On a possible subtraction for the Lorentz non-invariant model

Nuclear Physics B242 (1984) 542-546 © North-Holland Publishing Company

ADDENDUM H.B. N i e l s e n a n d I. Picek, Lorentz n o n - i n v a r i a n c e , Nucl. Phys. B211 (1983) 269.

On a possible subtraction for the Lorentz non-invariant model Received 19 December 1983 We define a subtraction of infinities appearing in the Lorentz non-invariant (LNI) renormalization of the electron kinetic term. This makes our previous model of LNI more precise. A relic of LNI appearing in the low-energy electron propagation is calculated, which modifies the Dirac equation as suggested in our previous paper.

1. Introduction A l t h o u g h the s t a n d a r d G l a s h o w - W e i n b e r g - S a l a m (GWS) model might provide a n effective theory rather t h a n the basic q u a n t u m field theory, it still can serve as a testing g r o u n d for some new ideas. Recently we i n v o k e d the simplest Lorentz n o n - i n v a r i a n t (LNI) couplings in such a theory in order to study possible deviations from the Lorentz i n v a r i a n t laws of n a t u r e , originating at the scale O ( 1 / r n w , z ) 10 16 cm. In fact we studied the model characterized by weak interactions having a special metric. While our first a c c o u n t [1] has b e e n restricted to a tree level, in the second one (to which this a d d e n d u m applies) the loop diagrams also come into the play. T h e r e b y a simplified estimate of their finite c o n t r i b u t i o n to the electron kinetic energy r e n o r m a l i z a t i o n has b e e n given, which now we w o u l d like to improve. As s h o w n in what follows, an e v a l u a t i o n of the q u a n t i t y at h a n d , the renormalization of the kinetic energy of the electron, introduces divergences which should be treated properly. Let us recall our L N I ansatz,

g~( D~q~)t(D~)-+ h'~( D~o)*(D~q~),

(1)

a c c o r d i n g to which the s t a n d a r d Higgs term is modified in a gauge i n v a r i a n t way. Let us stress that there is only a formal covariance in (1), since the c o u p l i n g constants h "" are not to be t r a n s f o r m e d u n d e r Lorentz t r a n s f o r m a t i o n . F u r t h e r m o r e , in order to have a small d e v i a t i o n from Lorentz symmetry, we i n t r o d u c e a p e r t u r b a t i o n e x p a n s i o n in the d e v i a t i o n from Lorentz i n v a r i a n c e h "v = g ' ~ - X "~.

(2)

Thus g'" is our e x p a n s i o n p a r a m e t e r of relevance, a n d in order to predict a n y physical L N I q u a n t i t y one should decide at what energy scale it s h o u l d be evaluated. 542

543

H . B . N i e l s e n , I. Picek / A d d e n d u m

This fact represents an additional arbitrariness similar to the arbitrariness in the definition of the other basic parameters of the standard model*. We choose a requirement, compatible with our ansatz (1), that at TeV (~mw.z) scale there be LNI only in the Higgs sector. An indirect justification of such a subtraction requirement will be provided by the explicit calculation of the LNI renormalization of the electron propagator, which exhibits a rapid m o m e n t u m 2 dependence only for small momenta (qZ< mw.z), and a slow (logarithmic) for a large momenta. Thus, there is no LNI in electron propagation at the TeV scale, but there will be some LNI effect induced at the very low energies.

2. Spreading of the imposed LNI through the bosonic sector of the G W S model

Once we impose L N I in the Higgs kinetic term of the GWS model, such a change permeates all sectors of the model. In particular it should manifest itself through the generation of masses of the heavy gauge bosons. Let us first fix the notation by writing down the bosonic fields of the theory [2], the physical gauge bosons,

W : : ~/~ (A2 ~ iA~) , Z u = A~ cos Ow + B . sin Ow, A . = - A 3 sin 0 w + B . cos 0w,

(3a)

and the SU(2) doublet Higgs field expressed as

q~ =

4)0

=

(

)

1 ~ (v + H + i4)z) "

(3b)

Then, the introduced L N I enters the propagators of the listed fields. For extracting these propagators it is very useful to adopt a 't Hooft gauge [3]. Namely, when calculating the expression (1) with q~ defined by (3b), one generates annoying gauge b o s o n - H i g g s mixing terms of the form

t•m w W . ( O " 4) +) - i m w W+(O"4) )+mzZu(aU4)z) -X"~{irnw W~(O~4) +) - imw(O.4)-) W + +½mz[Z.(O.4)z) + (O.4)z)Z~]},

(4)

which can be canceled in the 't Hooft gauge by the gauge fixing term 1

1

z2

1

~.~G-- 2 a ( G A ) 2 - - - ~ ( G ) --~ G - G + ,

(5a)

where G A =

O~'A. - X ~ ( g u A . ) ,

G ~ = o'~Z. - x"~(o.Z.)

G~=Ot'W~-x

~v

- amz4)~,

± (OuW~)

~

. ~-. ~tmw4)

(5b)

* F o r t h e s e m a t t e r s we w o u l d like to r e f e r to the e x h a u s t i v e p a p e r s b y S a k a k i b a r a [2], w h o s e n o t a t i o n we f o l l o w in the rest o f the p a p e r .

544

ll.B. Nielsen, L Picek / Addendum

At the same time eqs. (5) determine the F a d d e e v - P o p o v ghost piece o f the lagrangian and the additional L N I part quadratic in the gauge b o s o n fields. The latter, in the 't H o o f t - F e y n m a n gauge a = ~: = 1, acquires the form K ~ ( k) = g " ~ ( k : - m 2) + m 2 x ~ - ( k ~ k , x 0~ + x"Pkok~) ,

(6)

accounting for W, Z or a p h o t o n by putting m to row, mz or zero, respectively. The p r o p a g a t o r is given by the inverse of (6), iDF(k),~=-[K~"(k)] i[

'

g"~

m2

k " k p x ~ + x~Pk'k~] j.

Lk _m

-

(7)

In the a d o p t e d 't H o o f t - F e y n m a n gauge ch± and 4'~ a p p e a r as propagating fields with respective mass mw and mz, idly(k) =

(g

..

-X

i ~,~ )k,k,-

2

mw,z

- k 2_

1

2

mw,z

{ \1

+

x~k~k~3 2 2 k-row.z]

'

(8a)

while the observable scalar field has the p r o p a g a t o r iDH(k) - ( g ~ - X

, ~ )k.k~-mH 2-k

1 2_m 2

1+ k-m~/

.

(8b)

Let us notice that the last term in the bracket of eq. (7), originating from the guage-fixing, has been neglected in the approximate expression (A.3). It is justified in the case o f the tree diagrams, where it represents O ( l / m 4 , z ) correction at low energies. However, in the lo0p diagrams the last term in (7) induces divergences which require a proper treatment.

3. S u b t r a c t i o n c h o i c e and a relic o f L N I in the l o w - e n e r g y e l e c t r o n p r o p a g a t i o n

In fig. 1 we kinetic term*. the difference given entirely and R - h a n d e d

depict the contributions to the L N I renormalization o f the electron In fact, the actual quantity having p h e n o m e n o l o g i c a l implications is of the L N I renormalization for the L- and R - h a n d e d electron. It is by the diagrams (a)-(c), since diagrams (d)-(f) contribute to the Lelectron symmetrically. Symbolically, ~ L N I e L z P )x , XLLN'(P) = ZLwN"e~(P) +ZzLN~'*~(P) +":~,+ii+,~:t

(9a)

, ~ L N I ( p ) ~---,~77.Nl,eR(p) q_ ~ L N I *R" " - t - ~ ENI eR " x .

(9b)

,~ ' tP)

~+~+¢2tP)

Furthermore, it is in compliance with our model o f L N I to require that at a TeV scale there is L N I only in the Higgs sector. Thus, there should be no L N I in the electron p r o p a g a t o r for the electron m o m e n t u m in the TeV (p 2 = row.z) 2 region. Accordingly, we perform the subtraction of the divergent quantities (9) at this energy * From now on we restrict ourselves to LNI part of the electron self-energy, 12- ~;~ov.r,.., +XLN~.

H.B. Nielsen, I. Picek / Addendum

,d, J ' %

,o, Z/

eL

545

Z

eL

,~

x %

f-x ~ !

(b)

l

./

/

(c)

eR

t

(e)

.Z

Z

t

/ ~x-,~.(j6 ', u

/

(f)

~ eR

(g)

g

!

",~ H \

\

./

,.. ~ x - ,.,~Z \

m ' ,~'

,,P

- - x

Fig. 1. The six diagrams (a)-(f) contributing to LNI (indicated by a cross) renormalization of the electron kinetic term, and the corresponding counterterm (g). scale (~'LLNI)Ren. = ,f~'LNI __ ( ~ z L N I ~L

6ZLLNI = E[Nl(p2 = m 2w,z) ;

(10a)

(.~'LNI)Ren. = ~ ' L N I __ ( ~ z L N I ,

8Z~NI = - R ,Cc~l~ t P 2 = r o w2, z ) .

(10b)

S u b t r a c t i n g this w a y r e p r e s e n t s a m o d e l a s s u m p t i o n which m a k e s o u r p r e v i o u s L N I m o d e l m o r e precise. This s u p p l e m e n t s the s u b t r a c t i o n s c h e m e o f the s t a n d a r d G W S m o d e l , given for e x a m p l e [2] by the on-shell s u b t r a c t i o n s c h e m e with {a, mw, mz, mH, me} as a set o f i n d e p e n d e n t p a r a m e t e r s o f the m o d e l . H o w e v e r , the s u b t r a c t i o n choice (10) will be legitimate if the L N t q u a n t i t y turns out not to be very sensitive u p o n the p a r t i c u l a r value at which the s u b t r a c t i o n is p e r f o r m e d . F o r our set o f d i a g r a m s (fig. I) it will turn out to be the case. F o r illustraton, let us write d o w n the result for the d i a g r a m (a), o b t a i n e d by the d i m e n s i o n a l r e g u l a r i z a t i o n [4] in 26o d i m e n s i o n s (e = 2 - to):

,~LwNI,~L(p)=3~2X,,~T,,p~(2fO' d X X m 2 - p 2m2w ( 1 - x) +

dx 2 x ( 1 - x ) + 2 x ( l - x ) 2 ~

l-~(1-x) mw

_ l n (m2wx(1

p2

-Y5

546

H.B. Nielsen, L Picek / Addendum

2 The L N I self-energy (11) has rapid variation with p for the small m o m e n t a (p2 ~ rnw), and rather slow (logarithmic) for the big momenta (p2_~ rn~v). This fact justifies the subtraction chosen in (10). Now it is straightforward to extract the relic of L N I appearing in the low-energy electron propagation, corresponding to formula (A.8),

p~

PeR

(w+z)

(z)

, y " p ~ [ g , v + ( 3 . 1 0 2 x l 0 3)X~,~],

, y'p~[g,~ +(0.335

x 10-3)X~,,].

The brackets ( W + Z ) and (Z) are here in order to 2 2 energy, p 2 ~ me.~ row,z, only diagrams (a) and (b) our conjecture. However, in order to perform the we define the model more precisely. Up to a sign, are of the same order of magnitude as before. Let coefficients,

~3.102

remind us that at the very low contribute effectively*, proving calculation more exactly, here the L N I coefficients in eq. (12) us remind ourselves that these

10-3X~

for

ec

aL'R~ (0.335 x 10 3 ~ /

for

eR '

×

(12)

represent the modification of the Dirac equation for the L- and R-handed electrons. Then by matching it with the existing phenomenology, we obtain the bound on L N I which remains the same also when introducing the improvement of the present paper.

References [1] [2] [3] [4]

H.B. Nielsen and I. Picek, Phys. Lett. lI4B (1982) 141 S. Sakakibara, Phys. Rev. D24 (1981) 1149:MZ-TH/83-07 preprint (Sept. 1983) G. 't Hooft, Nucl. Phys. B35 (1971) 167 G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189

* Note the huge suppression factor m J2m w2 in diagram (c).