A Higgs-Higgs bound state due to New Physics at a TeV
UCSD/PTH 07-03
A Higgs-Higgs bound state due to New Physics at a TeV.
Benjamin Grinstein and Michael Trott1
1Department of Physics, University of California at San Diego,
La Jolla, CA, 92093
Abstract
We examine the effects of new physics on the Higgs sector of the standard model, focusing on
the effects on the Higgs self couplings. We demonstrate that a low mass higgs, mh < 2mt, can
have a strong effective self coupling due to the effects of a new interaction at a TeV. We investigate
the possibility that the first evidence of such an interaction could be a higgs-higgs bound state.
To this end, we construct an effective field theory formalism to examine the physics of such a low
mass higgs boson. We explore the possibility of a non relativistic bound state of the higgs field
(Higgsium) at LHC and construct a non relativistic effective field theory of the higgs sector that
is appropriate for such studies (NRHET).
http://arxiv.org/abs/0704.1505v3
I. INTRODUCTION
Currently, global fits to all precision electro weak give the higgs mass to be 113+56
with an upper bound given by mh < 241GeV at 95% CL (see, e.g., J. Erler and P. Langacker
in Sec. 10 of Ref. [1]). LEP has also placed a lower bound limit of mh > 114.4GeV [2].
Assuming the Standard Model (SM) of electroweak interactions, one expects that the higgs
will soon be found at LHC.
However, there are at least two reasons why the SM with a single higgs doublet is ex-
pected to be incomplete. The first is the triviality problem. This asserts that the higgs self
interaction, and hence its mass, must vanish unless the theory has a finite cut-off. Triviality
has been rigorously established only for simpler models, but it is widely believed to hold for
the SM higgs. The other is the hierarchy problem that quadratic divergences need to be
finely tuned to keep the scale of electroweak breaking smaller than the natural cut-off of the
theory (which in the absence of new physics would be the Planck scale).
In this paper, we investigate the effects that new physics, invoked to cure these problems,
may have on the higgs sector of the SM. We assume that the scale of the masses of new
quanta, M, is sufficiently higher than the scale of electro-weak symmetry breaking (v ∼
246GeV) so that the quanta of the unknown new physics can be integrated out. As we want
this new physics to address the hierarchy and triviality problems, and for phenomenological
reasons, we are interested in new physics where M ∼ TeV. The resulting low energy
effective theory is the one higgs standard model supplemented with non-renormalizable local
operators, of dimension D > 4, which are constructed of standard model fields invariant
under the SU(3) × SU(2) × U(1) gauge symmetry. This approach has been applied to
precision electroweak observables[3, 4, 5, 6, 7, 8] and has recently been the subject of further
investigations[9, 10, 11, 12, 13, 14, 15]. The advantage of this approach is that it is model
independent: any new physics scheme that results in a low energy spectrum coinciding
with the SM’s can be described in this way. The disadvantage is that the new physics is
parametrized in terms of several arbitrary parameters, the coefficients of higher dimension
operators, and nothing is known a priori about these coefficients.
For a particular extension to the standard model, consistency requires that fits such as [1]
be reconsidered with the new operators, severely relaxing the constraint on the higgs mass.
In fact, it has been shown that the effect of higher dimension operators [16, 17, 18] can
elminate the mass limit on the higgs. While more exotic possibilities are tantalizing, in this
paper, we focus on the possibility that the new physics integrated out is strongly interacting
and effecting the higgs sector above the scale M while the higgs itself has a relatively low
mass mh <∼ 2mt.
Various bounds can be placed on M from low energy experiments. In particu-
lar, flavor changing neutral current bounds such as those arising from K0 − K̄0 mix-
ing impose strong constraints, M ≥ 104TeV. These bounds can be relaxed by re-
stricting the higher dimensional operator basis through adopting the MFV hypothesis
[19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. This allows one to consider M to be a few TeV
while naturally suppressing FCNC.
However, even utilizing the MFV hypothesis to justify new physics at a TeV, higher
dimensional corrections to the standard model could exist that modify the relation mW =
mZ cos θW . This relationship is experimentally required to be respected to a fraction of a
percent. The PDG quotes ρ0 = 1.0002
+0.0007
−0.0004 for the global fit [29] of precision electro-weak
observables. This fact motivates the consideration of new physics being integrated out that
preserves ρ0 ≈ 1 naturally, even with possible strong dynamics effecting the higgs at the
scale M.
This can be accomplished assuming an approximate custodial SU(2)C symmetry [30, 31,
32], where the weak SU(2) gauge vector bosons transform as a triplet and the higgs field
transforms as a triplet and a singlet. The higgs vacum expectation value is in the singlet
representation of SU(2)C , so the approximate symmetry is explicitly realized, and is ex-
plicitly broken only by isospin splitting of fermion yukawa couplings and by hypercharge.
We require that the operator extensions to LSM respect this SU(2)C symmetry up to hyper-
charge and Yukawa coupling violations, as in the standard model. Operators that break the
custodial symmetry are allowed but their coefficients are taken to be naturally suppressed.
In the SM, the higgs cubic and quartic couplings are not independent parameters, but
given in terms of the higgs vacuum expectation value v and massmh. The obvious immediate
effect of D > 4 operators is to shift all these quantities in independent ways, so that
effectively the higgs cubic and quartic couplings become independent parameters. Of course,
the shift from the SM values is somewhat restricted, of order (v2/M2)C, where C is a
dimensionless coefficient, C ∼ 1. The effect on single higgs production rates of modifications
to the coupling of the higgs to weak vector bosons or to itself was investigated in Ref. [11].
The modification of higgs decay widths and this general class of models was also examined
in [33].
In this paper we address the question of whether a bound state of higgs particles can
form. In the SM a higgs bound state forms only if the higgs is very heavy[34, 35]. There is a
competition between the repulsive interaction of the quartic coupling, λ1, and the attractive
interaction of the higgs exchange (between higgs particles) which is determined by the cubic
coupling, λ1v. For large enough coupling the exchange interaction is strong enough to
produce binding, but since the mass,
λ1v, is also given in terms of the coupling, the higgs
mass is large. In the effective theory context the three parameters (mass and cubic and
quartic couplings), are independent and a bound state is possible for smaller higgs mass.
The question of detail becomes, how is the bound on the higgs mass for a bound state to
form relaxed by the coefficients of D = 6 operators? Can one have a bound state of light
higgses? We find that the effect of these operators can be significant, allowing for bound
states for much lighter higgs particles. Discovery of such bound states would give valuable
information on the scale of new physics.
There is no know solution to the bound state problem for identical scalar particles inter-
acting via cubic and quartic interactions. The higgs bound state problem has been addressed
using different approximations, the N/D method is Ref. [34] and a truncated version of the
homogeneous Bethe-Salpeter equation in Ref. [35]. Our aim here is to find a necessary con-
dition on the coupling for which a non relativistic (NR) bound state may form. To this end
we introduce a new method. We propose to study the formation of the bound state in a
non-relativistic effective theory for higgs-higgs interactions.
We begin by listing the D = 6 operators of the effective theory. We take two approaches.
In the first, linear realization, we consider operators that can be built out of the higgs
doublet and the fields in the gauge sector of the SM. Our primary interest here is in the
higgs sector per se, so we focus on higgs self interactions. The second approach, uses a
non-linear realization of the symmetry. Since the higgs field is intimately connected to the
symmetry breaking of the SM gauge symmetry, it is natural to expect that below the scale of
new physics the effects of symmetry breaking are already apparent. Were the higgs mass as
large as the scale of new physics, the SM would be supplemented not with a higgs doublet
but with a triplet of would-be goldstone bosons that are eaten by the W and Z vector
bosons. The higgs, if somewhat lighter than the scale M, would appear as a singlet under
the gauge symmetry.
We then proceed to construct the effective theory at low energies. If mh <∼ 2mt, one can
incorporate the virtual effects of the top by integrating it out and constructing a top-less
effective theory. In order to investigate the minimal coupling for which a NR higgs-higgs
bound state may form we then construct a non-relativistic higgs effective theory, and proceed
to determine this condition.
II. HIGGS EFFECTIVE FIELD THEORY: LINEAR REALIZATION
A. The D = 6 Custodial SU(2) Higgs Sector
The Lagrangian density of the standard model containing the higgs field1 is given by
L4φ = (Dµ φ)
(Dµ φ)− V (φ) (1)
where φ is the higgs scalar doublet. The covariant derivative of the φ field is given by
Dµ = 1 ∂µ − i
Bµ − i g2
W Iµ (2)
where σI are the pauli matrices, W Iµ , Bµ, are the SU(2) and U(1) SM gauge bosons and the
hypercharge of 1/2 has been assigned to the higgs. The higgs potential at tree level is given
V (φ) = −m2 φ† φ+ λ1
. (3)
No dimension five operator can be constructed out of higgs fields and covariant derivatives
that satisfies Lorentz symmetry and the standard model’s gauge symmetry.2 Utilizing the
equation of motion of the higgs field and partial integration the number of dimension six
operators is reduced. The effective Lagrangian density of the extended standard model is
given by
Lφ = L4φ +
+O( v
), (4)
1 We have omitted Yukawa interactions with fermions here.
2 To satisfy Lorentz invariance an even number of covariant derivatives are required. To be invariant under
the SU(2)×U(1) gauge group the operator must be bilinear in φ† and φ.
where the dimension six operators that preserve the symmetries of the standard model and
custodial SU(2)C in the Higgs sector are given by
L6φ = C1φ ∂µ (φ† φ)∂µ (φ† φ) + C2φ
(Dµ φ)
(Dµ φ)− λ2
. (5)
Note that the operators considered here preserve custodial symmetry and can result
from tree level topologies in the underlying theory.[36] As such, these operators need not be
suppressed by loop factors of 1/16π2 or proportional to a small custodial symmetry breaking
parameter. For these reasons these operators are expected to have the dominant effects on
the higgs self couplings and we take their coefficients to be O(1). There is only one operator
in the Higgs sector that violates custodial symmetry and could come from an underlying tree
topology, (φ†Dµ φ)2. The underlying topology in this case determines that the symmetry
breaking parameter is given by g21. The coefficient of this operator has been determined [37]
to be C < 4× 10−3 where we have used Λ = 1TeV. We neglect this operator.
We expand the higgs field about its vacuum expectation value with 〈h(x)〉 = 0 and treat
v2/M2 as a small perturbation. We expand the field as usual around a vacuum expectation
value v so that
φ(x) =
U(x)√
v + h(x)
 . (6)
Here U(x) = ei ξ
a(x)σa/v and the would-be goldstone boson fields of the broken symmetry are
ξa. In unitary gauge, the gauge transformation can be used to remove the goldstone boson
fields. We then redefine the higgs field (h) so that the kinetic term is normalized to 1/2,
using the field redefinition
(1 + 2CKh )
, (7)
where CKh = (v
2/M2)(C1φ + 14C
φ). The effective Lagrangian density is given, in terms of the
rescaled field, by
L4φ +
∂µ h′ ∂µ h
′ − Veff(h′) + C i,jh′ O
h′ + CW W OW W + CZ Z OZ Z
h′ W W O
h′ W W + C
h′ Z Z O
h′ Z Z , (8)
summed over i, j such that i+ j = 2 , where
(h′)i vj
∂µ h′ ∂µ h
OW W = W
µ , OZ Z = Z
h′ W W =
(h′)i vj
W+µ W
µ , O
h′ Z Z =
(h′)i vj
Z0µ Z
µ. (9)
The coefficients are given by
h′ = 0, C
4C1φ + C
C1φ +
, CW W = m
1 + C2φ
CZ Z =
1 + C2φ
h′ W W = m
C2φ − 2C1φ +
h′ W W = m
C2φ − 2C1φ +
3 ,−1
h′ W W = 2m
4 ,−2
h′ W W =
m2W C
h′ Z Z =
C2φ − 2C1φ +
h′ Z Z =
C2φ − 2C1φ +
3 ,−1
h′ Z Z = m
4 ,−2
h′ Z Z =
C2φ. (10)
The effective potential is
Veff (h
h′3 +
h′4 +
30 λ2
5 !M2
v h′5 +
30 λ2
6 !M2
h′6, (11)
which is written in terms of the rescaled mass term and the effective couplings, which are
given by
1− 2CKh
), (12)
3 = 3 λ1
1− 3CKh
), (13)
4 = 3 λ1
1− 4CKh
+O( v
). (14)
We will suppress the prime superscript on the higgs field for the remainder of the paper.
B. D = 6 SM Field Strength Operators
The operators that can be constructed out of the higgs scalar doublet and the field
strengths (or duals) of the standard model are as follows. We restrict our attention to those
operators listed in [6, 15] that preserve the SU(2)C custodial symmetry:
L6φ,V
= − cG g
GAµ ν G
Aµν − cW g
W Iµ ν W
I µ ν − cB g
Bµ ν B
− c̃G g
G̃Aµ ν G
Aµν − c̃W g
W̃ Iµ ν W
I µ ν
− c̃B g
B̃Aµ ν B
µν . (15)
Here GAµν , W
µ ν and Bµ ν stand for the field strength tensors of the SU(3) × SU(2) × U(1)
gauge bosons, and a tilde denotes the dual field strengths, F̃µν = ǫµ ν λσ F
λσ /2. Note that
the operator that is proportional to the S parameter given by
− cW B g1 g2
φ† σI φ
Bµ ν WI µ ν (16)
violates custodial symmetry and is naturally suppressed in our approach 3.
C. D = 6 Fermion Sector
Operators of dimension 5 and higher that couple the higgs to fermions, or purely fermionic
operators, can give rise to unacceptably large flavor changing neutral currents (FCNC). If
the coefficient of such operators are generically of order 1 the scale of new physics must be
taken to be M >∼ 104 TeV in order to suppress FCNC effects. We adopt the Minimal Flavor
Violation hypothesis (MFV) to naturally suppress the dangerous operators while maintaining
a low scale of new physics, M >∼ 1 TeV. In the absence of quark and lepton masses the SM
has a large flavor symmetry group, GF = SU(3)
5. The MFV asserts that there is a unique
source of breaking of this symmetry. All operators that break the symmetry must transform
precisely the same way under GF . As a result FCNC operators are suppressed by the
familiar factors of the Kobayashi-Maskawa (CKM) matrix in the quark sector and by the
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix and small neutrino masses in the lepton
sector.
Since the effects of fermionic operators are not needed for the rest of this investigation,
we do not list the operators. The interested reader can find a complete description of the
operators and their effects in [6].
3 See Appendix A
III. HIGGS EFFECTIVE FIELD THEORY: NON LINEAR REALIZATION
The construction in the previous section assumes that the field content of the effective
theory includes a higgs doublet. This is not necessary. If the electroweak symmetry is
spontaneously broken by a strong interaction the spectrum below the scale of this new
physics does not have to be described by a higgs doublet field, beyond the SM fields. Only
fields describing the would-be goldstone bosons need be introduced. Such higgs-less theories
have been discussed in the literature[38]. However, if the higgs particle is somewhat lighter
than the scale of new physics it has to be incorporated in the low energy description and
symmetry alone does not dictate that it appears as a member of an iso-doublet. It is sufficient
to have the goldstone bosons realize the broken symmetry non-linearly, and the higgs field
is then a singlet under the symmetry.
The situation is entirely analogous to the case of π’s and the σ in QCD. A phenomeno-
logical Lagrangian density describing π and σ interactions does not have to be a linear
realization of the chiral SU(2) × SU(2) symmetry. Instead, the π-fields have a better de-
scription through a non-linear chiral Lagrangian. Then the σ can be included through
interactions that satisfy the non-linearly realized symmetry and the usual rules for naive
dimensional analysis [39].
In the non-linear realization, the Lagrangian density in Eq. (1) is replaced by
LNL = 14v
2TrDµU
†DµU + 1
µh− V (h), (17)
where the would-be goldstone bosons ξa appear through the matrix U(x) = ei ξ
a(x)σa/v that
transforms under SU(2)L × SU(2)R linearly, U → LUR†, and h is a singlet field, describing
the higgs particle. Custodial symmetry SU(2)C is the diagonal subgroup of SU(2)L×SU(2)R
and the higgs field is invariant under it. 4
This Lagrangian is supplemented by higher order terms suppressed by powers of M. In
the case of the higgs potential, this can be included simply as
V (h) = M4f(h/M), (18)
where f(x) is an arbitrary function with a minimum at zero. The mass and couplings of the
4 The custodial symmetry is discussed in more detail in Appendix A.
higgs are given in terms of this dimensionless function by
m2h = M2f (′′)(0), (19)
3 = Mf (′′′)(0), (20)
4 = f
(iv)(0). (21)
It is not a surprise that in the non-linear realization of the symmetry the couplings and mass
are completely independent, and that they are all naturally of order 1 times the appropriate
power of the dimensionfull scale, M. The natural scale for the higgs mass is M, and we
are considering here the class of theories for which f (′′)(0) happens to be small, while higher
derivatives may remain of order 1. We stress that the natural scale for the cubic coupling is
M. Unless the mechanism (or numerical accident) that keeps the higgs mass small compared
to M also acts to suppress the cubic coupling, one must naturally expect λeff3 ∼ M/v ≫ 1.
We will also need the corrections to the derivative interactions. We write, generally,
L = 1
1 + c
∂µ h ∂µ h−
m2h h
2 − v λ
h3 − λ
h4 + · · · (22)
In the linear realization the derivative interaction couplings are related, 1
CKh = (v
2/M2)(C1φ + C2φ/4), but in the non-linear realization they are independent. And,
as in the case with λ
3 naive dimensional scaling gives an enhancement of c
1 that could
arise from the non-perturbative dynamics of the symmetry breaking sector. Naively, c
(v/M), which is enhanced over the linear realization value by a power of (M/v).
As we mentioned earlier, non-linear realizations have been extensively studied for higgs-
less theories, but have been neglected in studies including a light higgs. There are two
important consequences of the non-linear realization outside the pure higgs sector that we
point out here. It has been noted that significant corrections to the coupling of a higgs to
gluons are possible from D > 4 operators. The modifications can be large because there
is no SM contribution at tree level. In the linear realization there is a D = 6 operator
that contributes at tree level, and therefore competes with the SM one loop, top mediated
amplitude:
GaµνG
µν , (23)
Note that the linear realization implies a relation between the one and two higgs couplings
to two gluons. However, in the non-linear realization the two couplings are completely
independent,
GaµνG
µν . (24)
In Ref. [40] it was noted that a heavy quark with Yukawa coupling λ → ∞ produces a
coupling of two gluons to one or more higgs particles that cannot be described by the
effective theory operator in (23). Instead a nonpolinomial interaction was introduced to
describe this effect,
)GaµνG
There is no problem accommodating such interactions in the non-linear realization, by
ln(1 + h/v)GaµνG
[h/v + (h/v)2 + · · · ]GaµνGaµν . (25)
In much of what follows we implicitly assume the linear realization. However, results in
terms of the arbitrary parameters mh, λ
3 and λ
4 can be interpreted readily as arising
from the non-linear realization.
IV. A LOW ENERGY EFFECTIVE THEORY FOR THE HIGGS
In this section we will construct an effective theory for the light higgs, integrating out
momentum modes heavier than the higgs. This is useful in discussing physical effects with
a typical energy of order of the higgs mass. In particular, we integrate out the top quark.
As the coupling of the top quark to the higgs is fairly large, we would like to estimate the
effects of the top quark on the possibility of forming a higgs bound state. If the top quark
mass is much heavier than the higgs it is appropriate and convenient to describe the higgs
self interactions in a top-less theory. When the top quark has been integrated out, its effects
are accounted for through modifications of coupling constants and mass of the higgs.
While this is clearly appropriate when the top quark is much larger than the higgs mass,
we use this approximation even when the higgs is slightly heavier than the top. For mh <
2mt the approximation is known, ipso facto, to work better than one would expect. This is
due, in part, to the fact that there is no non-analytic dependence on the mass since the higgs
is the pseudo-goldstone boson of spontaneously broken scale invariance[41, 42, 43]. It is also
known that soft gluon effects are large and correctly reproduced by the effective theory [44].
For single and double higgs production, comparisons between the the full theory calcu-
lation and the effective top-less theory find that the latter is a good approximation for the
total rate for mh <∼ 2mt. For example, with the appropriate K factor, the resulting topless
effective field theory calculated to two loops is known to accurately describe the full NLO
result for g g → h to better than 5% accuracy in the full range 0 < mh < 2mt [44].
As another concrete example, consider the higgs mass dependence in the higgs IPI self-
energy. The first graph of Fig. 1 is the contribution of the top quark to the IPI self-energy
which we label −iΠ(p2). At 1-loop we find
Π(p2) =
1− x(1− x)
1 + 3 log
m2t − x(1− x)p2
, (26)
where we have used the MS subtraction scheme. The quantity 1− Π(0)/Π(p2) (at µ = mt)
never exceeds 30% when
p2 ranges from zero to 2mt.
For these reasons we consider it appropriate to integrate out the top quark for mh <∼ 2mt
in this initial study. When mh >> mt, these corrections should be taken only as indications
of the size of virtual top effects. While the approximation of neglecting higher order terms
in the p2/m2t expansion is known to work better than expected for the applications we will
consider, there is no guarantee that it will work well for processes not considered here.
[45, 46, 47, 48, 49]
A. Running to mt
The coefficients of the D = 6 operators at the scale M are unknown. We are assuming
that the new physics couples to the higgs field and is strongly interacting at the scale M.
In this context, it is natural to take
C iφ (M) , λ2 (M) ∼ 1. (27)
Similarly it is natural to assume that the coefficients of the D > 4 operators that couple the
higgs to other fields, like those in Eq. (15) or those that couple the higgs to quarks while
satisfying the MFV hypothesis, are all order unity.
The anomalous dimensions of the extended operator basis can be determined system-
atically. This is beyond the scope of this paper. But the effect of the running is easy to
understand. With minimal subtraction the calculation of the running of coefficients of higher
dimension operators can be done in the symmetric, massless phase. There is operator mixing
among the D = 6 operators with common quantum numbers. The anomalous dimension
matrix is a function of the relevant couplings (λ1, g1, g2 and the top quark Yukawa, λt).
The running is always proportional to these coefficients so the effect is roughly of the form
C iφ (mt) ∼ C iφ (M)
, (28)
where mixing is implicit, and c1α stands for a linear combination of λ1 and the squares of
g1, g2 and λt.
Since log (mt/M) ∼ 1 and the coefficients c1 ∼ 1 the running produces a small, calculable
shift in the unknown coefficients. Hence, we continue to take the unknown Wilson coefficients
at the scale mt to be ∼ 1 .
At mt the top quark is integrated out and this produces a different effect, a shift in the
C iφ(mt) by a C
φ-independent amount. This can be numerically significant, and we estimate
this next. Note that once the top is integrated out we continue to run down to the mass of
the higgs scalar mh. The effect of the running of these coefficients from mt to mh is again
small, so we take
C iφ (mh) , λ2 (mh) ∼ 1. (29)
B. Integrating out the top quark
Integrating out the top leads to further corrections to the higgs sector of the standard
model. The top mass is a result of symmetry breaking, so the resulting effective theory is
better presented in unitary gauge, as in (8) and (11). In unitary gauge, the top mass term
and coupling to the higgs is given by
LY = −mt q̄t qt
. (30)
We begin by considering effects on the higgs self-couplings. Figure 1 show the Feynman
graphs that contribute to modifications of the higgs self-couplings. The solid line denotes a
top quark, the dashed external lines denote the higgs.
We perform the calculation to lowest order in p2/m2t . Some details of the computation
are given in the appendix. The effect of these corrections is to further modify the effective
potential of the higgs scalar field h. The effective couplings and mass term of Eqs. (12)–(14)
FIG. 1: Integrating out the top quark.
are shifted by these corrections, and are now are given by
1− 2CKh
m2t m
), (31)
3 = 3 λ1
1− 3CKh
m2t m
), (32)
4 = 3 λ1
1− 4CKh
− 4Nc
+O( v
m2t m
). (33)
As emphasized above, these corrections are not multiplicative, that is, they are present
even for λ2 = C
h = 0. Whether they are important depends on the scale and strength of
the new physics. The condition
. (34)
is satisfied for λ2 ≈ 1 when M ≈ 2πv = 1.6 TeV. So the corrections are numerically
comparable to these new physics terms. Similarly, for λ1 ∼ 1 the condition
CKh λ1 =
(C1φ +
C2φ)λ1 ∼
. (35)
still requires M ≈ 1.6 TeV for C1φ + 14C
φ ∼ 1.
C. Corrections to Field Strength operators
Integrating out the top quark also results in effective operators of the higgs field and
the SM field strengths. The dominant SM production mechanisms for the higgs at LHC is
the gluon fusion process g g → h. We restrict our attention to such operators that effect
the production processes of the higgs through gluon fusion. Figure 2 shows the 1-loop
Feynman diagram for the top contribution to g g → h. For a higgs with mh < 2mt, the
expected production cross section of the g g → h process has been determined up to NNLO
[50, 51, 52]. For SM gluon fusion, the single higgs production mechanism is given by the
mt → ∞ effective Lagrangian density comprised of a dimension five operator
Lmt = C1GGh (αs)
Gaµ ν G
a , (36)
where the coefficient is given in the MS scheme, in terms of αs for five active flavors, by
[53, 54, 55, 56, 57, 58]
C1GGh (αs) =
11α2s
48 π2
+O(α3s). (37)
FIG. 2: The gluon fusion g g → h production process. The production process through the effective
local operators in shown in the second column. The effective local operators come from integrating
out the top quark and new physics at M.
The interactions in the effective Lagrangian of Eq. (15) also contribute to single higgs pro-
duction through gluon fusion. Combining results, at the scale mh, the effective Lagrangian
density for single higgs production is given by
Leff = CeffGGh
GAµν G
Aµν + C̃
G̃Aµ ν G
Aµν , (38)
where
GGh = C
GGh − 2 π αs cG
, (39)
GGh = −2 π αs c̃G
. (40)
Assuming that the new physics degrees of freedom carry the SU(3) gauge charge, the
Wilson Coefficients cG, c̃G will be approximately the same size as the coefficients C
φ, λ2 we
are interested in. If the new physics degrees of freedom are charged under SU(2)×U(1) but
not SU(3), below the scales M, mt effective local operators of this form will still be induced.
However, the corresponding Wilson Coefficients will be suppressed by factors of 16 π2.
The effect of these interactions on higgs production rates was examined in [14]. Note that
in the standard model, contributions to the operator G̃Aµν G
Aµν are highly suppressed[59]
and therefore neglected.
FIG. 3: The gluon fusion g g → hh production process and the effective local operators.
The production process of two higgs in the standard model is shown in Fig. 3. In analogy
with the single higgs production case we characterize the process in the effective theory by
Leff = CeffGGhh
GAµν G
Aµν + C̃
G̃Aµν G
Aµν , (41)
where the coefficients are given by
GGhh = C
GGhh −
cG π αs v
, (42)
GGhh = −
c̃G π αs v
. (43)
Here the top quark contribution is[56]
C1GGhh(µ
2) = −
11α2s
48 π2
+O(α3s). (44)
The Wilson coefficients for two higgs production in the effective theory is not suppressed
relative to the corresponding Wilson coefficient for single higgs production. Note that unlike
the case of single higgs production the expansion in p2/m2t does not, in general, have kine-
matics such that p2/m2t ∼ m2h/m2t . In two higgs production, higher order terms in p2/m2t
have p2 = s, t, u and in general (s, t, u)/m2t is not small. We calculate the next order in
the expansion of p2/m2t in Appendix B. These terms are neglected, and our application of
the expansion is valid for finite values of mt due to our interest in establishing a necessary
condition for a NR bound state to form. The kinematics for the production of a NR bound
state at threshold dictate (s, t, u)/m2t ∼ m2h/m2t .
V. PHENOMENOLOGY OF HIGGS EFFECTIVE THEORY
A. The Magnitude of Self Couplings
The effect of the D = 6 operators in the effective potential cause corrections to the
three and four point contact interactions and mh. To illustrate that the induced effects on
the higgs sector are under control, consider extending the effective potential with a single
dimension eight term. We find the following while neglecting the effects of integrating out
the top quark
1− 2CKh
+ 3× 10−2 λ2
1− 2CKh
+ 4.5× 10−4 λ3, (45)
3 = 3 λ1
1− 3CKh + 7.5 (CKh )2
+ 1.5× 10−1 λ2
1− 3CKh
+ 3.1× 10−3 λ3, (46)
4 = 3 λ1
1− 4CKh + 12 (CKh )2
+ 4.5× 10−1 λ2
1− 4CKh
+ 1.6× 10−2 λ3. (47)
From which one sees we are examining the potential of the theory in a controlled expansion,
even for M ∼ 1TeV.
Eliminating the self-coupling λ1 in favor of the higgs mass, we can write for the effective
cubic and quartic higgs-self couplings,
3 = 3
1− CKh
) m2h
− 7Nc
, (48)
4 = 3
1− 2CKh
) m2h
+ 6λ2
− 19Nc
. (49)
With v = 246GeV, mt = 174GeV and M = 1 TeV, and taking mh = v/2, these are
3 = 0.62− 0.05(C1φ + 14C
φ) + 0.06λ2, (50)
4 = 0.39− 0.09(C1φ + 14C
φ) + 0.36λ2. (51)
For negative λ2 of order one one can greatly reduce the repulsive contact interaction, λ
in a putative higgs-higgs bound state. Of course, this comes at the price of reducing the
attractive interaction, governed by λ
B. g g → hh Production
From our results in Section IVC, the production of two higgs in our effective theory
framework is straightforward to write down. The contributions to the amplitude are shown
in Fig. 4.
h h h h
FIG. 4: The two higgs production process in the effective theory.
The amplitude for two higgs production, to O(αs), is given by
〈h h|i A|Aα(P1)Aβ(P2)〉 = 〈h h|i A1|Aα(P1)Aβ(P2)〉+ 〈h h|i A2|Aα(P1)Aβ(P2)〉 (52)
where we have
〈i A1〉αβ = 2 i CF (CeffGGh)
fαβ(P1, P2)
(P1 + P2)2 −m2h + i ǫ
P 23 + P
4 + (P1 + P2)
〈i A2〉αβ = 4 i CF (CeffGGhh) f
αβ(P1, P2), (53)
where
fαβ(P1, P2) ≡ P α1 P
2 + P
2 − 2 gαβ P1 · P2. (54)
Using two higgs production as a test of the cubic self coupling of the higgs has been
examined in [60] where testing for the MSSM with this signal was investigated. Ref. [40]
advocated the examination of g g → h h to compare the one and two higgs production
coefficients in Eq. (24) since the naive relation between the two coefficients could be upset
by the presence of novel operators like that in (25). As we have discussed, in the non linear
realization of broken electro-weak symmetry, the relationship between g g → h h production
and g g → h production is not fixed as in the linear realization. Any deviation from the SM
value for g g → h h must be interpreted with care. The g g → h h production rate in our
effective theory construction (in the linear realization) depends on at least six unknowns,
namely, M, λ2, C1φ, C2φ, cG, c̃G. The effects of the operator advocated in [40] increase the
number of unknown parameters still further.
Clearly two higgs production is an important signal to test the higgs mechanism in the
standard model. The cross section of g g → h h is suppressed compared to the cross section
of g g → h by a factor of 1000, due to the effects of parton distribution functions and
phase space suppression[60]. The cross section falls from 50 fb to 10 fb as the higgs ranges
in mass from 100GeV to 200GeV. Thus once LHC enters its high luminosity running of
100 fb−1/Year one can expect roughly 1000 events per year. A significant excess or deficit of
this signal should be observable. However, the reconstruction of exactly what form of new
physics is present requires more information.
One could obtain more information on the unknown parameters involved by further
probes of the physics of the self interaction of the higgs. In the remainder of the paper
we examine the sensitivity of a higgs bound state (Higgsium) in the appropriate low energy
effective field theory on TeV scale physics to these parameters. If a bound state forms, one
can use the properties of the bound state such as its binding energy, as an a probe of the
physics above the scale M.
C. Higgsium: Production and Decay time
To get some rough understanding of the conditions under which a higgs-higgs bound state
may form, consider the non-relativistic Schrodinger equation
−∇2r + V (r)− E
ψ(r) = 0, (55)
with the potential from a yukawa exchange and a contact interaction,
V (r) = −
e−mh r
+ κ δ3(r). (56)
We are interested in the case g ∼ λeff3 and κ ∼ λ
4 as a non relativistic approximation of
the higgs self interactions. Neglect for now the contact interaction. It is well known that
the Yukawa potential produces bound states provided
>∼ 1.7, (57)
Neglecting new physics effects,
3 ≈ 3
, (58)
non-relativistic bound states could be expected for
mh >∼ 1.2v. (59)
The effect of TeV scale new physics changes the relationship between the mass and the
coupling. The above Yukawa bound state condition is modified to
1.54 + 0.09(C1φ +
C2φ)− 0.02 λ2
2 v. (60)
This demonstrates the point that if the higgs self coupling is significantly stronger due to
strong TeV scale new physics that contributes large Wilson coefficients, then a low energy
signal of this higher scale physics might be a NR bound state formed by two higgs.
However, one can see that it is difficult to realize the NR bound state condition when
we identify the couplings in this Schrodinger equation with our effective couplings. This
identification is in fact incorrect. We will demonstrate in Section VI that the correct NR
limit of the higgs sector is described by a Lagrangian containing only contact interactions
and higher derivative operators.
The formation time of the bound state can be approximated by the ratio of 4R0/u where
R0 is the characteristic radius of the NR bound state and u is the relative velocity of the
two higgs. This is roughly the period of oscillation for S wave states [61].
For a NR bound state we can approximate the relative momenta of the two higgs by
p ∼ mh u so that
. (61)
The SM higgs decays predominantly via h → b b̄ pairs through Yukawa interactions if
114.4 < mh ≪ 2MZ . We take these decays as dictating the decay width of Higgsium.
Neglecting the effects of our new operators, this decay has the decay width
. (62)
This gives an approximate decay time
. (63)
The condition that the bound state has time to form is that τf < τb which can be satisfied
. (64)
Thus a non relativistic bound state has time to form before it decays. Above 135GeV and
below the threshold of W+W− production, the dominant decay of the higgs is through a
virtual W pair, h → W W ⋆. Above mh > 2mW the decay into W+W− predominates and
the decay width is given by
1− aW
4− 4 aW + 3 a2W
, (65)
where aW = 4m
h using the notation of [29]. Comparing the formation and decay time
for a higgs whose mass is above the threshold of W+W− production we find
η(mh, mw)
, (66)
where η(mh, mw) ∼ 1.
The lower bounds on u in either case are compatible with the NR approximation for the
full range of higgs masses we consider. A relativistic bound state is also possible in either
case, although an approximation scheme that can estimate its formation time is lacking. In
the remainder of the paper we focus on the possibility of a NR bound state being formed by
a relatively light higgs, mh < 2mt, due to our treatment of the top quark. We also note that
a NR bound state may also have observable effects on the spectrum of two higgs production
even if a bound state does not fully form as in the case of top quark pair production near
threshold in e+e− collisions [61].
VI. NON RELATIVISTIC HIGGS EFFECTIVE THEORY
If the two higgs are created with small relative velocity and form a non relativistic bound
state it is appropriate to describe the physics of this state with a non relativistic effective
field theory of the higgs sector. We refer to our effective theory derived in Section I through
Section IVC as higgs Effective Theory (HET) and now match onto a non-relativistic version
of this theory (NRHET) where we take the c → ∞ limit of the scalar field Lagrangian
density of HET. Recall the Lagrangian density is of the form
1 + c
∂µ h ∂µ h−
m2h h
h4 +O
. (67)
We wish to construct the non relativistic limit of this Lagrangian density systematically,
retaining h̄ = 1 and making factors of c explicit with [c] ∼ [x]/[t]. The dimensionful
quantities can be expressed in units of length [x] and time [t]. As h̄ = 1, we still have
[E] ∼ 1/[t] and [p] ∼ 1/[x]. As the action S =
dt d3xL is dimensionless, we have [L] ∼
[x]−3 [t]−1. For the time and spatial derivatives to have the same units in L we take
, (68)
and so ∂µ ∼ 1/[x]. This gives [h] ∼ 1/
[x] [t]. We require [mh c
2] ∼ [E] ∼ 1/[t], so that
we have [mh] ∼ [t]/[x]2, and choose the electroweak symmetry breaking expectation value
to have the same dimensions as the field h, [v] ∼ 1/
[x] [t]. The Lagrangian density with
these unit conventions is given by
L = 1
1 + c
∂µh ∂µh−
m2h c
2 h2 − v λ
3 ! c
h3 − λ
4 ! c
h4 +O
Now consider the non-relativistic limit of this theory. The interaction terms will be
determined below by matching. Consider first the theory of a free real scalar field of mass
mh given by
L = 1
∂µϕ∂µϕ−
m2h c
2 ϕ2. (70)
The field ϕ must also be expanded in the c → ∞ limit. We remove a large energy scale
2 from this field with a field redefinition
ϕ(x) = e−imh c
2 r·x ϕ+(x) + e
imh c
2 r·x ϕ−(x), (71)
where r = (1, 0) and ϕ+(x), ϕ−(x) correspond to the creation and annihilation components
of the scalar field ϕ(x). Expanding the Lagrangian density in terms of ϕ±(x) we neglect
terms multiplied by factors of
en imh c
2t, (n 6= 0, n ∈ I). (72)
These are terms in the Lagrangian density where some of the fields are far off shell. Their
effect is only to modify coefficients of local operators in the effective action.
With this substitution we find
L = ∂0 ϕ− ∂0 ϕ+ − ∂i ϕ− ∂i ϕ+ + imh c
0 ϕ+ − ϕ+ ∂0 ϕ−
. (73)
The first term, with two time derivatives, is suppressed by 1/c2 and is suppressed in the
c→ ∞ limit. Integrating by parts the remaining kinetic terms and rescaling h± =
2mh ϕ±
gives
L0NR = h−
h+. (74)
One can extend this effective Lagrangian by adding interactions, including higher order
terms suppressed by |u|/c and v2/M2 where u is the relative velocity of the two higgs in a
non-relativistic bound state. Scattering is described by a contact interaction which can be
parameterized by a coupling CNR,
LNR = h−
+. (75)
There is no cubic interaction because this necessarily involves at least one far off shell
particle. The effect of the cubic interaction in the HET is incorporated in the coupling CNR,
and we will compute this in terms of the parameters of the HET below, in Sec. VIA.
In this effective theory, the energy and the momenta of the system are given by
mh |u|2, (76)
q = mh u, (77)
where |u| ≪ c is the relative velocity of the two higgs. It is advantageous to have power
counting rules in |u| that are as manifest as possible in the Lagrangian density as demon-
strated in [62]. We rescale so that the natural sizes of the coordinates are given by the above
energy and momentum and define a new field H±(x) and new, dimensionless coordinates X
and T by
x = λx X, t = λt T, h±(x) = λhH±(x). (78)
To ensure the rescaled energy and momenta are of order unity we have λt = mhλ
x and
mh |u|
, (79)
mh |u|2
, (80)
λh = m
h |u|
3/2, (81)
mh |u|2
, (82)
mh |u|
. (83)
The form of the Lagrangian density when we implement these re-scalings and introduce an
appropriately rescaled contact coupling, ĈNR = 4m
hCNR is given by
LNRH = H−
i ∂0 +
. (84)
This form of the Lagrangian makes power counting explicit in the small parameter u/c.
Physical quantities, such as the energy of bound states, can be equally calculated from the
theories in Eqs. (75) or (84). Which is used is a matter of convenience: the former has
familiar dimensions while the latter has explicit power counting.5
A. Matching onto NRHET
To determine the matching coefficient CNR we take the non relativistic limit of the h h →
h h scattering determined in HET. We neglect the running from m2t down to our matching
scale µ2 = m2h in this initial study, and perform the matching at tree level only.
1. Linear Realization
The HET contact interaction is given by
AL0 = −3 λ1 + 20 λ1CKh −
. (85)
5 The c → ∞ limit of NR effective field theories was studied in [63]. The reader interested in bound
states at threshold in NRHET would also profit from an examination of the treatment of bound states at
threshold in NN effective field theory, reviewed in [64].
The Yukawa exchange feynman diagrams, shown in Fig. 5, give the amplitude
iALy (s, t, u) = i(AL1 (t) + AL1 (u) + AL1 (s)), (86)
where s, t, u are the usual Mandelstam variables, and
AL1 (x) =
−3v2 λ1
x−m2h + i ǫ
3 λ1 + 5 λ2
− 4 (x+ 2m2h)
− 2Nc
− 18 λ1CKh
.(87)
A1(s) A1(u) A1(t) A0
FIG. 5: Tree level hh → hh scattering in the extended higgs theory. Time flows left to right.
The total amplitude for h h→ h h scattering is given by
ALhh→hh(s, t, u) = AL0 +ALy (s, t, u). (88)
To perform the matching we take the momenta of the higgs particles to be off-shell by a
small residual momenta p̃ with energy and momenta that scale as p̃0 ∼ mhu2 and p̃ ∼ mhu.
The momenta of the higgs are decomposed as (recall r = (1, 0))
p = mh r + p̃, k = mh r + k̃,
p′ = mh r + p̃′, k
′ = mh r + k̃′. (89)
This gives, in the center of mass frame
s = 4m2h + 4 |q|2,
t = − |q|2 (1− cos(θ)) , (90)
u = − |q|2 (1 + cos(θ)) ,
with q ∼ mhu. In the non relativistic limit we retain the lowest order in |u| and we have
ALNR = AL0 +AL1 (4m2h) + 2AL1 (0)
= 12 λ1 + 10 λ2
− 64 λ1CKh −
. (91)
To determine the coupling C4NR in the NRHET Lagrangian, Eq. (75), we compute the four
point amplitude and insist that it equals ANR. Inserting four factors of
2mh to account
for relativistic normalization of states, we finally arrive at
(2mh)
2CLNR = Ĉ
NR = 12 λ1 + 10 λ2
− 64 λ1CKh −
. (92)
2. Non-Linear Realization
For a non-linear realization we find the following for the HET contact interaction
ANL0 = −λ
4 + 4
2 . (93)
The yukawa exchange diagrams give
ANL1 (x) =
x−m2h + i ǫ
2m2h + x
. (94)
The matching is performed as in a linear realization and we find
ANLNR = ANL0 +ANL1 (4m2h) + 2ANL1 (0)
2 − λeff4 − 2c
2 − (c
) m2h
. (95)
This gives the effective HET coupling in the non-linear realization
(2mh)
2CNLNR = Ĉ
2 − λeff4 − 2c
2 − (c
) m2h
. (96)
B. NRHET Bound State Energy
To find the approximate bound state energy of the higgs, we calculate the bubble sum
in our NRHET theory and interpret the pole in the re-summed bubble chain as the bound
state energy of Higgsium. Note that this calculation is formally justified in the large N limit
[65] where the higgs sector is equivalent to and O(4) theory [66]. The Feynman rules for the
NRHET Lagrangian in (75) are shown in Fig. 6.
The bubble sum is straightforward to calculate in NRHET. The leading order term is
directly obtained from the Feynman rules, we use the Lagrangian given by Eqs. (75) in the
following. The leading bubble graph is given by
iA1-loop = (i CNR)2
dk0 ddk
(2π)d
(E + k0)− k2/2mh + iǫ
−k0 − k2/2mh + i ǫ
P0 − P2/ 2
FIG. 6: Feynman rules for NRHET.
We have chosen to work in the center of mass frame, and E = P 01 +P
2 stands for the center
of mass energy. Performing the first integral by residues and the remaining integrations with
dimensional regularization, we find
iA1−loop = −i
mh(CNR)
(−mhE)1/2. (98)
FIG. 7: The bubble sum of graphs leading to the bound state pole in NRHET.
The terms in the bubble sum of diagrams shown in Fig. 7 are given by the geometric
series
1− mhCNR
(−mhE)1/2 +
mhCNR
(−mhE)1/2
+ · · ·
i CNR
1 + mhCNR
(−mhE)1/2
This result agrees with [62, 67] and indicates a bound state with a bound state for CNR > 0
with binding energy
mhCNR
. (99)
There is an implicit renormalization condition introduced by dimensional regularization.
The integral has no pole as d → 3, so it is interesting to ask what subtraction has been
made. This is easily understood by performing the d = 3 integration with a momentum
cut-off |k| < Λ in terms of the bare coupling C0NR:
iAΛ1−loop = imh(C0NR)2
(−mhE)1/2
. (100)
The renormalized coupling CNR(µ) can be defined as the amplitude at a fixed energy E = −µ
[68]. Then the combination
CNR(µ)
(mhµ)
(101)
is renormalization group invariant. This is precisely the coupling that appears in (98).
It would appear that for any positive value of CNR we have bound states. However for
our NR description to be self consistent we require that the binding energy of the bound
state satisfy Eb < mh, that is,
ĈNR > 16π. (102)
1. Linear Realization
In the case of a linear realization a heavy higgs seems necessary for the bound state
to form, but the new physics effects may allow significantly smaller masses for the bound
state. If we neglect the effects of new physics (λ2 and C
h ) the bound (102) translates into
mh > 2.0 v. Retaining the effectsof λ2 and C
h one determines a condition for the NRHET
calculation of the bound state energy to be self consistent
16π − 4 λ2 v
+ 51Nc
12− 40CKh
. (103)
Alternatively, for a given value of the higgs mass, say mh = ξ v, the self-consistency
condition implies a constraint on the coefficients of the higher dimension operators:
12 ξ2 + 4
λ1 − 10 ξ2 (C1φ + 14C
> 16 π +
(104)
Using M = 1 TeV and the PDG value for the top quark mass, this condition simplifies to
1.2 ξ2 + 0.024λ1 − 0.24 ξ2 (C1φ + 14C
φ) > 5.1 (105)
So, for example, for |C1φ + 14C
φ| = 1 or 5 the minimal higgs mass for a NR bound state is
reduced by 6% or 28%, respectively. Near the limit of validity of our calculation mh ∼ 2mt,
for negative values of C1φ +
C2φ we find that a bound state is possible for O(1) wilson
coefficients as we illustrate in Fig 7.
-6.5 -6 -5.5 -5 -4.5
�������
M = 1 TeV, mh = 350 GeV
FIG. 8: In the linear realization the allowed parameter space for NR bound state formation is
above the line.
2. Non-Linear Realization
In the nonlinear realization this condition is easily satisfied even for a light higgs, mh < v.
Recall that λ
3 and c
1 are both enhanced by powers of M/v. Taking, for example,
mh = 120GeV and M = 1TeV, neglecting the contribution of ceff2 , the NR bound state
condition is
)2 − 2 c̃1eff λ̃3
eff −
eff)2 > 16 π + λ4, (106)
where
1 . (107)
Note that asM grows larger the region that satisfies the NR bound state condition grows.
This is due to the fact that the attractive interaction given by λ
3 ∼ M/v is a relevant
operator. We find that as mh grows and as M is larger the allowed parameter space of the
NR bound state condition is significant, demonstrating that a bound state is likely to form
in the non linear realization.
-10 -5 0 5 10
mh = 120 GeV
FIG. 9: In the Non-linear realization, holding mh fixed and set λ4 = 0 as it is O(1) and suppressed
by 16π. We vary M for the values M = 1TeV (dotted line), M = 3TeV (dashed line) and
M = 10TeV (solid line). The region above (the upper) and below (the lower) hyperbolic curves
satisfy NR the bound state condition.
-10 -5 0 5 10
M = 1 TeV
FIG. 10: In the Non-linear realization, holding M fixed and set λ4 = 0 as it is O(1) and suppressed
by 16π. We vary mh = 300GeV (solid line), mh = 200GeV (dashed line) and mh = 100GeV
(dotted line). The region above (the upper) and below (the lower) hyperbolic curves satisfy the
NR bound state condition.
VII. SUMMARY AND CONCLUSIONS
If a new strong interaction is responsible for electroweak symmetry breaking but a higgs
particle, the pseudo-goldstone boson of broken scale invariance, remains unnaturally light,
the self-interactions of this higgs particles could be quite strong. If strong enough these
self-interactions could bind two higgs particles.
To study these questions we formulated two different effective theories of the light, self-
interacting higgs below the scale M of the new physics. In the first, the symmetry is realized
linearly and the higgs field is described as one component of an SU(2)L doublet, just as in
the standard model of electroweak interactions. In the second approach the symmetry is
realized non-linearly: the triplet of would-be goldstone bosons and the higgs field are not in
a common multiplet. We note that operators of dimension 3 in the effective Lagrangian in
the non-linear realization are naturally expected to be enhanced by a power of M/v relative
to their linear realization counterparts.
In order to study how large these couplings need be, we have studied the case of non-
relativistic bound states. To this end we constructed a non-relativistic higgs effective theory
(NRHET) describing self-interacting higgs particles in the rest frame of the bound state, in
the non-relativistic limit.
The effects of the top quark are small but non-negligible. We estimated them by including
the virtual top quark effects as a modification to the couplings in the NRHET.
Our results show, perhaps not surprisingly, that in the non-linear realization it is quite
easy to form light Higgsium, as we call the higgs-higgs bound state. For natural couplings
in the linear realization a bound state is only likely to form for mh ∼ v. Relativistic bound
states are possible in both the linear and nonlinear realizations.
There are many questions that we have not addressed. The most immediate one is how to
search for Higgsium. Assuming a light higgs is found, one could imagine strategies involving
invariant mass distributions of higgs-pair production. A dedicated study is required to
determine if this or other strategies are viable. Another, related question is whether the
effects of a short lived bound state could be seen indirectly, much like would-be toponium
affecting the line shape in top quark pair production near threshold in e+e− collisions.
It would also be interesting to solve the bound state equation in the more general, fully
relativistic case. We hope to return to these problems in the future.
Acknowledgments
Work supported in part by the US Department of Energy under contract DE-FG03-
97ER40546.
APPENDIX A: CUSTODIAL SYMMETRY AND THE S PARAMETER
There is some confusion in the literature regarding custodial symmetry and the operator
− cW B g1 g2
φ† σI φ
Bµ ν WI µ ν (A1)
which corresponds to the S parameter. Consider the matrix representation of this operator
[69] where the Higgs doublet field is given by
 . (A2)
Then ǫ φ⋆ is also an SUL(2) doublet with components
ǫ φ⋆ =
 , (A3)
where φ− = φ+ ⋆. The Higgs bi-doublet field is given by
(ǫ φ⋆, φ) ,
φ0 ⋆ φ+
−φ− φ0
 . (A4)
The SUL(2)×UY(1) gauge symmetry acts on the Higgs bi-doublet as
SUL(2) : Φ → Lφ (A5)
UY (1) : Φ → Φ e−iσ3 θ/2. (A6)
In the limit that hyper charge vanishes the Lagrangian also has the following global symmetry
SUR(2) : Φ → φR†. (A7)
When the Higgs acquires a vacuum expectation value, both SUL(2) and SUR(2) are broken,
however the subgroup SUL=R(2) is unbroken, ie
L 〈Φ〉L† = 〈Φ〉. (A8)
This is explicitly the custodial symmetry, and the corresponding transformation of the Higgs
bi-doublet under this symmetry. It is easy to see that
− cW B g1 g2
Φ† σI WI µ νΦ
Bµ ν (A9)
is invariant under this symmetry. The Higgs bi-doublet transforms as above and the field
strength σI WI µ ν transforms as
σI WI µ ν → LσI WI µ ν L†. (A10)
However, it is also easy to see that this representation of the operator vanishes by explicitly
performing the trace; one finds
Φ† σI WI µ ν Φ
= 0. (A11)
The non trivial representation of the operator in terms of the bi-doublet is given by
Φ† σI Φσ3
. (A12)
With this factor of σ3, required for a non trivial representation in terms of the Higgs bi-
doublet, one finds that this operator violates custodial symmetry.
APPENDIX B: TOP QUARK OPE
As an example of the effect of the neglected terms in in the top quark OPE, consider the
OPE corrections to the four point function of the higgs. The amplitude is given by
i A4(s, t, u) = −6NC
(2π)d
(k/+mt)
k2 −m2t
(k/+ a/ +mt)
(k + a)2 −m2t
(k/+ b/ +mt)
(k + b)2 −m2t
(k/+ c/+mt)
(k + c)2 −m2t
We find the leading order in p2/m2t → 0 the amplitude is given by
i A04(s, t, u) = −24NC
(2π)d
(m4t + 6 k
2m2t + k
(k2 −m2t )
= − i Nc
16 π2
− 64 + 24 log
. (B1)
The leading order matching gives a factor of −4NC m4t/v4.
Consider performing the top quark OPE to higher orders. We find for the next order in
p2/m2t
i A14(s, t, u) = −
16 π2
80m2t
a2 + b2 + c2 + a · b+ 6 a · c+ b · c
. (B2)
The invariants of the external momenta a, b, c averaged over the sum of all A4 diagrams can
be expressed in the Mandelstam variables. We find that our momenta expressed in terms of
these variables are
〈a2〉 = 4 !m2h,
〈b2〉 = 8 (s+ t+ u) ,
〈c2〉 = 4 !
(s+ t + u)− 3m2h
〈a · b〉 = 4 (s+ t + u) ,
〈a · c〉 = 8
s+ t+ u− 3m2h
〈b · c〉 = 16
s+ t+ u− 3m2h
. (B3)
With these substitutions, the next order in the expansion gives
i A14(s, t, u) = −
16 π2
s+ t+ u
16 π2
. (B4)
Where in the last expression we simplified with s + t + u = 4m2h. This term matches onto
the operator
∂µ h ∂µ h, (B5)
with a Wilson coefficient that contains contributions from the integrating out TeV scale new
physics and the top quark. At the scale µ2 = m2t the Wilson coefficient is
t ) =
4CKh (m
t ) +
20 π2
. (B6)
The later term in the Wilson Coefficient is an example of a term that is neglected in our
calculation. Corrections of this form can be systematically included by taking the top quark
OPE to next order in p2/m2t .
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	Introduction
	Higgs Effective Field Theory: Linear Realization
	The D= 6 Custodial SU(2) Higgs Sector
	D=6 SM Field Strength Operators
	D=6 Fermion Sector
	Higgs Effective Field Theory: Non Linear Realization
	A Low Energy Effective Theory for the higgs
	Running to mt
	Integrating out the top quark
	Corrections to Field Strength operators
	Phenomenology of Higgs Effective Theory
	The Magnitude of Self Couplings
	g  g h  h Production
	Higgsium: Production and Decay time
	Non Relativistic higgs Effective Theory
	Matching onto NRHET
	Linear Realization
	Non-Linear Realization
	NRHET Bound State Energy
	Linear Realization
	Non-Linear Realization
	Summary and Conclusions
	Acknowledgments
	Custodial Symmetry and the S parameter
	Top Quark OPE
	References
