$ρ$ Parameter in the Vector Condensate Model of Electroweak Interactions

3 Pages·0.05 MB·English

by  G. Cynolter

ρ PARAMETER IN THE VECTOR CONDENSATE MODEL
OF ELECTROWEAK INTERACTIONS
G.Cynolter, E.Lendvai and G.P´ocsik
Institute for Theoretical Physics
Eo¨tv¨os Lora´nd University, Budapest
4
9
9 ABSTRACT
1
Inthe standardmodelofelectroweakinteractionsthe Higgsdoublet
n
a is replaced by a doublet of vector bosons and the gauge symmetry is
J
broken dynamically. This generates masses for the gauge bosons and
6
fermions,aswellasitfixestheinteractionsinthemodel. Themodelhas

1
a low momentum scale. In this note we show that the model survives
v
2 the test of the ρ parameter, and to each momentum scale ρ chooses a
1
possible range of vector boson masses.
2
1
0
4
9 Although the Higgs mechanism is a very simple realisationof symmetry breaking, alternative realisations
/
h [1] are also important, since the Higgs has not yet been seen.
p
- Recently,amodelofsymmetrybreakinghasbeenintroduced[2]insuchawaythatinthe standardmodel
p
e of electroweak interactions the Higgs doublet was replaced by a Y=1 doublet of vector fields,
h
:
v B(+)
B = µ , (1)
Xi µ (cid:18)Bµ(0) (cid:19)
r
a and B(0) forms a nonvanishing condensate,
µ
B(0)+B(0) =g d, d=0,
D µ ν E0 µν 6
B(+)+B(+) =0. (2)
µ ν
D E0
This generates mass terms for W and Z, the tree level ρ parameter remains 1 and the photon is massless.
Assuming a quartic self-interaction for B , the B+,0 particles become massive and the ratio of their bare
µ
masses is (4/5)1/2.
Fermions can easily be made massive by introducing Ψ B Ψ B(0)ν+ + h.c. type interactions. Also the
L ν R
Kobayashi-Maskawa mechanism can be built in. Low energy charged current phenomenology shows that
√ 6d=246 GeV. The fermion-B coupling is 3(2)−1/2m G , a G12 factor weaker than the coupling to the
− f F F
Higgs. The model can be considered as a low energy effective model valid up to about Λ 2.6 TeV, and
≤
m 43 GeV [2].
B
≥
1
The aim of this paper is to show that oblique radiative corrections [3] due to B-loops can give arbitrarily
smallcontributionsto theρ parameter,providedthe B+,0 massesaresuitably chosen. ThecutoffΛ remains
unrestricted.
The contribution ∆ρ due to B-loops to ρ is
∆ρ=αT =ε (3)
1
whereT(orε )isoneofthe threeparameters[3]constrainedbyprecisionexperiments. TheanalysisinRef.
1
[4] finds for beyond the standard model ∆ρ= (0.009 0.25)10−2 at m =130 GeV, m =m .
t H Z
− ±
The parameter T is defined by
e2
αT = Π (0) Π (0) (4)
s2c2m2Z(cid:16) WW − WW (cid:17)
with s = sinθ ,c = cosθ is calculated in one-B-loop order. Π is expressed by the g terms of the
W W ik µν
vacuum polarization contributions Π due to B-loops as
ik
e2 e2
Π = Π , Π = Π (5)
ZZ s2c2 ZZ WW s2 WW
The interactions giving rise to W and Z self energies follow from the starting gauge invariant Lagrangianof
the B+,0 fields [2]. From these only the trilinear ones are important to one-loop order,
L B0 = ig∂µB(0)ν+ Z B(0) Z B(0) +h.c.,
2c (cid:16) µ ν − ν µ (cid:17)
(cid:0) (cid:1)
L B+B++Z = C L B0 B+ , C =c2 s2, (6)
(cid:16) (cid:17) − · (cid:0) → (cid:1) −
L B0B+W = ig ∂µB(+)ν+ W+B(0) W+B(0) +
√2(cid:20) (cid:16) µ ν − ν µ (cid:17)
(cid:0) (cid:1)
+∂µB(0)ν+ W−B(+) W−B(+) +h.c. .
(cid:16) µ ν − ν µ (cid:17)(cid:21)
In a renormalizable theory T is finite. In the present model, however, it is a function of the cutoff Λ which
cannot be removed. After lengthy calculations we get
5 1 C 11 17 Λ2
64π2Π (0)= + Λ4+ (1+C)Λ2 m2ln 1+
− ZZ − 4(cid:18)m20 m2+(cid:19) 2 − 2 0 (cid:16) m20(cid:17)−
(7)
17 Λ2 3m4 3Cm4
Cm2 ln 1+ +3(m2+Cm2) 0 +
− 2 + (cid:16) m2+(cid:17) 0 + − Λ2+m20 − Λ2+m2+
and
1 1 1 −1 (m2 m2)2
+ Π (0)= 5Λ4+11(m2+m2)Λ2 0− + Λ2
−(cid:20)256π2 (cid:18)m2 m2 (cid:19)(cid:21) WW − 0 + − m2+m2 −
0 + 0 +
1 Λ2 Λ2
17(m2+m2)2+3(m2 m2)2 ln 1+ 1+ +
−4 0 + 0− + (cid:18) m2(cid:19)(cid:18) m2 (cid:19) (8)
(cid:2) (cid:3) 0 +
1 15(m2+m2)(m2 m2)+ 1(m20−m2+)3 3(m20+m2+)3 ln 1+ mΛ220
2(cid:20)− 2 0 + 0− + 2 (m2+m2) − (m2 m2) (cid:21) 1+ Λ2
0 + 0− + m2
+
2
where m (m ) is the physical mass of B+(B0).
+ 0
From (3), (4), (7), (8) it follows that ∆ρ cannot be small for Λ2 m2,m2 and there is no compensation
≫ 0 +
for Λ2 m2 m2 or Λ2 m2 m2 . There exists, however, a compensation between the vacuum
∼ 0 ≫ + ∼ + ≫ 0
polarizations if Λ, m , m , are not very different. This is seen in Fig.1 showing ∆ρ as the function of
0 +
m at various Λ’s for m = 0.8m . Similar curves can be drawn for not very different m /m , but at
0 + 0 + 0
higher Λ the sensitivity to m /m increases (Fig.2). For instance, if Λ=2 TeV and m =0.9m 10 % ,
+ 0 + 0
±
then m = 1557 50 GeV at ∆ρ = 0. We have checked numerically the compensation up to Λ = 15 TeV.
0
±
Increasing Λ m (∆ρ = 0) grows and the m interval corresponding to ∆ρ shrinks. For instance, for
0 0 exper.
m = 0.9m and at Λ = (0.8,1.4,2,4) TeV m (∆ρ = 0) = (623,1090,1557,3115)GeV and from ∆ρ
+ 0 0 exper.
the m interval is (619- 630, 1088-1094,1556-1560,3114-3116)GeV, etc.
0
Inconclusion,the vectorcondensatemodeldoes notcontradictto the experimentalρ parameterandeven
∆ρ =0 can be included, but this is possible only if the B particle is heavy.
exper.
This work is supported in part by OTKA I/3, No. 2190.
REFERENCES
1. ProceedingsofLHCWorkshop,Aachen1990,CERN90-10,Eds. G.JarlskogandD.Rein, Vol. 2,p.757.
M.Lindner, Int. Journal of Mod. Phys. A8 (1993) 2167. R.Casalbuoni, S.De Curtis and M.Grazzini, Phys.
Lett. B 317 (1993) 151.
2. G.P´ocsik, E.Lendvai and G.Cynolter, Acta Phys. Polonica B 24 (1993) 1495.
3. D.Kennedy and B.W.Lynn, Nucl. Phys. B 322 (1989) 1. M.E.Peskin and T.Takeuchi, Phys. Rev.
Lett. 65(1990)964. G.AltarelliandR.Barbieri,Phys. Lett. B253(1990)161. M.E.PeskinandT.Takeuchi,
Phys. Rev. D 46 (1992) 381.
4. J.Ellis, G.L.Fogli and E.Lisi, Phys. Lett. B 292 (1992) 427.
FIGURE CAPTIONS
Fig. 1 ∆ρ vs. m at various Λ’s for m =0.8m .
0 + 0
Fig. 2. ∆ρ vs. m at various Λ’s for m =m .
0 + 0
3