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ELECTROWEAK RADIATIVE CORRECTIONS
J. Wheater
To cite this version:
J. Wheater. ELECTROWEAK RADIATIVE CORRECTIONS. Journal de Physique Colloques, 1982,
43 (C3), pp.C3-305-C3-309. �10.1051/jphyscol:1982360�. �jpa-00221915�
ELECTROWEAK R A D I A T I V E CORRECTIONS
J. F. Wheater
Department o f !7'heoreticaZ Physics, Oxford University, 2 Keble Road, Oxford OX1 3RH, U.K.
The advent of r e n o r m a l i z a b l e gauge t h e o r i e s of t h e electroweak i n t e r a c t i o n s h a s made p o s s i b l e t h e c a l c u l a t i o n of f i n i t e r a d i a t i v e c o r r e c t i o n s t o t r e e graph amplitudes.
We a r e a b l e t o probe t h e s t r u c t u r e of t h e s t a n d a r d SU2xU1 model of electroweak i n t e r - a c t i o n s a t t h e l e v e l of one loop quantum c o r r e c t i o n s ; i f t h e quantum f i e l d t h e o r y i s c o r r e c t then we should f i n d t h a t t h e r e s u l t s c a l c u l a t e d t o one loop f i t e x p e r i - mental d a t a b e t t e r than does t h e Born approximation. This i s t h e analogue of what happens i n QED where e f f e c t s such a s t h e Lamb s h i f t and anomalous magnetic moments a r e found e x p e r i m e n t a l l y although they a r e n o t p r e d i c t e d by t h e Dirac e q u a t i o n ; we a r e compelled t o i n c l u d e t h e e f f e c t s of second q u a n t i z a t i o n i n o r d e r t o g a i n agree- ment between theory and experiment. I n t h e c a s e of electroweak t h e o r i e s , i t i s t h e r e l a t i o n s between the p h y s i c a l masses of t h e @,Z gauge bosons and o t h e r measurable q u a n t i t i e s which should d i f f e r e n t i a t e between t r e e graph and one loop r e s u l t s [I].
R a d i a t i v e c o r r e c t i o n s a r e a l s o important i n t e s t i n g grand u n i f i e d t h e o r i e s i n which t h e v a l u e s of low energy parameters a r e p r e d i c t e d . I n the r e s t of t h i s a r t i c l e I d i s c u s s b r i e f l y t h e e x p e c t a t i o n s from GUTS and t h e n t h e experiments which measure s i n 2 8 , t h e electroweak mixing parameter, t h e p r e d i c t i o n s we make a t p r e s e n t f o r
%,Mz and f i n a l l y t h e p r o s p e c t s f o r a convincing t e s t of t h e f i e l d t h e o r e t i c a s p e c t of t h e SU2xU1 model.
The u s u a l p o i n t of comparison between grand u n i f i e d t h e o r i e s and low energy d a t a i s the v a l u e of s i n 2 8 . Beyond t h e t r e e graph l e v e l , t h e v a l u e of t h i s parameter deduced from e i t h e r GUT o r experiment depends upon t h e d e f i n i t i o n and r e n o r m a l i z a t i o n scheme adopted. There a r e an i n f i n i t e number of t h e s e , d i f f e r i n g from one another by f i n i t e c a l c u l a b l e amounts. I w i l l use MS r e n o r m a l i z a t i o n w i t h a s c a l e y =
% -
h e r e a f t e r t h a t i s t h e d e f i n i t i o n of s i n 2 8 . The scheme h a s t h e v i r t u e s of b e i n g gauge independent, a u t o m a t i c a l l y s a t i s f y i n g a l l Ward i d e n t i t i e s and being e a s y t o c a l c u l a t e with. I n electroweak t h e o r i e s where a l l coupling c o n s t a n t s a r e s m a l l t n e r e i s no advantage t o be gained by choosing a n y t h i n g more complicated i n o r d e r t o minimise t h e s i z e of c o r r e c t i o n s which a r e i n any c a s e small ( t h i s i s i n c o n t r a s t t o t h e c a s e of QCD which has l a r g e couplings and p o t e n t i a l l y l a r g e c o r r e c t i o n s ) . Ultimately, of c c x r s e , a l l our r e s u l t s may be w r i t t e n a s r e l a t i o n s i n v o l v i n g o n l y p h y s i c a l measurable q u a n t i t i e s
(u
l i f e t i m e , v c r o s s - s e c t i o n s , %,lulZ e t c . ) ; such r e l a t i o n s a r e independent of t h e r e n o r m a l i z a t i o n schemes. The p r e d i c t i o n of minimal SU5 i s C21The v a r i a t i o n i n s i n 2 0 p r e d i c t e d from non-minimal GUTS i s q u i t e l a r g e , % 2.02 [ 3 1 , because of t h e l a r g e numbers of e x t r a p a r t i c l e s i n such models.
A t p r e s e n t , t h e experiments which provide measurements of s i n 2 8 w i t h the b e s t s t a t i s t i c s a r e
i ) v-hadron s c a t t e r i n g C41.
i i ) e-d asymmetry experiment a t SLAC 151.
Both types of experiment a r e l i m i t e d by our l a c k of understanding of hadronic
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982360
C3-306 JOURNAL DE PHYSIQUE
systems. Unfortunately, p u r e l y l e p t o n i c p r o c e s s e s which a r e p r e f e r a b l e f o r study- i n g t h e f i e l d t h e o r y of electroweak i n t e r a c t i o n s a r e a t p r e s e n t e i t h e r i n s e n s i t i v e t o s i n 2 e (e+e- a n n i h i l a t i o n ) o r have very low e v e n t r a t e s and l a r g e backgrounds (ve s c a t t e r i n g ) . The lepton-hadron s c a t t e r i n g experiments a r e a n a l y s e d a t t r e e graph l e v e l i n t h e electroweak i n t e r a c t i o n s i n c l u d i n g QCD s c a l i n g v i o l a t i o n e f f e c t s , a n t i p a r t o n s e a e t c . C61; t h e electroweak r a d i a t i v e c o r r e c t i o n s have been c a l c u l a t e d i g n o r i n g s c a l i n g v i o l a t i o n i n t h e p a r t o n d i s t r i b u t i o n s . The e r r o r a s s o c i a t e d w i t h such a procedure may be e s t i m a t e d by the s e n s i t i v i t y of t h e r e s u l t s t o v a r i a t i o n of tne p a r t o n d i s t r i b u t i o n and t u r n s o u t t o be s m a l l . I n d i s c u s s i n g t h e r a d i a t i v e c o r r e c t i o n s I w i l l t a k e t h e n e u t r i n o experiments a s a n example b u t s i m i l a r consider- a t i o n s a p p l y t o the e-d experiment.
I n t h e Born approximation ( f i g . 1 ) t h e r a t i o of n e u t r a l t o charged c u r r e n t c r o s s s e c t i o n f o r s c a t t e r i n g o f f a n i s o s c a l a r t a r g e t i s given by
b u t r e a l l y t h e r e a r e many more graphs a t one loop o r d e r ( f i g . 2) and a l s o
bremss t r a h l u n g ( f i g . 3) t o be considered. These c o n t r i b u t e c o r r e c t i o n s a t 0 (orem) and O(aSU2) s o t h a t we o b t a i n
Fig. 1.
Fig. 2.
Fig. 3 .
by i n c l u d i n g F, we w i l l f i n d a d i f f e r e n t v a l u e f o r s i n 2 8 . I n g e n e r a l , F w i l l c o n t a i n terms
which could e a s i l y produce a s h i f t of f .02 i n s i n 2 @ . The l o g a r i t h m i c terms a r e n o t s u f f i c i e n t l y l a r g e r than t h e r e s t f o r a l e a d i n g l o g a r i t h m c a l c u l a t i o n t o be u e a n i n g f u l ; i t i s n e c e s s a r y t o do a f u l l c a l c u l a t i o n C71 a f t e r which i t is found i n some c a s e s t h a t the l o g a r i t h m i c terms alone C81 do give a reasonably a c c u r a t e r e s u l t . This i s presumably f o r t u i t o u s
-
however most l e a d i n g l o g c a l c u l a t i o n s do i n f a c t i n c l u d e t h e $..rr2 terms a s w e l l . The c a l c u l a t i o n of t h e bremsstrahlung i s s e n s i t i v e t o the e x p e r i n e n t a l c u t s i n x and y which have t o be taken i n t o account;t h e low y c u t produces a p o t e n t i a l l y l a r g e e f f e c t because of t h e r e s u l t i n g incom- p l e t e c a n c e l l a t i o n of t h e l o g a r i t h m i c muon mass s i n g u l a r i t y .
The r e s u l t s f o r t h e v a r i o u s measurements of s i n 2 8 a r e shown i n Table 1. The l a r g e s t c o n t r i b u t i o n t o F i n the experiments comes from t h e p u r e l y e l e c t r o m a g n e t i c c o r r e c t i o n s t o charged c u r r e n t s c a t t e r i n g which accounts f o r about 90X of t h e s h i f t i n s i n 2 0 . I n e-d s c a t t e r i n g about h a l f t h e s h i f t comes from p u r e l y e l e c t r o m a g n e t i c e f f e c t s and most 3f t h e r e s t from y-z mixing through fermion loops. From t a b l e 1 i t can be s e e n t h a t
i ) The r a d i a t i v e c o r r e c t i o n s improve t h e agreement between experimental r e s u l t s and t n e minimal SU5 p r e d i c t i o n
-
t h e s i g n i f i c a n c e of t h i s i s ~t c l e a r .i i ) T h e i n c l u s i o n of r a d i a t i v e c o r r e c t i o n s t o d i f f e r e n t e x p e r i u e n t s does n o t i n t r o - duce massive d i s c r e p a n c i e s n o t seen a t t h e t r e e y a p h l e v e l ; t h i s i n t e r n a l c o n s i s t e n c y encourages us i n t h e b e l i e f t h a t s i n 8 i s indeed a parameter of a renormalized Lagrangian.
Table 1.
v
c
.229*.018
s i n 2 8 .215+ .015 .221?.018
C l e a r l y , t h e p r i n c i p a l i n t e r e s t i n r a d i a t i v e c o r r e c t i o n c a l c u l a t i o n s i s t h e p r e d i c t i o n of t h e p h y s i c a l masses of t h e W,Z bosons. If t h e f u l l propagator i s
t h e n the p h y s i c a l mass M i s given by
de can always w r i t e C i n terms of o t h e r p h y s i c a l q u a n t i t i e s bqt t h e form of t h e f u n c t i o n depends on t h e o r d e r of p e r t u r b a t i o n theory t o which we c a l c u l a t e . I f t h e quantum f i e l d t h e o r y and i t s p e r t u r b a t i o n s e r i e s a r e v a l i d , then t h e one loop r e l a t i o n should f i t t h e d a t a b e t t e r t h a n t h e lowest o r d e r one. This i s t h e weak i n t e r a c t i o n e q u i v a l e n t of t h e Lamb s h i f t e t c . i n QED. Unfortunately t h e one loop r e s u l t s depend upon q u a n t i t i e s t h a t we do n o t know; t h e masses of t h e Higgs boson, Ohe t quark and any o t h e r a s y e t unknown fermion g e n e r a t i o n s . I f we assume MIi = MZ, mt = 20 GeV and no f u r t h e r g e n e r a t i o n s then, using R,, t o determine s i n 2 0 we f i n d t h e r e s u l t s shown i n t a b l e 2 C71. It i s f o r t u n a t e t h a t t h e mass s h i f t s a r e
C3-308 JOURNAL DE PHYSIQUE
r e a s o n a b l y l a r g e s i n c e t h e phenomenal a c c u r a c y o f t h e low e n e r g y p u r e QED e x p e r i m e n t s i s n o t a t t a i n a b l e a t h i g h e n e r g i e s . The r e s u l t s a r e i n s e n s i t i v e t o
%
which i s a n example of t h e " s c r e e n i n g theorem" of Veltman; changing Mx from 1 0 t o 500 GeV s h i f t s t h e W mass by +.23 GeV and t h e Z mass by +.34 GeV. Large mass s p l i t t i n g s i n fermion d o u b l e t s [ 9 1 c a n produce much l a r g e r changes; t h e e f f e c t of a d o u b l e t(i:)
.mul>>md1 ,M w i s 2 m.. 1Table 2.
I f t h e Z mass were t o t u r n o u t s m a l l e r t h a n t h a t p r e d i c t e d from t h e Born approxima- t i o n i t might i n d i c a t e t h e e x i s t e n c e of s u c h a d o u b l e t
-
b u t i t might i n d i c a t e t h a t t h e t h e o r y i s wrong. A change i n Rv of -.0016 (about 0.5%) produces a +.4 G e V s h i f t i n Mz which means t h a t t h e r e s u l t s a r e v u l n e r a b l e t o h i g h e r t w i s t e f f e c t s i n s c a t t e r i n g o f f h a d r o n i c t a r g e t s .The u n c e r t a i n t i e s u s i n g R, a r e p r o b a b l y i r r e d u c i b l e and i t c a n be s e e n t h a t t h e y a r e a l i t t l e t o o l a r g e t o d i f f e r e n t i a t e d e f i n i t i v e l y between z e r o and one l o o p s even i f God t o l d u s t h e e x a c t v a l u e s o f M,, MH and mt. I n o r d e r t o r e d u c e t h e e r r o r s a d e t e r m i n a t i o n of s i n 2 0 u s i n g a p u r e l y l e p t o n i c p r o c e s s i s r e q u i r e d . A r e a s o n a b l e e v e n t r a t e d i c t a t e s t h e use of t h e Z resonance where, f o r example, t h e p o l a r i z a t i o n o f t a u p a i r s o r t h e forward-backward asymmetry of muon p a i r s may b e measured
[lo].
The e x p e c t e d v a l u e of N, can b e c a l c u l a t e d and compared w i t h t h e measured v a l u e . I n t h i s way t h e SU2xUI t h e o r y s h o u l d be t e s t e d t o h i g h p r e c i s i o n . An a l t e r n a t i v e method i s t o d e t e r m i n e t h e v a l u e s of Mz and
%
a c c u r a t e l y ( t h e measurement of EI, i s d i f f i c u l t bucmay u l t i m a t e l y b e a c h i e v e d ) . The r e l a t i o n between t h e two i s d i f f e r e n t depending upon whether z e r o o r one l o o p s a r e i n c l u d e d t h u s p r o v i d i n g a n o t h e r t e s t of t h e f i e l d theory.One l o o p .215 s i n 2 8
I n c o n c l u s i o n we s e e t h a t r a d i a t i v e e f f e c t s produce s i g n i f i c a n t c o r r e c t i o n s t o p r e s e n t day e x p e r i m e n t s and t h a t t h e i r i n c l u s i o n does n o t d e s t r o y t h e agreement between t h e d i f f e r e n t e x p e r i m e n t s . However, t h e e r r o r s and u n c e r t a i n t i e s i n e x p e r i m n t s i n v o l v i n g h a d r o n s a r e t o o l a r g e f o r a c o n v i n c i n g check of t h e quantum f i e l d t h e o r y . We need LEP t o p r o v i d e s u f f i c i e n t l y a c c u r a t e measurements of Mz and s i n 2 0 f o r a r e a l t e s t o f t h e c o n s i s t e n c y o f t h e t h e o r y .
No l o o p s .227
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&
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30
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