arXiv:0704.1900v3  [hep-ph]  15 Nov 2007
Preprint typeset in JHEP style - HYPER VERSION
Causal vs. Analytic constraints on anomalous quartic
gauge couplings
L. Vecchi
INFN, Sezione di Trieste and
Scuola Internazionale Superiore di Studi Avanzati
via Beirut 4, I-34014 Trieste, Italy
Abstract: We derive one loop constraints on the anomalous quartic gauge couplings
using a general non-forward dispersion relation for the elastic scattering amplitude of two
longitudinally polarized vector bosons. We show that for exactly chiral theories more
stringent bounds can be obtained by the assumption that the underlying theory satisfies
the causality principle of Special Relativity.
Keywords: Spontaneous Symmetry Breaking, Chiral Lagrangians, Beyond the Standard
Model.
http://arxiv.org/abs/0704.1900v3
http://jhep.sissa.it/stdsearch
http://jhep.sissa.it/stdsearch
Contents
1. Introduction 1
2. Analytical bounds 2
2.1 Application to the gauged chiral lagrangian 3
2.2 Derivation of the analytical bounds 3
3. Causal bounds 5
4. Conclusions 7
1. Introduction
The general structure of an effective lagrangian is dictated by the interplay between quan-
tum mechanics, Poincaré invariance, and internal symmetries. Its coefficients are not con-
strained by the symmetries and must be determined by experiments. Unitarity usually
sets an upper bound on the energy scale below which a perturbative effective approach is
reliable.
We can interpret the standard model (SM) as an effective theory extending its la-
grangian to include new non-renormalizable operators with unknown coefficients. Some of
them enter the scattering amplitudes of longitudinally polarized vector bosons. These are
called anomalous quartic gauge couplings since they measure the deviation from the SM
predictions. These coefficients are necessarily connected with the not yet observed Higgs
sector. In the case the Higgs boson is not a fundamental state, or even no Higgs boson
will be observed, they provide important informations on the nature of the electro-weak
symmetry breaking sector. Whereas there are no significant experimental bounds on them
at the moment [1], theoretical arguments can reduce significantly their allowed range and
can serve as a guide for future experiments.
The authors of [2] have noticed that the coefficients of a general effective lagrangian
may be constrained by requiring the S-matrix of the full theory respects some desirable
property such as analyticity, crossing symmetry, Lorentz invariance and unitarity.
We follow these authors and consider the SM SU(2)× U(1) → U(1) breaking pattern
in the case there exists a light Higgs-like boson as well as in the case no Higgs boson
can propagate under the cut off of the effective theory. We show that a general non-
forward dispersion relation leads to a less constraining bound than the one derived by the
request the UV completion respects the causality principle of Special Relativity. This is not
surprising because it is commonly believed that the analytical properties of the S-matrix
are a consequence of its causal nature.
– 1 –
2. Analytical bounds
We briefly review an analytic tool which has been used in the context of the chiral la-
grangian of QCD to constrain some effective coefficients.
Consider a multiplet of scalar particles, which to be definite we call pions πa, having
mass m. Assume they are lighter than any other quanta and that they have appropriate
quantum numbers to forbid the transition 2π → π. The other states can be general unstable
quanta of complex masses M much greater than 2m. The S-matrix element for a general
transition 2π → 2π is a Lorentz scalar function of the Mandelstam variables s, t, u and of
the mass m2.
We study the amplitude for the elastic scattering πaπb → πaπb and assume it can be
analytically continued to the complex variables s, t. We denote this analytical function
by F (s, t) and require that its domain of analyticity be dictated entirely by the optical
theorem and the crossing symmetry. More precisely, we assume that the singularities come
from simple poles in the correspondence of the physical masses of the quantum states which
can be produced in the reaction, and branch cuts in the real axis starting at the threshold
of multi-particle production.
Since no mass-less particle exchange is included in F (s, t), the analytical amplitude
satisfies a twice subtracted dispersion relation for a variety of complex t [3]. For any non
singular complex point s, t we can write:
d2F (s, t)
+ P =
ImF (x+ iε, t)
(x− s)3
ImFu(x+ iε, t)
(x− u)3
(2.1)
where we defined u = 4m2 − s− t and used the crossing symmetry to write the amplitude
in the u-channel as Fu(x, t) = F (4m
2 − x− t, t).
The P on the left hand side of (2.1) denotes the second derivative of the residues.
By the analyticity assumption this term comes entirely from the complex simple poles
produced by the exchange of unstable states. In our discussion the pole term can be
neglected since its contribution turns out not to be relevant .
In the case of forward scattering (t = 0) the imaginary part ImF (x, 0) is proportional
to the total cross section of the transition 2π →’everything’ and is therefore non negative.
The crossing symmetry leads to a similar result for the u-channel. We conclude that
F ′′(s, 0) is a strictly positive function for any real center of mass energy s in the range
0 ≤ s ≤ 4m2.
The analyticity assumption can be used to generalize the domain of positivity of the
imaginary part of the amplitude. This can be seen by expanding ImF (x+ iε, t) in partial
waves in the physical region and observing that, due to the optical theorem and the prop-
erties of the Legendre polynomials, any derivative with respect to t at the point x ≥ 4m2,
t = 0 is non negative. The Taylor series of ImF (x+ iε, t) for t ≥ 0 is therefore greater than
zero. Since an analog result holds for the u-channel, we conclude that the second derivative
F ′′(s, t) is strictly positive (and analytical) for any real kinematical invariant belonging to
the triangle ∆ =
s, t, u| 0 ≤ s, t, u ≤ 4m2
In QCD, the scattering of pions at a scale comparable with their masses is very well
described by the chiral lagrangian. The 4 pion operators produce order s2 corrections to
– 2 –
the scattering amplitude and eq. (2.1) implies positive bounds on some combination of
their coefficients (see [4], for example).
2.1 Application to the gauged chiral lagrangian
We can think of the SM as an effective theory and extend its action to include non renor-
malizable operators in the standard way [5].
The anomalous quartic gauge couplings enter the scattering amplitude of two longi-
tudinally polarized gauge bosons at order s2. We expect that the method outlined in the
previous section may be used to bound these coefficients.
There exists, however, a fundamental difference from the QCD case. The assumptions
made to derive the relation (2.1) are the analytic, Lorentz and crossing symmetric nature
together with the asymptotic behavior of the amplitude F (s, t). A sufficient condition
for the latter hypothesis to hold is that no massless particle exchange contribute to F
(Froissart bound). In the electroweak case this latter assumption is not natural because of
the presence of the electromagnetic interactions.
Although we may consider only amplitudes with no single photon exchange (like
W±Z0 → W±Z0 for example), there is still an operative difficulty due to the fact that the
amplitude F is generally dominated by the SM graphs at low energy scales. These latter
give rise to positive contributions to F (s, t), since the SM is well defined even for vanishing
coefficients, and one is lead to conclude that eq. (2.1) implies that the effective operators
involved cannot produce a ”too large and negative” contribution to the amplitude F (s, t)
and that, as a consequence, no significant bound can be derived in the gauged theory.
Notice that this is also true in the absence of a light Higgs boson as far as the CM energy
is of the order of the Z0 mass.
One way to overcome these apparent complications is considering amplitudes with no
single photon exchange and evaluating them at a high scale s ≫ m2Z with the equivalence
theorem (ET). In this case one has to prove the positivity of the second derivative of the
amplitude is guaranteed in the energy regime in which the approach is defined [6].
Another way, which we decide to follow, is working in the global limit. The crucial
observation in order to justify this assumption is that in the matching between the effective
lagrangian and the UV theory the transverse gauge bosons contribute, because of their
weak coupling, in a subdominant way to the effective coefficients of our interest. An
accurate estimate of them, and the respective bounds, can therefore be obtained neglecting
completely the gauge structure and studying the coefficients of the global theory.
Using this conceptually different (though operationally equivalent) perspective we can
study any two by two elastic scattering amplitude and generalize the analysis of [6] to
non-forward scattering.
2.2 Derivation of the analytical bounds
We first specialize to the case there appears no Higgs-like boson under a cut off Λ.
In this context the basic tool is a non linearly realized effective lagrangian for the break-
ing pattern SU(2)× U(1) → U(1) written in terms of a SU(2) matrix U = exp(iπaσa/v),
where σa are the three Pauli matrices with a = 1, 2, 3 and v ≃ 250 GeV is the EW vacuum.
– 3 –
As usual, under a global SU(2)L × U(1)Y transformation U → LUR†, where L ∈ SU(2)L
and R ∈ U(1)Y ⊂ SU(2)R.
Assuming m2Z ≪ Λ2 and working at energies comparable with the Z0 mass, the most
general lagrangian respecting the above symmetries and up to O(s2) is given in reference [7].
The globally symmetric version is:
LEWChL = −
Tr (VµV
2v2[Tr(TVµ)]
+ α4[Tr(VµVν)]
2 + α5[Tr(VµV
µ)]2 + α6Tr(VµVν)Tr(TV
µ)Tr(TV ν)
+ α7Tr(VµV
µ)Tr(TVν)Tr(TV
α10[Tr(TVµ)Tr(TVν)]
2, (2.2)
where Vµ = (∂µU)U
† and T = Uσ3U †.
We stress that in this idealized scenario the πa are exact Goldstone bosons. To avoid
any complication with the asymptotic behavior of the amplitude we can introduce by hand
a πa mass and proceed as in QCD. This mass is actually the consequence of an explicit
symmetry breaking term in the UV theory. Being interested in constraining the underlying
symmetric theory we are forced to take m2 ≪ m2Z , s. The bounds we derive differ from the
QCD ones for this very reason.
Although no mass gap is present in this context, an approximate positive constraint for
F ′′(s, t) can be derived. This we do by noticing that a general dispersion relation like (2.1)
can be used to bound the anomalous quartic couplings only if the O(s3) contribution to
F (s, t) is negligible. In this regime the second derivative F ′′(s, t) is dominantly s indepen-
dent and, for a small non vanishing imaginary part for s, the dispersion relation can be
approximated as:
d2F (s, t)
ImF (x+ iε, t)
ImFu(x+ iε, t)
(2.3)
where the limit m2/s → 0 was assumed and the resonant pole term has been neglected.
Eq. (2.3) shows that, as far as O(s3) are negligible compared to O(s2), the second derivative
of the amplitude is strictly positive.
Before evaluating the bounds we notice that the smallness of the EW precision tests T
parameter [8] is conveniently achieved by assuming the existence of an approximate global
SU(2)C custodial symmetry under which the Goldstone boson matrix transforms as the
adjoint representation. The dominant coefficients associated to anomalous quartic gauge
operators are α4 and α5 and any π
aπb → πcπd scattering amplitude can be written in
terms of a function A(s, t, u). The relevant processes turn out to be:
A(π0π0 −→ π0π0) = A(s, t, u) +A(t, s, u) +A(u, t, s)
A(π±π0 −→ π±π0) = A(t, s, u), (2.4)
where, at one loop level and in the limit m2/s → 0, we have [9]
A(s, t, u) =
2α5(µ)s
2 + α4(µ)(t
2 + u2) +
(4π)2
10s2 + 13(t2 + u2)
96π2v4
t(t− u) log
+ u(u− t) log
+ 3s2 log
.(2.5)
– 4 –
Notice that we have chosen to work with the renormalized coefficients α4,5(µ) as defined by
the modified minimal subtraction scheme, rather than using the non standard normalization
of [9].
We can now derive (2.4) twice with respect to s and evaluate the result at s + iε, t,
where 0 < s, t ≪ Λ2. It is convenient to choose a different representation for the kinematical
invariants in order to eliminate the logarithms in the final result. We define a scale w =
s(s+ t) =
−su > s and obtain:
α4(w) + α5(w) > −
(4π)2
α4(w) >
(4π)2
. (2.6)
For t = 0 we have α4 + α5 & −0.40 × 10−3 and α4 & −0.35 × 10−3 at an arbitrary scale
w = s ≪ Λ2. This result coincides with the one obtained in [6], as expected.
In the case of non-forward scattering, the bound on α4(w) cannot get arbitrarily large
(large w or, equivalently, large t) because at some unknown scale, much smaller than Λ2,
the O(s3) corrections become relevant in the determination of the amplitude and the bound
would not apply. Without a detailed knowledge of the perturbative expansion in the weak
coupling s/Λ2, (that is, of the full theory!) we cannot realistically tell which is the strongest
bound derived by this analysis.
What we can certainly do is to compare (2.6) with the well known constraints on
the corresponding parameters l1 = 4α5 and l2 = 4α4 of QCD. Strong bounds on these
coefficients have been evaluated in the triangle ∆ [10]. We may interpret our analysis as a
study of the axiomatic constraints on the two pion amplitudes in the complementary region
m2 ≪ s ≪ Λ2. Using the notation introduced in [9] we translate (2.6) into 2l̄1 + 4l̄4 & 3
and l̄2 & 0.3. These constraints are compatible with the experimental observations [11] but
are less stringent than those obtained in [10].
We conclude that our analysis does not lead to an improvement of the bounds on l̄1,2.
If the chiral symmetry is exact, on the other hand, eqs. (2.6) represent stringent bounds
on the anomalous quartic couplings implied by the assumptions of analyticity, crossing
symmetry, unitarity and Lorentz invariance of the S-matrix.
Eq. (2.3) is not rigorous if a light state enters the processes under consideration and
therefore (2.6) are not valid if a Higgs-like scalar propagates under the cutoff. In the next
paragraph we discuss an approach which works in this context as well, provided the chiral
symmetry is exact.
3. Causal bounds
Given a general solution of the equations of motion derived from (2.2) we can study the
oscillations around it. Consistency with Special Relativity requires the oscillations to prop-
agate sub-luminally. This request may be expressed as a constraint on the same coefficients
which enter the elastic scattering of two Goldstone bosons because the dynamics of the os-
cillation on the background can be interpreted as a scattering process on a macroscopic
– 5 –
‘object‘. If the background has a constant gradient, the presence of super-luminal propa-
gations sum up and can in principle become manifest in the low energy regime [2].
A constant gradient solutions admitted by the lagrangian (2.2) is defined by π0 =
µ, where σ is a generic isospin direction and the constant vector Cµ is fine-tuned in
order to satisfy C2 ≪ v4. The quadratic lagrangian for the oscillations δπ = π−π0 around
the background have the general form:
L = δπ
δπ, (3.1)
with α = α4, α4 + α5. In the evaluation of (3.1) we neglected O(Cx/v) terms. We can
imagine in fact the non trivial background to be switched on in a finite space-time domain
so that the latter approximation is seen as a consequence of the fine-tuning of the parameter
A perturbative study of the interacting field δπ is in principle possible for energies
under a certain scale (to be definite we call this scale the cut-off of the effective theory).
By assumption, this cut off is arbitrarily close to Λ as C2/v4 goes to zero and, having this
fact in mind, we simply denote it as Λ.
A necessary condition for such a perturbative study to make any sense is that the
quadratic lagrangian be well defined. This is the case for (3.1) only if α ≥ 0. In fact, the
field δπ has velocity dE/dp = E/p (where pµ = (E, p̄) and |p̄| = p) and for α < 0 its quanta
propagate super-luminally.
It is important to notice that the presence of super-luminal modes is not the conse-
quence of a bad choice of the vacuum. The quadratic hamiltonian is stable in any vacuum
(parametrized by Cµ) if α is ’sufficiently small’ but generally leads to violations of the
causality principle of Special Relativity when α < 0. In the latter hypothesis then different
inertial frames may not agree on the physical observations and, for example, the quadratic
hamiltonian may appear unbounded from below to a general Lorentzian frame boosted
with a sufficiently high velocity.
We finally interpret the constraint α ≥ 0 as a causal bound.
The effective coefficients α which appear in the perturbative analysis are actually
the renormalized couplings so that the above bound can be extended to all energy scales
w < Λ2, where the perturbative study is assumed to be meaningful, after taking into
account the running effect:
α4(w) + α5(w) ≥
(4π)2
α4(w) ≥
(4π)2
. (3.2)
This approach may be applied even to scenarios in which a scalar Higgs, composite or
fundamental, can propagate under the cut off. In this latter case the causal constraints
read α4 ≥ 0 and α4 + α5 ≥ 0 but now the coefficients do not have any scale dependence
because the theory has no extra-SM divergences at order s2. Therefore, the possibility
α4 = α5 = 0 can not and must not be excluded (consider the particular example of the
– 6 –
SM). The analytical bounds, which would imply a strict inequality, do not apply as already
noticed.
The bounds 3.2 cannot be compared to the QCD ones because π0 does not solve the
equations of motion when m 6= 0.
4. Conclusions
We have derived general bounds on the anomalous quartic gauge couplings using two
distinct approaches. The causal one relies on the absence of superluminal propagations.
The analytical one relies on the assumption of analyticity, crossing and Lorentz symmetry
together with a good behavior at infinity of the scattering amplitude F (s, t). The latter
method works in the context of a strongly coupled theory with no Higgs propagating at
low energy only. In this scenario (2.6) can be compared to (3.2). We see that the bound
on α4 +α5 is clearly dominated by the causal result and that this is also the case for α4 if,
roughly, the ratio (w/s)2 does not exceed 16 log(Λ/
w). We cannot tell if the analytical
bound still apply up to this scale
More importantly, if the fermionic effects are considered separately from α4,5, a realistic
estimate of the constraints should take the fermions couplings to the Goldstone bosons into
account. It is easy to see that the one loop effect induced by the SM fermions gives rise
to a positive contribution to the second derivative of the amplitude. This of course lowers
the analytical bounds while the causal argument remains valid and (3.2) is not altered.
The bound (3.2) for the higgsless scenario, together with the constraint α4 ≥ 0 and
α4+α5 ≥ 0 for the light Higgs-like scenario provide the most stringent and reliable bounds
on the effective coefficients α4,5.
In order to have a rough estimate of (3.2) we assume Λ ∼ 1 TeV and get α4 + α5 &
3.8×10−3, α4 & 2.5×10−3 at the Z0 pole. These values lie inside the very wide experimental
bounds −0.1 . α4,5 . 0.1. Eqs. (3.2) significantly reduce the allowed range.
The experimental constraints are extremely weak since they have been derived by
estimating the loop corrections induced by α4,5 on the electroweak precision parameters [1].
A direct measurement of the anomalous gauge couplings turns out to be of fundamental
importance in order to have some insight on the actual nature of the electroweak breaking
sector [13]. LHC may improve the bounds [1] by an order of magnitude but the linear
collider seems far more appropriate to resolve the coefficients [12]. The measurement of a
negative value of α4 and α4+α5 at the next linear collider would therefore signal a breaking
of causality, irrespective of the presence of a light scalar state like the Higgs boson. This
seems a rather unlikely possibility because it would require too drastic a modification of
our physical understanding. A more conservative point of view consists in interpreting the
bounds (3.2) as theoretical constraints on the full theory.
Acknowledgments
This work is partially supported by MIUR and the RTN European Program MRTN-CT-
2004-503369.
– 7 –
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– 8 –

  We derive one loop constraints on the anomalous quartic gauge couplings using
a general non-forward dispersion relation for the elastic scattering amplitude
of two longitudinally polarized vector bosons. We compare this result with
another one derived by the assumption that the underlying theory satisfies the
causality principle of Special Relativity and show that this latter is more
constraining.

Introduction 1
2. Analytical bounds 2
2.1 Application to the gauged chiral lagrangian 3
2.2 Derivation of the analytical bounds 3
3. Causal bounds 5
4. Conclusions 7
1. Introduction
The general structure of an effective lagrangian is dictated by the interplay between quan-
tum mechanics, Poincaré invariance, and internal symmetries. Its coefficients are not con-
strained by the symmetries and must be determined by experiments. Unitarity usually
sets an upper bound on the energy scale below which a perturbative effective approach is
reliable.
We can interpret the standard model (SM) as an effective theory extending its la-
grangian to include new non-renormalizable operators with unknown coefficients. Some of
them enter the scattering amplitudes of longitudinally polarized vector bosons. These are
called anomalous quartic gauge couplings since they measure the deviation from the SM
predictions. These coefficients are necessarily connected with the not yet observed Higgs
sector. In the case the Higgs boson is not a fundamental state, or even no Higgs boson
will be observed, they provide important informations on the nature of the electro-weak
symmetry breaking sector. Whereas there are no significant experimental bounds on them
at the moment [1], theoretical arguments can reduce significantly their allowed range and
can serve as a guide for future experiments.
The authors of [2] have noticed that the coefficients of a general effective lagrangian
may be constrained by requiring the S-matrix of the full theory respects some desirable
property such as analyticity, crossing symmetry, Lorentz invariance and unitarity.
We follow these authors and consider the SM SU(2)× U(1) → U(1) breaking pattern
in the case there exists a light Higgs-like boson as well as in the case no Higgs boson
can propagate under the cut off of the effective theory. We show that a general non-
forward dispersion relation leads to a less constraining bound than the one derived by the
request the UV completion respects the causality principle of Special Relativity. This is not
surprising because it is commonly believed that the analytical properties of the S-matrix
are a consequence of its causal nature.
– 1 –
2. Analytical bounds
We briefly review an analytic tool which has been used in the context of the chiral la-
grangian of QCD to constrain some effective coefficients.
Consider a multiplet of scalar particles, which to be definite we call pions πa, having
mass m. Assume they are lighter than any other quanta and that they have appropriate
quantum numbers to forbid the transition 2π → π. The other states can be general unstable
quanta of complex masses M much greater than 2m. The S-matrix element for a general
transition 2π → 2π is a Lorentz scalar function of the Mandelstam variables s, t, u and of
the mass m2.
We study the amplitude for the elastic scattering πaπb → πaπb and assume it can be
analytically continued to the complex variables s, t. We denote this analytical function
by F (s, t) and require that its domain of analyticity be dictated entirely by the optical
theorem and the crossing symmetry. More precisely, we assume that the singularities come
from simple poles in the correspondence of the physical masses of the quantum states which
can be produced in the reaction, and branch cuts in the real axis starting at the threshold
of multi-particle production.
Since no mass-less particle exchange is included in F (s, t), the analytical amplitude
satisfies a twice subtracted dispersion relation for a variety of complex t [3]. For any non
singular complex point s, t we can write:
d2F (s, t)
+ P =
ImF (x+ iε, t)
(x− s)3
ImFu(x+ iε, t)
(x− u)3
(2.1)
where we defined u = 4m2 − s− t and used the crossing symmetry to write the amplitude
in the u-channel as Fu(x, t) = F (4m
2 − x− t, t).
The P on the left hand side of (2.1) denotes the second derivative of the residues.
By the analyticity assumption this term comes entirely from the complex simple poles
produced by the exchange of unstable states. In our discussion the pole term can be
neglected since its contribution turns out not to be relevant .
In the case of forward scattering (t = 0) the imaginary part ImF (x, 0) is proportional
to the total cross section of the transition 2π →’everything’ and is therefore non negative.
The crossing symmetry leads to a similar result for the u-channel. We conclude that
F ′′(s, 0) is a strictly positive function for any real center of mass energy s in the range
0 ≤ s ≤ 4m2.
The analyticity assumption can be used to generalize the domain of positivity of the
imaginary part of the amplitude. This can be seen by expanding ImF (x+ iε, t) in partial
waves in the physical region and observing that, due to the optical theorem and the prop-
erties of the Legendre polynomials, any derivative with respect to t at the point x ≥ 4m2,
t = 0 is non negative. The Taylor series of ImF (x+ iε, t) for t ≥ 0 is therefore greater than
zero. Since an analog result holds for the u-channel, we conclude that the second derivative
F ′′(s, t) is strictly positive (and analytical) for any real kinematical invariant belonging to
the triangle ∆ =
s, t, u| 0 ≤ s, t, u ≤ 4m2
In QCD, the scattering of pions at a scale comparable with their masses is very well
described by the chiral lagrangian. The 4 pion operators produce order s2 corrections to
– 2 –
the scattering amplitude and eq. (2.1) implies positive bounds on some combination of
their coefficients (see [4], for example).
2.1 Application to the gauged chiral lagrangian
We can think of the SM as an effective theory and extend its action to include non renor-
malizable operators in the standard way [5].
The anomalous quartic gauge couplings enter the scattering amplitude of two longi-
tudinally polarized gauge bosons at order s2. We expect that the method outlined in the
previous section may be used to bound these coefficients.
There exists, however, a fundamental difference from the QCD case. The assumptions
made to derive the relation (2.1) are the analytic, Lorentz and crossing symmetric nature
together with the asymptotic behavior of the amplitude F (s, t). A sufficient condition
for the latter hypothesis to hold is that no massless particle exchange contribute to F
(Froissart bound). In the electroweak case this latter assumption is not natural because of
the presence of the electromagnetic interactions.
Although we may consider only amplitudes with no single photon exchange (like
W±Z0 → W±Z0 for example), there is still an operative difficulty due to the fact that the
amplitude F is generally dominated by the SM graphs at low energy scales. These latter
give rise to positive contributions to F (s, t), since the SM is well defined even for vanishing
coefficients, and one is lead to conclude that eq. (2.1) implies that the effective operators
involved cannot produce a ”too large and negative” contribution to the amplitude F (s, t)
and that, as a consequence, no significant bound can be derived in the gauged theory.
Notice that this is also true in the absence of a light Higgs boson as far as the CM energy
is of the order of the Z0 mass.
One way to overcome these apparent complications is considering amplitudes with no
single photon exchange and evaluating them at a high scale s ≫ m2Z with the equivalence
theorem (ET). In this case one has to prove the positivity of the second derivative of the
amplitude is guaranteed in the energy regime in which the approach is defined [6].
Another way, which we decide to follow, is working in the global limit. The crucial
observation in order to justify this assumption is that in the matching between the effective
lagrangian and the UV theory the transverse gauge bosons contribute, because of their
weak coupling, in a subdominant way to the effective coefficients of our interest. An
accurate estimate of them, and the respective bounds, can therefore be obtained neglecting
completely the gauge structure and studying the coefficients of the global theory.
Using this conceptually different (though operationally equivalent) perspective we can
study any two by two elastic scattering amplitude and generalize the analysis of [6] to
non-forward scattering.
2.2 Derivation of the analytical bounds
We first specialize to the case there appears no Higgs-like boson under a cut off Λ.
In this context the basic tool is a non linearly realized effective lagrangian for the break-
ing pattern SU(2)× U(1) → U(1) written in terms of a SU(2) matrix U = exp(iπaσa/v),
where σa are the three Pauli matrices with a = 1, 2, 3 and v ≃ 250 GeV is the EW vacuum.
– 3 –
As usual, under a global SU(2)L × U(1)Y transformation U → LUR†, where L ∈ SU(2)L
and R ∈ U(1)Y ⊂ SU(2)R.
Assuming m2Z ≪ Λ2 and working at energies comparable with the Z0 mass, the most
general lagrangian respecting the above symmetries and up to O(s2) is given in reference [7].
The globally symmetric version is:
LEWChL = −
Tr (VµV
2v2[Tr(TVµ)]
+ α4[Tr(VµVν)]
2 + α5[Tr(VµV
µ)]2 + α6Tr(VµVν)Tr(TV
µ)Tr(TV ν)
+ α7Tr(VµV
µ)Tr(TVν)Tr(TV
α10[Tr(TVµ)Tr(TVν)]
2, (2.2)
where Vµ = (∂µU)U
† and T = Uσ3U †.
We stress that in this idealized scenario the πa are exact Goldstone bosons. To avoid
any complication with the asymptotic behavior of the amplitude we can introduce by hand
a πa mass and proceed as in QCD. This mass is actually the consequence of an explicit
symmetry breaking term in the UV theory. Being interested in constraining the underlying
symmetric theory we are forced to take m2 ≪ m2Z , s. The bounds we derive differ from the
QCD ones for this very reason.
Although no mass gap is present in this context, an approximate positive constraint for
F ′′(s, t) can be derived. This we do by noticing that a general dispersion relation like (2.1)
can be used to bound the anomalous quartic couplings only if the O(s3) contribution to
F (s, t) is negligible. In this regime the second derivative F ′′(s, t) is dominantly s indepen-
dent and, for a small non vanishing imaginary part for s, the dispersion relation can be
approximated as:
d2F (s, t)
ImF (x+ iε, t)
ImFu(x+ iε, t)
(2.3)
where the limit m2/s → 0 was assumed and the resonant pole term has been neglected.
Eq. (2.3) shows that, as far as O(s3) are negligible compared to O(s2), the second derivative
of the amplitude is strictly positive.
Before evaluating the bounds we notice that the smallness of the EW precision tests T
parameter [8] is conveniently achieved by assuming the existence of an approximate global
SU(2)C custodial symmetry under which the Goldstone boson matrix transforms as the
adjoint representation. The dominant coefficients associated to anomalous quartic gauge
operators are α4 and α5 and any π
aπb → πcπd scattering amplitude can be written in
terms of a function A(s, t, u). The relevant processes turn out to be:
A(π0π0 −→ π0π0) = A(s, t, u) +A(t, s, u) +A(u, t, s)
A(π±π0 −→ π±π0) = A(t, s, u), (2.4)
where, at one loop level and in the limit m2/s → 0, we have [9]
A(s, t, u) =
2α5(µ)s
2 + α4(µ)(t
2 + u2) +
(4π)2
10s2 + 13(t2 + u2)
96π2v4
t(t− u) log
+ u(u− t) log
+ 3s2 log
.(2.5)
– 4 –
Notice that we have chosen to work with the renormalized coefficients α4,5(µ) as defined by
the modified minimal subtraction scheme, rather than using the non standard normalization
of [9].
We can now derive (2.4) twice with respect to s and evaluate the result at s + iε, t,
where 0 < s, t ≪ Λ2. It is convenient to choose a different representation for the kinematical
invariants in order to eliminate the logarithms in the final result. We define a scale w =
s(s+ t) =
−su > s and obtain:
α4(w) + α5(w) > −
(4π)2
α4(w) >
(4π)2
. (2.6)
For t = 0 we have α4 + α5 & −0.40 × 10−3 and α4 & −0.35 × 10−3 at an arbitrary scale
w = s ≪ Λ2. This result coincides with the one obtained in [6], as expected.
In the case of non-forward scattering, the bound on α4(w) cannot get arbitrarily large
(large w or, equivalently, large t) because at some unknown scale, much smaller than Λ2,
the O(s3) corrections become relevant in the determination of the amplitude and the bound
would not apply. Without a detailed knowledge of the perturbative expansion in the weak
coupling s/Λ2, (that is, of the full theory!) we cannot realistically tell which is the strongest
bound derived by this analysis.
What we can certainly do is to compare (2.6) with the well known constraints on
the corresponding parameters l1 = 4α5 and l2 = 4α4 of QCD. Strong bounds on these
coefficients have been evaluated in the triangle ∆ [10]. We may interpret our analysis as a
study of the axiomatic constraints on the two pion amplitudes in the complementary region
m2 ≪ s ≪ Λ2. Using the notation introduced in [9] we translate (2.6) into 2l̄1 + 4l̄4 & 3
and l̄2 & 0.3. These constraints are compatible with the experimental observations [11] but
are less stringent than those obtained in [10].
We conclude that our analysis does not lead to an improvement of the bounds on l̄1,2.
If the chiral symmetry is exact, on the other hand, eqs. (2.6) represent stringent bounds
on the anomalous quartic couplings implied by the assumptions of analyticity, crossing
symmetry, unitarity and Lorentz invariance of the S-matrix.
Eq. (2.3) is not rigorous if a light state enters the processes under consideration and
therefore (2.6) are not valid if a Higgs-like scalar propagates under the cutoff. In the next
paragraph we discuss an approach which works in this context as well, provided the chiral
symmetry is exact.
3. Causal bounds
Given a general solution of the equations of motion derived from (2.2) we can study the
oscillations around it. Consistency with Special Relativity requires the oscillations to prop-
agate sub-luminally. This request may be expressed as a constraint on the same coefficients
which enter the elastic scattering of two Goldstone bosons because the dynamics of the os-
cillation on the background can be interpreted as a scattering process on a macroscopic
– 5 –
‘object‘. If the background has a constant gradient, the presence of super-luminal propa-
gations sum up and can in principle become manifest in the low energy regime [2].
A constant gradient solutions admitted by the lagrangian (2.2) is defined by π0 =
µ, where σ is a generic isospin direction and the constant vector Cµ is fine-tuned in
order to satisfy C2 ≪ v4. The quadratic lagrangian for the oscillations δπ = π−π0 around
the background have the general form:
L = δπ
δπ, (3.1)
with α = α4, α4 + α5. In the evaluation of (3.1) we neglected O(Cx/v) terms. We can
imagine in fact the non trivial background to be switched on in a finite space-time domain
so that the latter approximation is seen as a consequence of the fine-tuning of the parameter
A perturbative study of the interacting field δπ is in principle possible for energies
under a certain scale (to be definite we call this scale the cut-off of the effective theory).
By assumption, this cut off is arbitrarily close to Λ as C2/v4 goes to zero and, having this
fact in mind, we simply denote it as Λ.
A necessary condition for such a perturbative study to make any sense is that the
quadratic lagrangian be well defined. This is the case for (3.1) only if α ≥ 0. In fact, the
field δπ has velocity dE/dp = E/p (where pµ = (E, p̄) and |p̄| = p) and for α < 0 its quanta
propagate super-luminally.
It is important to notice that the presence of super-luminal modes is not the conse-
quence of a bad choice of the vacuum. The quadratic hamiltonian is stable in any vacuum
(parametrized by Cµ) if α is ’sufficiently small’ but generally leads to violations of the
causality principle of Special Relativity when α < 0. In the latter hypothesis then different
inertial frames may not agree on the physical observations and, for example, the quadratic
hamiltonian may appear unbounded from below to a general Lorentzian frame boosted
with a sufficiently high velocity.
We finally interpret the constraint α ≥ 0 as a causal bound.
The effective coefficients α which appear in the perturbative analysis are actually
the renormalized couplings so that the above bound can be extended to all energy scales
w < Λ2, where the perturbative study is assumed to be meaningful, after taking into
account the running effect:
α4(w) + α5(w) ≥
(4π)2
α4(w) ≥
(4π)2
. (3.2)
This approach may be applied even to scenarios in which a scalar Higgs, composite or
fundamental, can propagate under the cut off. In this latter case the causal constraints
read α4 ≥ 0 and α4 + α5 ≥ 0 but now the coefficients do not have any scale dependence
because the theory has no extra-SM divergences at order s2. Therefore, the possibility
α4 = α5 = 0 can not and must not be excluded (consider the particular example of the
– 6 –
SM). The analytical bounds, which would imply a strict inequality, do not apply as already
noticed.
The bounds 3.2 cannot be compared to the QCD ones because π0 does not solve the
equations of motion when m 6= 0.
4. Conclusions
We have derived general bounds on the anomalous quartic gauge couplings using two
distinct approaches. The causal one relies on the absence of superluminal propagations.
The analytical one relies on the assumption of analyticity, crossing and Lorentz symmetry
together with a good behavior at infinity of the scattering amplitude F (s, t). The latter
method works in the context of a strongly coupled theory with no Higgs propagating at
low energy only. In this scenario (2.6) can be compared to (3.2). We see that the bound
on α4 +α5 is clearly dominated by the causal result and that this is also the case for α4 if,
roughly, the ratio (w/s)2 does not exceed 16 log(Λ/
w). We cannot tell if the analytical
bound still apply up to this scale
More importantly, if the fermionic effects are considered separately from α4,5, a realistic
estimate of the constraints should take the fermions couplings to the Goldstone bosons into
account. It is easy to see that the one loop effect induced by the SM fermions gives rise
to a positive contribution to the second derivative of the amplitude. This of course lowers
the analytical bounds while the causal argument remains valid and (3.2) is not altered.
The bound (3.2) for the higgsless scenario, together with the constraint α4 ≥ 0 and
α4+α5 ≥ 0 for the light Higgs-like scenario provide the most stringent and reliable bounds
on the effective coefficients α4,5.
In order to have a rough estimate of (3.2) we assume Λ ∼ 1 TeV and get α4 + α5 &
3.8×10−3, α4 & 2.5×10−3 at the Z0 pole. These values lie inside the very wide experimental
bounds −0.1 . α4,5 . 0.1. Eqs. (3.2) significantly reduce the allowed range.
The experimental constraints are extremely weak since they have been derived by
estimating the loop corrections induced by α4,5 on the electroweak precision parameters [1].
A direct measurement of the anomalous gauge couplings turns out to be of fundamental
importance in order to have some insight on the actual nature of the electroweak breaking
sector [13]. LHC may improve the bounds [1] by an order of magnitude but the linear
collider seems far more appropriate to resolve the coefficients [12]. The measurement of a
negative value of α4 and α4+α5 at the next linear collider would therefore signal a breaking
of causality, irrespective of the presence of a light scalar state like the Higgs boson. This
seems a rather unlikely possibility because it would require too drastic a modification of
our physical understanding. A more conservative point of view consists in interpreting the
bounds (3.2) as theoretical constraints on the full theory.
Acknowledgments
This work is partially supported by MIUR and the RTN European Program MRTN-CT-
2004-503369.
– 7 –
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