- Email: [email protected]
Effective operator contributions to the oblique parameters
30 July 1998
Physics Letters B 432 Ž1998. 383–389
Effective operator contributions to the oblique parameters G. Sanchez-Colon ´ ´ 1, J. Wudka Physics Department, UniÕersity of California RiÕerside, RiÕerside, CA 92521-0413, USA Received 6 January 1998 Editor: M. Dine
Abstract We present a model and process independent study of the contributions from non-Standard Model physics to the oblique parameters S, T and U. We show that within an effective lagrangian parameterization the expressions for the oblique parameters in terms of observables are consistent, while those in terms of the vector-boson vacuum polarization tensors are ambiguous. We obtain the constraints on the scale of new physics derived from current data on S, T and U and note that deviations in U from its Standard Model value would favor a scenario where the underlying physics does not decouple. q 1998 Published by Elsevier Science B.V. All rights reserved.
The oblique parameters S, T and U w1x are known to be sensitive probes of non-Standard Model physics. Because of this they are often used in deriving bounds on the scale Žand other properties. of new physics form the existing and expected experimental bounds w2x. These parameters are often defined in terms of the vector-boson vacuum polarization tensors w1x, but a practical definition requires them to be expressed in terms of direct observables w3x. Thus, whenever the contributions from new interactions to the oblique parameters are calculated, the modifications to all observable quantities involved should be included. When dealing with specific models the calculation of the oblique parameters is a straight-forward exercise. In contrast, when considering the same calculation using a model-independent Žeffective Lagrangian. parameterization of the new-physics effects, some subtleties arise. The reason is that the effective Lagrangian parameterization is not unique in the sense that one can change the effective Lagrangian without affecting the S matrix w4x, yet these modifications do alter the vector boson vacuum polarization and, with this, the corresponding definition of the oblique parameters. Because of this the definition of S, T and U in terms of the the vector-boson vacuum polarization tensors is ambiguous. We will show that this problem can be avoided by defining S, T and U in terms of observables Žwhich unfortunately is seldom the case w5,6x.. We will obtain the complete expressions for the contributions from non-Standard Model physics to the oblique parameters within an effective Lagrangian parameterization. From these expressions unambiguous limits on the scale of new physics can be derived. Finally we will also argue that accurate measurements of the U parameter will provide information on whether the heavy physics decouples. 1 Permanent address: Departamento de Fısica Aplicada. Centro de Investigacion A.P. ´ ´ y de Estudios Avanzados del IPN. Unidad Merida. ´ 73, Cordemex. Merida, Yucatan ´ ´ 97310, Mexico.
0370-2693r98r$ – see frontmatter q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 6 8 2 - 0
G. Sanchez-Colon, ´ ´ J. Wudkar Physics Letters B 432 (1998) 383–389
384
Within the Standard Model the oblique parameters vanish at tree-level, but they are non-zero at one loop w7x; new physics will also, in general, generate non-vanishing contributions w2x. To lowest order we expect S s drad S q D S Žwith similar expressions for T and U ., where drad S denotes the radiative Standard Model contributions and D S the contributions generated by the heavy physics. The quantities drad Ž S,T,U . are well known and have been studied extensively w7x. In this paper we concentrate on DŽ S,T,U . keeping in mind that these quantities denote the deviations from the Standard Model predictions with radiative corrections included; in calculating D S, DT, DU we will ignore all Standard Model loop effects. The oblique parameters can be expressed w3x in terms of the fine-structure constant a Žmeasured at the Z mass., the vector boson masses MZ and MW , the Fermi constant GF , and the width G Ž Z ™ lq ly . and forward-backward asymmetry, AFB Ž Z ™ lq ly ., for the decay of the Z into charged leptons. From AFB we obtain g Vrg A Žthe vector-coupling to axial-coupling ratio of the Z to the charged leptons.; with this result and using the other observables the oblique parameters are obtained from
G Ž Z™l l . s MW2 MZ2
Ž
1 y s02
.
GF MZ3 24'2 p
s1q
ž
1q
1 y s02
g V2 g A2
/
aTy
1 y 2 s02
1
Ž1qa T . ,
4
aS 2Ž
1 y 2 s02
ž
1q
aU q
.
4 s02
gV gA
/
s s02 y
s02 Ž 1 y s02 . 1 y 2 s02
aTq
aS 4 Ž 1 y 2 s02 .
,
,
Ž 1.
where s02 Ž 1 y s02 . s
pa
'2 GF MZ2 .
Ž 2.
To obtain the heavy physics contributions to the oblique parameters it is then necessary to determine the contributions to the observables used in the above definitions. In this paper we will use an effective Lagrangian parameterization of the heavy physics w6,8x which has the advantage of being model and process independent. The detailed form of the effective Lagrangian depends crucially on the low energy spectrum, we will include three families of fermions as well as the usual Standard Model gauge bosons. For the scalars we will consider two possibilities: in the first, which we label the linear case w9x, we assume a single light scalar doublet; in the second, which we call the chiral case w10x, we assume that there are no light physical scalars. In both cases we denote the scale of new physics by L. For the linear case the part of the effective Lagrangian which contributes to the oblique parameters takes the form w9,11x 1 1 Leff s 2 Ý bi Oi q O , Ž 3. L i L3
ž /
where
2
Of W s 12 Ž f †f . WmnI W I mn , OfŽ1. s Ž f †f .
Ž Dm f .
†
Of B s 12 Ž f †f . Bmn B mn ,
D mf ,
OfŽ3.l s i Ž f †t I D mf . l t Igm l ,
ž
/
OfŽ3. s Ž f †D mf .
OW B s Ž f †t If . WmnI B mn ,
Ž Dm f .
†
Of e s i Ž f †D mf . Ž egm e . ,
f ,
OfŽ1.l s i Ž f †D mf . l gm l ,
O lŽ3.l s
ž
1 2
Ž l t Ig m l . ž l t Igm
/ l /.
Ž 4.
The coefficients bi parameterize all heavy physics contributions to the oblique parameters. The choice of operators is, however, not unique w4x; we will discuss this issue below. 2 We use the following conventions w11x: I, J, K denote SUŽ2. indices, the Pauli matrices are denoted by t I; WmI and Bm denote the SUŽ2. X and UŽ1. gauge fields, and WmnI and Bmn the corresponding curvatures; the gauge coupling constants are denoted by g and g respectively. Left-handed quark and lepton doublets are denoted by q and l respectively; right-handed up and down-type quarks correspond to u and d, while the right-handed charged lepton corresponds to e; all fermion fields have implicit family indices. The scalar doublet is denoted by f and the covariant derivative by Dm . The scalar vacuum expectation value is denoted by Õ defined so that Õ , 246 GeV.
G. Sanchez-Colon, ´ ´ J. Wudkar Physics Letters B 432 (1998) 383–389
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When there are no-light scalars Žchiral case. the effective Lagrangian can be obtained from Eq. Ž3. by replacing
f ™ fchir s S 0Õ ;
ž /
S †PSs1
Ž 5.
Ž . where Õ , 246 GeV. In this case the effects of the operators Of W , Of B and Of 1 can be absorbed by an appropriate renormalization of the Standard Model parameters. Therefore, in the chiral case we set bf W s bf B s Ž . bf 1 s 0. A consistent expansion of the chiral effective Lagrangian w6x requires that we include all operators if f fermion fields and d derivatives are such that 3 d q fr2 F 4, hence we must also include the operator w12x † † OW W s 12 Ž fchir t Ifchir .Ž fchir t Jfchir . WmnI W J mn
Ž 6.
Žwhich does not appear in Eq. Ž4. since in the linear case it corresponds to a dimension 8 operator which will generate subdominant contributions to the oblique parameters.. Using Eq. Ž3., Eq. Ž4. Žtogether with Eq. Ž5., Eq. Ž6. in the chiral case. we obtain the heavy physics contributions to the Z couplings and mass, the W mass, a and GF . We first provide the expressions in terms of the SUŽ2. and UŽ1. gauge coupling constants, g and g X respectively, and the vacuum expectation value Õ; and then express these in terms of direct observables. We consider first the case where there is a single light scalar doublet Žthe linear case.. The Z axial and vector couplings to the charged leptons equal, respectively g A s gR y gL ,
gV s gR q gL ,
Ž 7.
where g L s y 12 q x y 12 Ž 1 y x . Ž 1 q 2 x . bf W y Ž x q 2 y 2 . bf B y 2 y Ž 1 y 2 x . b W B q bfŽ1.l q bfŽ3.l e q O Ž e 2 . ; g R s x y 12 bf e q 2 y 2 bf W y 2 x Ž 2 y x . bf B y 4 y Ž 1 y x . b W B e q O Ž e 2 . ;
Ž 8.
and
es
Õ2
L2
,
xs
gX 2 g 2 q gX 2
,
ys
gX g g 2 q gX 2
.
Ž 9.
For the remaining observables we find gÕ MW s 1 q Ž bf W q 12 bfŽ1. . e q O Ž e 2 . , 2 gÕ MZ s 1 q xbf B q Ž 1 y x . bf W q 2 yb W B q bfŽ1. q 12 bfŽ3. e q O Ž e 2 . , 2 '1 y x
½
GF s
as
1
'2 Õ 2
g2x 4p
5
Ž3. Ž1. 2 1 q Ž 2 b Ž3. l l q 4 bf l y bf y 4 bf W . e q O Ž e . ,
1 q 2
xbf W q Ž 1 y x . bf B y 2 yb W B e 4 q O Ž e 2 . ;
Ž 10 .
which can be used to express x, g, e , etc. in terms of observables. These expressions do not contain the Standard Model radiative corrections since, as discussed above, we expect the corresponding contributions to
3
This is a generalization of the derivative expansion when fermions are present.
G. Sanchez-Colon, ´ ´ J. Wudkar Physics Letters B 432 (1998) 383–389
386
the oblique parameters to be additive and we are interested only in the contributions generated by the heavy physics. Substituting Eq. Ž7. –Eq. Ž10. into Eq. Ž1. we get
D lin T s y D lin S s D lin U s
4p g2 x
8p
ybf e q 2 x Ž bfŽ1.l q bfŽ3.l . q 4 yb W B e q O Ž e 2 . ,
g2 x 16p g2
Ž 2 bŽ3.l l q 2 bfŽ3.l y 2 bfŽ1.l q bfŽ3. q 2 bf e y 4 bf W . e q O Ž e 2 . ,
Ž 2 bŽ3.l l q 2 bfŽ3.l y 2 bfŽ1.l q bf e y 4 bf W . e q O Ž e 2 . .
Ž 11 .
In the chiral case, using Eq. Ž5. and Eq. Ž6. we obtain
Dchir T s y Dchir S s Dchir U s
4p
Ž 2 bŽ3.l l q 2 bfŽ3.l y 2 bfŽ1.l q bfŽ3. q 2 bf e . e q O Ž e 2 . ,
g2 x
8p
ybf e q 2 x Ž bfŽ1.l q bfŽ3.l . q 4 yb W B e q O Ž e 2 . ,
g2 x 16p g2
Ž 2 bŽ3.l l q 2 bfŽ3.l y 2 bfŽ1.l q bf e y 2 bW W . e q O Ž e 2 . ,
Ž 12 .
where b W W is the coefficient of OW W in Eq. Ž6. Žwhich has dimension 4.. The above expressions can be re-written in terms of observables using the tree-level relations g 2 s 4'2 GF MW2 ,
2
x s 1 y Ž MW rMZ . ,
es
1
'8 GF L2 .
Ž 13 .
These expressions were obtained using the effective Lagrangian Eq. Ž3., Eq. Ž4., together with Eq. Ž5., Eq. Ž6. in the chiral case. But it is well known w4x that there is no unique choice of operators in an effective Lagrangian parameterization. Given two operators O1 and O2 such that O1 y O2 vanishes when the classical equations of motion are imposed, then the term b 1 O1 q b 2 O2 in the effective Lagrangian generates modifications to the S matrix which depend only on b 1 q b 2 w4x, but not on b 1 and b 2 independently. Our expressions for D S, DT and DU satisfy this property. As an example consider the operator I
I
ODW s Ž DmWnr . Ž D m W nr . ,
Ž 14 .
which, up to terms which vanish when the classical equations of motion are imposed, satisfies ODW s 2 g OW q
g2 2
2
Ž3. 6 OfŽ1. q 2 m2 Ž f †f . y 6 l Of q 4 OfŽ3.l q 4 OfŽ3.q q 2 O lŽ3.l q 2 O lŽ3. q q 2 Oq q ,
Ž 15 .
where m denotes the scalar mass, l the scalar self-coupling and where the operators not defined in Eq. Ž4. are 3
Of s 13 Ž f †f . , I m O lŽ3. q s Ž lt g l
OfŽ3.q s i Ž f †t I D mf . Ž qt Igm q . ,
. Ž qt Igm q . ,
1 I m I OqŽ3. q s 2 Ž qt g q . Ž qt gm q . ,
OW s ´ I JK WmIn Wn J lWlK m .
Ž 16 .
G. Sanchez-Colon, ´ ´ J. Wudkar Physics Letters B 432 (1998) 383–389
387
It then follows that the replacement Leff ™ Leff q
bDW
L2
Ž 17 .
ODW
is equivalent to bi ™ bi q d bi where 1 3
Ž3. Ž3. d bfŽ1. s y 13 d bf s 12 d bfŽ3.l s 12 d bfŽ3.q s d b Ž3. l l s d b l q s d bq q s
1
le
dl s bDW g 2 ,
Ž 18 .
which, in fact, leave DŽ S,T,U . invariant. This result can also be obtained without using the equations of motion on ODW . Adding a term bDW ODW rL2 to the effective Lagrangian generates a quadratic term in the vector bosons, bDW WmIE 2 W I m . When the quadratic part of the vector-boson Lagrangian is re-diagonalized the W and Z masses and the vacuum expectation value Õ are modified, d MW2 rMW2 s d MZ2rM Z2 s d ÕrÕ s g 2 bDW e . The Fermi constant is unaffected, d GF s 0, and the coupling of the Z to the left-handed fermionic current JL becomes yŽ1 q g 2 bDW e . g 2 q g X 2 JL P Z. It is a tedious exercise Žfor which we used Ref. w11x after correcting a few typographical errors. to show that these modifications correspond to Eq. Ž18.. This illustrates the fact that Eq. Ž11. are consistent definitions of the heavy physics to the oblique parameters. In contrast, the naive definition of the oblique parameters in terms of the vacuum polarization tensors, are not invariant under the replacement Eq. Ž17.. Indeed, using an SUŽ2. = UŽ1. basis,
(
S vac .pol .s y
Uvac .pol .s
8p MZ2
16p MW2
P 3Y Ž MZ2 . y P 3Y Ž 0 . , Tvac .pol .s
P 11 Ž MW2 . y P 11 Ž 0 . y
16p MZ2
16p 2
sin Ž 2 u W . MZ2
P 11 Ž 0 . y P 33 Ž 0 . ,
P 33 Ž MZ2 . y P 33 Ž 0 . ,
Ž 19 .
from which, using Eq. Ž3., we obtain
D lin Tvac .pol .s y
4p 2
g x
afŽ3.e q O Ž e 2 . ,
D lin S vac .pol .s
32p y g2 x
bW B e q O Ž e 2 . ,
D lin Uvac .pol .s O Ž e 2 . ,
Ž 20 . but in this case Eq. Ž17. does not leave Uvac.pol. invariant,
D lin Uvac .pol .™ 16p g X 2 bDW e q O Ž e 2 . ,
Ž 21 .
which illustrates the importance of using the definitions Eq. Ž1. for the oblique parameters. It must be noted that the operator ODW generates a p 4 contribution to the vacuum polarizations P Ž p ., and within the linear approximation w1,3x in p 2 , this operator will not affect the oblique parameters. This does not mean that the effective lagrangian contributions to S vac.pol., Tvac.pol. and Uvac.pol. within the linear approximation are unambiguous. Consider, for example, the operator Ž f †D mf . E n Bnm which contributes to D lin S vac.pol.; using the equations of motion this operator is equivalent to Ž ig Xr2. Ž 2 OfŽ3. q OfŽ1. . – plus a string of operators Ž . involving fermions which do not contribute to Eq. Ž19. – and Of 3 contributes to D lin Tvac.pol. only. In contrast, the definitions Eq. Ž1. present no such ambiguity and should be used whenever an effective Lagrangian computation is performed.
G. Sanchez-Colon, ´ ´ J. Wudkar Physics Letters B 432 (1998) 383–389
388
Using Eq. Ž11. or Eq. Ž12. and currently available data we can derive limits on the scale of new physics L. Ž . The operators Of l 1,3 and Of e modify the Z coupling to the fermions and the corresponding coefficients can be bounded using data from LEP1 w13x,
e bfŽ1,2. l - 0.0016, the operator O l l
Ž 3.
e bf e - 0.0014;
Ž 22 .
contributes to eq ey™ mq my and it coefficient can be correspondingly bounded 4 ,
y0.105 - e b Ž3. l l - 0.056.
Ž 23 .
Finally the limits on the oblique parameters are w14x y0.0414 -
g2 x 8p
S - 0.0060,
y0.0875 -
g2 x 4p
T - 0.0951,
y0.0072 -
g2 16p
U - 0.0020.
Ž 24 .
Ž . Ž . Note however that the operators O l l 3 , Of l 3 and Of W also contribute to GF and this can be used to impose better bounds on the corresponding coefficients Žagain assuming no cancellations.. Using GF , a and MZ as Ž3. y4 input parameters the uncertainity in the predictions of MW requires e b Ž3. l l , e bf l , e bf W Q 5 = 10 . In the linear case we then have, to a good approximation,
D lin T , y D lin S , D lin U ,
4p g2 x
8p g2 x 16p g2
Ž y2 bfŽ1.l q bfŽ3. q 2 bf e . e q O Ž e 2 . ,
Ž ybf e q 2 xbfŽ1.l q 4 ybW B . e q O Ž e 2 . , Ž y2 bfŽ1.l q bf e . e q O Ž e 2 . .
Ž 25 .
In the chiral case
Dchir T s y Dchir S s Dchir U s
4p g2 x
8p g2 x 16p g2
Ž y2 bfŽ1.l q bfŽ3. q 2 bf e . e q O Ž e 2 . ,
Ž ybf e q 2 xbfŽ1.l q 4 ybW B . e q O Ž e 2 . , Ž y2 bfŽ1.l q bf e y 2 bW W . e q O Ž e 2 . .
Ž 26 .
Using these expressions and the above experimental constraints we find the following bounds
e bfŽ3. Q 0.1,
e b W B Q 0.02,
e b W W Q 0.006
Ž 27 .
Žwhere the last is relevant only for the chiral case.. In the linear case the natural size w15x for the coefficients are bfŽ3. Q 1 and b W B Q gg XrŽ4p . 2 . The limit on Ž 3. Ž . bf implies L R 550 GeV while, from b W B , L R 50 GeV. This disparity is due to the fact that Of 3 can be generated at tree level by the heavy physics, while OW W is necessarily loop generated w15x. The 550 GeV limit refers to the mass of a heavy scalar or vector boson whose interactions violate the custodial symmetry w16x. For the chiral case the natural sizes w17x are bfŽ3. Q 1, b W B Q g g X , and b W W Q g 2 ; moreover we also have L ; 4p Õ ; 3 TeV so that e Q 1rŽ4p . 2 . The above limits are not sufficiently precise to provide useful information in this case; for example, the limit on b W W implies L R 1.5 TeV. 4
There are many other operators that contribute to this reaction, we assume there are no significant cancellations
G. Sanchez-Colon, ´ ´ J. Wudkar Physics Letters B 432 (1998) 383–389
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Ž . Finally we note a peculiarity of the parameter U: the heavy physics contributions generated by Of l 1 and Of e were measured at LEP1 and are known to be small; this means that in the linear case current data implies DU ; 0 ŽU ; e 2 Q 0.01 for L ) 550 GeV.. In contrast there are no severe bounds on the contributions generated by OW W ; in the chiral case we therefore have < Dchir U < ; 32p < b W W < e Q 2rp . Should a future measurement produce a deviation of order 0.1 in the measurement of U, this observation would not only indicate the presence of new physics, but would strongly disfavor the existence of light Higgs-like scalars. Note, however, that a bound DU Q 0.1 does not imply the presence of light scalars since this could also occur within the chiral case for a sufficiently large L.
Acknowledgements This work was partially supported by CONACyT ŽMexico ´ . and by the US Department of Energy under contract DE-FG03-94ER40837.
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Physics Letters B 432 Ž1998. 383–389
Effective operator contributions to the oblique parameters G. Sanchez-Colon ´ ´ 1, J. Wudka Physics Department, UniÕersity of California RiÕerside, RiÕerside, CA 92521-0413, USA Received 6 January 1998 Editor: M. Dine
Abstract We present a model and process independent study of the contributions from non-Standard Model physics to the oblique parameters S, T and U. We show that within an effective lagrangian parameterization the expressions for the oblique parameters in terms of observables are consistent, while those in terms of the vector-boson vacuum polarization tensors are ambiguous. We obtain the constraints on the scale of new physics derived from current data on S, T and U and note that deviations in U from its Standard Model value would favor a scenario where the underlying physics does not decouple. q 1998 Published by Elsevier Science B.V. All rights reserved.
The oblique parameters S, T and U w1x are known to be sensitive probes of non-Standard Model physics. Because of this they are often used in deriving bounds on the scale Žand other properties. of new physics form the existing and expected experimental bounds w2x. These parameters are often defined in terms of the vector-boson vacuum polarization tensors w1x, but a practical definition requires them to be expressed in terms of direct observables w3x. Thus, whenever the contributions from new interactions to the oblique parameters are calculated, the modifications to all observable quantities involved should be included. When dealing with specific models the calculation of the oblique parameters is a straight-forward exercise. In contrast, when considering the same calculation using a model-independent Žeffective Lagrangian. parameterization of the new-physics effects, some subtleties arise. The reason is that the effective Lagrangian parameterization is not unique in the sense that one can change the effective Lagrangian without affecting the S matrix w4x, yet these modifications do alter the vector boson vacuum polarization and, with this, the corresponding definition of the oblique parameters. Because of this the definition of S, T and U in terms of the the vector-boson vacuum polarization tensors is ambiguous. We will show that this problem can be avoided by defining S, T and U in terms of observables Žwhich unfortunately is seldom the case w5,6x.. We will obtain the complete expressions for the contributions from non-Standard Model physics to the oblique parameters within an effective Lagrangian parameterization. From these expressions unambiguous limits on the scale of new physics can be derived. Finally we will also argue that accurate measurements of the U parameter will provide information on whether the heavy physics decouples. 1 Permanent address: Departamento de Fısica Aplicada. Centro de Investigacion A.P. ´ ´ y de Estudios Avanzados del IPN. Unidad Merida. ´ 73, Cordemex. Merida, Yucatan ´ ´ 97310, Mexico.
0370-2693r98r$ – see frontmatter q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 6 8 2 - 0
G. Sanchez-Colon, ´ ´ J. Wudkar Physics Letters B 432 (1998) 383–389
384
Within the Standard Model the oblique parameters vanish at tree-level, but they are non-zero at one loop w7x; new physics will also, in general, generate non-vanishing contributions w2x. To lowest order we expect S s drad S q D S Žwith similar expressions for T and U ., where drad S denotes the radiative Standard Model contributions and D S the contributions generated by the heavy physics. The quantities drad Ž S,T,U . are well known and have been studied extensively w7x. In this paper we concentrate on DŽ S,T,U . keeping in mind that these quantities denote the deviations from the Standard Model predictions with radiative corrections included; in calculating D S, DT, DU we will ignore all Standard Model loop effects. The oblique parameters can be expressed w3x in terms of the fine-structure constant a Žmeasured at the Z mass., the vector boson masses MZ and MW , the Fermi constant GF , and the width G Ž Z ™ lq ly . and forward-backward asymmetry, AFB Ž Z ™ lq ly ., for the decay of the Z into charged leptons. From AFB we obtain g Vrg A Žthe vector-coupling to axial-coupling ratio of the Z to the charged leptons.; with this result and using the other observables the oblique parameters are obtained from
G Ž Z™l l . s MW2 MZ2
Ž
1 y s02
.
GF MZ3 24'2 p
s1q
ž
1q
1 y s02
g V2 g A2
/
aTy
1 y 2 s02
1
Ž1qa T . ,
4
aS 2Ž
1 y 2 s02
ž
1q
aU q
.
4 s02
gV gA
/
s s02 y
s02 Ž 1 y s02 . 1 y 2 s02
aTq
aS 4 Ž 1 y 2 s02 .
,
,
Ž 1.
where s02 Ž 1 y s02 . s
pa
'2 GF MZ2 .
Ž 2.
To obtain the heavy physics contributions to the oblique parameters it is then necessary to determine the contributions to the observables used in the above definitions. In this paper we will use an effective Lagrangian parameterization of the heavy physics w6,8x which has the advantage of being model and process independent. The detailed form of the effective Lagrangian depends crucially on the low energy spectrum, we will include three families of fermions as well as the usual Standard Model gauge bosons. For the scalars we will consider two possibilities: in the first, which we label the linear case w9x, we assume a single light scalar doublet; in the second, which we call the chiral case w10x, we assume that there are no light physical scalars. In both cases we denote the scale of new physics by L. For the linear case the part of the effective Lagrangian which contributes to the oblique parameters takes the form w9,11x 1 1 Leff s 2 Ý bi Oi q O , Ž 3. L i L3
ž /
where
2
Of W s 12 Ž f †f . WmnI W I mn , OfŽ1. s Ž f †f .
Ž Dm f .
†
Of B s 12 Ž f †f . Bmn B mn ,
D mf ,
OfŽ3.l s i Ž f †t I D mf . l t Igm l ,
ž
/
OfŽ3. s Ž f †D mf .
OW B s Ž f †t If . WmnI B mn ,
Ž Dm f .
†
Of e s i Ž f †D mf . Ž egm e . ,
f ,
OfŽ1.l s i Ž f †D mf . l gm l ,
O lŽ3.l s
ž
1 2
Ž l t Ig m l . ž l t Igm
/ l /.
Ž 4.
The coefficients bi parameterize all heavy physics contributions to the oblique parameters. The choice of operators is, however, not unique w4x; we will discuss this issue below. 2 We use the following conventions w11x: I, J, K denote SUŽ2. indices, the Pauli matrices are denoted by t I; WmI and Bm denote the SUŽ2. X and UŽ1. gauge fields, and WmnI and Bmn the corresponding curvatures; the gauge coupling constants are denoted by g and g respectively. Left-handed quark and lepton doublets are denoted by q and l respectively; right-handed up and down-type quarks correspond to u and d, while the right-handed charged lepton corresponds to e; all fermion fields have implicit family indices. The scalar doublet is denoted by f and the covariant derivative by Dm . The scalar vacuum expectation value is denoted by Õ defined so that Õ , 246 GeV.
G. Sanchez-Colon, ´ ´ J. Wudkar Physics Letters B 432 (1998) 383–389
385
When there are no-light scalars Žchiral case. the effective Lagrangian can be obtained from Eq. Ž3. by replacing
f ™ fchir s S 0Õ ;
ž /
S †PSs1
Ž 5.
Ž . where Õ , 246 GeV. In this case the effects of the operators Of W , Of B and Of 1 can be absorbed by an appropriate renormalization of the Standard Model parameters. Therefore, in the chiral case we set bf W s bf B s Ž . bf 1 s 0. A consistent expansion of the chiral effective Lagrangian w6x requires that we include all operators if f fermion fields and d derivatives are such that 3 d q fr2 F 4, hence we must also include the operator w12x † † OW W s 12 Ž fchir t Ifchir .Ž fchir t Jfchir . WmnI W J mn
Ž 6.
Žwhich does not appear in Eq. Ž4. since in the linear case it corresponds to a dimension 8 operator which will generate subdominant contributions to the oblique parameters.. Using Eq. Ž3., Eq. Ž4. Žtogether with Eq. Ž5., Eq. Ž6. in the chiral case. we obtain the heavy physics contributions to the Z couplings and mass, the W mass, a and GF . We first provide the expressions in terms of the SUŽ2. and UŽ1. gauge coupling constants, g and g X respectively, and the vacuum expectation value Õ; and then express these in terms of direct observables. We consider first the case where there is a single light scalar doublet Žthe linear case.. The Z axial and vector couplings to the charged leptons equal, respectively g A s gR y gL ,
gV s gR q gL ,
Ž 7.
where g L s y 12 q x y 12 Ž 1 y x . Ž 1 q 2 x . bf W y Ž x q 2 y 2 . bf B y 2 y Ž 1 y 2 x . b W B q bfŽ1.l q bfŽ3.l e q O Ž e 2 . ; g R s x y 12 bf e q 2 y 2 bf W y 2 x Ž 2 y x . bf B y 4 y Ž 1 y x . b W B e q O Ž e 2 . ;
Ž 8.
and
es
Õ2
L2
,
xs
gX 2 g 2 q gX 2
,
ys
gX g g 2 q gX 2
.
Ž 9.
For the remaining observables we find gÕ MW s 1 q Ž bf W q 12 bfŽ1. . e q O Ž e 2 . , 2 gÕ MZ s 1 q xbf B q Ž 1 y x . bf W q 2 yb W B q bfŽ1. q 12 bfŽ3. e q O Ž e 2 . , 2 '1 y x
½
GF s
as
1
'2 Õ 2
g2x 4p
5
Ž3. Ž1. 2 1 q Ž 2 b Ž3. l l q 4 bf l y bf y 4 bf W . e q O Ž e . ,
1 q 2
xbf W q Ž 1 y x . bf B y 2 yb W B e 4 q O Ž e 2 . ;
Ž 10 .
which can be used to express x, g, e , etc. in terms of observables. These expressions do not contain the Standard Model radiative corrections since, as discussed above, we expect the corresponding contributions to
3
This is a generalization of the derivative expansion when fermions are present.
G. Sanchez-Colon, ´ ´ J. Wudkar Physics Letters B 432 (1998) 383–389
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the oblique parameters to be additive and we are interested only in the contributions generated by the heavy physics. Substituting Eq. Ž7. –Eq. Ž10. into Eq. Ž1. we get
D lin T s y D lin S s D lin U s
4p g2 x
8p
ybf e q 2 x Ž bfŽ1.l q bfŽ3.l . q 4 yb W B e q O Ž e 2 . ,
g2 x 16p g2
Ž 2 bŽ3.l l q 2 bfŽ3.l y 2 bfŽ1.l q bfŽ3. q 2 bf e y 4 bf W . e q O Ž e 2 . ,
Ž 2 bŽ3.l l q 2 bfŽ3.l y 2 bfŽ1.l q bf e y 4 bf W . e q O Ž e 2 . .
Ž 11 .
In the chiral case, using Eq. Ž5. and Eq. Ž6. we obtain
Dchir T s y Dchir S s Dchir U s
4p
Ž 2 bŽ3.l l q 2 bfŽ3.l y 2 bfŽ1.l q bfŽ3. q 2 bf e . e q O Ž e 2 . ,
g2 x
8p
ybf e q 2 x Ž bfŽ1.l q bfŽ3.l . q 4 yb W B e q O Ž e 2 . ,
g2 x 16p g2
Ž 2 bŽ3.l l q 2 bfŽ3.l y 2 bfŽ1.l q bf e y 2 bW W . e q O Ž e 2 . ,
Ž 12 .
where b W W is the coefficient of OW W in Eq. Ž6. Žwhich has dimension 4.. The above expressions can be re-written in terms of observables using the tree-level relations g 2 s 4'2 GF MW2 ,
2
x s 1 y Ž MW rMZ . ,
es
1
'8 GF L2 .
Ž 13 .
These expressions were obtained using the effective Lagrangian Eq. Ž3., Eq. Ž4., together with Eq. Ž5., Eq. Ž6. in the chiral case. But it is well known w4x that there is no unique choice of operators in an effective Lagrangian parameterization. Given two operators O1 and O2 such that O1 y O2 vanishes when the classical equations of motion are imposed, then the term b 1 O1 q b 2 O2 in the effective Lagrangian generates modifications to the S matrix which depend only on b 1 q b 2 w4x, but not on b 1 and b 2 independently. Our expressions for D S, DT and DU satisfy this property. As an example consider the operator I
I
ODW s Ž DmWnr . Ž D m W nr . ,
Ž 14 .
which, up to terms which vanish when the classical equations of motion are imposed, satisfies ODW s 2 g OW q
g2 2
2
Ž3. 6 OfŽ1. q 2 m2 Ž f †f . y 6 l Of q 4 OfŽ3.l q 4 OfŽ3.q q 2 O lŽ3.l q 2 O lŽ3. q q 2 Oq q ,
Ž 15 .
where m denotes the scalar mass, l the scalar self-coupling and where the operators not defined in Eq. Ž4. are 3
Of s 13 Ž f †f . , I m O lŽ3. q s Ž lt g l
OfŽ3.q s i Ž f †t I D mf . Ž qt Igm q . ,
. Ž qt Igm q . ,
1 I m I OqŽ3. q s 2 Ž qt g q . Ž qt gm q . ,
OW s ´ I JK WmIn Wn J lWlK m .
Ž 16 .
G. Sanchez-Colon, ´ ´ J. Wudkar Physics Letters B 432 (1998) 383–389
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It then follows that the replacement Leff ™ Leff q
bDW
L2
Ž 17 .
ODW
is equivalent to bi ™ bi q d bi where 1 3
Ž3. Ž3. d bfŽ1. s y 13 d bf s 12 d bfŽ3.l s 12 d bfŽ3.q s d b Ž3. l l s d b l q s d bq q s
1
le
dl s bDW g 2 ,
Ž 18 .
which, in fact, leave DŽ S,T,U . invariant. This result can also be obtained without using the equations of motion on ODW . Adding a term bDW ODW rL2 to the effective Lagrangian generates a quadratic term in the vector bosons, bDW WmIE 2 W I m . When the quadratic part of the vector-boson Lagrangian is re-diagonalized the W and Z masses and the vacuum expectation value Õ are modified, d MW2 rMW2 s d MZ2rM Z2 s d ÕrÕ s g 2 bDW e . The Fermi constant is unaffected, d GF s 0, and the coupling of the Z to the left-handed fermionic current JL becomes yŽ1 q g 2 bDW e . g 2 q g X 2 JL P Z. It is a tedious exercise Žfor which we used Ref. w11x after correcting a few typographical errors. to show that these modifications correspond to Eq. Ž18.. This illustrates the fact that Eq. Ž11. are consistent definitions of the heavy physics to the oblique parameters. In contrast, the naive definition of the oblique parameters in terms of the vacuum polarization tensors, are not invariant under the replacement Eq. Ž17.. Indeed, using an SUŽ2. = UŽ1. basis,
(
S vac .pol .s y
Uvac .pol .s
8p MZ2
16p MW2
P 3Y Ž MZ2 . y P 3Y Ž 0 . , Tvac .pol .s
P 11 Ž MW2 . y P 11 Ž 0 . y
16p MZ2
16p 2
sin Ž 2 u W . MZ2
P 11 Ž 0 . y P 33 Ž 0 . ,
P 33 Ž MZ2 . y P 33 Ž 0 . ,
Ž 19 .
from which, using Eq. Ž3., we obtain
D lin Tvac .pol .s y
4p 2
g x
afŽ3.e q O Ž e 2 . ,
D lin S vac .pol .s
32p y g2 x
bW B e q O Ž e 2 . ,
D lin Uvac .pol .s O Ž e 2 . ,
Ž 20 . but in this case Eq. Ž17. does not leave Uvac.pol. invariant,
D lin Uvac .pol .™ 16p g X 2 bDW e q O Ž e 2 . ,
Ž 21 .
which illustrates the importance of using the definitions Eq. Ž1. for the oblique parameters. It must be noted that the operator ODW generates a p 4 contribution to the vacuum polarizations P Ž p ., and within the linear approximation w1,3x in p 2 , this operator will not affect the oblique parameters. This does not mean that the effective lagrangian contributions to S vac.pol., Tvac.pol. and Uvac.pol. within the linear approximation are unambiguous. Consider, for example, the operator Ž f †D mf . E n Bnm which contributes to D lin S vac.pol.; using the equations of motion this operator is equivalent to Ž ig Xr2. Ž 2 OfŽ3. q OfŽ1. . – plus a string of operators Ž . involving fermions which do not contribute to Eq. Ž19. – and Of 3 contributes to D lin Tvac.pol. only. In contrast, the definitions Eq. Ž1. present no such ambiguity and should be used whenever an effective Lagrangian computation is performed.
G. Sanchez-Colon, ´ ´ J. Wudkar Physics Letters B 432 (1998) 383–389
388
Using Eq. Ž11. or Eq. Ž12. and currently available data we can derive limits on the scale of new physics L. Ž . The operators Of l 1,3 and Of e modify the Z coupling to the fermions and the corresponding coefficients can be bounded using data from LEP1 w13x,
e bfŽ1,2. l - 0.0016, the operator O l l
Ž 3.
e bf e - 0.0014;
Ž 22 .
contributes to eq ey™ mq my and it coefficient can be correspondingly bounded 4 ,
y0.105 - e b Ž3. l l - 0.056.
Ž 23 .
Finally the limits on the oblique parameters are w14x y0.0414 -
g2 x 8p
S - 0.0060,
y0.0875 -
g2 x 4p
T - 0.0951,
y0.0072 -
g2 16p
U - 0.0020.
Ž 24 .
Ž . Ž . Note however that the operators O l l 3 , Of l 3 and Of W also contribute to GF and this can be used to impose better bounds on the corresponding coefficients Žagain assuming no cancellations.. Using GF , a and MZ as Ž3. y4 input parameters the uncertainity in the predictions of MW requires e b Ž3. l l , e bf l , e bf W Q 5 = 10 . In the linear case we then have, to a good approximation,
D lin T , y D lin S , D lin U ,
4p g2 x
8p g2 x 16p g2
Ž y2 bfŽ1.l q bfŽ3. q 2 bf e . e q O Ž e 2 . ,
Ž ybf e q 2 xbfŽ1.l q 4 ybW B . e q O Ž e 2 . , Ž y2 bfŽ1.l q bf e . e q O Ž e 2 . .
Ž 25 .
In the chiral case
Dchir T s y Dchir S s Dchir U s
4p g2 x
8p g2 x 16p g2
Ž y2 bfŽ1.l q bfŽ3. q 2 bf e . e q O Ž e 2 . ,
Ž ybf e q 2 xbfŽ1.l q 4 ybW B . e q O Ž e 2 . , Ž y2 bfŽ1.l q bf e y 2 bW W . e q O Ž e 2 . .
Ž 26 .
Using these expressions and the above experimental constraints we find the following bounds
e bfŽ3. Q 0.1,
e b W B Q 0.02,
e b W W Q 0.006
Ž 27 .
Žwhere the last is relevant only for the chiral case.. In the linear case the natural size w15x for the coefficients are bfŽ3. Q 1 and b W B Q gg XrŽ4p . 2 . The limit on Ž 3. Ž . bf implies L R 550 GeV while, from b W B , L R 50 GeV. This disparity is due to the fact that Of 3 can be generated at tree level by the heavy physics, while OW W is necessarily loop generated w15x. The 550 GeV limit refers to the mass of a heavy scalar or vector boson whose interactions violate the custodial symmetry w16x. For the chiral case the natural sizes w17x are bfŽ3. Q 1, b W B Q g g X , and b W W Q g 2 ; moreover we also have L ; 4p Õ ; 3 TeV so that e Q 1rŽ4p . 2 . The above limits are not sufficiently precise to provide useful information in this case; for example, the limit on b W W implies L R 1.5 TeV. 4
There are many other operators that contribute to this reaction, we assume there are no significant cancellations
G. Sanchez-Colon, ´ ´ J. Wudkar Physics Letters B 432 (1998) 383–389
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Ž . Finally we note a peculiarity of the parameter U: the heavy physics contributions generated by Of l 1 and Of e were measured at LEP1 and are known to be small; this means that in the linear case current data implies DU ; 0 ŽU ; e 2 Q 0.01 for L ) 550 GeV.. In contrast there are no severe bounds on the contributions generated by OW W ; in the chiral case we therefore have < Dchir U < ; 32p < b W W < e Q 2rp . Should a future measurement produce a deviation of order 0.1 in the measurement of U, this observation would not only indicate the presence of new physics, but would strongly disfavor the existence of light Higgs-like scalars. Note, however, that a bound DU Q 0.1 does not imply the presence of light scalars since this could also occur within the chiral case for a sufficiently large L.
Acknowledgements This work was partially supported by CONACyT ŽMexico ´ . and by the US Department of Energy under contract DE-FG03-94ER40837.
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