Contributions of effective theories with linearly realized and strongly interacting Higgs sectors to the process $e^+e^- \to W^+W^-$

14 Pages·0.15 MB·English

by  Rob Szalapski

KEK-TH-466
KEK Preprint 95-194
CONTRIBUTIONS OF EFFECTIVE-THEORIES WITH
LINEARLY REALIZED AND STRONGLY INTERACTING
HIGGS SECTORS TO THE PROCESS e+e W+W
− −∗
→
6
9 R. Szalapski
9
1 Theory Group, KEK, Tsukuba, Ibaraki 305, Japan

n
a e-mail: [email protected]
J

2
1

1 Abstract
v
1
5
2 A future linear collider will address several important issues. One topic of
1
0 particular interest is the mechanism of electroweak-symmetry breaking. First
6
9
it is necessary to establish the existence or non-existence of a light physical
/
h
p Higgs boson, and if such a particle exists it is necessary to compare its prop-
-
p
erties to the predictions of the Standard Model (SM). Because the massive
e
h
: vector bosons obtain their masses via the symmetry-breaking mechanism, a
v
i
X study of the Higgs sector is related to the study of the properties of massive
r
a gauge bosons. This talk is primarily concerned with the latter, particularly
through the study of the process e+e− W+W−.
→
In order to compare experiment with the predictions of the SM it is nec-
essary to parameterize deviations from the SM in a convenient manner. The
parameterization shouldbesufficientlygeneral,yetmanageable. Anespecially
powerful tool is the effective theory. In general one assumes that the effects
of new physics characterized by a high-energy scale may be observed at low
∗Talk presented at the 1995 Workshop on Physics and Experiments at Linear Colliders, Morioka-
Appi, Iwate, Japan, 8-12 September 1995.
1
energies only through quantum corrections. These quantum corrections may
be described by effective operators which are constructed from the fields of
thelow-energy theory; thecoefficients of theseoperators may bemeasuredvia
experiment, butatheoretical determination isonlypossibleoncethecomplete
theory is known.
Because the existence or non-existence of a light physical Higgs boson has
not yet been established the inclusion or exclusion of the Higgs field in the
construction of the effective Lagrangian is open to debate. For this reason
we discuss both the linearly realized effective Lagrangian, which includes a
light physical Higgs boson, and the chiral Lagrangian, where the Higgs boson
has been removed by a formal integration. In either realization seven CP
conserving operators contribute to the process e+e− W+W−. However,
→
a subset of these is already constrained by the low-energy and Z-pole data,
hence the available parameter-space is manageable.
I. THE LINEAR REPRESENTATION AND ENERGY-DIMENSION-SIX
OPERATORS
If the scale of new physics, Λ, is large compared to the vacuum expectation value (vev)
of the the Higgs field, v = 246.22GeV, then the effective Lagrangian may be expressed as the
SM Lagrangian plus terms with energy dimension greater than four suppressed by inverse
powers of Λ, i.e.
(n) (n)
f
= + i Oi . (1.1)
Leff LSM Λn−4
n≥5 i
XX
The energy dimension of each operator is denoted by n, and the index i sums over all
(n)
operators of the given energy dimension. The coefficients f are free parameters.
i
In this section we assume that the low-energy theory, i.e.the SM, contains a light physical
scalar Higgs particle which is the remnant of a complex Higgs-doublet field. We assume that
the couplings of the new physics to fermions are suppressed, and we only consider opera-
tors which conserve CP. Upon restricting the analysis to operators not exceeding energy-
dimension six we find that twelve operators form a basis set; all are dimension-six and
2
separately conserve C and P [1]. They are summarized in Table I. For convenience when
defining the normalizations of the individual operators we use the ‘hatted’ field strength
tensors defined according to
D ,D = Wˆ +Bˆ = igTaWa +ig′YB . (1.2)
µ ν µν µν µν µν
h i
Combining the twelve operators of Table I with Eqn. (1.1) completes the construction of the
effective Lagrangian in the linear representation.
Also in Table I we indicate those vertices to which each operator contributes with an
‘X’ in the appropriate box. First, observe that four of the operators, , , and
DW DB BW
O O O
, contribute to gauge-boson two-point-functions at the tree level. For this reason their
Φ,1
O
respective coefficients, f , f , f and f , are strongly constrained by LEP/SLC and
DW DB BW Φ,1
low-energy data [2], and these constraints will be improved by data at higher energy lepton
colliders. (This will be discussed in greater detail in Sec. III.) These four operators will
contribute to the process e+e− W+W− through corrections to the charge form-factors,
→
e¯2(q2), s¯2(q2), g¯2(q2) and g¯2(q2), and through the W-boson wave-function-renormalization
Z
1/2
factor, Z .
W
The operators and also make a direct contribution to WWγ and WWZ
DW BW
O O
vertices. Three additional operators contribute as well. They are , and ; their
WWW W B
O O O
respective coefficients are f , f and f . The non-zero direct contributions may be
WWW W B
summarized by
q2 +2mˆ2
∆gγ (q2) = 2gˆ2 Wf , (1.3a)
1,direct Λ2 DW
q2 +2mˆ2 1mˆ2
∆gZ (q2) = 2gˆ2 Wf + Zf , (1.3b)
1,direct Λ2 DW 2 Λ2 W
q2 +2mˆ2 1mˆ2
∆κ (q2) = 2gˆ2 Wf + W f 2f +f , (1.3c)
γ,direct Λ2 DW 2 Λ2 W − BW B
(cid:16) (cid:17)
q2 +2mˆ2 1mˆ2
∆κ (q2) = 2gˆ2 Wf + Z cˆ2f +2sˆ2f sˆ2f , (1.3d)
Z,direct Λ2 DW 2 Λ2 W BW − B
(cid:16) (cid:17)
∆λ (q2) = ∆λ (q2)
γ,direct Z,direct
3 mˆ2
= gˆ2 4f +f W . (1.3e)
2 − DW WWW Λ2
(cid:16) (cid:17)
Electromagnetic gauge invariance requires that ∆gγ should vanish for on-shell photons, in
1
apparent contradiction with (1.3a); the apparent contradiction will be resolved when all
effects are included.
3
W
Z A A
(6) WW ZZ AZ AA WZ WA WW WZ WZ WA ZZZ
Oi W W W W W Z
W
= Tr D ,Wˆ Dµ,Wˆ νρ X X X X X X X X X X
DW µ νρ
O
(cid:18)h ih i(cid:19)
= g′2 ∂ B ∂µBνρ X X X
ODB − 2 µ νρ
(cid:16) (cid:17)(cid:16) (cid:17)
= Φ†Bˆ Wˆ µνΦ X X X X X
BW µν
O
†
= D Φ Φ Φ† DµΦ X
Φ,1 µ
O
(cid:20)(cid:16) (cid:17) (cid:21) (cid:20) (cid:16) (cid:17)(cid:21)
= Tr Wˆ Wˆ νρWˆ µ X X X X X X
WWW µν ρ
O
(cid:16) (cid:17)
= Φ†Wˆ Wˆ µνΦ O O O O O O O O O O
WW µν
O
= Φ†Bˆ BˆµνΦ O O O O O O O O O O
BB µν
O
†
= D Φ Wˆ µν D Φ X X X X X
W µ ν
O
(cid:16) (cid:17) (cid:16) (cid:17)
†
= D Φ Bˆµν D Φ X X
B µ ν
O
(cid:16) (cid:17) (cid:16) (cid:17)
= 1∂ Φ†Φ ∂µ Φ†Φ
OΦ,2 2 µ
(cid:16) (cid:17) (cid:16) (cid:17)
3
= 1 Φ†Φ
OΦ,3 3
(cid:16) (cid:17)
†
= Φ†Φ D Φ DµΦ O O
Φ,4 µ
O
(cid:16) (cid:17)(cid:20)(cid:16) (cid:17) (cid:16) (cid:17)(cid:21)
TABLE I. Energy-dimension-six operators in the linear representation of the Higgs mechanism.
The contribution of an operator to a particular vertex is denoted by an ‘X’ . In some cases an
operator naively contributes to a vertex, yet that contribution does not lead to observable effects.
In such cases the ‘X’ is replaced by an ‘O’ .
Naively one would expect contributions to (1.3) and to the gauge-boson two-point-
functions from and . However, their contributions may be completely absorbed
WW BB
O O
by a redefinition of SM fields and gauge couplings, leading to a null contribution. For this
reason an ‘O’ is used for these operators in Table I. Additionally contributes to the W-
Φ,4
O
and Z-mass terms, while contributes to the Z-mass term only. Hence violates the
Φ,1 Φ,1
O O
4
custodial symmetry, and the T parameter is explicitly dependent upon f . On the other
Φ,1
hand, the contributions from exactly cancel in the calculation of T, hence it does not
Φ,4
O
contribute.
II. THE NON-LINEAR REALIZATION AND OPERATORS OF THE
ELECTROWEAK CHIRAL LAGRANGIAN.
It is possible that there is no physical Higgs boson, and the mechanism of spontaneous
symmetry breaking is not part of the SM, but a part of its extension. In such a scenario the
full Lagrangian may be written as
= + + . (2.1)
eff SM i
L L L ···
i
X
In contrast to the linearly realized Lagrangian (1.1), the leading correction terms in the
chiral Lagrangian are not suppressed by inverse powers of some high scale.
In order to proceed it is convenient to introduce some additional notation. The charge-
conjugate Higgs field may be written Φc = iτ2Φ∗. Then
√2
U Φc,Φ 1 , (2.2a)
≡ v −→
(cid:16) (cid:17)
D U = ∂ U +igTaWaU ig′UT3B igTaWa ig′T3B , (2.2b)
µ µ µ − µ −→ µ − µ
T 2UT3U† 2T3 , (2.2c)
≡ −→
V (D U)U† D U . (2.2d)
µ µ µ
≡ −→
The arrows indicate the unitary-gauge expression. In the notation of Appelquist and Wu
[3] we present a list of chiral operators through energy-dimension four which conserve CP.
There are twelve such operators given by
′ = β1v2 Tr(TV ) 2 , (2.3a)
L1 4 µ
α ggh′ i
= 1 B Tr TWµν , (2.3b)
1 µν
L 2
iα g′ (cid:16) (cid:17)
= 2 B Tr T[Vµ,Vν] , (2.3c)
2 µν
L 2
(cid:16) (cid:17)
= iα gTr W [Vµ,Vν] , (2.3d)
3 3 µν
L
(cid:16) 2 (cid:17)
= α Tr(V V ) , (2.3e)
4 4 µ ν
L
h i
5
2
= α Tr(V Vµ) , (2.3f)
5 5 µ
L
h i
= α Tr V V Tr TVµ Tr TVν , (2.3g)
6 6 µ ν
L
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
= α Tr V Vµ Tr TV Tr TVν , (2.3h)
7 7 µ ν
L
α8g2(cid:16) (cid:17) (cid:16)2 (cid:17) (cid:16) (cid:17)
= Tr(TW ) , (2.3i)
8 µν
L 4
h i
iα g
= 9 Tr TW Tr T[Vµ,Vν] , (2.3j)
9 µν
L 2
α10 (cid:16) (cid:17) (cid:16) 2 (cid:17)
= Tr(TV )Tr(TV ) , (2.3k)
10 µ ν
L 2
h i
= α gǫµνρσTr TV Tr V W . (2.3l)
11 11 µ ν ρσ
L
(cid:16) (cid:17) (cid:16) (cid:17)
The last operator, , violates parity, P. The operators are summarised in Table II, which
11
L
employs the same format as Table I. Additionally Table II pairs each chiral operator with
its linearly realized counterpart, four of which appear in Sec. I. The remainder, which occur
at the energy-dimension-eight, -ten and -twelve level may be found elsewhere [4].
Three of the chiral operators, ′, and , contribute to gauge-boson two-point-
L1 L1 L8
functions. Like , ′ contributes only to the Z-mass term but not to the W-mass term
OΦ,1 L1
and leads to a violation of the custodial symmetry. Through contributions to the charge
form-factors e¯2(q2), s¯2(q2), g¯2(q2) and g¯2(q2) these three operators will contribute to the
Z
process e+e− W+W−. None of the operators contributes to the WW two-point-function,
→
hence, in contrast to the linear realization, no contribution will be made via the W-boson
wave-function-renormalization factor, and the t-channel terms are not modified.
In total six of the operators, , , , , and , contribute directly to three-
1 2 3 8 9 11
L L L L L L
gauge-boson vertices. The non-zero corrections are
∆gZ,nl = gˆ2α , (2.4a)
1,direct Z 3
∆κnl = gˆ2 α +α +α α +α , (2.4b)
γ,direct − 1 2 3 − 8 9
(cid:16) (cid:17)
∆κnl = gˆ2sˆ2 α α +gˆ2 α α +α , (2.4c)
Z,direct Z 1 − 2 3 − 8 9
(cid:16) (cid:17) (cid:16) (cid:17)
∆gZ,nl = gˆ2α . (2.4d)
5,direct Z 11
Notice that here, unlike Eqn. (1.3a), ∆gγ,nl is trivially zero. Furthermore notice that
1,direct
∆λnl and ∆λnl are identically zero; these couplings are associated with energy-
γ,direct Z,direct
dimension-six operators, which are higher order corrections in the present scheme.
6
III. FOUR-FERMION PROCESSES
First we present the contributions of the linearly realized operators of Table I to the
oblique parameters. Due to the presence of (q2)2 terms in the gauge-boson two-point-
W
Z A A
(n) W Z Z A WZ WA W WZ WZ WA ZZ
Lchiral Olinear W Z A A W W W W W W ZZ
W
′ 4β1 X
L1 − v2 OΦ,1
4α1 X X X X X
L1 v2 OBW
8α2 X X
L2 v2 OB
8α3 X X X X X
L3 v2 OW
4α4 (8) X X X
L4 v4 O4
16α5 (8) X X X
L5 v4 O5
64α6 (10) X X
L6 − v6 O6
64α7 (10) X X
L7 − v6 O7
4α8 (8) X X X X X X
L8 − v4 O8
16α9 (8) X X X
L9 − v4 O9
128α10 (12) X
L10 v8 O10
8α11 (8) X X
L11 v4 O11
TABLE II. Column one lists energy-dimension-four operators in the non-linear representation.
The linear-representation counterparts appear in the second column. For the definitions of the
operators (n) the reader is referred to the text. An ‘X’ is used to indicate the the contribution of
Oi
an individual operator to a particular vertex.
7
functions the original derivative-based definitions of S, T and U [5] are not convenient. A
more appropriate scheme for present purposes is where the derivative is replaced by a finite
difference [6] of the form
ΠAB(q2) ΠAB(m2 )
ΠAB(q2) = T − T V . (3.1)
T,V q2 m2
− V
Then
v2
∆S 16π e ∆Π3Q(m2) ∆Π33 (0) = 4π f , (3.2a)
≡ R T,γ Z − T,Z − Λ2 BW
(cid:20) (cid:21)
4√2G 1 v2
F 33 11
∆T e ∆Π (0) ∆Π (0) = f , (3.2b)
≡ α R T − T −2αΛ2 Φ,1
(cid:20) (cid:21)
m2 m2
∆U 16π e ∆Π33 (0) ∆Π11 (0) = 32π Z − Wf , (3.2c)
≡ R T,Z − T,W Λ2 DW
(cid:20) (cid:21)
where S = S + ∆S, T = T + ∆T and U = U + ∆U. Because the f and f
SM SM SM Φ,1 Φ,4
contributions to the two-point functions are independent of q2 they may contribute only to
T. The f contributions exactly cancel, as was ‘predicted’ during the discussion of Table I.
Φ,4
The (q2)2 terms in the two-point functions also lead to a non-standard running of the SM
charge form-factors. The combination of S, T and U with the non-standard running leads
to the convenient expressions
q2
∆α(q2) = 8παˆ2 f +f , (3.3a)
− Λ2 DW DB
(cid:16) (cid:17)
q2 1 v2
∆g2(q2) = 2gˆ4 cˆ4f +sˆ4f gˆ2 f , (3.3b)
Z − ZΛ2 DW DB − 2 ZΛ2 Φ,1
(cid:16) (cid:17)
sˆ2cˆ2 m2 m2 1 v2
∆s2(q2) = − 8παˆ Z f +f + Zf f
cˆ2 sˆ2" Λ2 DW DB Λ2 BW − 2Λ2 Φ,1#
− (cid:16) (cid:17)
q2 m2
+8παˆ − Z cˆ2f sˆ2f , (3.3c)
Λ2 DW − DB
(cid:16) (cid:17)
m2 ∆s2(m2) 1 v2 q2
∆g2 (q2) = 8παˆgˆ2 Zf gˆ2 Z gˆ4 f 2gˆ4 f . (3.3d)
W − Λ2 DB − sˆ2 − 4 Λ2 BW − Λ2 DW
The ‘hatted’ couplings satisfy the tree-level relationships eˆ gˆsˆ gˆ sˆcˆand eˆ2 4παˆ.
Z
≡ ≡ ≡
The calculations may be repeated for the non-linear representation. In this case the
dependence on q2 is at most linear, hence there is no non-standard running of the charge
form-factors. The oblique parameters are given by
∆S = 16πα , (3.4a)
1
−
2
∆T = β , (3.4b)
1
α
∆U = 16πα , (3.4c)
8
−
8
which agrees with [3]. The contributions to the charge form-factors may be written
∆α(q2) = 0 , (3.5a)
∆g2(q2) = 2gˆ2β , (3.5b)
Z Z 1
cˆ2sˆ2
∆s2(q2) = 2β +gˆ2α , (3.5c)
−cˆ2 sˆ2 1 Z 1
− (cid:16) (cid:17)
∆s2(m2)
∆g2 (q2) = gˆ2 Z gˆ4 α +α . (3.5d)
W − sˆ2 − 1 8
(cid:16) (cid:17)
The low-energy data severely constrain those parameters of the chiral Lagrangian which
contribute to S, T and U; these constraints may be improved via improved measurements
at low energies. The corresponding parameters of the linearly realized Lagrangian are less
stringently constrained at present. However, because some of their contributions to observ-
ables are enhanced by higher powers of q2 these constraints may be expected to improve
significantly at LEP2 and beyond [2].
IV. CORRECTIONS TO e+e− W+W−
→
The most general amplitude for e+e− W+W−, depicted in Figure 1 may be written
→
e−(k,τ) W−α(p,λ)
e+(−k,−τ) W+β(p−,−λ)
FIG. 1. The process ee W+W− with momentum and helicity assignments. All momenta are
→
defined flowing to the right. In the massless-electron limit τ = τ.
−
9
(k,k¯,τ;p,p¯,λ,λ¯) = F (s,t)j (k,k¯,τ)Tµαβǫ∗(p,λ)ǫ∗(p¯,λ¯) , (4.1)
M i,τ µ i α β
i=1
X
where all dynamical information is contained in the scalar form-factors F (s,t). The other
i,τ
factors in Eqn. (4.1) are of a purely kinematical nature; ǫ∗(p,λ) and ǫ∗(p¯,λ¯) are the polar-
α β
ization vectors for the W− and W+ bosons respectively, and j (k,k¯,τ) is the fermion current
µ
for massless electrons. It is necessary to include nine tensors, Tµαβ, to be completely general.
i
One possible choice is
Tµαβ = Pµgαβ , (4.2a)
1
1
Tµαβ = − Pµqαqβ , (4.2b)
2 m2
W
9
Tµαβ = qαgµβ qβgαµ , (4.2c)
3 −
Tµαβ = i qαgµβ +qβgαµ , (4.2d)
4
(cid:16) (cid:17)
Tµαβ = iǫµαβρP , (4.2e)
5 ρ
Tµαβ = ǫµαβρq , (4.2f)
6 − ρ
1
Tµαβ = − Pµǫαβρσq P , (4.2g)
7 m2 ρ σ
W
Tµαβ = Kβgαµ +Kαgµβ , (4.2h)
8
i
Tµαβ = Kαǫβµρσ +Kβǫαµρσ q P , (4.2i)
9 m2 ρ σ
W(cid:16) (cid:17)
¯
where P = p p¯and K = k k. This completely determines the kinematics, hence we focus
− −
on the form factors F (s,t).
i,τ
We include only the tree-level contributions of the effective Lagrangians from Sec. I
and Sec. II. If we choose the renormalization conditions sˆ2 = s¯2(q2) and eˆ2 = e¯2(q2) the
expressions for the F (s,t) are somewhat simplified. We write
i,τ
1 1 1
F (s,t) = Qeˆ2fγ + (I sˆ2Q)cˆ2gˆ2fZ + I gˆ2ft . (4.3)
i,τ s i,τ s m2 +isΓZ Θ(s) 3 − Z i,τ 2t 3 i,τ
− Z mZ
The amplitudes for e+e− W+W− are completely determined once the lower-case form-
→
factors, fX, are specified. The results for the SM, which are particularly simple, may be
i
found in Table III.
i = 1 i = 2 i = 3 i = 4 i = 5 i = 6 i = 7 i = 8 i = 9
fγ 1 0 2 0 0 0 0 0 0
i,±
fZ 1 0 2 0 0 0 0 0 0
i,±
ft 1 0 2 0 1 0 0 1 0
i,−
TABLE III. Explicit values for the form factors of the SM at the tree level.
These values guarantee the gauge cancellations which prevent unitarity violations in ampli-
tudes which involve one or more longitudinally polarized W boson.
We may calculate the form-factors with effective-Lagrangian contributions. In general we
expect results that will spoil the gauge cancellations of the SM. The non-zero form-factors
in the linear representation, for the photonic contributions, may be written
10