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LEP data and the light gluino window
Volume 262, number 1
PHYSICS LETTERS B
13 June 1991
LEP data and the light gluino window I. A n t o n i a d i s
a, John Ellis b and D.V. Nanopoulos c,
a CentredePhysique Th~orique, EcolePolytechnique, F-91128Palaiseau, France b TheoryDivision, CERN, CH-1211 Geneva 23, Switzerland Center for Theoretical Physics, Department of Physics, Texas A & M University, College Station, TX 77843-4242, USA and Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Woodlands, TX 77831, USA
Received 28 March 1991
LEP data on hadronic final states are known to be consistent with the asymptotic freedom expected for QCD. We use this consistency to constrain the number of light strongly-interacting degrees of freedom. In particular, we use DELPHI data to exclude light gluinos weighing < 5 GeV, and determine the number of light quarks to be 5.3 +_1.5 on the basis of a comparison between LEP, deep inelastic and l" decay data. The absence of a light gluino means that supersymmetric model-builders must incorporate a primordial gaugino mass.
N o new particles have yet been discovered at LEP [ 1 ], but the precision o f the data obtained so far has exceeded expectations. They have tested the radiative corrections whose calculability is the litmus test o f the standard model. Already precision electroweak data have been used to predict the mass o f the top quark [ 2 ] and constrain models of spontaneous symmetry breaking via the radiative corrections they induce [ 3 ]. However, the electroweak data are not sensitive to the possible existence of massive supersymmetric particles [4 ]. In parallel to these electroweak studies, the point was soon made that LEP data on hadronic final states, principally jet cross sections and energy-energy correlations, were consistent with the asymptotic freedom expected for Q C D [ 5 ], which is also an effect o f radiative corrections. The agreement between the values o f the Q C D scale A ~ extracted from low- [6,7 ] and high-energy data has encouraged extrapolations [8,9 ] to even higher energies. This extrapolation depends on the assumed number of active strongly-integrating degrees of freedom. The comparison with supersymmetric grand unified theories ( G U T s ) works for sin20w far better than that with non-supersymmetric G U T s [ 8,9 ], encouraging the belief that sparticles have relatively light Supported in part by DOE grant DE-AS05-81ER40039.
masses < l TeV [ 9 ]. The experimental value o f the bottom quark mass can also be predicted successfully if there are only three generations [ 10 ]. However, we are unaware o f any quantitative analysis o f the number o f light strongly-interacting degrees o f freedom permitted by the consistency o f LEP and lower-energy data. One particular point o f interest is the possibility of a light gluino g. Many fixed-target and accelerator experiments have looked for direct manifestations o f light gluinos ( m I £ 5 GeV), but have not yet been able to exclude this possibility completely [ 11 ]. As we will review in more detail shortly, the possibility that m~< 1 GeV, or that m~~ 3 and 4 GeV for certain ranges of lifetime, had not been excluded until recently, although a recent H E L I O S search [ 12 ] for missing-energy events apparently excludes the possibility that m~< 1 GeV. We point out in this paper that the consistency between D E L P H I data on a s ( m z ) and the na'fve extrapolation of low-energy data may be used to exclude m~< 5 GeV, thus closing the last remaining window for a light gluino, and c o m m e n t at the end o f this paper on the implications of this observation for model-building. The LEP data correspond to N just N f = 5.3 + 1.6 quark flavours being effective between
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
109
Volume 262, number I
PHYSICS LETTERS B
scales Q = 5 GeV and mz, which is comforting though hardly surprising. We start by reviewing the previous status of the light gluino window. We see in fig. 1, adapted from ref. [ 11 ], that the window actually contains three panes: a region I where m i < 0.7 GeV, essentially independently of the gluino lifetime zs, a region II where 2 . 7 < m s < 4 GeV and zs< 10-nns, and a possible region III where m s > 2 . 7 GeV and 10-s>~zs~> 10-1°s, the possibility of whose existence depends [ 14 ] on the difficult-to-estimate gluino-hadron scattering cross section. We are aware of one relevant new search since ref. [ 11 ], namely that for missing-energy events in 450 GeV pN collisions by the HELIOS Collaboration [ 12 ]. They give an upper limit on new particle production as a function o f mass and lifetime that we have compared with the predictions o f a phenomenological model due to Bourquin and Gaillard [ 13 ]. The HELIOS limit [ 12 ] in the (m s, zs) plane is shown as the dashed line in fig. 1. It seems to suggest that m s >~1-2 GeV, although the limit has an uncertainty inherent to calculations in this non-perturbative region where one cannot trust perturbative cross section estimates. Nevertheless, we interpret the HELIOS data as excluding the very light gluino region I. A more rigorous exclusion o f this region as well as 10 4
,
,03.
~ t ~ , S T A ~ E I 'ARTICLE S E A R C I I ~ I ~
.e i i i J
/
,I III
>~ toE
4
,o
/11
,/t~
.
UA1
"
;IV
a
101 0
as(5 GeV) = 0 . 1 8 0 + 0.013,
i 2
t 3
( 1)
whereas the Y data give [ 6 ] or,(5 GeV) = 0 . 1 7 9 + 0.009.
(2)
Both o f these estimates are subject to additional, unknown, systematic errors due for example to higher orders o f perturbation theory, and to non-perturbative effects such as uncertainties in the choice o f parametrization o f higher-twist effects and corrections to the non-relativistic quarkonium potential model. The allowance to be made for these systematic errors is largely a matter of taste. The uncertainty due to higher-order perturbative effects can perhaps be gauged by comparing the effective number of light quark degrees o f freedom Nf inferred by using leading-order and next-to-leading order formulae for ot~(Q) to compare LE and HE data. We can mention in anticipation that the shift in Nf will be much smaller than the quoted errors, so we are not greatly concerned about the higher-order perturhative effects. We can only ignore possible non-perturbative systematic errors. Doing so, we average the determinations ( 1 ) and (2) to obtain oq(5 GeV) = 0 . 1 7 9 + 0.007,
4
i 5
m~ (GeV)
Fig. 1. Region of gluino and squark masses excluded by different experiments, as well as curves of equal gluino lifetime, adapted from ref. [ 11] which also contains originalreferences. The dashed line is obtained from the results of ref. [ 12] using the model of ref. [ 13], and apparently excludes region I. Region II and much of region III is apparently excluded by DELPHI data on oq [ 9 ]. We thank J.-M. Gaillard for help in preparing this figure.
110
of region II and much of region IIl, is in principle possible by a quantitative analysis of the asymptotic freedom of Q C D using low-energy (LE) non-LEP and high-energy (HE) LEP data. W e take as our reference LE scale Q = 5 GeV. The two most precise determinations of a, (5 GeV) seem to bc those from a recent global analysis of structure function data [7 ], and from Y decays [6 ]. Other LE data are consistent with these determinations, but have largererrors [6 ]. O n the basis of the structure function data, the estimate A~--sfs =4 = 185 + 50 M e V has been given [7 ], corresponding to
.,~/,
CUSB
i 1
13 June 1991
(3)
to be compared with the HE LEP value. Opinions on the right HE value o f a , ( m z ) vary, with different LEP Collaborations and rapporteurs emphasizing different jet measurements and quoting different systematic errors. For definiteness, we will use the value quoted recently by some members o f the D E L P H I Collaboration [ 9 ]: a s ( m z ) =0.108 + 0.005.
(4)
A caveat is in order at this point. The effective mo-
Volume 262, number I
PHYSICS LETTERSB
mentum scale Q at which oq ( Q ) is measured in e ÷ e collisions may be somewhat less than mz, and the value (4) has been obtained assuming the two-loop Nf= 5 formula for as (Q), so we are in danger of using what we want to prove in order to prove what we want to prove [ 15 ]. Therefore we will quote results for Q=mz and mz/2, using (4) or the corresponding value o q ( m z / 2 ) =0.1 lo+o.oo7 x Y __ 0 . 0 0 6
(6)
where 0= 1 if Q > mg, or 0 if Q < m~, and Nf is the number of active quark flavours with mq < Q. Comparing ot(Q z) with or(Q22) then gives 33-60-2Nf
×
°q(Q~)-°q(Q2) oq(Q2)ots(Q2 ) In(Q2/Q2) ,
5 GeV versus mz :Nf + 30=4.6 + 1.6, 5 GeV versus mz/2: Nf + 30= 4.5 _+2.3.
ln[as(Q~)/ots(Q2) ] oq(Q 2) -oq(Q2)~ oq(Q2)ots(Q2) ln(Q2/Q 2) ,]'
where N o is the number indicated by the tree-level analysis (8). Using the values (3) and (4) or (5) we find: 5 GeV versus mz :Nf= 5.3 _ 1.5,
( 1 la)
5 GeV versus mz/2: Nf = 5.3 +_2.2,
( 11b )
again in good agreement with the expectation Nf= 5. We note, moreover, that going from the leading-order formula to the higher-order formula changes the central value of Nf by less than half a standard deviation. As for the possible existence of light gluinos, we can
(7)
in leading order. Using the value ( 3 ) and (4) or ( 5 ) we find:
use
12n 2 2 - 6 0 = In(Q2/Q 2)
(cts(Q 2) - a s ( Q 2) × \ a~---~l~
(8a) (8b)
These determinations are comfortably consistent with the expected Nf= 5 and 0= 0, with the presence of light gluinos ( 0 = 1 ) excluded at better than the 90% confidence level if we compare ot~(5 GeV) with ors(Mz). A more complete analysis must use the next-toleading order formula
×
6(58-7200 ) (23-600) 2
ln [cq ( QE) / ots(Q2) ] ors(Q21)-oq ( Q2) ~ oq(Q2)oq(Q2) In(Q2/Q 2) / ' (12)
together with the values (3) and (4) and (5) to obtain: 5 GeV versus mz : 0= 0.1 _ 0.5,
(13a)
5 GeV versus mz/2:0=0.1 _+0.7,
(13b)
indicating that gluinos are excluded at almost the 2 - a level if we compare oq ( 5 GeV) with as (mz). An alternative way to restate this result is to use the full next-to-leading order formula (5) with Nf= 5 and c~s( 5 GeV) = 0.179 _ 0.007 to predict that at Q = mz:
12n
ors(Q) = ( 3 3 - 2 N f - 6 0 ) ln(Q2/A 2) ×
(as(QE)-a~(Q 2) 6 ( 1 5 3 - 19N ° ) × \ as(Q2)oq(Q~) - (33_2NO)2
(10)
12n
=12n
12n 3 3 + 2 N f = ln(QZ2/Q2)
( 5 )
at the lower renormalization scale, which is related to (4) by the two-loop Nf= 5 formula for as(Q). To compare the LE and HE values of oq, we first use the leading-order formula
as(Q) = ( 3 3 _ 6 0 _ 2 N s ) ln(QE/A2 )
13 June 1991
(1 -- 6(153-- 1 9 N f - 7 2 0 ) lnln(QE/A2)) \ (33-2Nf-60) 2 ln(QE/A 2) /" (9)
In this case, one can legitimately approximate the number of effective quark flavours Nf allowed by measurements of ors (Q2) and as(Q 2) i f 0 = 0 by
without/with light gluinos: ots(mz) =0.107_+ 0.003, a I~a+o.oo3
~ v- t z,J--O.004
,
(14) Ill
Volume 262, number 1
PHYSICS LETTERS B
to be c o m p a r e d with the LEP value a s ( m z ) = 0.108 + 0.005, or at Q = m z / 2 : w i t h o u t / w i t h light gluinos: ots(mz/2) = 0 . 1 1 8 + 0 . 0 0 3 , _ / ' ~ 1"1"1+0.003
.......
o.oos,
( 15 )
to be c o m p a r e d with the LEP value o t s ( m z / 2 ) = 0.1,1 o+o.oov~, oo6 _ • The agreement between the LE and HE d a t a is always perfect with N f = 5 and no light gluinos, whereas including light gluinos always increases the predicted HE value o f ot~ unacceptably. We conclude that the LEP d a t a rule out light gluinos in the regions I, II, and in much o f region III o f fig. 1. We make some final c o m m e n t s about the implications o f these results for modelbuilding. Supersymmetry breaking is usually p a r a m e t r i z e d p h e n o m e n o logically in terms o f a universal b a r gaugino mass mi/2, a universal scalar mass mo, a n d certain other soft supersymmetry-breaking p a r a m e t e r s [ 16 ]. The leading logarithms o f p e r t u r b a t i o n theory, res u m m e d using the renormalization group, give m o d ified effective soft s u p e r s y m m e t r y breaking p a r a m e ters at low scales: 2 mca,¢= mo2 + C~,¢mZ/z ,
M~,w,i~ + a~,2,1 O/GU T
ml/2.
(16)
M a n y theoretical and phenomenological arguments swirl about the relative magnitudes o f mo a n d m~/2. The case m~/2=0 was already disfavoured by LEP [ 17 ] a n d other data, but this p a p e r furnishes another argument. In the case m~/2, the d o m i n a n t contribution to mg comes from one-loop diagrams involving the t o p - s t o p supermultiplet [ 18 ] : O~s
rn~ = -~n mtF( mt, m ~ , m~2 ) ,
(17)
where
{ b2 "~ b2 ( c2 ) c2 F(a, b, c) =- - ~a-T-S~_b2)In ~-5 + ~ In ~ .
(18) It is easy to verify that even i f mt "~ 160 GeV, the maxi m u m allowed by precision electroweak d a t a [ 2,4], the m a x i m u m possible value o f rn~ according to eq. (17) is about 3 GeV. Thus m~ would be in the range 112
13 June 1991
excluded by our analysis. We conclude that m l/2 # O. This conclusion should still be regarded as preliminary, p e n d i n g further reduction o f the experimental a n d theoretical uncertainties. Nevertheless, we hope that this analysis convinces our colleagues that the effort required to reduce these errors would b e a r dividends in terms o f interesting new constraints on physics b e y o n d the s t a n d a r d model. We thank U. A m a l d i a n d S. Kelley for clarifying a n d encouraging discussions.
References [ 1] M. Green, Review talk Royal Society Discussion meeting (London, 1991 ). [2] J. Ellis and G.L. Fogli, Phys. Lett. B 249 (1990) 543. [ 3 ] M. Peskin and T. Takeuchi, Phys. Rev. Lett. 65 (1990) 964. [4] A. Bilal, J. Ellis and G.L. Fogli, Phys. Lett. B 246 (1990) 441. [ 5 ] For a recent review, see S. Bethke, CERN preprint PPE/9136 (1991). [ 6 ] Particle Data Group, J.J. Hermindez et al., Review of particle properties, Phys. Lett. B 239 (1990) 1. [7]A.D. Martin, R.G. Roberts and W.J. Stifling, Durham University preprint DTP/90/76 (1990). [8 ] J. Ellis, S. Kelley and D.V. Nanopoulos, Phys. Lett. B 249 (1990) 441; CERN preprint CERN-TH. 5943/90 ( 1990 ). [ 9 ] U. Amaldi, W. De Boer and H. Fiirstenau, CERN preprint PPE/91-44 ( 1991 ). [ 10] M.S. Chanowitz, J. Ellis and M.K. Gaillard, Nucl. Phys. B 128 (1977) 506; A.J. Buras, J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B 135 (1978) 66; D.V. Nanopoulos and D.A. Ross, Nucl. Phys. B 157 ( 1979 ) 273; Phys. Lett. B 108 (1982) 351; B 118 (1982) 99; J. Ellis, S. Kelley and D.V. Nanopoulos, Phys. Lett. B 249 (1990) 441. [ 11 ] UA1 Collab., C. Albajar et al., Phys. Lett. B 198 (1987) 261, and references therein. [ 12] HELIOS Collab., T. Akesson et al., CERN preprint PPE/ 90-186 (1990). [ 13 ] M. Bourquin and J.-M. Gaillard, Nucl. Phys. B 114 ( 1976 ) 334. [14] NA3 Collab., J.P. Dishaw et al., Phys. Lett. B 85 (1979) 142. [ 15 ] M. Gibbon, private communication to J. Ellis (1960). [ 16 ] For a review, see A.B. Lahanas and D.V. Nanopoulos, Phys. Rep. 145 (1987) 1. [ 17 ] J. Ellis, G. Ridolfi and F. Zwirner, Phys. Lett. B 237 (1990) 423. [ 18] R. Barbieri, L. Girardello andA. Masiero, Phys. Lett. B 127 (1983) 429.
PHYSICS LETTERS B
13 June 1991
LEP data and the light gluino window I. A n t o n i a d i s
a, John Ellis b and D.V. Nanopoulos c,
a CentredePhysique Th~orique, EcolePolytechnique, F-91128Palaiseau, France b TheoryDivision, CERN, CH-1211 Geneva 23, Switzerland Center for Theoretical Physics, Department of Physics, Texas A & M University, College Station, TX 77843-4242, USA and Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Woodlands, TX 77831, USA
Received 28 March 1991
LEP data on hadronic final states are known to be consistent with the asymptotic freedom expected for QCD. We use this consistency to constrain the number of light strongly-interacting degrees of freedom. In particular, we use DELPHI data to exclude light gluinos weighing < 5 GeV, and determine the number of light quarks to be 5.3 +_1.5 on the basis of a comparison between LEP, deep inelastic and l" decay data. The absence of a light gluino means that supersymmetric model-builders must incorporate a primordial gaugino mass.
N o new particles have yet been discovered at LEP [ 1 ], but the precision o f the data obtained so far has exceeded expectations. They have tested the radiative corrections whose calculability is the litmus test o f the standard model. Already precision electroweak data have been used to predict the mass o f the top quark [ 2 ] and constrain models of spontaneous symmetry breaking via the radiative corrections they induce [ 3 ]. However, the electroweak data are not sensitive to the possible existence of massive supersymmetric particles [4 ]. In parallel to these electroweak studies, the point was soon made that LEP data on hadronic final states, principally jet cross sections and energy-energy correlations, were consistent with the asymptotic freedom expected for Q C D [ 5 ], which is also an effect o f radiative corrections. The agreement between the values o f the Q C D scale A ~ extracted from low- [6,7 ] and high-energy data has encouraged extrapolations [8,9 ] to even higher energies. This extrapolation depends on the assumed number of active strongly-integrating degrees of freedom. The comparison with supersymmetric grand unified theories ( G U T s ) works for sin20w far better than that with non-supersymmetric G U T s [ 8,9 ], encouraging the belief that sparticles have relatively light Supported in part by DOE grant DE-AS05-81ER40039.
masses < l TeV [ 9 ]. The experimental value o f the bottom quark mass can also be predicted successfully if there are only three generations [ 10 ]. However, we are unaware o f any quantitative analysis o f the number o f light strongly-interacting degrees o f freedom permitted by the consistency o f LEP and lower-energy data. One particular point o f interest is the possibility of a light gluino g. Many fixed-target and accelerator experiments have looked for direct manifestations o f light gluinos ( m I £ 5 GeV), but have not yet been able to exclude this possibility completely [ 11 ]. As we will review in more detail shortly, the possibility that m~< 1 GeV, or that m~~ 3 and 4 GeV for certain ranges of lifetime, had not been excluded until recently, although a recent H E L I O S search [ 12 ] for missing-energy events apparently excludes the possibility that m~< 1 GeV. We point out in this paper that the consistency between D E L P H I data on a s ( m z ) and the na'fve extrapolation of low-energy data may be used to exclude m~< 5 GeV, thus closing the last remaining window for a light gluino, and c o m m e n t at the end o f this paper on the implications of this observation for model-building. The LEP data correspond to N just N f = 5.3 + 1.6 quark flavours being effective between
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
109
Volume 262, number I
PHYSICS LETTERS B
scales Q = 5 GeV and mz, which is comforting though hardly surprising. We start by reviewing the previous status of the light gluino window. We see in fig. 1, adapted from ref. [ 11 ], that the window actually contains three panes: a region I where m i < 0.7 GeV, essentially independently of the gluino lifetime zs, a region II where 2 . 7 < m s < 4 GeV and zs< 10-nns, and a possible region III where m s > 2 . 7 GeV and 10-s>~zs~> 10-1°s, the possibility of whose existence depends [ 14 ] on the difficult-to-estimate gluino-hadron scattering cross section. We are aware of one relevant new search since ref. [ 11 ], namely that for missing-energy events in 450 GeV pN collisions by the HELIOS Collaboration [ 12 ]. They give an upper limit on new particle production as a function o f mass and lifetime that we have compared with the predictions o f a phenomenological model due to Bourquin and Gaillard [ 13 ]. The HELIOS limit [ 12 ] in the (m s, zs) plane is shown as the dashed line in fig. 1. It seems to suggest that m s >~1-2 GeV, although the limit has an uncertainty inherent to calculations in this non-perturbative region where one cannot trust perturbative cross section estimates. Nevertheless, we interpret the HELIOS data as excluding the very light gluino region I. A more rigorous exclusion o f this region as well as 10 4
,
,03.
~ t ~ , S T A ~ E I 'ARTICLE S E A R C I I ~ I ~
.e i i i J
/
,I III
>~ toE
4
,o
/11
,/t~
.
UA1
"
;IV
a
101 0
as(5 GeV) = 0 . 1 8 0 + 0.013,
i 2
t 3
( 1)
whereas the Y data give [ 6 ] or,(5 GeV) = 0 . 1 7 9 + 0.009.
(2)
Both o f these estimates are subject to additional, unknown, systematic errors due for example to higher orders o f perturbation theory, and to non-perturbative effects such as uncertainties in the choice o f parametrization o f higher-twist effects and corrections to the non-relativistic quarkonium potential model. The allowance to be made for these systematic errors is largely a matter of taste. The uncertainty due to higher-order perturbative effects can perhaps be gauged by comparing the effective number of light quark degrees o f freedom Nf inferred by using leading-order and next-to-leading order formulae for ot~(Q) to compare LE and HE data. We can mention in anticipation that the shift in Nf will be much smaller than the quoted errors, so we are not greatly concerned about the higher-order perturhative effects. We can only ignore possible non-perturbative systematic errors. Doing so, we average the determinations ( 1 ) and (2) to obtain oq(5 GeV) = 0 . 1 7 9 + 0.007,
4
i 5
m~ (GeV)
Fig. 1. Region of gluino and squark masses excluded by different experiments, as well as curves of equal gluino lifetime, adapted from ref. [ 11] which also contains originalreferences. The dashed line is obtained from the results of ref. [ 12] using the model of ref. [ 13], and apparently excludes region I. Region II and much of region III is apparently excluded by DELPHI data on oq [ 9 ]. We thank J.-M. Gaillard for help in preparing this figure.
110
of region II and much of region IIl, is in principle possible by a quantitative analysis of the asymptotic freedom of Q C D using low-energy (LE) non-LEP and high-energy (HE) LEP data. W e take as our reference LE scale Q = 5 GeV. The two most precise determinations of a, (5 GeV) seem to bc those from a recent global analysis of structure function data [7 ], and from Y decays [6 ]. Other LE data are consistent with these determinations, but have largererrors [6 ]. O n the basis of the structure function data, the estimate A~--sfs =4 = 185 + 50 M e V has been given [7 ], corresponding to
.,~/,
CUSB
i 1
13 June 1991
(3)
to be compared with the HE LEP value. Opinions on the right HE value o f a , ( m z ) vary, with different LEP Collaborations and rapporteurs emphasizing different jet measurements and quoting different systematic errors. For definiteness, we will use the value quoted recently by some members o f the D E L P H I Collaboration [ 9 ]: a s ( m z ) =0.108 + 0.005.
(4)
A caveat is in order at this point. The effective mo-
Volume 262, number I
PHYSICS LETTERSB
mentum scale Q at which oq ( Q ) is measured in e ÷ e collisions may be somewhat less than mz, and the value (4) has been obtained assuming the two-loop Nf= 5 formula for as (Q), so we are in danger of using what we want to prove in order to prove what we want to prove [ 15 ]. Therefore we will quote results for Q=mz and mz/2, using (4) or the corresponding value o q ( m z / 2 ) =0.1 lo+o.oo7 x Y __ 0 . 0 0 6
(6)
where 0= 1 if Q > mg, or 0 if Q < m~, and Nf is the number of active quark flavours with mq < Q. Comparing ot(Q z) with or(Q22) then gives 33-60-2Nf
×
°q(Q~)-°q(Q2) oq(Q2)ots(Q2 ) In(Q2/Q2) ,
5 GeV versus mz :Nf + 30=4.6 + 1.6, 5 GeV versus mz/2: Nf + 30= 4.5 _+2.3.
ln[as(Q~)/ots(Q2) ] oq(Q 2) -oq(Q2)~ oq(Q2)ots(Q2) ln(Q2/Q 2) ,]'
where N o is the number indicated by the tree-level analysis (8). Using the values (3) and (4) or (5) we find: 5 GeV versus mz :Nf= 5.3 _ 1.5,
( 1 la)
5 GeV versus mz/2: Nf = 5.3 +_2.2,
( 11b )
again in good agreement with the expectation Nf= 5. We note, moreover, that going from the leading-order formula to the higher-order formula changes the central value of Nf by less than half a standard deviation. As for the possible existence of light gluinos, we can
(7)
in leading order. Using the value ( 3 ) and (4) or ( 5 ) we find:
use
12n 2 2 - 6 0 = In(Q2/Q 2)
(cts(Q 2) - a s ( Q 2) × \ a~---~l~
(8a) (8b)
These determinations are comfortably consistent with the expected Nf= 5 and 0= 0, with the presence of light gluinos ( 0 = 1 ) excluded at better than the 90% confidence level if we compare ot~(5 GeV) with ors(Mz). A more complete analysis must use the next-toleading order formula
×
6(58-7200 ) (23-600) 2
ln [cq ( QE) / ots(Q2) ] ors(Q21)-oq ( Q2) ~ oq(Q2)oq(Q2) In(Q2/Q 2) / ' (12)
together with the values (3) and (4) and (5) to obtain: 5 GeV versus mz : 0= 0.1 _ 0.5,
(13a)
5 GeV versus mz/2:0=0.1 _+0.7,
(13b)
indicating that gluinos are excluded at almost the 2 - a level if we compare oq ( 5 GeV) with as (mz). An alternative way to restate this result is to use the full next-to-leading order formula (5) with Nf= 5 and c~s( 5 GeV) = 0.179 _ 0.007 to predict that at Q = mz:
12n
ors(Q) = ( 3 3 - 2 N f - 6 0 ) ln(Q2/A 2) ×
(as(QE)-a~(Q 2) 6 ( 1 5 3 - 19N ° ) × \ as(Q2)oq(Q~) - (33_2NO)2
(10)
12n
=12n
12n 3 3 + 2 N f = ln(QZ2/Q2)
( 5 )
at the lower renormalization scale, which is related to (4) by the two-loop Nf= 5 formula for as(Q). To compare the LE and HE values of oq, we first use the leading-order formula
as(Q) = ( 3 3 _ 6 0 _ 2 N s ) ln(QE/A2 )
13 June 1991
(1 -- 6(153-- 1 9 N f - 7 2 0 ) lnln(QE/A2)) \ (33-2Nf-60) 2 ln(QE/A 2) /" (9)
In this case, one can legitimately approximate the number of effective quark flavours Nf allowed by measurements of ors (Q2) and as(Q 2) i f 0 = 0 by
without/with light gluinos: ots(mz) =0.107_+ 0.003, a I~a+o.oo3
~ v- t z,J--O.004
,
(14) Ill
Volume 262, number 1
PHYSICS LETTERS B
to be c o m p a r e d with the LEP value a s ( m z ) = 0.108 + 0.005, or at Q = m z / 2 : w i t h o u t / w i t h light gluinos: ots(mz/2) = 0 . 1 1 8 + 0 . 0 0 3 , _ / ' ~ 1"1"1+0.003
.......
o.oos,
( 15 )
to be c o m p a r e d with the LEP value o t s ( m z / 2 ) = 0.1,1 o+o.oov~, oo6 _ • The agreement between the LE and HE d a t a is always perfect with N f = 5 and no light gluinos, whereas including light gluinos always increases the predicted HE value o f ot~ unacceptably. We conclude that the LEP d a t a rule out light gluinos in the regions I, II, and in much o f region III o f fig. 1. We make some final c o m m e n t s about the implications o f these results for modelbuilding. Supersymmetry breaking is usually p a r a m e t r i z e d p h e n o m e n o logically in terms o f a universal b a r gaugino mass mi/2, a universal scalar mass mo, a n d certain other soft supersymmetry-breaking p a r a m e t e r s [ 16 ]. The leading logarithms o f p e r t u r b a t i o n theory, res u m m e d using the renormalization group, give m o d ified effective soft s u p e r s y m m e t r y breaking p a r a m e ters at low scales: 2 mca,¢= mo2 + C~,¢mZ/z ,
M~,w,i~ + a~,2,1 O/GU T
ml/2.
(16)
M a n y theoretical and phenomenological arguments swirl about the relative magnitudes o f mo a n d m~/2. The case m~/2=0 was already disfavoured by LEP [ 17 ] a n d other data, but this p a p e r furnishes another argument. In the case m~/2, the d o m i n a n t contribution to mg comes from one-loop diagrams involving the t o p - s t o p supermultiplet [ 18 ] : O~s
rn~ = -~n mtF( mt, m ~ , m~2 ) ,
(17)
where
{ b2 "~ b2 ( c2 ) c2 F(a, b, c) =- - ~a-T-S~_b2)In ~-5 + ~ In ~ .
(18) It is easy to verify that even i f mt "~ 160 GeV, the maxi m u m allowed by precision electroweak d a t a [ 2,4], the m a x i m u m possible value o f rn~ according to eq. (17) is about 3 GeV. Thus m~ would be in the range 112
13 June 1991
excluded by our analysis. We conclude that m l/2 # O. This conclusion should still be regarded as preliminary, p e n d i n g further reduction o f the experimental a n d theoretical uncertainties. Nevertheless, we hope that this analysis convinces our colleagues that the effort required to reduce these errors would b e a r dividends in terms o f interesting new constraints on physics b e y o n d the s t a n d a r d model. We thank U. A m a l d i a n d S. Kelley for clarifying a n d encouraging discussions.
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