q4 )-terms. In view of this we neglect the F QF and retain N -dependence only: - 3363 ) + (-
280 3

At 4-loops it is result- e.g. the and (N + 3 F contribution of 3 = (
304 27 1 256

(-

1261 8

+ 5523 - 4805 )N +
128 9 3

- 1603 + 964 )N 2 + N 3 (-

664 81

)N

3

+ (64 - 4803 ) N (lig ht - by - lig ht - scattering term) Notice the appearance of 3 in 3 - in QCD this term is multiplied 4 by CF .
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CONCLUSION: For 1) For careful study of H - + is worth to take into account 2-loop running of mass and 1 -loop coefficient function. At 2-loop we may sum 2 -terms (Gorishny,Kataev,Larin (84), Bakulev et al(08-09) 2) The structure of RG functions m in the previous THREE coefficients this 3 did not appear, but appear at the 4-loop levelrather similar to maximal transcendendality property, which appear in conformal-invariant theories. a) Is this 4-loop property similar to the one, observed in QED at 5-loop level only ? Note, that in Yang-Mills theory with matter these parts of l - loop contributions to the RG functions are l proportional th CF . In case of -function conformal limit. b) Next question- what is the origin of appearance of 4 -term in 4 ? We hope to clarify these question at Bogolubov Conference (09)
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