- Email: [email protected]
The effect of a strongly interacting Higgs sector within an SU(2) × U(1) gauge theory
Nuclear Ph,,slcs B197 (1982) 385-398 , North-Holland P u b h s h m g C o m p a n )
T H E EFFECT OF A S T R O N G L Y I N T E R A C T I N G H I G G S S E C T O R W I T H I N AN SU(2) X U(I) G A U G E T H E O R Y * Kirk OI Y N Y K
Department of l'hl ~tcs. Umt er~ttl o / ( ahfotma, Ios 4neeh's C,I 90024, L $4 Rctcl~ ed 22 September 1981
The lov,-energ,, sensltl,,lty of the Wemberg-Salam model t<5 a strongl'., lnterat.tmg Hlggs bo,,on sector is mve,,tlgated In order to t.atalog all possible hea~', thggs bo,,on cffc~.ts, a gauged non-hnear o-model is used. m which the l h g g s ma',s of the linear thcor', ( ~I H ), assume.', the role of a t.utoff Radlatl',e t.orrechons t<5 several experimental o b s e ~ a b l e s arc cah.ulated Hov, c,,cr, the effects are small, the leading ,,cns~t~'~tlc,, arc proportional to g 'In( A,lll)
I. Introduction
The Welnberg-Salam model of weak and electromagnetic interactions [1] has gained increasing acceptance during the past ten years [2]. The minimal model spontaneously breaks the SU(2) I × U(I) R symmetry by introducing a complex scalar doublet, producing three Goldstone bosons and a single massive scalar, the Hlggs particle. Little is known about the fundamental nature of the symmetry-breaking mechanism. However, we may hypothesize that such a mechanism has a mass scale anywhere between 300 GeV and 1 TeV, depending upon the dimensional argument used [3]. Further. at experimental energies below the threshold for Hlggs production. It is possible to discern the impact of the heavy Higgs sector on the rest of the theor3,** To extract the heavy Higgs effects, we start with the results and notation of ref. [3]. We then generalize the notation and extend the results to the context of a spontaneously broken SU(2)t, × U(I) R gauge theory. This analysis assumes that the Higgs theory is an effective low-energy model of the symmetry-breaking process. Further, the Hlggs boson mass Mtt, becomes a cutoff for this model's sensitivity to the strongly interacting Hlggs sector generated through radiative corrections. We first treat the M H ~ ~ limit in order to preserve the symmetry of the theory in an explicit manner. The result is a gauged non-linear o-model ( G N L o ) which is non-renormahzable. We deduce radiative corrections, producing new cutoff-depen* **
Thl.s v,ork is supported m part b~ the National S~.lent.e Foundation (US) Ref [4] gl~es an m,.omplete hst of references deahng '~,lth radlaw, c corrections of the standard model 385
386
K Olvnvh / Strongl~ interacting Htgg~ ~e(tor
dent structures. These we systematically tabulate and summarize by means of gauge-invariant terms added to the lagrangian. We interpret the cutoff as dependent upon the Higgs mass of the linear model. Sect. 2 sets forth a description of the GNLo model, our method of analys~s and a detailed explanation of the relationship of the G N L o model to the standard model. Sect. 3 outlines the systemauc procedure for listing, and estimating the cutoff dependence, of counterterms generated m the loop expansion. First, we generalize the counterterms calculated tn ref. [3]. Then we list all other structures generated at the one-loop level. Sect. 4 sets forth the calculation of the new cutoff-dependent structures listed in sect. 3. In sect. 5 we first obtain from the cutoff-dependent counterterms radiative corrections to the various couphng strengths and masses of the theory. We then correlate these radiative corrections to natural relations wtuch may be experimentally verified. Much of this analysis has been also carried out by Longhitano [5]; the calculation presented here seems considerably simpler. In addition, we treat some different physical effects.
2. The model In the Weinberg-Salam model of weak-electromagnetic interactions, the symmetry of the vacuum state is broken vm the presence of a complex Higgs doublet. This paper does not address the fundamental nature of these fields, rather, the Hlggs fields wdl be considered as an effective low-energy model. A convenient representation of our scalar fields ~s the matrix
M(x)=o(x)+t~..~r(x):~.~[_dp~cb_
q'-]q~o"
(2.1)
which transforms under the local gauge groups as.
M'(x) : L ( x ) M ( x ) R * ( x ) = e '('/2' °'~)M(x)e ,~,,/2)o,,~,) where
(2.2)
O(x) and Oo(x) are real. The scalar potential is V( ~, ) = ~,[ " Tr M t M _ f 212 = M~, ( o2 + ,,2 _ f 2 )~, -
8f2
(2.3)
"
w h e r e f is the vacuum expectation value of the scalar field. The gauge fields are also conveniently cast into a matrix representation gwen by
_ 1 W,(x) =T,
=lzaB;,(x ),
(2.4)
K Oh Io'k / Strongly mtera~ tmg Ihggv se~ tor
387
which transform according to the SU(2) L and U(1)R groups, respectwely. The gauge-mvanant lagrangian is* 1
2
,Nv = :Tr(F..) +
i
2
M2
+ ~TrD~M(D,M) ~ + - ~ ] 5 [ ~ T r M ' M - f2]2,
(2.5)
where
D),M=O~,M+gW.M-g'MY~,.
(2.6)
For ease of calculation we quantize our theory in the Landau gauge by adding to the lagrangaan a gauge-fixing term
(
)2.
(2.7)
In the limit ~, ~' --, ~ (the Landau gauge) the Fadeev-Popov ghost term takes the form**
GC + gO,CX c . w . - do.Co.
(2.8)
The theory presented to this point is a hnear o-model under the local gauge group SU(2) L × U(1) R, a renormahzable theory. If the Higgs mass becomes very large ( M H ~ oo) the scalar potentml becomes a constraint,
MtM = MM* = f 2
(2.9)
and our theory has been transformed from a hnear to a non-linear o-model, a non-renormahzable theory. This means that when one calculates a physmal process dwergent terms not of the form of the tree-level lagrangmn, ~'tree, could arise. It ~s possible to render the theory flmte for an arbitrary number of loops by adding the a p p r o p n a t e counterterms to the tree-level lagrangian; however, some of these counterterms will have structures not seen m ~,r~" The only requirement of these new counterterms is that they must respect both gauge and Lorentz lnvanance. " W e a d o p t the We~t Coast m e t n c ( + . - - - ) ** T h e choice of the L a n d a u gauge m,,ure,, that calculatmn,, p e r f o r m e d m this p a p e r will not include gauge held propagator,, T h e r e f o r e ghost held,, ',~111 not a p p e a r m our computation,,
388
K Olynyk / Strongly mtera~tmg thgg* ~ector
These reqmrements arise because the constraint (2.9) satisfies the same criteria. The cutoff dependence of these new terms can be calculated and then assocmted with M n of the linear theory [3]. Once armed with these new structures and their cutoff dependence, one may read off physical effects dependent upon the presence of a heavy Higgs boson sector in the Weinberg-Salam model. 3. Structure of counterterms in the non-linear model
This method of analysis has heretofore been applied to a model with a local SU(2)L gauge invanance. In this paper, we generalize this previous calculation by adding another local gauge lnvariance, U(1)R. The two theories will be related in that, if we suppress the local U(1)R symmetry in our results, the results of the SU(2) L theory must be regained. This relationship can be represented symbolically, l,m SU(2)L × U(1)R
g'~0
----
SU(2)L.
(3.1)
The S U ( 2 ) L theory has an additional global S U ( 2 ) R symmetry (see ref. [3]) which severely limits the form of the counterterms. Thus, counterterms in the S U ( 2 ) L X U ( I ) R theory which do not have this symmetry in the limit g ' - , 0 must have coeffioents which vanish as g ' - , 0. This relationship between the two models wdl provide a tremendous simplification to our computation. The divergent counterterms must be both Lorentz and gauge invanant. We may systematically list all poss,ble structures satisfying these criteria by using power counting arguments [3]. Defining a dimensionless unitary matrix
U~ M/f,
(3.2)
where U is a funcnon of dimensionless scalar fields, _,tL-- ~r/f,
U= ~/1-~_Z(x) + t'r'~_(x),
(3.3)
the effective lagranglan for the non-linear model is then,
= ~f2TrDuU(D~'U) * + ~'o,
(3.4)
where ~o = ½TrF~ F ~" ± ~ Tr Y.. Y"" + ~;E + ~ r l,.
(3.5)
Defining d as the dimension of the counterterm which operates at 1 loops, d will count one for every Lorentz index that the counterterm carries*. A power-counting * The scalar fields _~ arc damens~onle~s, thcrcfore, theF do not contnbutc to the d~menslonahty of the countcrterms onc may form from them
K Oh'n) k / Strongly interacting Htggs sector
389
analys~s shows that d = 2 + 2 1 - r - 2b,
(3.6)
where b is the number of pure gauge propagators minus the number of pure gauge vertmes and r ~s the number of powers of a dimensional regulator A, which accompany each counterterm. If we consider graphs that have no internal gauge or ghost lines (b = 0), we see that the &vergent counterterms with the greatest &mension are generated by logarithmic (r = 0) divergences. We shall consider the leadmg divergences at one loop of counterterms up to dimensmn four. There is only one possible gauge-invariant term at d = 0, T r U t U = 1,
(3.7)
implying that there cannot be any quadratic divergences. For d = 2, the only possible mvariant structures are
£~= floTrD~U( D~U) *,
~; = BI(TrU*D~U'q ) z.
(3.8)
The term E6 is seen in the tree-level lagrangian so dwergences such as these can be absorbed into the original lagrangian and are physmally unobservable. The other term, t~'l, is not of the form of Etree and thus leads to physically observable effects. At d = 4 a convenient object to use in the enumeration of the possible counterterms is
v. =-(by)u* = g w . - g'Vry* + (O.v)v*.
(3.9)
V~ has dimension one, it is a U(1) R invariant and transforms under SU(2)L as
V~ --, V; = L( x )V~( x )Lt( x ).
(3.10)
Covariant derivatives of these objects take the form
]
(3.11)
It is important to note that m the limit g'--, 0, both these objects become their counterparts introduced by Appelqmst and Bernard in their calculatmn for a spontaneously broken SU(2)L gauge theory. We now have at our disposal the machinery to construct all gauge-invariant objects of dimension four. There are many such objects, but fortunately they need not all be considered. In order that (3.1) hold, in the limit g ' ~ 0 the only remaining counterterms must be exactly those of the SU(2)L theory, with the identical coefficients. The only other counterterms we must consider are those that explicitly vanish m the same limit.
K Olvn) l~ / Strongly interacting Ihggs ,ettor
390
From the preceding argument we may tmmedtately write down all the & m e n s l o n four counterterms that regain the global SU(2) R symmetry m the hm~t g ' ~ 0. These counterterms are simply the SU(2) c × U(1) R generalization of the ones m the SU(2) x calculation. These objects are
~,=a,[TrVuV ,u] 2 , t~2 = a2[Tr V~V~]2,
~.~ = g%Tr( Fu~[ V",V~]) ~,., = - ( ~ ' , -
l)gO, W.. w " ×
w"
~ 2 -- - ~ ~ - l g~( ~ × w~)
~,,~ = - , ( ~ - i)(a.w. - a,w.) 2,
(3.12)
where V ~ is defined in (3.9). It is the SU(2)I. × U(1)R generalization of the object used originally m the SU(2) t calculation. Further. at one loop, the coefficients must be the same as those calculated m the SU(2)I theory, namely* --1
1
-1
16~r2 1 2 t '
±
l
g 2 ( ~ 3 - - 1 ) =-g-S(~" , -- 1 ) = ~
1
1
16~r 2 6 t "
l
) ~'3
1
16~.2 12e
-l
,
1 -- 16~-2 6 e '
(3.13)
where e = 4 - n in a dimensional regulanzation scheme. The coefficients ~'l and ~'3 are not equal to their corresponding coefficients of the linear model, Z t and Z 3, but contain additional logarithmic divergences that appear only as M H becomes large. Using the identity
% v . - % v . = [ v., v.] + g F ~ - g V r . y ,
(3.14)
one may deduce that the only independent counterterms which do not regain a " T h e s t r u c t u r e of the c o u n t e r t e r m s insures that, at o n e loop, the coefficients c a n n o t c o n t a i n e x t r a t e r m s t h a t v a m s h m the h r m t g ' ~ 0 S u c h a satuat~on c o u l d arise o n l y at t w o l o o p s a n d b e y o n d
K Oh'nvk / Strongh interacting lhggs w~tor
391
global SU(2) R symmetry in the limit g' ~ 0 are
f4=a,g'Tr(UY~,~U*[V",V"]),
(3.15) The symmetry reqmrement forces the structures to have exphc~t factors of g' m front ((~" - 1) ~s proporuonal to g,2). It remains our task to determine the coefficients of these three counterterms.
4. Computation of divergence~ Power-counting arguments show the leading divergences at the one-loop level to be logarithmtc. Tlus allows us to regularize d~menstonally and then, at the end, interpret I/E as ln(MH/Mw) in the hnear model. The one-loop dwergence associated with the B self-energy shown m fig. 1 has the value
-1 tg '2 16--ff2 6e ["l[g~,~k2_k~,k~].
(4.1)
The only contribution of the counterterms to this process comes from E~; whose value is
-,(~"-
1)[g,~k2-k~k~].
(4.2)
-i
(4.3)
From (4.1) and (4.2) we conclude g-1'2(~'' - 1)
1
16~r 2 6e"
The one-loop contribution to the ~rWB vertex shown in fig. 2 has the value,
1
I gg,e3,~b[g~,~(q2 +
k 2) _
q~,q~_ k~,k~]
16~r 2 6e
B
~
~
Fig 1 B~elf-energy
Bu
(4.4)
K OI)nvk / Strongl~ mteractmg Ihgg~ aector
392
;°
w /
Bv
b
+
Bv
Eb
Wp.°
+
w2
8v
Fig 2 One-loop conttabutlon to the rrWB vertex T h e c o u n t e r t e r m c o n t r l b u U o n to the s a m e vertex is
-- 2tgg'e3"h[ a3( g ~ ' k 2 -- k~'k " ) + aa( g " ' q 2 - q~q" ) + (a 3 + a 4 + as)(g~'k'q-
(4.5)
q~k')].
T h e terms in (4.4) a n d (4.5) m u s t s u m to zero, thus
0/3 --
1
I
16~r 2 12e
,
~t4 --
1
1
16~r 2 12e
,
% -
-1
1
16,r 2 6 t
.
(4.6)
W e have n o w solved for all the d = 4 coefficients b y the c o n s i d e r a t i o n of j u s t two Green functions*. T h e d = 2 c o u n t e r t e r m is d e t e r m i n e d b y the ~'-field self-energy; the result is [6]
fl, _ 1 1 3M~vtan20 ' 16~r 2 e 4
(4.7)
where tan 0 =- g ' / g .
5. P h y s i c a l
effects
Each c o u n t e r t e r m is the n e g a t i v e of a d i v e r g e n c e seen at the o n e - l o o p level in a c o m p u t a t i o n ; thus, the physical effect of the p r e s e n c e of the heavy Higgs sector m a y be read directly f r o m o u r c o u n t e r t e r m s a n d their coefficients. W e m a k e the c o n n e c t i o n with the l i n e a r t h e o r y b y i d e n t i f y i n g o u r d i m e n s i o n a l r e g u l a t o r with the mass of * The same results are obtamed by constdermg all possible SU(2)I X U(I)R mvanant counterterms generated by all processes with a single I-hggs loop [5]
393
K Ol~nvA / Strongly mteractmg Htggs sector = ~+"-"(3"'~
Fig 3 The full propagator for thc neutral vector boson
the H~ggs particle,
l - m l e
4-n
~ln
MH
(5.1)
M '
w h e r e M is a low e n e r g y scale. W e a s s u m e the mass of a n y f e r m i o n is small c o m p a r e d to M; therefore, we neglect the c o u p l i n g of the f e r m i o n s to the Higgs. However, o u r n o w massless f e r m i o n s d o c o u p l e to the gauge b o s o n s a n d thus p r o v i d e a p r o b e for the i n v e s u g a t i o n of heavy Higgs effects. T a k i n g the o n e - l o o p d i v e r g e n c e s f r o m the c o u n t e r t e r m s , we c a l c u l a t e the full p r o p a g a t o r s for the p h o t o n , the n e u t r a l v e c t o r b o s o m a n d the c h a r g e d vector b o s o n . F o r e x a m p l e , the full p r o p a g a t o r for the n e u t r a l p a r u c l e , as s h o w n in fig. 3, ts f o u n d to be, A z z ( _ q2g~,, + q~,q~/q2 )
t.t,u
DFULL(q ) =
(5.2)
q2 _ ( A z z / A z ) M . ~ + ,E
A similar c a l c u l a t i o n yields the a n a l o g o u s effecnve* r e n o r m a h z a t i o n c o n s t a n t s for the p h o t o n a n d c h a r g e d vector b o s o n . T h e results are
8A ww = A ww - 1 = 8~"3 - 8 Z 3, ~AAA ----AAA - - 1 = (8~3 --
8Z3)smZO+ (8~" -
~ Z ' ) c o s 2 0 --[-
2asgg'smOcosO,
8 A z z = A z z - I = (8~"3 - 8Z3)cos2O + ( 8~' - 8Z')sin20 - 2asgg'sinOcosS, 8Aw =---Aw - ! = 0 , 8AA=--AA - 1 = 0 ,
g2
8A z --=A z -- I = 1 - 213) M 2 c o s 2 0 ,
(5.3)
a n d their physical masses, as d e f i n e d as the p o s i t r o n of the poles m the p r o p a g a t o r s , * The factors of 8Z, wltban the effecuve renormahzauon constants, 8A,, are counterterms of the hnear theory These hnear counterterms are added to the lagrangmn to subtract ultraviolet d~vergences arasmg m the hnear theory
K 01~ nvk / Strongly mtera~tmg Htgg, *e~tor
394
are (we denote physical quantities by a bar) /~2
AAA
=--~-A M2 = O,
M z 2= - _ A_ ~zz M] = ( 1 + 6A z / - 8A z ) g ~ ,
--~
A ww
9
M~v= Aw M f f = ( l + S A w w - S A w ) M 2 w .
(5.4)
With these results we rescale the coupling constants of the theory to g~ve Hlggs-corrected physical values,
= A~/2e,
AIz/z2G,
g =
2 A'i wg.
(5.5)
From the radiatwe correctzons hsted m (5.2)-(5.4) we may determine the M u dependent corrections to physical processes. At low experimental energies* measurement of natural relations provide a check of the results of our calculation. The only natural relauon mvestigated expenmentally is p = ( M w / M z c o S S ) 2 [7], which is predicted to be umty at the tree level. If we define the electric charge, ~, by measurements of Thompson scattering near zero momentum exchange, and g~ as the on-mass-shell couphng of a W to two fermions, we have an experimental defimtion of the Weinberg angle, sin~,) = ~/~.
(5.6)
Another way to define the Wemberg angle would be to compare the on-mass-shell couplings of the W and Z particle to ferrmons, cos
-
7/o.
(5.7)
Comparing ~ to the coefficient of Z-decay yields a third value, sin ~3)cos t~3~= ~ / G .
(5.8)
Usmg our physical masses M w and M z we obtain three corresponding predictions * "Low experimental energjes" refers to energies below the threshold for vector meson production
395
K Olvnvlt / Strongli interacting lhggs sector
o f * O: - 1 + 3 A w w + 3A z - 8 A 7 7 + Ian20(SAAA -- 3 A w w )
= 1 - 2flt-~
o
= !_ ~
p(2) ~"
4- 2 a s g 2 t a n 2 O
(M,,)
({;t)secZOln
-if-,
o
1 - ~--~ (3)sec2Oln
o ,
0~3~= 1 - ~
(5.9)
(~,)sec2Osec 28( 11 cos20 - 9 sin20 )In
w h e r e * * a = e2/4rr. Closely related to thts natural relation ts the natural relation c o m p a r i n g the values of the W e m b e r g angle measured by different schemes. W e find f r o m (5.5)-(5.8) that
COS2~1) COS2~2)
- - -
cosZ~l)
1 -2asg2tan20
o
= 1 +~-~(~)sec2Oln
sin40
a
- - 9 ---
1 + 2asg 2
COS2~2) - GOS2~3)
1 + 2asg2sm2Osec20 = 1
COS-~3 )
(%MH)- ,
l )tan2Osec201n (
COS20COS20 o
~----~( { ) s e c 2 0 1 n
(MH )
(5.10)
Just two of these natural relations are independent. Finally, consider the m v a r i a n t amplitudes associated wtth a neutral-current a n d a charged current process,
82 ~ ,
(5.11)
_ . q2 _ M ]
(5.12)
1N(q2)=---q2~M
~2 Ic(q 2)~
* For P~l)(012)) the same result a~ gxven m ref [8] (ref [9]) ** Note that due to the nature of natural relauons, they cannot depend upon the 8Z,',, or 3~','s The natural rclauons cannot be ultrawolet cutoff dependent
K OIvnvk / Strongly interacting Htgg~ wctor
396
Experimental informaUon about these funcUons would be obtained from neutralcurrent processes such as: ue + ~--* X + v,, or ve + e - -, u~ + e - ; and charge-current processes such as: B-decay,/~-decay or v¢ + ~ -, X 4- e - , etc. At present, accelerators can probe these functions only at low energies, so we may measure low m o m e n t u m expansions,*
IN(O)--
82 M-~ '
d 2)q: o = i ~ ( 0 ) dq 2 IN( q
82 .~ '
Ic(O) --
~~2 2'
d ic(q2) q 2 = o _ l ~ ( O ) _ dq2
g2w . )~r4
(5.13)
From these values we may define experimental values for the vector boson masses,
--
g,2
I~(0) [
I~(0) ]-' 1
/~2w=
It(0)
1
(0---~ I~.
,
(5.14)
A natural relation would be the ratio of these "low-energy" mass definitions to their physical mass counterparts,
ME _ 1 -- a c s c 2 0 ( ~ _ 3tan20)l n
"M2w-1-
ot
MH ( ~ ) c s c 2 Oln(---~)
(5.15)
We have not discussed corrections involving E ~, ~2, ~3 and E4. The reason for this omission is that these counterterms contain divergences of processes involving at least three gauge bosons. Such processes have not been investigated experimentally and so are of less interest. It should be emphasized that our counterterms summarize all possible physical effects of the Hlggs sector at one loop. These effects may be extracted m the same manner we have demonstrated. '~ Note that Pt2)[eq (5 9)] is equal to IN(O)/I¢(O ) Therefore, we need not produce vector bosons in order to measure all natural relataons
K Oh m k / Strongly interacting, lhgg~ set tor
397
6. Conclusion We have catalogued the leading sensltiwty of the standard model to the Hlggs sector up to one loop m the expansion. This was accomplished by considering the large-M H iirmt of the standard model without fermions, a non-hnear o-model. W~thin this Iinut, cutoff-dependent terms are generated which demonstrate the sensitivity of the model to the heavy physical Hlggs parucles. This sensiuvity ~s summarized in the form of gauge mvariant counterterms added to the lagran~an, so as to subtract out these aforementtoned cutoff-dependent terms seen at one loop in the expansion. From these counterterms all leading, Higgs-dependent, physical effects can be extracted in a simple manner. The counterterms were calculated by generalizing to the gauge group, SU(2) L × U(1)R of the standard model, the results of the prewous calculation of an SU(2) L gauge model by Appelqmst and Bernard. In addition, we calculated the cutoff-dependent coefficients of the three independent counterterms which have no counterpart in their SU(2) L theory. However, it should be noted that in the linut M H--. 1 TeV, the expansion parameter,
g2M _ 8,rr 2 M 2
2~-2f 2
will become large and so the effect of the Higgs sector is, at most, ( g 2/192 ~r2 )In( M~/MZw) times a factor of order one. Therefore, although this factor rmght possibly be calculated accurately in a strong couphng expansion scheme, the results from our G N L o model calculation would be only a rough grade to the sensitivity of the standard model to the heavy H~ggs sector. The conclusions reached herein will be most rehable if M H is less than 1 TeV but greater than M w. In sect. 5 we continued the analysis by calculating Higgs-dependent corrections to natural relations of the standard model. Thus far, however, the only experimentally measured natural relation is p. We gave three numerically different predictions, based on three different definitions for the Weinberg angle. Investigation of the other natural relations considered must await the production of the vector mesons. I would like to thank C. Bernard for suggesting the problem and for his many helpful discussions.
References [I] S Wemberg, Phys Rev Lett 19 (1967) 1264; A Salam, in Elementary particle physics, relativistic groups and analyUoty (Nobel Symp. no 8), ed N Svartholm (Almqwst and Wlksell, Stockholm, 1968) p 367 [2] J J Sakurat, m Current trends m the theory of fields, ed J E Lannutu and P K Wdhams (A/P, New York, 1978) p 38 [3] T Appelqmst and C Bernard, Phys Rev D22 (1980) 200
398
K Olvnyk / Strongly interacting thgg~ ~ector
[4] G Passanno and M Veltman, Nucl Phys BI60 (1980) 151, M Consoh, Nucl Phys BI60 (1980) 208, M Lcmome and M Veltman, Nucl Phys B164 (1980) 445, M Green and M Veltman, Nucl Phys B169 (1980) 137 (E. B175 (1980) 547), F Antonelh, M Con,,oh and G Corbo. Phys Left 91B (1980) 90, M Veltman, Phys Lett 91B (1980) 95, A Swim. Phys Rev D22 (1980) 971 [5] A Longhatano, Nucl Phy,, B188(1981)118. K Olynyk, unpubhshed [6] A C Longhltano, Phys Rev D22 (1980)1166 [7] P Langacker et al. Proc Neutnno '79, vol 1, p 276 [8] W J Maroano. Phys Rev D22 (1979) 274 [9] W J Maroano and A S, rhn, Phys Rev D22 (1980) 2695
T H E EFFECT OF A S T R O N G L Y I N T E R A C T I N G H I G G S S E C T O R W I T H I N AN SU(2) X U(I) G A U G E T H E O R Y * Kirk OI Y N Y K
Department of l'hl ~tcs. Umt er~ttl o / ( ahfotma, Ios 4neeh's C,I 90024, L $4 Rctcl~ ed 22 September 1981
The lov,-energ,, sensltl,,lty of the Wemberg-Salam model t<5 a strongl'., lnterat.tmg Hlggs bo,,on sector is mve,,tlgated In order to t.atalog all possible hea~', thggs bo,,on cffc~.ts, a gauged non-hnear o-model is used. m which the l h g g s ma',s of the linear thcor', ( ~I H ), assume.', the role of a t.utoff Radlatl',e t.orrechons t<5 several experimental o b s e ~ a b l e s arc cah.ulated Hov, c,,cr, the effects are small, the leading ,,cns~t~'~tlc,, arc proportional to g 'In( A,lll)
I. Introduction
The Welnberg-Salam model of weak and electromagnetic interactions [1] has gained increasing acceptance during the past ten years [2]. The minimal model spontaneously breaks the SU(2) I × U(I) R symmetry by introducing a complex scalar doublet, producing three Goldstone bosons and a single massive scalar, the Hlggs particle. Little is known about the fundamental nature of the symmetry-breaking mechanism. However, we may hypothesize that such a mechanism has a mass scale anywhere between 300 GeV and 1 TeV, depending upon the dimensional argument used [3]. Further. at experimental energies below the threshold for Hlggs production. It is possible to discern the impact of the heavy Higgs sector on the rest of the theor3,** To extract the heavy Higgs effects, we start with the results and notation of ref. [3]. We then generalize the notation and extend the results to the context of a spontaneously broken SU(2)t, × U(I) R gauge theory. This analysis assumes that the Higgs theory is an effective low-energy model of the symmetry-breaking process. Further, the Hlggs boson mass Mtt, becomes a cutoff for this model's sensitivity to the strongly interacting Hlggs sector generated through radiative corrections. We first treat the M H ~ ~ limit in order to preserve the symmetry of the theory in an explicit manner. The result is a gauged non-linear o-model ( G N L o ) which is non-renormahzable. We deduce radiative corrections, producing new cutoff-depen* **
Thl.s v,ork is supported m part b~ the National S~.lent.e Foundation (US) Ref [4] gl~es an m,.omplete hst of references deahng '~,lth radlaw, c corrections of the standard model 385
386
K Olvnvh / Strongl~ interacting Htgg~ ~e(tor
dent structures. These we systematically tabulate and summarize by means of gauge-invariant terms added to the lagrangian. We interpret the cutoff as dependent upon the Higgs mass of the linear model. Sect. 2 sets forth a description of the GNLo model, our method of analys~s and a detailed explanation of the relationship of the G N L o model to the standard model. Sect. 3 outlines the systemauc procedure for listing, and estimating the cutoff dependence, of counterterms generated m the loop expansion. First, we generalize the counterterms calculated tn ref. [3]. Then we list all other structures generated at the one-loop level. Sect. 4 sets forth the calculation of the new cutoff-dependent structures listed in sect. 3. In sect. 5 we first obtain from the cutoff-dependent counterterms radiative corrections to the various couphng strengths and masses of the theory. We then correlate these radiative corrections to natural relations wtuch may be experimentally verified. Much of this analysis has been also carried out by Longhitano [5]; the calculation presented here seems considerably simpler. In addition, we treat some different physical effects.
2. The model In the Weinberg-Salam model of weak-electromagnetic interactions, the symmetry of the vacuum state is broken vm the presence of a complex Higgs doublet. This paper does not address the fundamental nature of these fields, rather, the Hlggs fields wdl be considered as an effective low-energy model. A convenient representation of our scalar fields ~s the matrix
M(x)=o(x)+t~..~r(x):~.~[_dp~cb_
q'-]q~o"
(2.1)
which transforms under the local gauge groups as.
M'(x) : L ( x ) M ( x ) R * ( x ) = e '('/2' °'~)M(x)e ,~,,/2)o,,~,) where
(2.2)
O(x) and Oo(x) are real. The scalar potential is V( ~, ) = ~,[ " Tr M t M _ f 212 = M~, ( o2 + ,,2 _ f 2 )~, -
8f2
(2.3)
"
w h e r e f is the vacuum expectation value of the scalar field. The gauge fields are also conveniently cast into a matrix representation gwen by
_ 1 W,(x) =T,
=lzaB;,(x ),
(2.4)
K Oh Io'k / Strongly mtera~ tmg Ihggv se~ tor
387
which transform according to the SU(2) L and U(1)R groups, respectwely. The gauge-mvanant lagrangian is* 1
2
,Nv = :Tr(F..) +
i
2
M2
+ ~TrD~M(D,M) ~ + - ~ ] 5 [ ~ T r M ' M - f2]2,
(2.5)
where
D),M=O~,M+gW.M-g'MY~,.
(2.6)
For ease of calculation we quantize our theory in the Landau gauge by adding to the lagrangaan a gauge-fixing term
(
)2.
(2.7)
In the limit ~, ~' --, ~ (the Landau gauge) the Fadeev-Popov ghost term takes the form**
GC + gO,CX c . w . - do.Co.
(2.8)
The theory presented to this point is a hnear o-model under the local gauge group SU(2) L × U(1) R, a renormahzable theory. If the Higgs mass becomes very large ( M H ~ oo) the scalar potentml becomes a constraint,
MtM = MM* = f 2
(2.9)
and our theory has been transformed from a hnear to a non-linear o-model, a non-renormahzable theory. This means that when one calculates a physmal process dwergent terms not of the form of the tree-level lagrangmn, ~'tree, could arise. It ~s possible to render the theory flmte for an arbitrary number of loops by adding the a p p r o p n a t e counterterms to the tree-level lagrangian; however, some of these counterterms will have structures not seen m ~,r~" The only requirement of these new counterterms is that they must respect both gauge and Lorentz lnvanance. " W e a d o p t the We~t Coast m e t n c ( + . - - - ) ** T h e choice of the L a n d a u gauge m,,ure,, that calculatmn,, p e r f o r m e d m this p a p e r will not include gauge held propagator,, T h e r e f o r e ghost held,, ',~111 not a p p e a r m our computation,,
388
K Olynyk / Strongly mtera~tmg thgg* ~ector
These reqmrements arise because the constraint (2.9) satisfies the same criteria. The cutoff dependence of these new terms can be calculated and then assocmted with M n of the linear theory [3]. Once armed with these new structures and their cutoff dependence, one may read off physical effects dependent upon the presence of a heavy Higgs boson sector in the Weinberg-Salam model. 3. Structure of counterterms in the non-linear model
This method of analysis has heretofore been applied to a model with a local SU(2)L gauge invanance. In this paper, we generalize this previous calculation by adding another local gauge lnvariance, U(1)R. The two theories will be related in that, if we suppress the local U(1)R symmetry in our results, the results of the SU(2) L theory must be regained. This relationship can be represented symbolically, l,m SU(2)L × U(1)R
g'~0
----
SU(2)L.
(3.1)
The S U ( 2 ) L theory has an additional global S U ( 2 ) R symmetry (see ref. [3]) which severely limits the form of the counterterms. Thus, counterterms in the S U ( 2 ) L X U ( I ) R theory which do not have this symmetry in the limit g ' - , 0 must have coeffioents which vanish as g ' - , 0. This relationship between the two models wdl provide a tremendous simplification to our computation. The divergent counterterms must be both Lorentz and gauge invanant. We may systematically list all poss,ble structures satisfying these criteria by using power counting arguments [3]. Defining a dimensionless unitary matrix
U~ M/f,
(3.2)
where U is a funcnon of dimensionless scalar fields, _,tL-- ~r/f,
U= ~/1-~_Z(x) + t'r'~_(x),
(3.3)
the effective lagranglan for the non-linear model is then,
= ~f2TrDuU(D~'U) * + ~'o,
(3.4)
where ~o = ½TrF~ F ~" ± ~ Tr Y.. Y"" + ~;E + ~ r l,.
(3.5)
Defining d as the dimension of the counterterm which operates at 1 loops, d will count one for every Lorentz index that the counterterm carries*. A power-counting * The scalar fields _~ arc damens~onle~s, thcrcfore, theF do not contnbutc to the d~menslonahty of the countcrterms onc may form from them
K Oh'n) k / Strongly interacting Htggs sector
389
analys~s shows that d = 2 + 2 1 - r - 2b,
(3.6)
where b is the number of pure gauge propagators minus the number of pure gauge vertmes and r ~s the number of powers of a dimensional regulator A, which accompany each counterterm. If we consider graphs that have no internal gauge or ghost lines (b = 0), we see that the &vergent counterterms with the greatest &mension are generated by logarithmic (r = 0) divergences. We shall consider the leadmg divergences at one loop of counterterms up to dimensmn four. There is only one possible gauge-invariant term at d = 0, T r U t U = 1,
(3.7)
implying that there cannot be any quadratic divergences. For d = 2, the only possible mvariant structures are
£~= floTrD~U( D~U) *,
~; = BI(TrU*D~U'q ) z.
(3.8)
The term E6 is seen in the tree-level lagrangian so dwergences such as these can be absorbed into the original lagrangian and are physmally unobservable. The other term, t~'l, is not of the form of Etree and thus leads to physically observable effects. At d = 4 a convenient object to use in the enumeration of the possible counterterms is
v. =-(by)u* = g w . - g'Vry* + (O.v)v*.
(3.9)
V~ has dimension one, it is a U(1) R invariant and transforms under SU(2)L as
V~ --, V; = L( x )V~( x )Lt( x ).
(3.10)
Covariant derivatives of these objects take the form
]
(3.11)
It is important to note that m the limit g'--, 0, both these objects become their counterparts introduced by Appelqmst and Bernard in their calculatmn for a spontaneously broken SU(2)L gauge theory. We now have at our disposal the machinery to construct all gauge-invariant objects of dimension four. There are many such objects, but fortunately they need not all be considered. In order that (3.1) hold, in the limit g ' ~ 0 the only remaining counterterms must be exactly those of the SU(2)L theory, with the identical coefficients. The only other counterterms we must consider are those that explicitly vanish m the same limit.
K Olvn) l~ / Strongly interacting Ihggs ,ettor
390
From the preceding argument we may tmmedtately write down all the & m e n s l o n four counterterms that regain the global SU(2) R symmetry m the hm~t g ' ~ 0. These counterterms are simply the SU(2) c × U(1) R generalization of the ones m the SU(2) x calculation. These objects are
~,=a,[TrVuV ,u] 2 , t~2 = a2[Tr V~V~]2,
~.~ = g%Tr( Fu~[ V",V~]) ~,., = - ( ~ ' , -
l)gO, W.. w " ×
w"
~ 2 -- - ~ ~ - l g~( ~ × w~)
~,,~ = - , ( ~ - i)(a.w. - a,w.) 2,
(3.12)
where V ~ is defined in (3.9). It is the SU(2)I. × U(1)R generalization of the object used originally m the SU(2) t calculation. Further. at one loop, the coefficients must be the same as those calculated m the SU(2)I theory, namely* --1
1
-1
16~r2 1 2 t '
±
l
g 2 ( ~ 3 - - 1 ) =-g-S(~" , -- 1 ) = ~
1
1
16~r 2 6 t "
l
) ~'3
1
16~.2 12e
-l
,
1 -- 16~-2 6 e '
(3.13)
where e = 4 - n in a dimensional regulanzation scheme. The coefficients ~'l and ~'3 are not equal to their corresponding coefficients of the linear model, Z t and Z 3, but contain additional logarithmic divergences that appear only as M H becomes large. Using the identity
% v . - % v . = [ v., v.] + g F ~ - g V r . y ,
(3.14)
one may deduce that the only independent counterterms which do not regain a " T h e s t r u c t u r e of the c o u n t e r t e r m s insures that, at o n e loop, the coefficients c a n n o t c o n t a i n e x t r a t e r m s t h a t v a m s h m the h r m t g ' ~ 0 S u c h a satuat~on c o u l d arise o n l y at t w o l o o p s a n d b e y o n d
K Oh'nvk / Strongh interacting lhggs w~tor
391
global SU(2) R symmetry in the limit g' ~ 0 are
f4=a,g'Tr(UY~,~U*[V",V"]),
(3.15) The symmetry reqmrement forces the structures to have exphc~t factors of g' m front ((~" - 1) ~s proporuonal to g,2). It remains our task to determine the coefficients of these three counterterms.
4. Computation of divergence~ Power-counting arguments show the leading divergences at the one-loop level to be logarithmtc. Tlus allows us to regularize d~menstonally and then, at the end, interpret I/E as ln(MH/Mw) in the hnear model. The one-loop dwergence associated with the B self-energy shown m fig. 1 has the value
-1 tg '2 16--ff2 6e ["l[g~,~k2_k~,k~].
(4.1)
The only contribution of the counterterms to this process comes from E~; whose value is
-,(~"-
1)[g,~k2-k~k~].
(4.2)
-i
(4.3)
From (4.1) and (4.2) we conclude g-1'2(~'' - 1)
1
16~r 2 6e"
The one-loop contribution to the ~rWB vertex shown in fig. 2 has the value,
1
I gg,e3,~b[g~,~(q2 +
k 2) _
q~,q~_ k~,k~]
16~r 2 6e
B
~
~
Fig 1 B~elf-energy
Bu
(4.4)
K OI)nvk / Strongl~ mteractmg Ihgg~ aector
392
;°
w /
Bv
b
+
Bv
Eb
Wp.°
+
w2
8v
Fig 2 One-loop conttabutlon to the rrWB vertex T h e c o u n t e r t e r m c o n t r l b u U o n to the s a m e vertex is
-- 2tgg'e3"h[ a3( g ~ ' k 2 -- k~'k " ) + aa( g " ' q 2 - q~q" ) + (a 3 + a 4 + as)(g~'k'q-
(4.5)
q~k')].
T h e terms in (4.4) a n d (4.5) m u s t s u m to zero, thus
0/3 --
1
I
16~r 2 12e
,
~t4 --
1
1
16~r 2 12e
,
% -
-1
1
16,r 2 6 t
.
(4.6)
W e have n o w solved for all the d = 4 coefficients b y the c o n s i d e r a t i o n of j u s t two Green functions*. T h e d = 2 c o u n t e r t e r m is d e t e r m i n e d b y the ~'-field self-energy; the result is [6]
fl, _ 1 1 3M~vtan20 ' 16~r 2 e 4
(4.7)
where tan 0 =- g ' / g .
5. P h y s i c a l
effects
Each c o u n t e r t e r m is the n e g a t i v e of a d i v e r g e n c e seen at the o n e - l o o p level in a c o m p u t a t i o n ; thus, the physical effect of the p r e s e n c e of the heavy Higgs sector m a y be read directly f r o m o u r c o u n t e r t e r m s a n d their coefficients. W e m a k e the c o n n e c t i o n with the l i n e a r t h e o r y b y i d e n t i f y i n g o u r d i m e n s i o n a l r e g u l a t o r with the mass of * The same results are obtamed by constdermg all possible SU(2)I X U(I)R mvanant counterterms generated by all processes with a single I-hggs loop [5]
393
K Ol~nvA / Strongly mteractmg Htggs sector = ~+"-"(3"'~
Fig 3 The full propagator for thc neutral vector boson
the H~ggs particle,
l - m l e
4-n
~ln
MH
(5.1)
M '
w h e r e M is a low e n e r g y scale. W e a s s u m e the mass of a n y f e r m i o n is small c o m p a r e d to M; therefore, we neglect the c o u p l i n g of the f e r m i o n s to the Higgs. However, o u r n o w massless f e r m i o n s d o c o u p l e to the gauge b o s o n s a n d thus p r o v i d e a p r o b e for the i n v e s u g a t i o n of heavy Higgs effects. T a k i n g the o n e - l o o p d i v e r g e n c e s f r o m the c o u n t e r t e r m s , we c a l c u l a t e the full p r o p a g a t o r s for the p h o t o n , the n e u t r a l v e c t o r b o s o m a n d the c h a r g e d vector b o s o n . F o r e x a m p l e , the full p r o p a g a t o r for the n e u t r a l p a r u c l e , as s h o w n in fig. 3, ts f o u n d to be, A z z ( _ q2g~,, + q~,q~/q2 )
t.t,u
DFULL(q ) =
(5.2)
q2 _ ( A z z / A z ) M . ~ + ,E
A similar c a l c u l a t i o n yields the a n a l o g o u s effecnve* r e n o r m a h z a t i o n c o n s t a n t s for the p h o t o n a n d c h a r g e d vector b o s o n . T h e results are
8A ww = A ww - 1 = 8~"3 - 8 Z 3, ~AAA ----AAA - - 1 = (8~3 --
8Z3)smZO+ (8~" -
~ Z ' ) c o s 2 0 --[-
2asgg'smOcosO,
8 A z z = A z z - I = (8~"3 - 8Z3)cos2O + ( 8~' - 8Z')sin20 - 2asgg'sinOcosS, 8Aw =---Aw - ! = 0 , 8AA=--AA - 1 = 0 ,
g2
8A z --=A z -- I = 1 - 213) M 2 c o s 2 0 ,
(5.3)
a n d their physical masses, as d e f i n e d as the p o s i t r o n of the poles m the p r o p a g a t o r s , * The factors of 8Z, wltban the effecuve renormahzauon constants, 8A,, are counterterms of the hnear theory These hnear counterterms are added to the lagrangmn to subtract ultraviolet d~vergences arasmg m the hnear theory
K 01~ nvk / Strongly mtera~tmg Htgg, *e~tor
394
are (we denote physical quantities by a bar) /~2
AAA
=--~-A M2 = O,
M z 2= - _ A_ ~zz M] = ( 1 + 6A z / - 8A z ) g ~ ,
--~
A ww
9
M~v= Aw M f f = ( l + S A w w - S A w ) M 2 w .
(5.4)
With these results we rescale the coupling constants of the theory to g~ve Hlggs-corrected physical values,
= A~/2e,
AIz/z2G,
g =
2 A'i wg.
(5.5)
From the radiatwe correctzons hsted m (5.2)-(5.4) we may determine the M u dependent corrections to physical processes. At low experimental energies* measurement of natural relations provide a check of the results of our calculation. The only natural relauon mvestigated expenmentally is p = ( M w / M z c o S S ) 2 [7], which is predicted to be umty at the tree level. If we define the electric charge, ~, by measurements of Thompson scattering near zero momentum exchange, and g~ as the on-mass-shell couphng of a W to two fermions, we have an experimental defimtion of the Weinberg angle, sin~,) = ~/~.
(5.6)
Another way to define the Wemberg angle would be to compare the on-mass-shell couplings of the W and Z particle to ferrmons, cos
-
7/o.
(5.7)
Comparing ~ to the coefficient of Z-decay yields a third value, sin ~3)cos t~3~= ~ / G .
(5.8)
Usmg our physical masses M w and M z we obtain three corresponding predictions * "Low experimental energjes" refers to energies below the threshold for vector meson production
395
K Olvnvlt / Strongli interacting lhggs sector
o f * O: - 1 + 3 A w w + 3A z - 8 A 7 7 + Ian20(SAAA -- 3 A w w )
= 1 - 2flt-~
o
= !_ ~
p(2) ~"
4- 2 a s g 2 t a n 2 O
(M,,)
({;t)secZOln
-if-,
o
1 - ~--~ (3)sec2Oln
o ,
0~3~= 1 - ~
(5.9)
(~,)sec2Osec 28( 11 cos20 - 9 sin20 )In
w h e r e * * a = e2/4rr. Closely related to thts natural relation ts the natural relation c o m p a r i n g the values of the W e m b e r g angle measured by different schemes. W e find f r o m (5.5)-(5.8) that
COS2~1) COS2~2)
- - -
cosZ~l)
1 -2asg2tan20
o
= 1 +~-~(~)sec2Oln
sin40
a
- - 9 ---
1 + 2asg 2
COS2~2) - GOS2~3)
1 + 2asg2sm2Osec20 = 1
COS-~3 )
(%MH)- ,
l )tan2Osec201n (
COS20COS20 o
~----~( { ) s e c 2 0 1 n
(MH )
(5.10)
Just two of these natural relations are independent. Finally, consider the m v a r i a n t amplitudes associated wtth a neutral-current a n d a charged current process,
82 ~ ,
(5.11)
_ . q2 _ M ]
(5.12)
1N(q2)=---q2~M
~2 Ic(q 2)~
* For P~l)(012)) the same result a~ gxven m ref [8] (ref [9]) ** Note that due to the nature of natural relauons, they cannot depend upon the 8Z,',, or 3~','s The natural rclauons cannot be ultrawolet cutoff dependent
K OIvnvk / Strongly interacting Htgg~ wctor
396
Experimental informaUon about these funcUons would be obtained from neutralcurrent processes such as: ue + ~--* X + v,, or ve + e - -, u~ + e - ; and charge-current processes such as: B-decay,/~-decay or v¢ + ~ -, X 4- e - , etc. At present, accelerators can probe these functions only at low energies, so we may measure low m o m e n t u m expansions,*
IN(O)--
82 M-~ '
d 2)q: o = i ~ ( 0 ) dq 2 IN( q
82 .~ '
Ic(O) --
~~2 2'
d ic(q2) q 2 = o _ l ~ ( O ) _ dq2
g2w . )~r4
(5.13)
From these values we may define experimental values for the vector boson masses,
--
g,2
I~(0) [
I~(0) ]-' 1
/~2w=
It(0)
1
(0---~ I~.
,
(5.14)
A natural relation would be the ratio of these "low-energy" mass definitions to their physical mass counterparts,
ME _ 1 -- a c s c 2 0 ( ~ _ 3tan20)l n
"M2w-1-
ot
MH ( ~ ) c s c 2 Oln(---~)
(5.15)
We have not discussed corrections involving E ~, ~2, ~3 and E4. The reason for this omission is that these counterterms contain divergences of processes involving at least three gauge bosons. Such processes have not been investigated experimentally and so are of less interest. It should be emphasized that our counterterms summarize all possible physical effects of the Hlggs sector at one loop. These effects may be extracted m the same manner we have demonstrated. '~ Note that Pt2)[eq (5 9)] is equal to IN(O)/I¢(O ) Therefore, we need not produce vector bosons in order to measure all natural relataons
K Oh m k / Strongly interacting, lhgg~ set tor
397
6. Conclusion We have catalogued the leading sensltiwty of the standard model to the Hlggs sector up to one loop m the expansion. This was accomplished by considering the large-M H iirmt of the standard model without fermions, a non-hnear o-model. W~thin this Iinut, cutoff-dependent terms are generated which demonstrate the sensitivity of the model to the heavy physical Hlggs parucles. This sensiuvity ~s summarized in the form of gauge mvariant counterterms added to the lagran~an, so as to subtract out these aforementtoned cutoff-dependent terms seen at one loop in the expansion. From these counterterms all leading, Higgs-dependent, physical effects can be extracted in a simple manner. The counterterms were calculated by generalizing to the gauge group, SU(2) L × U(1)R of the standard model, the results of the prewous calculation of an SU(2) L gauge model by Appelqmst and Bernard. In addition, we calculated the cutoff-dependent coefficients of the three independent counterterms which have no counterpart in their SU(2) L theory. However, it should be noted that in the linut M H--. 1 TeV, the expansion parameter,
g2M _ 8,rr 2 M 2
2~-2f 2
will become large and so the effect of the Higgs sector is, at most, ( g 2/192 ~r2 )In( M~/MZw) times a factor of order one. Therefore, although this factor rmght possibly be calculated accurately in a strong couphng expansion scheme, the results from our G N L o model calculation would be only a rough grade to the sensitivity of the standard model to the heavy H~ggs sector. The conclusions reached herein will be most rehable if M H is less than 1 TeV but greater than M w. In sect. 5 we continued the analysis by calculating Higgs-dependent corrections to natural relations of the standard model. Thus far, however, the only experimentally measured natural relation is p. We gave three numerically different predictions, based on three different definitions for the Weinberg angle. Investigation of the other natural relations considered must await the production of the vector mesons. I would like to thank C. Bernard for suggesting the problem and for his many helpful discussions.
References [I] S Wemberg, Phys Rev Lett 19 (1967) 1264; A Salam, in Elementary particle physics, relativistic groups and analyUoty (Nobel Symp. no 8), ed N Svartholm (Almqwst and Wlksell, Stockholm, 1968) p 367 [2] J J Sakurat, m Current trends m the theory of fields, ed J E Lannutu and P K Wdhams (A/P, New York, 1978) p 38 [3] T Appelqmst and C Bernard, Phys Rev D22 (1980) 200
398
K Olvnyk / Strongly interacting thgg~ ~ector
[4] G Passanno and M Veltman, Nucl Phys BI60 (1980) 151, M Consoh, Nucl Phys BI60 (1980) 208, M Lcmome and M Veltman, Nucl Phys B164 (1980) 445, M Green and M Veltman, Nucl Phys B169 (1980) 137 (E. B175 (1980) 547), F Antonelh, M Con,,oh and G Corbo. Phys Left 91B (1980) 90, M Veltman, Phys Lett 91B (1980) 95, A Swim. Phys Rev D22 (1980) 971 [5] A Longhatano, Nucl Phy,, B188(1981)118. K Olynyk, unpubhshed [6] A C Longhltano, Phys Rev D22 (1980)1166 [7] P Langacker et al. Proc Neutnno '79, vol 1, p 276 [8] W J Maroano. Phys Rev D22 (1979) 274 [9] W J Maroano and A S, rhn, Phys Rev D22 (1980) 2695