Heavy top and heavy Higgs control in electroweak observables

Volume 260, number 1,2

PHYSICS LETTERS B

9 May 1991

Heavy top and heavy Higgs control in electroweak observables F.M. Renard Laboratoire de Physique Math~matique, USTL, 1;-34095 Montpellier Cedex 5, France

and C. Verzegnassi 2 Laboratoire d'Annecy-le- Vieux de Physique des Particules, 1N2P3-CNRS, B.P. 110, F- 74941 Annecy-le- Vieux Cedex, France Received 21 January 1991

We point out how the availabilityof different types of LEP observables may allow one to separate the virtual effects of a heavy top quark from those of a heavy Higgs boson and also to separate both from possible New Physics effects.

An embarrassing picture that might show up in the next few years, if the results o f direct searches at C D F and LEP2 would turn out to be negative, is one where both the top and the Higgs would be sufficiently heavy for not being p r o d u c e d before the beginning o f the giant pp colliders operative era. I f this were the case, the role o f precision m e a s u r e m e n t s o f electroweak observables, where the virtual effects o f these particles can be sizeable at one loop in the M S M framework [ 1 ], would b e c o m e even m o r e essential. Unfortunately, as a general rule, the top and the Higgs seem to have a vicious tendency to interfere destructively in the most c o m m o n l y considered observables, which renders the chances o f an indirect search from very high precision measurements even more problematic. The p o i n t that we want to stress in this short p a p e r is that the viciousness o f this interference is not a general and u n a v o i d a b l e feature, and that with a ( m o d e s t ) theoretical effort it is possible to define observables where the top a n d Higgs effects can be fully controlled, i.e. either separately isolated or completely and simultaneously washed out, leading to a self-consistent prescription for their possible successUnit6 associ6e au CNRS no. URA 768, PM/90-38. On leave of absence from Department of Theoretical Physics, Trieste, and INFN, Sezione di Trieste, Trieste, Italy

ful identification or for the search o f a d d i t i o n a l nons t a n d a r d effects. To make the previous statements more clear, a short review o f the virtual effects o f the top and Higgs at one loop is opportune. Although this subject was already exhaustively treated in previous references [2,3 ], we s u m m a r i z e here briefly in the first o f this letter, for sake o f completeness, the m a i n ( k n o w n ) results, that will be utilized to pursue the real program o f our work. The most c o m m o n top and Higgs effects occur at one loop in v a c u u m polarizations ( " o b l i q u e " corrections in the notation o f ref. [ 2 ] ). In particular if one defines the general transverse v a c u u m polarization a m p l i t u d e Huj >(q2) where i,j refer to the particular gauge boson, 11(i,j) (q2) - II(i,j) (0) -I-q2H'(i,j) (q2) , H' (qZ) -

H(q z) -11(0) q2

(1)

Then, for the most general four-fermion process at given q2 ( a n d fixed M z , GF) one only has to retain f o u r gauge-invariant and finite c o m b i n a t i o n s o f vacu u m polarization functions [ 2 ]. In particular, to describe physics on Z resonance one needs the three

0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )

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quantities (in the notation of ref. [2], but not with that metric)

A . ( M 2) =_H}ry,(O ) - H } v v ) ( M 2) Ap(0)-

Hczz)(0) M2z

(2)

H~ww)(0) M2w ,

(3)

e2 ZI 3Q (M 2 ) =

\

S2C2

H3Q(M2)) M~ ]'

Mz

(4)

while a fourth combination is requested to write e.g. the theoretical prediction for Mw, i.e. e2

9 May 1991

having used the normalization given in a recent review paper on this subject [ 5 ]. Eqs. ( 6 ) - (9) display the typical potential virtual effects that a heavy top and Higgs can produce in the MSM. To visualize this effect better, we write now an expression for various observables where the Ap(0), .... A~,v content is made explicit. This can be done in a simple way starting from an "improved" Born approximation computed in the philosophy of the starred system of Kennedy and Lynn [6]. In this scheme, the expression e.g. for the partial width Z-~ffcan be written as ( f ~ b )

Fe

N a'Z*

Mz - 48 s*2c .2 [1 + ( I - 4 I Q f l s*z)z] ,

(10)

where N is the usual color factor. Here a * =

A,Q(M~) = s2c2

a ( M z ) ~ - l/128.8,

×{H~ (M2w) - H , l (0) -c2H3Q(M2z)'~

),

(5)

Z*=

(11)

I--A3Q ,

and where c z =M2w/ M~. In practice, A,~(M~ ) can always be reabsorbed in the redefinition of the electric charge e.g. in a suitable "improved" Born approximation, which leaves three independent oblique corrections affected by top and Higgs effects. In the limiting situation m,, mH > Mz that we are here concerned with, the analytic expression of the effect is rather simple, and can be written to good approximation as [4]

am 2 A~t'm (0) - n M ~

a,

M~

~ m ~z-z '

(6)

s'2~$2[ 1 --8C2So2(zJp(O) -[-A3Q) ] , with sg=l(l__

H) ( M z2 ) . ,o ( M z2 ) - C 2Z~Q

A(H)

(7)

226

+ --ln775-~2 '

zJ'

/t2= (38.45 GeV) z .

The expressions of the various widths can now be easily derived. They read [ neglecting systematically terms of order (OWo), Vo= 1 - 4 s 2 << 1 ] re+ e _

OZ*

- 48s~c~ [ l + A p ( 0 ) ] ,

(14)

Fv~ 2a* Mz - 48sgcg [ 1 +Ao(0) ] ,

(15)

Fu. 5a* Mz - 72sgcg [1 +~AA0) +~,J3Q],

(16)

Faa _

(17)

(8)

Oblique corrections are not the only quantities that are sensitive to heavy particle virtual effects in the MSM. There is, in fact, a Zbb vertex correction that also exhibits a quadratic top mass dependence (but no Higgs dependence). This was first recognized in ref. [3], where the expression of such a term was computed. In the limit of large top mass, one can write to a good approximation A l ' v - ~ - -19 -n\~z

~1__4/'~2"~

(13)

Mz a M~ 4A(H)ta~ A ~ ) ( M 2 ) = - ~ l n ~ z - z =~=p , ~ , ,

(12)

(9)

Mz

13a* 19 6 1 "1-3"~Z~p(0) +~A3Q] 144sgcg [

In the case of Z-,bl3, the extra vertex correction eq. (9) has to be added. This gives 13a* 19 t 6 Mz - 144S~Coz [ 1 +i3(Ap(0) +Abv) +rgAaQ] • /b6

(18) To the previous observables, most of which have been

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PHYSICS LETTERSB

already measured with great accuracy at LEP 1, one can add a number of quantities to be (hopefully) measured in the near future. Particularly relevant from the theoretical point of view is the longitudinal polarization asymmetry [ 7 ]: trL --aR _~2( 1 --4S 2) + 3 [Ap(0) +A3Q ] .

ALR = O'L q- O"R

(19) As was stressed in several papers [ 8 ], ALR is theoretically equivalent to (at least) two different measurable asymmetries, i.e. to the final x polarization asymmetry A~ and to the forward-backward asymmetry for b13 production (or c~ production, although the experimental conditions should be worse) A~hc), according to the empirical prescription (20)

hLR ~z~lx---~'9-z'l . 13xbFB "

Also, A 2R is ~ theoretically equivalent to A ~-a since (21)

A ~ B ~ 3 ~ALR 2 •

A final observable that we will consider is the W mass, or equivalently the ratio M w 2 / M z2, for which the approximate expression can be written as M2

s2

2 2 ~---~ |

Mzco

1-2s~

e.w. dr

'

(22)

where 2 ~2 __S 2 ArE-W. = __ S ~ [ A f ) ( 0 ) "[-Z~3Q] -~t-t'Os2~0 Z~IQ

¢2 [ A p ( O ) -+-282A3Q]

+

~ooA.Q .

(23)

Eqs. ( 1 4 ) - ( 2 3 ) summarize the potential information on m,, m . that can be derived from the measurements of conventional observables on Z resonance and of the W mass. Similar expressions can be easily derived for v-e [2] or v - N [9] scattering; for our specific purposes, though, the requested experimental accuracy, although rather good, is still not fully adequate, and we shall therefore ignore neutrino scattering at least in this short paper. A few comments on eqs. ( 1 4 ) - ( 2 3 ) are opportune. First, the three independent quantities Ap(0),

9 May 1991

A3Q, AIQ contribute to the various observables in different and characteristic ways. More precisely, the two leptonic widths Fe, Fv only depend on a single variable, i.e. on Ap(0). Similarly, all asymmetries and all ratios of partial widths Ff/Ff, f, f' ~ b, only depend on the block [Ap(0)+A3Q] (the exception is represented by the ratio Fv/Fe~-constant). Finally, the variable ~ [eq. (22)] is the only one that also depends on AIQ. From the previous classification the important feature emerges that in all the previously mentioned observables the heavy top and heavy Higgs effects are

always interfering viciously, i.e. in a destructive way. This is simply due to the fact that for both do(0) and (Ap + A3Q) [ eqs. ( 6 ) - ( 8 ) ] the two contributions are of opposite sign. The same property is retained by the W - Z mass ratio since the zJiO contribution, that would interfere constructively, is highly depressed when compared to the Higgs terms in Ap(0) and A3q. A contradiction to the previous general statement is represented by ratios of widths that contain the partial width of Z into b13 (for example, Fb/Fe or Fh/ Fe = R k ). In this case, the negative top contribution from A[,v either cancels (practically) exactly the top effect (in the ratio Rk ), or changes its sign (in Fb/ F~) compared to the positive Ao(0) top effect, leaving therefore a residual contribution where a typically constructive (additive) top-Higgs interference is at work. Unfortunately, the surviving term is too small to produce a measurable effect, even at the aimed experimental accuracies for the two ratios [ 8 ]. In Fb/Fe the heavy Higgs contribution is given by 6A(H) ~----~(a/4n) l n M 2 / M 2, i.e. about ~19A(H)-1p -- "[3~3Q 1% for a Higgs of 1 TeV, which is just the highest level of accuracy one could reach with 108 Z. The situation is similar with R [ that receives a Higgs contribution - ( 6 5 a / 3 5 4 n ) In M2/M~ which can reach the 0.3% level corresponding to the best accuracy one can expect on its measurement. Still, this shows that it must be possible to generate in a more systematic way observables where the top-Higgs interference can be suitably adjusted. To illustrate this statement with the simplest impressive example, we have chosen two observables whose measurements represent, in a sense, the goal of the future LEPI and LEP2 phases. These are the W mass and the longitudinal polarization asymmetry ALR (or, equivalently, A~ and/or A~B ). Rather than 227

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Mw one can choose the variable ~ [eq. (22)] and write from eqs. (22), ( 2 3 ) a n d (6), (8) ~--1+

cg

2~

Co ~ S o

x ~'+~"'+A~' ~ l+ ~

c~

c°-s°~I~' ~g

(a~" + ~ " ~ ) .

(24)

CO - - S O

The analogous expression for ALR is

8s~c~

ALR~2(1--4S~)+ ~

(A~t ) + ~ A ~ n ) ) .

(25)

9 M a y 1991

250 GeV is equal to 0.009. This effect should be compared with the expected overall theoretical and experimental error on ~ that, at the aimed accuracies for Mw and ALR [ 8 ], should be about + 0.003. Thus, under those conditions, the special combination ~t, would be indeed able to "see" a heavy top in a clean and unbiased way, totally independent of the Higgs mass. The obvious and tempting next step is to look for the orthogonal combination to eq. (26) i.e., the one that would react to a heavy Higgs, but not to a heavy top. From eqs. (24), (25) one would guess that this combination should not be far from the quantity

CO - - S O

If the measurements of ~ and ALR were accurate enough, eqs. (24) and (25 ) could be solved for A~t) and A~H) leading to a determination of mt and mH. However, because of the experimental uncertainties and of the fact that the corrections are nearly proportional ( ~ versus ~ ) , the situation is more intricate. As a first exercise, we want to generate an observable that reacts to a heavy top, but not to a heavy Higgs. We expect that, if our expressions eqs. (24), ( 25 ) are sufficiently accurate, this observable is given by the following combination: ~t----~--~ALR.

(26)

In fact, we also expect some (suitably) small deviation from this prediction, owing to the approximations that we made [e.g. we neglected systematically terms of O (voAo) or of O (Vo2 ) ]. Thus, a rigorous numerical check of our proposal is required. We performed the latter using the available programs ~l for 100~
--~

t mt L0"947 = ~ 0.956

0.947 0.956 "

As one sees, the variation of ~t when mH varies from 100 to 1000 GeV (these two values are quoted in the equation) is indeed "practically" vanishing. On the contrary, the variation when mt ranges from 100 to ~t We t h a n k W. Hollik for m a k i n g his n u m e r i c a l programs available to us.

228

1 ~ - - 8-'~02ALR " ~ - - 0 . 5 4 A L R

.

(27)

Again, we have used the available programs (see footnote l ) to identify a combination "close" to eq. (27) that meets the previous requirement. In fact, we have found the good candidate to be the observable ~H=~--0.63ALR

(28)

.

As one sees, the simple approximation eq. (27) is reasonable, but not outstandingly good. The reason is probably due to the fact that, as we explicitly checked, a more rigorous expansion including e.g. terms of O(voA o) should be used now to cancel exactly the dominant top term and leave the smaller Higgs effect alone. When this is done, one gets an expression that is closer to the final empirical form eq. (28). The variations of ~" when (rot, mH) vary in the range 100-250 GeV and 100-1000 GeV respectively are m H

--~

H mt[0.919 = + 0.919

0.922 0.922 "

As one sees, ~H is indeed "blind" to a heavy top. It does react, though, to a heavy Higgs; its variation when 100 ~
(29)

and, although this is at the limit of the highest conceivable precision in ~H [8 ], it shows that the program of isolating the Higgs contribution is, in principle, not an utopia for future LEP experiments.

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PHYSICS L E T T E R S B

An additional and independent constraint can be provided by the measurement of Fe+e-, or equivalently the "reduced" leptonic width:

48S2oC~ ee+e- _ 1 +,j~, ) +A~H). Ye --

OL*

(30)

Mz

As one can see the combination of 3~ t) and A~") is again slightly different from that in eqs. (24) and (25). This offers an alternative and interesting possibility of isolating the effects of the Higgs sector. From the (approximate) eqs. (14), (19) one easily sees that it must be possible to generate a linear combination of ye and Of ALR that is only affected by J3Q and not by Jp(0) (or, obviously, by J1o). To guess the right combination requires, as in the previous case o f ~ " [eq. (28) ], a more rigorous expansion than that ofeqs. (14), (19), i.e. one where e.g. the next terms of O (VoJo) are retained. If one does that, one is led to the prediction that the Jp-free quantity should be "close" to the combination ~H=~__

l+3V°A

3 U-~---O -- 5

LR

--~I+A3Q ,

Vo--~0.08

(31)

In fact, a more accurate numerical check confirms that the approximation is satisfactorily good, since the top effect in eq. (31 ) is practically vanishing, while the Higgs effect is ( 100 GeV ~
0.003,

(32)

identical to that of eq. (29). To be sensitive to this, the leptonic width should be thus measured to a relative accuracy of less than a few per mille. This might become realistic in a future LEP phase with very high luminosity, whose features have been exhaustively discussed elsewhere [8]. Note that, as previously stressed, the combination eq. (31 ) is only sensitive to 33Q. This will become relevant in the second part of this letter. It is not difficult to generate now the orthogonal combination of y and A LRthat would only react to the top, and not to the Higgs. A straightforward application of the rules of our game and of eqs. (14), ( 19 ) and (6), (7) would indicate the combination 4 top . ytxy--3A-~ 1+T330

(33)

The correct combination, that can be found numerically, is not far from eq. (33). It shows a top effect of the order of ~ 0.004, i.e. not much higher than the

9 May 1991

Higgs effect in the orthogonal combination. This is a consequence of the fact that the overall oblique corrections in y and ALR are less nearly proportional than the case of ~ and ALR, and differ more sizeably in the Higgs content, which in principle would be good for a separate identification of the two effects. So far we have sticked to the MSM and shown that, for the specific purpose of fully isolating e.g. the top contribution, it is rather essential to take correctly into account all the three independent radiative corrections (including the usually neglected ones, i.e. A3Q and, more specifically, 310). When one abandons the MSM scheme and looks for virtual signals of New Physics, the same feature becomes even more important, and this point has been stressed in a number of very recent papers [ 10-12 ]. In particular, in ref. [ 10 ] the remarkable fact has been shown that the quantity called S ~ A3Q(0) can be large in technicolor theories. On the contrary, a certain combination ~ [33Q-- ( 1/ Co 2 )Ato] called e2 in ref. [ 11 ] would be blind to the Higgs as from eq. (8), but would be particularly sensitive to anomalous 7WW and ZWW couplings, showing the relevance of the various radiative corrections with respect to different manifestations of New Physics. An orthogonal problem that we want to face now is that of finding special observables that are vice versa completely free both of heavy top [Ap(0) and /l~,v ] andofheavy Higgs (A3Qand AlO) effects. These variables would have the two following special features: (I) In the MSM, they would be a generalization of the simpler top-free variables proposed in recent papers [ 13,14 ]. The numerical value of the related theoretical predictions would be therefore fixed in the MSM with the smallest conceivable theoretical uncertainty. This might become a relevant feature for the next-to-come experimental phases at LEP, whose accuracy is moving towards the final goal, i.e. well below the relative one percent level. (II) For indirect searches of New Physics, these variables would be blind, not only to all the models entering the redefinitions of3p (0) ~ Vpand 3[,v ~ Vbv and reviewed e.g. in ref. [14] (supersymmetry, charged Higgses, anomalous neutral Higgses .... ), but also to the extra models considered e.g. in refs. [ 10,11 ] (technicolor, anomalous three boson couplings .... ). As such, they would provide a more gen229

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eral and powerful way of isolating effects coming from additional pieces in the Z current, for example those due to the mixing with one extra neutral Z, that have been reviewed very recently [ 15 ]. The general strategy to generate such observables is simple. It stems from the fact, already stressed in this paper, that asymmetries and ratios Ff/Ff, (f, f' ¢ b) on Z resonance only depend on the block Ao = A o ( 0 ) +A3Q .

(34)

Therefore, the same "twiddling" operation introduced in ref. [ 13 ] that washes out the Ao(0) component, automatically washes out the A3Q (and, trivially, the AIQ already absent) component as well for observables of the previous kind. On the contrary, any combination of such observables with ~= 2 2 2 will retain a contribution ~ [A3Q-Mw/Mzco ( 1/c~ )AIo] that cannot be killed by the "twiddling" operation. However, as shown in ref. [11] this remaining quantity vanishes in the standard model and could receive contributions from anomalous threeboson couplings which are probably very small. From this more general point of view, such models would show their most unbiased effects in variables that contain neither the ratio M2w/M~ nor the partial width Fb, and are made of ratios of cross sections or of asymmetries. This requires, unavoidably, a very accurate measurement of Fb or of Fb/Fe (to be subtracted away) and of one of the three asymmetries in eq. (20). In particular, we can form in this way the combination "4

4

o_= Fe -- 5ALR,

(36)

whereas in the case of the right-left symmetric models one gets

5D=~OM(OtLR + a~R) .

(37'

Using the expected experimental accuracies [ 8 ] this 230

shows that one is sensitive to Z-Z' mixing angles down to [0M I = 0.005. An alternative possibility is to consider a variable where only SUSY effects (in the previous set of models) could be at work. This would actually be the "magic" combination M already considered in the literature [ 14,15 ], whose expression reads

3(rb

26~b "~

M = i3\~ee - ~ F a ]

19 t = [ O ( 1 ) l (1 +gAbv)

(38)

which is Ao(0), A3Q,and AIQ free. This variable would be a very genuine and unbiased top indicator in the MSM and beyond. In a minimal SUSY model [ 16 ], in fact, its top effect might be drastically enhanced, as stressed in ref. [ 14 ], without any possible confusion with the other competitor models listed in this paper. In conclusion, we have shown that in the most general treatment that retains all the three relevant oblique electroweak corrections at one loop, LEP physics has the remarkable feature that a MSM heavy top and a heavy Higgs effect can always be separately isolated. For indirect searches of New Physics beyond the MSM, these effects can be completely washed out, together with those of a sufficiently wide set of models, if unbiased searches of a new Z must be performed. Alternatively, a genuine top effect that might only be enhanced by SUSY effects can also be generated. The latter possibilities would only be feasible, though, if a high luminosity LEP phase became soon available.

(35)

that is completely Ao(O), A~v, A3Q and A~Q free. It should be considered therefore as a very unbiased Z' indicator measurable at future LEP phases. For example [ 15 ] a Z' originating from E6 would give 6D= 20M(23@6 COSfl+ 3@2 sin fl) ,

9 May 1991

References [ 1 ] A recent updated review of the subject has been given in: G. Altarelli, R. Kleiss and C. Verzegnassi, eds., Z physics at LEP 1, CERN report CERN-89-08. [2] B.W. Lynn, M.E. Peskin and R.G. Stuart, CERN Yellow Book 86-02, eds. J. Ellis and R. Peccei, p. 90. [ 3 ] A.A. Akhundov, D.Y. Bardin and T. Riemann, Nucl. Phys. B276 (1986) 1; W. Beenakker and W. Hollik, Z. Phys. C 40 (1988) 141. [4]See, for instance, M. Kuroda, G. Moultaka and D. Schildknecht, CERN preprint CERN-TH-5818/90. [5] A. Djouadi, G. Girardi, W. Hollik, F.M. Renard and C. Verzegnassi, preprint LAPP-TH-286/90. [6] D.C. Kennedy and B.W. Lynn, Nucl. Phys. B 322 (1989) 1.

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[7] B.W. Lynn and C. Verzegnassi, Phys. Rev. D 35 (1987) 3326. [ 8 ] See, for a recent review, the CERN report High luminosities at LEP, to appear soon as a CERN Yellow Book. [ 9 ] CHARM Collab., J.V. Allaby et al., Phys. Lett. B 177 (1986) 446, Z. Phys. C 36 (1987) 611; CDHS Collab., A. Blondel et al., CERN preprint CERNEP/89-101; A. Blondel, CERN preprint CERN-EP/89-84. [ 10] M.E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65 (1990) 964; see also B.W. Lynn et al., in Physics at LEP, eds. J. Ellis and R. Peccei, CERN report CERN-86-02 (1986).

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[ 11 ] G. Altarelli and R. Barbieri, CERN preprint CERN-TH5863/90. [ 12] A. Dobado et al., CERN preprint CERN-TH-5785/90. [ 13] F. Boudjema, F.M. Renard and C. Verzegnassi, Nucl. Phys. B314 (1989) 301. [ 14] J. Layssac, F.M. Renard and C. Verzegnassi, preprint LAPPTH-290/90. [ 15 ] F. Boudjema, A. Djouadi and C. Verzegnassi, Phys. Lett. B 238 (1990) 423. [ 16] For a review and a list of references, see H. Nilles, Phys. Rep. l0 (1984) l; see also R. Barbieri, Nuovo Cimento I l (1988).

231