Scalar fields, hierarchical UV/IR mixing and the Weak Gravity Conjecture
JHE
Scalar fields, hierarchical UV/IR mixing and the
Dieter Lu¨st 0 1 2
Eran Palti 0 1
0 Fohringer Ring 6 , 80805 Mu ̈nchen , Germany
1 Theresienstrasse 37 , 80333 Mu ̈nchen , Germany
2 ArnoldSommerfeldCenter for Theoretical Physics , LudwigMaximiliansUniversita ̈t
The Weak Gravity Conjecture (WGC) bounds the mass of a particle by its charge. It is expected that this bound can not be below the ultraviolet cutoff scale of the effective theory. Recently, an extension of the WGC was proposed in the presence of scalar fields. We show that this more general version can bound the mass of a particle to be arbitrarily far below the ultraviolet cutoff of the effective theory. It therefore manifests a form of hierarchical UV/IR mixing. This has possible implications for naturalness. We also present new evidence for the proposed contribution of scalar fields to the WGC by showing that it matches the results of dimensional reduction. In such a setup the UV/IR mixing is tied to the interaction between the WGC and nonlocal gauge operators.
Black Holes; Models of Quantum Gravity; Effective Field Theories; Gauge

HJEP02(
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Weak
Symmetry
1 Introduction 2 3 4
5
The Weak Gravity Conjecture and UV/IR mixing
2.1
The WGC with scalars and dimensional reduction
The UV cutoff scale
4.1
Relation to a scalar Weak Gravity Conjecture
Quantum corrections and naturalness
5.1
5.2
Quantum corrections to mφ
Quantum corrections to β
5.2.1
5.2.2
A scalar WGC particle
A fermion WGC particle
5.3
Quantum corrections in the 5dimensional model
6
Discussion
1
Introduction
conjectured property of quantum gravity is the Weak Gravity Conjecture (WGC) [3]. It is
fields [4] it is, at least in principle, possible. Specifically, we propose that quantum gravity
physics can restrict the mass of a particle to an IR scale far below the UV cutoff scale
of an effective theory if the particle couples to gauge fields and massless scalar fields with
almost precisely equal strength.
The rest of the introduction is dedicated to expanding on this and forms an overview
of some of the main points of this note. Consider a theory with gravity, a U(
1
) with gauge
coupling g, a charged scalar h with charge q = 1 and mass m, and a neutral scalar φ with
mass mφ
L =
− Dh2 − 2
(∂φ)2 − m2h∗h − 2
1
m2φφ2 − 2mµφh ∗
h + . . . .
(1.1)
Here µ is a dimensionless parameter which parameterises the coupling of φ to h. The . . .
denote arbitrary further terms in the Lagrangian. All the quantities in the Lagrangian (1.1)
are the quantum corrected expressions having integrated out all the UV physics. The UV
completion of the theory is taken to be as follows. There is a scale ΛUV ∼ gMp above
which the theory can not be completed by a quantum field theory, so it is the scale where
quantum gravity related physics is reached. Below this scale, but above m and mφ, there
may be other scales where new degrees of freedom appear but for each one the theory can
be completed by a quantum field theory. The new degrees of freedom are constrained such
that the particle with the largest chargetomass ratio with respect to the U(
1
) remains h.
Then the claim is that in the limit mφ → 0, this theory must satisfy [4]
pg2 − µ 2Mp ≡ βMp ≥ m .
(1.2)
We also impose g ≥ µ . For finite mφ 6= 0 the expression (1.2) will receive corrections
suppressed by mmφ .
β → β + O
We will estimate in section 5 that this modifies (1.2) such that
μ2mφ . We therefore observe a new mass scale in the theory βMp. We will
βm
argue that, assuming sufficiently small mφ, this mass scale can be separated arbitrarily far
from the quantum gravity UV scale of the theory m ≤ βMp ≪ ΛUV. Since (1.2) is tied
to quantum gravity physics, with an associated mass scale ΛUV, but is a constraint on an
arbitrarily low IR scale, it manifests a form of UV/IR mixing.
In other words, say the particle with the largest chargetomass ratio in the theory
happens to couple almost precisely the same to gauge fields and massless scalar fields.
From the perspective of quantum field theory this would not have any implications, but
from the perspective of quantum gravity we claim that this would imply that the mass of
the particle would have to be at an IR scale far below the UV scale of the theory. The
reason is that if m violates the bound (1.2) then the charged particle h would form a tower
of absolutely stable bound states of increasing charges. If m is increased adiabatically from
satisfying (1.2) to violating it, then upon crossing the bound a tower of at least βg such
stable states forms below the mass scale gMp. Since β and g are separate parameters this
implies a setup where the number of stable states below a fixed mass scale is arbitrary.
– 2 –
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Indeed, we can consider βg →
∞ inducing an infinite number of such states. We expect
that this should not be possible in quantum gravity, in analogy with the argument against
global symmetries. We discuss this in more detail in section 2.
There are two important and related points to state from the outset. The first is
that what we propose is UV/IR mixing is different from other known cases. Specifically,
one definition of UV/IR mixing is that it is a property of UV physics that naturally and
inevitably one recovers an IR mass scale from it. For example, Tduality in string theory
implies that going to the UV automatically recovers IR physics. This is not the case here.
Rather, we have to input an IR mass scale mφ into the theory, and further tune β ≪ g.
Nonetheless, (1.2) could only be deduced, to our knowledge, from quantum gravity physics.
This has an associated mass scale of the UV scale ΛUV which as βg → 0 becomes arbitrarily
far away from the IR scale where this relation holds. We therefore believe this should be
called UV/IR mixing.
The second point, which is a consequence of the first, is that the UV/IR mixing does
not form in itself a solution to the problem of the naturalness of a scalar mass. This is
because we have to put in an IR mass scale mφ by hand and tune β to obtain an IR bound
on the particle mass. A full solution to the naturalness problem from UV/IR mixing would
be one where the UV physics automatically gives an IR scalar mass. In other words, we
propose that if a particle couples almost equally to gauge and light scalar fields then its
mass is light. There is nothing stopping the particle being heavy and coupling to gauge
and scalar fields differently. Our results are therefore only a reformulation of the question
of the naturalness of a scalar mass. Interestingly, this reformulation is one which we believe
could only arise from quantum gravity and involves UV/IR mixing in the sense described
above. It therefore may have a role to play in a possible full solution to the naturalness
problem that involves UV/IR mixing in some way.1
Indeed, a striking property of (1.2) is that it is a bound on a dimensionful mass
parameter by dimensionless parameters. This is certainly encouraging in terms of technical
naturalness. Both g and µ are technically natural. The parameter β is not technically
natural in the sense of there being an enhanced symmetry when it vanishes, however, it
is very well protected. Consider the theory (1.1) but without any additional terms. Then
the 1loop correction to the gauge coupling diverges logarithmically but goes as g3. The
correction to µ is not divergent, and goes as µ 3 and g2µ . Then as long as g2
the hierarchy β ≪ g is not disturbed. This tells us that if we recouple new physics then it
does not need to disturb a hierarchy. Of course, it is easy to couple in new physics that
does spoil a hierarchy β ≪ g by loop corrections, but the hierarchy is stable with respect
∼ µ 2 . β,
to the physics needed for the bound (1.2).
We have also introduced a light neutral scalar φ by hand. While keeping it light after
coupling to new physics beyond (1.1) is generally difficult, within (1.1) the scalar only has
to couple to h, and so the contributions of loops of h to its mass will go as mφ ∼ µm . Since
µ is small the scalar remains light with respect to m. Of course, if φ couples directly to
1Indeed some interesting recent proposals that are potentially relevant for naturalness and which utilise
the WGC [32, 34, 35] assume that it must have some UV/IR mixing properties.
– 3 –
some new high mass scale then we expect it to gain a large mass, but no such coupling is
required for (1.2). Rather, it is a modelbuilding challenge to UV complete (1.1) in a way
that allows φ to couple much more strongly to h than to high scale UV physics.
The note is structured as follows. In section 2 we discuss generally UV/IR mixing and
the Weak Gravity Conjecture. In section 3 we develop a toy model based on dimensional
reduction of a 5dimensional theory. This serves the role of illustrating the relevant physics,
but also leads to new insights on the WGC. In section 4 we discuss the UV cutoff scale
associated to quantum gravity physics. In section 5 we discuss quantum corrections and
the implications for naturalness. We finish with a discussion in section 6.
2
The Weak Gravity Conjecture and UV/IR mixing
The Weak Gravity Conjecture (WGC) [3] (see [6–36] for some recent work on this) states
that given a U(
1
) symmetry with gauge coupling g, there must exist a particle of charge q
and mass m which satisfies the inequality
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(2.1)
(2.2)
(2.3)
gqMp ≥ m .
We will henceforth generally set q = 1. This bound is interesting in the sense that even
though it is based on quantum gravity physics, it can bound the mass of a particle to be
parametrically light compared to the Planck scale. It is therefore natural to consider if the
bound on the WGC particle mass could be manifesting UV/IR mixing. One property of
the WGC that could be in support of this is that it appears to be in tension with Wilsonian
naturalness. The left hand side of (2.1) runs logarithmically with energy scales, but if the
WGC particle is a scalar, then it is bounding a mass that runs linearly with the energy
scale. This means that the scalar mass m is bounded to be only logarithmically sensitive
to UV physics at energy scales above gMp.
However, we expect that (2.1) does not manifest UV/IR mixing, and for the same
reason is also consistent with the Wilsonian notion of naturalness [5]. The reason is that
it is expected that [3] the cutoff scale of the effective theory Λ is bounded by
gMp ≥ Λ .
Indeed, (2.2) can be argued for by reasoning unrelated to naturalness and is termed the
Magnetic WGC [3]. Therefore the WGC (2.1) only bounds the mass to be below the UV
cutoff scale of the effective theory. While it is very interesting that the cutoff scale Λ can
be far below the naive quantum gravity physics scale Mp, within the effective theory gMp
is the UV scale.
Recently a generalisation of the WGC in the presence of massless scalar fields was
proposed [4]. Restricted to a simplified case so as to be comparable with (2.1) it states that
Here µ is the coupling of the WGC particle to massless scalar fields. For example, if we
consider a single real canonically normalised massless scalar φ, and expand it about its
vacuum expectation value φ = hφi + δφ, then the coupling to a complex scalar WGC particle
Similarly, for a fermion WGC particle ψ the coupling is
L ⊃ 2mµδφh ∗h .
L ⊃ µδφ ψψ .
Note that this can also be written as µ = ∂φm, which is how it was presented in [4].
The key point of interest for this note is that in the presence of massless scalar fields
it is possible to bound the mass m to an IR mass scale by sending
β
2
≡ g2 − µ 2
→ 0 ,
(2.4)
(2.5)
(2.6)
while keeping g, and therefore µ , finite. If the UV scale associated to quantum gravity
physics is still only constrained by (2.2), then for β ≪ g the WGC particle mass is bound
far below the UV scale of the effective theory m
≪ Λ. Since the WGC is associated
to UV quantum gravity physics but is bounding a mass at an IR scale, this would be a
manifestation of UV/IR mixing.
There are two crucial questions with regards to this proposal. The first is whether (2.3)
is indeed a property of quantum gravity. The second is whether sending β → 0 does not
also modify the cutoff scale of the effective theory so that the quantum gravity UV scale
would be βMp rather than gMp. If that happened then we would be back to the case where
the bound on the mass is only the UV cutoff scale of the effective theory. In this note we
present new evidence that (2.3) is a property of quantum gravity and that sending β → 0
does not lower the quantum gravity cutoff of the effective theory. Note that it is possible
that β → 0 does lower the scale of some physics, like a finite number of particles, but we
claim that the scale at which one must utilise quantum gravity physics can remain finite.
In [4] the bound (2.3) was proposed based on a number of reasons. One is related
to N = 2 supersymmetry and is discussed below. A more general argument is that if we
consider the particle with the largest charge to mass ratio in the theory, then unless it
satisfies (2.3) it would form a tower of stable gravitationally bound states. Specifically, two
such particles would feel a mutual repulsive force due to the gauge field with strength g,
an attractive force due to gravity with strength m, and a further attractive force due to
the massless scalar fields with strength µ . The inequality (2.3) is the requirement that the
repulsive force beats the attractive forces so that the particles do not form a bound state.
Such a bound state would be completely stable by charge and energy conservation. We
could then keep building such stable states by adding more particles.
Now consider how many such bound states we could fit below a finite mass scale, say
Mp. We know that the particle mass must satisfy m ≥ βMp to form bound states. Then
we can fit at least β1 such states below Mp. But now send β → 0 and we find an infinite
number of states. By a species argument, as in [2] for example, we deduce that the cutoff
scale of the theory must go to zero in this limit. However, we have proposed, and will
present evidence for this in section 4, that the cutoff scale of the theory stays finite in this
limit. This therefore implies that the stable states must not be present to start off with,
– 5 –
and so (2.3) must be satisfied. Note that one way to avoid this is to say that we are free
to take m ≫ βMp. This is true, but nonetheless, we can imagine starting with m < βMp
and adiabatically increasing m past the threshold βMp turning a theory consistent with
quantum gravity into a theory inconsistent with quantum gravity and therefore βMp is
clearly an intrinsic quantum gravity scale.
We can reach the same conclusion by thinking about black hole remnants [4]. If an
extremal black hole sources scalar fields through its charge, then the relation between its
charge and ADM mass is modified in such a way that, in principle, the ADM mass could
be far smaller than the charge. The relation between the ADM mass and the charge is
the same as (2.3), at least for any black holes that can be described by a socalled ‘fake
superpotential’ [37], but with the scalar coupling µ replaced by an appropriate coupling to
the black hole. Schematically, if the gauge coupling depends on some massless scalar field
g (φ), and if we can write it as
g (φ)2 = W (φ)2 + (∂φW )2 ,
for some real function W (φ), then the black hole solution will be such that the ADM mass
is MADM = W (φ∞). Here φ∞ is the value of the field at spatial infinity. We can therefore
define
βBH ≡ g (φ∞)2 − (∂φW )2
2
φ=φ∞
,
(2.7)
(2.8)
and write MADM = βBHMp. Now one can consider how many black hole remnants can
1
fit below Mp and again we find βBH such states. By sending βBH → 0 we can reach an
infinite number of remnants but with a finite cutoff. This is inconsistent and therefore the
remnants must be able to decay.
While this argument tells us that there should be a particle in the theory that must
become arbitrarily superextremal in order for black holes to decay and avoid the remnants,
it does not quite imply (2.3) since we need some relation between βBH and β for this.
There is a nice way to see such a relation, and also to connect the remnant picture with
the bound states picture. It is possible to think of the extremality bound on a black hole
as the statement that the black hole mass should be at least the binding energy in bringing
in all the charge in the black hole from infinity [38]. In the presence of scalar fields this
binding energy decreases since the attractive force makes it easier to bring the charges
in. This is why the ADM mass of a black hole can be much less than its charge. It also
directly relates the bound states with the remnants, and justifies a relation between βBH
and β. While it is important to find a more quantitative version of this relation, we believe
that (2.3) is strongly motivated by this argument.2
Note that the remnants argument is in some sense stronger than the bound states
one, but requires the additional assumptions about the dependence of g (and thereby µ )
on φ. We can therefore define two versions of the general WGC (2.3). The first one is
assuming the structure of g (φ) (2.7) and is therefore motivated by both remnants and
bound states. The second one is a stronger statement which does not assume anything
2Quantitatively, we are assuming that qparticleβBH ≤ qBHβ, where q denotes the charge. The more
general expression is Q2particleβBH ≤ Q2BHβ2, with Q2 defined as in [4].
2
– 6 –
about the dependence of g on φ, and is only motivated by bound states. We denote the
former the Weak General WGC and the latter the Strong General WGC. Note that if one
adopts only the weak version, then the action (1.1) should be appropriately modified. This
can lead to an additional source of mass for the field φ and should be accounted for in
understanding mφ within a UV completion.
It is worth noting that these arguments are much stronger than those presented in [3]
for the original WGC. There the number of bound states, or remnants, below Mp was g1 .
This is consistent with a species argument as long as the cutoff of the theory went to
zero when g → 0, which is indeed the case. Therefore the presence of the bound states or
remnants was not excluded by any known arguments.
At this point it is worth making a further clarifying remark. In our analysis we will
assume that the WGC particle has charge of order one q ∼ 1. This is a strong version
of the WGC, and it avoids possible loopholes to do with increasing the charge and mass
of the particle so that it ends up above the cutoff scale of the effective theory. Such a
possibility for avoiding a strong statement by the WGC was pointed out already in [3].
It was also noted within the context of axion alignment [7, 8, 11], and its dual version of
higgsing a linear combination of U(
1
) groups [26]. The proposition that q ∼ 1 is supported
by evidence from string theory [14, 15, 23, 24]. Note though that a strict q = 1 condition is
violated in some cases [15, 23]. The arguments presented above also rule out the possibility
of parametrically increasing q in the sense that if the WGC particle is kept at the cutoff
scale of the theory, then we would still have an infinite number of bound states or remnants
in the β → 0 limit.
2.1
Relation to N = 2 supersymmetry
The general version of the WGC (2.3) has close ties to N = 2 supersymmetry. This is
because BPS states in N = 2 saturate the bound (2.3). Indeed, the simplest way to argue
for (2.3) is that in N = 2 supergravity supersymmetric black holes are themselves BPS
states. This means that they can only decay to other BPS states. So the WGC particle,
responsible for the decay of extremal black holes, must be a BPS particle and therefore
satisfies (2.3).
The N = 2 setting is also the simplest illustration of the UV/IR mixing property
of (2.3). Consider going to a point in moduli space where the mass of a BPS state is
vanishingly small m → 0 but where g remains finite. Since a BPS state saturates (2.3) this
is also a point in moduli space where β → 0 for that state. Now, consider just a pure gauge
theory with no charged matter at this point in moduli space. This would be perfectly
fine from the perspective of quantum field theory. However, the WGC would demand the
existence of a charged particle. The original version (2.1) would tell us that this charged
particle must have a mass below gMp. However, we know that the decay of black holes
requires a much stronger condition which is that the charged particle must actually be
BPS. This implies that its mass is at an IR scale far below gMp and therefore, as we argue
in section 4, far below the scale of quantum gravity physics. The general WGC (2.3) is not
quite as strong a statement as N = 2 would imply, since it could be satisfied in principle
by a nonBPS particle, but the UV/IR mixing aspect is the same.
– 7 –
There is a also a more practical relation between (2.3) and N = 2 supersymmetry
in that it is a setting that allows for protection against quantum corrections. Of course
with respect to the hierarchy problem this is not so interesting since supersymmetry also
protects a scalar mass. But from the perspective of trying to understand the microscopic
physics behind (2.3) it is a useful starting point.
3
The WGC with scalars and dimensional reduction
In this section we present new evidence for (2.3) based on dimensional reduction. The
point is that the WGC (2.1) in 5 dimensions leads, upon a classical dimensional reduction,
to the generalised version of the WGC (2.3) in 4 dimensions.3 In this section we will
consider a classical dimensional reduction. In general, as discussed in the introduction, at
the quantum level it is difficult to control β and the mass of the scalar mediators. This
example model is no exception to this, and quantum corrections can significantly modify
the scenario. However, we will be able to control them, at least to some extent, by utilising
some supersymmetry and will argue that in that case the key physics insights of the classical
results will remain. We return to a discussion of this as part of the general discussion on
quantum corrections in section 5.3.
We consider first the case where the WGC particle is a scalar and return to the fermion
case later. To simplify notation we henceforth work in units where Mp = 1. The
5dimensional theory of interest is
S5D =
Z
M4×S1
d5X
√
−G
Here, we have a higher dimensional gauge field AM , with gauge coupling g5, and a complex
scalar H. The scalar H is charged with charge q and acts as the WGC particle to make
this theory consistent. The bound on its mass from the 5dimensional WGC reads (see for
example [15])
zero modes
S40D =
If we reduce this theory on a circle, then we get an effective 4dimensional action for the
Z
M4
d x
4 √
−g
1
2
R+IIJ FμIν F J,μν
1
− 2
1
(∂ϕ)2 − 2r2g52 (∂a)2 −Dh2 −m2h∗h .
Here the 4dimensional fields are the dilaton ϕ, an axion a, a complex scalar h, the
graviphoton A0μ, and the zero mode of the gauge field A1 . The dilaton measures the circumference
μ
r of the extra dimension
The covariant derivative of the scalar field is given by
3See also [15, 16] for a dimensional reduction analysis of the WGC.
g5q ≥
MH .
r 2
3
r = e
−q 2 ϕ
Dμh = ∂μ − iqA1μ h ,
where q is its charge, and the mass is given by
The index I = 0, 1 runs over the two 4dimensional gauge fields AIμ and reads
,
−a
1
I
−1 IJ = − r3
8
a
1
The axion a originates from the extradimensional component of the gauge field a = RS1 A4.
It can be thought of as a Wilson line in the extra dimension and it enjoys a gauge discrete
shift symmetry. There are no fields charged under the graviphoton U(
1
)0 since we only
write the action for the zero modes. The WGC states for U(
1
)0 are KK modes. The lightest
field charged under U(
1
)1 is the scalar h and so it plays the role of the WGC particle.
To utilise the WGC (2.3) we need the general formulation for it presented in [4]. For
a Lagrangian (not including the WGC particle h)
with scalar fields φi and gauge fields AI , the generalisation of (2.3) is
For purely electric charges qI , we have Q
expression (3.9) to (3.3) we have
1
2 = − 2 qI I
−1 IJ qJ . Applying the general
Q ≥ m2 + gij ∂im∂j m .
2
Q ≥ m2 + µ 2a + µ 2ϕ ,
2
where we define µ 2a = gaa∂am∂am, and µ 2ϕ = gϕϕ∂ϕm∂ϕm. These measure the strength of
the force mediated by the axion and dilaton respectively. The full expressions are
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
Using these, the 4dimensional bound (3.10) reads
Q
2 =
µ 2a =
,
This precisely reproduces the 5dimensional bound (3.2).
We therefore find that the general WGC with scalar fields can be deduced from a
classical dimensional reduction of the WGC in the absence of scalar fields. This lends some
further weight to its validity. It is also interesting in the sense that it forms an example
where the bound (3.9) is not saturated.
– 9 –
For the case where the WGC particle is a fermion we can consider a 5dimensional Dirac
fermion Ψ, with 5dimensional mass MΨ and 4dimensional Dirac zeromode ψ.
Dimensional reduction leads to exactly the same results as (3.11), with MH → MΨ, and therefore
again matches the 5dimensional WGC. Since, in this respect, it is no different from the
scalar case we do not reproduce the calculations here. It is, however, worth commenting
about a point regarding the pseudoscalar nature of the axion a. In both the scalar and
fermion cases the expression for µ a, which is the nonrelativistic force mediated by the
axion, vanishes if a has a vanishing expectation value (in which case the nonrelativistic limit
requires MΨ, MH 6= 0). In the scalar case this is readily seen from the action (3.3) since
there is no cubic coupling for a = 0, as is guaranteed by parity conservation. In the fermion
case there is a cubic coupling in the action r− 23 qaψγ5ψ. However, the straight exchange
of the axion using this coupling does not lead to a long range force in the nonrelativistic
limit. Indeed, for a = 0, the leading force is a dipole force which is spindependent and
scales as l−4, with l the separation distance (see for example [48]). The expression for µ a
is capturing an exchange of the axion with an insertion of its vacuum expectation value at
each external leg. As in the case of a scalar, it is a long range nonrelativistic force that is
only present due to the spontaneous breaking of parity.
4
The UV cutoff scale
The results of [4], and those of section 3, lend support to the proposal that (2.3) is a
property of quantum gravity. The next question with regards to whether (2.3) can bound
m far below the UV cutoff scale is whether it is possible to send β → 0 without also
implying Λ → 0, where Λ is the cutoff scale of the effective theory. In this section we
present arguments that indeed this is possible.
Note that we take Λ as an enforced cutoff scale on any quantum field theory due
to the appearance of quantum gravity physics. Typically this would be the mass scale of
an infinite number of states. So we consider Λ as, for example, the string scale. It could
be that for a given quantum field theory there is a lower cutoff scale than Λ where new
physics appears ΛNP. But this lower cutoff could in principle be completed by some other
quantum field theory. We do not make the claim that ΛNP does not go to zero when β does.
It could well be that there must be new physics at the scale βMp for a given quantum field
theory. For example, supersymmetry may have to appear. But, although βMp is a scale
implied by quantum gravity, we claim that this new physics is not that of quantum gravity.
Sending g → 0 in (2.3) is clearly problematic in the sense that the gauge field decouples
and we recover an exact U(
1
) global symmetry. Note that taking multiple U(
1
)s, and
therefore using the general expression (3.9) where g2 → Q
2 does not help in this regard.
Since in Q
2 the charge vector q is contracted with the gauge kinetic matrix I, a vanishing
Q
2 would imply a vanishing determinant for I which would again decouple a gauge field
leading to a global symmetry. We expect this to not be possible in quantum gravity. The
bound (2.1), or (2.3), alone would only imply that a particle become massless in this limit.
This seems insufficient to block such a global symmetry, and it is therefore natural to
expect that the stronger condition (2.2) should be imposed. In contrast, the β → 0 limit
HJEP02(
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does not decouple the gauge field and therefore just a particle becoming massless appears
consistent from this perspective.
In the presence of N = 2 supersymmetry the WGC particle must be BPS, at least in
the sense that supersymmetric extremal black holes can only decay to BPS particles. BPS
particles satisfy the equality limit of (2.3). Therefore in this case m → 0 implies, rather than
is just implied by, β → 0. So if β → 0 would imply Λ → 0 then we would have a quantum
gravity obstruction to massless BPS states. This would be very strange. Indeed, the classic
example of BPS states in string theory are wrapped branes. If we consider compactifications
of type IIB string theory on CalabiYau manifolds, then D3 branes wrapped on 3cycles
lead to BPS particles in four dimensions. If we go to the conifold locus in moduli space then
one such BPS particle (hypermultiplet) becomes massless [45]. But only one BPS state is
becoming massless which can be understood by looking at its effect on the gauge coupling.
While the conifold locus is a somewhat exotic point in complexstructure moduli space,
there is no reason to expect that it is not possible to work with an effective theory with a
finite cutoff in its vicinity. This is in contrast to the g → 0 limit, where an infinite number of
states become massless sending Λ → 0 and matching general quantum gravity expectations.
In section 3 we showed that (2.3) can be understood from classical dimensional
reduction. In such a setting there are a number of UV cutoff scales. There is the scale where
the 5dimensional theory breaks down (see for example [47]), the KaluzaKlein scale, and
an expected 5dimensional magnetic WGC scale4
(4.1)
(4.3)
2
Λ5D ∼ r
1
We would like to consider how the parameter β2 in (2.6), which bounds the mass m2,
compares to these scales. Again we will focus on the classical results, and return to quantum
corrections in section 5.3. The explicit form is
Q − µ 2a − µ 2ϕ ≡ β2 =
2
3a4q4 + 6a2M H2 q2r2 + 6g52M H2 q2r4 − M H4 r4
3r3 a2q2 + M H2 r2
.
(4.2)
There are two ways in which we could choose the parameters in this model to obtain
a small β. One of them violates the WGC while the other does not. Therefore, the
difference between them must arise due to interaction with UV physics and so is associated
to UV/IR mixing.
Consider first the case which violates the WGC. Let us set a = 0 so that we have
β
2
a=0 =
6g52q2 − M H2 .
3r
and it leads to a large mass m2
Then if we forget about the 5dimensional WGC (3.2), we can take M H2 → 6g52q2. In that
case we have β → 0. This limit is perfectly fine from a quantum field theory perspective
→
6g5r2q2 . So from a quantum field theory perspective, there
4Note that in these estimates we have included an extra factor of r−1 relative to the versions of these
expressions in the literature, due to the change from the 5dimensional to 4dimensional Planck mass. Either
way, this extra factor is not important for the primary point.
is nothing that stops us from considering a particle which couples equally to gauge and
scalar fields, yet has a large mass. This exemplifies the expectation that the generality of
the connection between the coupling equality β = 0 and the mass of a particle is inherently
quantum gravitational in nature.
rameter values
Consider now the way to reach small β while respecting the WGC. We take the
paWe have that β2 and Q2 read in this case
r2g52q ≫ a2q ≫ g5MH r2 .
β
2
≃
a2q2
2
r3 , Q ≃
q2g52 .
r
for the WGC particle.
we have
Therefore, we find β2 ≪ Q2 in this regime. Since (4.4) is compatible with the WGC, we
indeed find that the mass of the WGC particle is bound far below any UV cutoff scale.
This is contrast to the field theory setting discussed above. It is also in contrast to the
scale Q2, which would bound the mass in the absence of scalar fields, and which can not be
separated from Λ2WGC,5. We therefore conclude that the UV/IR mixing properties of the
WGC impose that in a quantum theory of gravity having a small β implies a small mass
Let us return to the case a = 0. If we do impose the 5dimensional WGC (3.2) then
β
2
a=0 ≥
3g52q2
2r
This means that the WGC bound on the mass is tied to the UV scale and there is no
UV/IR mixing. This is interesting from the microscopic perspective since it implies that
the UV/IR mixing is tied to µ a which in this case vanishes. The contribution µ a due to the
axion arises in the 5dimensional theory as a Wilson line around the extra circle dimension.
A Wilson line is a nonlocal gauge operator. Therefore we see that the UV/IR mixing aspect
is manifesting due to the interaction of the WGC with nonlocal physics. This matches
the general expectation that UV/IR mixing should be tied to nonlocal physics. If we have
a 6= 0, then the UV/IR mixing is tied to the parameter range (4.4). This can be written as
aq ≫ MH r. If we were to take MH r very large the WGC particle would have a localised
Compton wavelength and would not sense the compactness of the circle direction or the
associated Wilson line. It would be sensitive to only local physics and accordingly we see
that UV/IR mixing would not be possible.
Note that the WGC bound (2.3) can only be understood in terms of classical long range
forces if the mass of the WGC particle is heavier than the mediator forces. But the β → 0
limit is the one where the WGC state becomes massless and therefore must be approached
with care. We should keep this in mind, but it does not form a fundamental obstruction
for any realistic scenario where we are concerned with β ≪ g rather than β = 0.
To summarise, in this section we presented evidence that the general formulation of
the WGC (2.3) can exhibit a bound on the WGC particle mass parametrically far below
the UV cutoff scale of the theory. In other words, that the WGC particle possesses the
property that if it couples almost precisely the same to gauge fields and scalar fields then
its mass is far below the UV scale of the theory.
(4.4)
(4.5)
(4.6)
In [4] it was shown that in an N = 2 supersymmetric setting the scalar forces act stronger
than gravity on the WGC particle. More precisely, for each scalar field there is one particle
on which the scalar force acts stronger than gravity. This was proposed as a possible Scalar
WGC. If we apply this to the zero mode h it reads
Utilising (3.11) this implies
µ 2a + µ 2ϕ Mp2 ≥ m2 .
Therefore, for sufficiently large MH we find that the Scalar WGC is not satisfied by the
WGC particle. So gravity is acting stronger than the scalar forces on the WGC particle.
A possible interpretation of this result is that it is evidence against a Scalar WGC. If
that is the case it could have interesting implications for large field excursions in quantum
gravity. In [4] it shown that a scalar WGC could be a possible explanation for the
exponentially decreasing mass of states for superPlanckian scalar field variations.5 Here it
appears that taking large MH could allow to violate this. As we will discuss in section 5, in
a quantum setting the mass MH should be tied to supersymmetry breaking. In this sense
it could be suggesting that looking at highly nonsupersymmetric situations could allow
for large field variations.
Having stated this, it is the case still that even in this model the Scalar WGC is
satisfied in the sense that there is still a particle on which gravity acts weaker than the
scalar forces. This is because any KaluzaKlein mode of the 5dimensional field H will
satisfy (4.7). This can be seen by noting that the replacement qa → qa + 2πn, where n is
an integer (the KaluzaKlein number), is a gauge symmetry of the theory. For any nonzero
n the inequality (4.8) can not be violated within the effective theory limit MH r ≪ 2π.
Yet another possibility is to require that the Scalar WGC applied to the zeromode h,
which is the WGC particle in the sense that it has the largest chargetomass ratio, should
be satisfied. Then this would be interpreted as a bound on the parameters (4.8). Or as a
statement that the original 5dimensional theory is not compatible with a quantum gravity
UV completion as it is.
With respect to the point of interest in this note. The Scalar WGC can also be used
to restrict the mass of a particle since it imposes µM p ≥ m. This appears in many ways
simpler than utilising (2.3). However, it is natural to expect that sending µ
Scalar WGC particle implies a vanishing cutoff Λ → 0 [49]. Further, if we accept that there
are no additional constraints of relevance on the 5dimensional theory or the parameter
MH , then the Scalar WGC particle must be a KaluzaKlein mode rather than the particle
with the largest chargetomass ratio. This tells us that we can not utilise the Scalar WGC
→ 0 for the
to constrain its mass to be below the UV cutoff scale of the theory.
5See [39–44] for recent work on this.
It is interesting to note that both the Scalar WGC µM p ≥ m and the original WGC
gMp ≥ m can be unified in a sense if we modify (2.3) to
is that if we send Mp → ∞ in (2.3) we seem to still have information left about the sign
5
Quantum corrections and naturalness
HJEP02(
18
)4
So far we have primarily discussed the evidence for UV/IR mixing in the WGC. In this
section we focus on quantum corrections and the implications of the WGC for naturalness.
There are two key questions in thinking about this.
The first is to do with the fact
that (2.3) is taken to apply in the presence of massless scalar fields, but at the quantum
level we expect these scalars to gain a mass. This can lead to corrections to (2.3) which
need to be quantified. The second question is how to obtain a hierarchy between β and g
that is protected from quantum corrections. Both of these are modeldependent questions,
but there are some general results that we discuss in this section.
5.1
Quantum corrections to mφ
Consider first the issue of the mass of the scalar force mediators. For ease of notation let us
consider a single scalar (or pseudoscalar) mediator φ with mass mφ. The WGC bound (2.3)
applies precisely only for mφ = 0. More generally, we can expect it to apply approximately
as long as mφ ≪ m. This is the regime where the scalar φ acts as a long range classical
force on the WGC particle. One way to ensure a mass separation at the quantum level is
to impose supersymmetry. This protects the scalar mass mφ from perturbative quantum
corrections, and as long as nonperturbative effects are small, this is sufficient. Of course,
the problem with this is that in protecting mφ we are also protecting m. This makes the
WGC bound on m less interesting, but it remains nontrivial. For example, it could be
used to explain the smallness of technically natural parameters, such as a supersymmetric
mass term.
To gain insights into naturalness we must consider a nonsupersymmetric setting. In
this case a massless, or at least arbitrarily light, scalar field at the quantum level is difficult
to implement but not impossible. For example, axions can have arbitrarily low masses.
More generally, if it couples only very weakly to other sectors its mass could be protected.
Also the scalar, or pseudoscalar, does not have to be fundamental, it could be a type of
pion. With this in mind we should recall that, as discussed in section 3, the pseudoscalar
interaction required is a long range spinindependent one, which means that it relies on
some breaking of parity.
The mass mφ is therefore modeldependent, and while it may or may not be difficult
to protect it, this is a question of how it couples to other fields. However, there is a
universal minimum bound on its coupling because in order to apply (2.3) we require that
the scalar field φ couples to the WGC particle with coupling µ that is not arbitrarily small.
Therefore, at the quantum level, we know that its mass at least receives corrections from
the WGC particle so that we expect it to be driven to at least mφ ∼ µm . This is the
generic expectation though and it may be possible to construct models where it is much
lighter. This is not reintroducing the full hierarchy problem because it is a question of
protecting mφ from m rather than from the full UV physics. We can generate a hierarchy
mφ ≪ m by taking µ
≪ 1. In general, we expect that we require to take µ and g very
small so as to reduce sensitivity to loops. However, we should not take µ as small as µ ∼ β
since then we are lowering the UV cutoff scale (2.2) all the way to m. In this sense there
is a modelbuilding challenge in coupling φ much more strongly to h than to UV physics.
An important question in a nonsupersymmetric context therefore appears to be what
are the quantitative corrections to the WGC (2.3) for a finite mφ but with mφ ≪ m. We do
not know, but one estimate could be as follows. Since mφ ≪ m we can still approximate the
effect of the scalar φ as a long range classical Yukawa force. The question of the presence of a
bound state is now dependent on the separation scale l between the WGC particles. At
separation scales mφl . 1 the scalar force is still significant, and we can take l as small as m−1.
At this scale the Yukawa exponential suppression gives a leading correction to the force
In the last equality we used the expected contribution to the scalar mass from loops of
the WGC particle as discussed above. Such a correction would not modify the bound as
long as µ 23 < β ≪ µ . This is a statement on the absence of a bound state at the smallest
distance scale at which the force behaves classically. This is a reasonable guess but, since it
is not clear what exactly is the constraint from quantum gravity on the presence of bound
states, the distance scale at which to apply it remains an even larger uncertainty. However,
this is a limitation on our calculation ability rather than a fundamental obstruction to a
small β. In principle, if we knew the corrected µ we could tune this against g to reach an
arbitrarily small β unrelated to µ . There is a weaker bound on β relative to µ coming from
quantum corrections to which we now turn.
5.2
Quantum corrections to β
In this subsection we will consider the quantum corrections to β. We are particularly
interested in whether a hierarchy β ≪ g could be protected against quantum corrections and
so could be technically natural. We will not focus on a possible mechanism to generate the
hierarchy in the first place. However, there are a some comments worth making regarding
some universal features of the effects of UV physics on β. The first is that since g is the
gauge coupling of a U(
1
), as long as this remains a U(
1
) and does not complete into a
nonAbelian symmetry, adding a heavy state and integrating it out can only decrease the
coupling g. The second point is that introducing new scalars that couple to the WGC
particle has a treelevel effect on β because the scalar force µ is a sum over all contributions.
A scalar field can only lead to an attractive force, and so can only increase µ .6 Therefore,
6If the new particle is very heavy there is no sense it which it acts as a classical long range force.
Nonetheless, it is natural to expect that integrating it out leads to a positive treelevel contribution to μ.
there is a sense in which adding new physics only serves to decrease β. This may be a
possible way to reach a hierarchically small β to start off with.
The primary effect we are interested in are the quantum corrections to β, which depend
on the UV completion. We will not study in detail the possibility that β could be protected
by a symmetry present in the UV theory that then ensures its technical naturalness. It
is not clear what type of symmetry could protect β. It would have to relate gauge fields
and scalar fields. Of course, N
= 2 supersymmetry is sufficient in this respect. The
interesting thing is that we only requires its action on the bosonic fields. Also the setting
of a higher dimensional compactification, as in section 3, offers some protection against
quantum corrections because the gauge and scalar fields are related by a higherdimensional
gauge symmetry. For example, deep UV physics above the KaluzaKlein scale, where the
gauge symmetry is restored, does not correct the mass ma. If the 5dimensional mass
MH or MΨ could be kept small, then deep UV physics would also not correct a small β.
In any case, however, threshold corrections from the KaluzaKlein scale are too large. A
symmetry that relates gauge fields and scalars on branes in string theory is Tduality. This
is interesting in the sense that this duality also ties to UV/IR mixing.
There are two ways in which the WGC bound on a particle mass could restrict the
mass of a scalar field. One is directly, so if the WGC particle is itself a scalar field, then
the WGC directly bounds its mass. The other is indirectly. Say the WGC particle is a
fermion that gains a mass from Yukawa coupling to a scalar. Then restricting the mass of
the fermion WGC particle restricts the vacuum expectation value of the scalar. Through
minimising the scalar potential this indirectly bounds its mass. We will consider each
possibility in turn.
A scalar WGC particle
Again, these do not disturb a hierarchy a long as µ 2 . β.
δµ ∼ µ 3 + µg 2
The case of a scalar WGC particle was discussed briefly in the introduction. Consider the
model (1.1). We will set mφ = 0 for simplicity, and consider no additional terms beyond
those in (1.1) for now. We are interested in the 1loop corrections to the gauge coupling
g and the parameter µ . Consider starting at some UV scale M , and running down to an
IR scale m, and evaluating the change in the IR values of the parameters. For the gauge
coupling this reads
δg ∼ g3 ln
M
m
Therefore, if we consider g2 . β, this variation will not disturb a hierarchy β ≪ g. In fact,
g can only ever decrease as long as UV physics does not involves massive vectors coupling
to it, which means that the UV corrections can only ever decrease β in any case.
Consider now the parameter µ . To understand its corrections we need to consider
the 1PI diagrams correcting the cubic coupling. There are two such diagrams, due to the
exchange of φ and the U(
1
) photon, and neither is divergent due to the fact that they
involve one loop momentum integral but three bosonic propagators. Schematically, for
m ≪ M , the correction takes the form
(5.2)
(5.3)
These results tell us that coupling the theory (1.1) to new physics does not need to
spoil a hierarchy β ≪ g. In this sense β is a technically natural parameter. It is of course
possible to couple in new physics that does spoil the hierarchy at 1loop. For example, say
we introduced a new scalar S with mass MS that couples to the WGC particle as
Then we would have
L ⊃ 2mµ S Sh∗h .
m
MS
2
We would therefore require µ S
m
MS
. β to maintain the hierarchy. Since we need µ ≫ β
this puts a limit on how strongly S can couple to h and on how light S could be.
Overall, the question of whether a hierarchically small β ≪ g can be protected against
UV physics is model dependent, but there is no serious obstruction to doing so as far as
we can see.
action
5.2.2
A fermion WGC particle
The case of a fermion WGC particle is similar. Consider a Dirac WGC particle ψ with
L =
for the case of a pseudoscalar coupling L ⊃ µφ ψγ5ψ.
Again we have δg ∼ g3 and δµ ∼ µ 3 + µg 2, and so β is protected. The same analysis holds
In this case the bound (2.3) is on a fermion mass. However, it can be turned into a
bound on a scalar mass. Consider setting m = 0 in (5.6), which is technically natural, and
instead adding a scalar h with a potential
Lh ⊃ − 2
1
(∂h)2 − µ hhψψ −
m2hh2 +
λh4
.
1
− 2
1
4
Then minimising the potential we obtain the new mass
We therefore obtain a bound on a scalar mass. Adding the scalar h modifies µ due to a
new 1loop diagram which means that
m = mh √µh .
λ
δµ ∼ µ 2
h
m
mh
2
We want to maintain m ∼ mh, which can be done by choosing λ ∼ µ 2h, and so to control
this correction to β we need to take µ h . β.
The analysis of the dimensional reduction of the 5dimensional theory studied in sections 3
and 4 was performed at the classical level. At the quantum level there are significant
changes which exemplify the general problems discussed in this section. The problems
begin already with the 5dimensional theory since it is nonrenormalisable, the coupling g5
should become strong and the mass MH is not protected from the UV scale. Even, if we
take the 5dimensional theory with small g5 and MH as a given starting point, then the
4dimensional scalars a and ϕ are expected to gain a mass near the KaluzaKlein scale from
loop effects. Also they would have a potential, in the case of a this would be periodic, which
would fix dynamically their values, rather than having them as the free parameters utilised
in section 4. This would imply that reaching a small β dynamically is difficult. Further, β
would receive quantum corrections that would be on top of this dynamical problem.
The point of sections 3 and 4 was not to argue for a mechanism that can address
naturalness questions, but rather to provide some evidence for UV/IR mixing in the WGC.
The question we are interested in is therefore if the physical conclusions reached through
the classical analysis could still hold at the quantum level. One way to ensure this is
to employ supersymmetry. If we consider N = 1 supersymmetry in the 5dimensional
theory then the quantum problems discussed above are all addressed. Consider the
5dimensional action (3.1). In the case of MH = 0 it forms the bosonic sector of an N = 1
supersymmetric theory. Therefore we could complete it to a supersymmetric theory and
the analysis of sections 3 and 4 would remain unchanged.
In such a supersymmetric setting the N = 1 5dimensional supersymmetry reduces on
a circle to N = 2 4dimensional supersymmetry. This explains the vanishing of (3.12) since
h becomes BPS. It is still the case that the WGC bounds the mass of the particle below the
UV scale of the theory, although this only amounts to the statement that it should be BPS.
Now consider still a supersymmetric 5dimensional action apart from turning on
MH 6= 0. This is a nonsupersymmetric mass term and so breaks supersymmetry
completely. However, since it is the only source of supersymmetry breaking it forms a control
parameter for its effects. This setup therefore allows us to perturb away from N = 2,
and we see this since (3.12) no longer vanishes. It is reassuring that under this
nonsupersymmetric perturbation we recover the 5dimensional WGC. From the 5dimensional
perspective a small MH is fine since it is technically natural. The 4dimensional scalars will
still obtain a potential, and β quantum corrections. However, we are not after a mechanism
that dynamically fixes a small β. We therefore assume that as long as the masses ma and
mϕ are sufficiently light we could imagine displacing the fields to the point of small β by
hand. So that it still makes sense to ask how the theory would behave at small β. With
this assumption, we are left with the question of the scalar masses. We require a hierarchy
ma ≪ m and mϕ ≪ m to be able to utilise the WGC. Such a hierarchy can be induced by
the general mechanism described in this section. Since their masses are induced by MH
as the source of supersymmetry breaking, they are of order µ aMH and µ ϕMH . While m
is at least of order MH . We therefore reach the situation described in (5.1). If indeed the
corrections to the WGC are as proposed in (5.1), then for sufficiently small couplings these
are sufficiently small to trust the results.
We therefore conclude that, under some reasonable assumptions, the classical analysis
of sections 3 and 4 is expected to capture the correct physics as long as µ a and µ ϕ are small.
6
In this note we proposed that in the presence of scalar fields the general version of the
WGC (1.2) can bound the mass of the WGC particle far below the UV cutoff scale of the
effective theory. We supported this by presenting new evidence for (1.2) and for the claim
that taking small β does not necessarily lower the cutoff scale of the effective theory. Such
a bound on an IR mass from UV quantum gravity physics is interesting in that it manifests
a form of UV/IR mixing. Of course, our results depend on the validity of (1.2) and while
we presented some new arguments for it, extending the work in [4], it is crucial to build
up more evidence for it.
A primary motivation for the interest in such a bound is that it could have potential
relevance to the question of the naturalness of a scalar mass. At a general level this could
be tied to UV/IR mixing in quantum gravity and so any insights into this are useful. In
this respect, it was interesting to see that in the toy model 5dimensional reduction on a
circle example, the UV/IR mixing was tied to the interaction of the WGC with a Wilson
line in the extra dimensions which is a nonlocal operator.
A direct application of our current understanding of the bound from the WGC to the
naturalness problem of a scalar mass requires addressing two key questions. The first is
how to protect the mass of the force mediator scalar mφ. The second is how to protect a
hierarchy β ≪ g. We presented arguments that while this is a modeldependent question,
there is no obstruction in principle to doing so in the sense that the ingredients that go
into the bound, as in the toy model (1.1), do not in themselves obstruct a small mφ and
β at the quantum level. Having stated this, there is a modelbuilding challenge in that φ
must couple much more strongly to h than to UV physics. Also, we only considered if a
hierarchy β ≪ g is technically natural, and did not study in detail how such a hierarchy
could be induced in the first place.
One of the largest uncertainties in our analysis is the magnitude of the correction
2
to (1.2) for finite but small mφ. We presented an estimate of this in (5.1), but further work
on understanding this effect is crucial. Our estimate, combined with a natural expectation
for the minimal value of mφ, led to a bound on how large g could be of g . β 3 . This
implies that the U(
1
) and scalar mediators must be very weakly coupled. It also implies
2
a relatively low UV scale ΛUV . β 3 Mp. However, these limitations were really due to
our lack of knowledge of the precise effect of a finite mφ. In principle, β could be tuned
accounting for this correction arbitrarily far below g.
implies g2 ∼ µ 2 . β. This relation implies the less strong bound ΛUV . √
βMp.
Another relation between g, µ and β comes from controlling quantum corrections which
We end with a few brief comments about the mechanism discussed in this note and
the actual hierarchy problem as applies to the Higgs in the Standard Model. Since the
WGC bound relies on coupling to a U(
1
) and light scalars, which should be in some hidden
sector, it is easier to imagine an indirect bound on the Higgs mass. So we can consider that
the WGC particle obtains some of its mass from the Higgs vacuum expectation value and
thereby restricts it. An attractive feature of the bound on the mass from the WGC is that
it involves only IR quantities that are potentially measurable by experiments. So say that
some particle had a light mass due to the WGC bound, it would imply a sharp prediction
that its couplings to massless scalar fields and gauge fields would have to be equal up to an
accuracy of the ratio of its mass to the UV scale. If the scalar fields are not massless but
still very light, then there would be small corrections to this equality. One would expect
a similar such striking relation between gauge and scalar couplings for some particle, not
necessarily the Higgs, possibly dark matter, if this mechanism plays a role in the hierarchy
problem of the Standard Model.
If the scalar force mediators have a significant mass then the required equality of the
couplings can be significantly corrected. More generally, the WGC argument is based on a
classical long range force picture, but it should be better thought of as an argument for a
more general microscopic property of quantum gravity that manifests UV/IR mixing and
most likely is present even when the long range classical force picture breaks down. In
this more general sense, one could imagine even a very massive scalar mediator leading to
the bound on the WGC particle mass. In particular, it could be applied to the Standard
Model matter fields with the Higgs as the scalar force mediator. While the mass of the
Higgs implies there is no classical long range force picture, the more general mechanism
could still manifest as a more complicated relation between the gauge and scalar couplings.
A quantitative understanding of this would require understanding the effects of a large mass
for the scalar force mediators in (2.3). Such a formulation could then be applied directly
to the Standard Model Yukawa and gauge couplings, with say the electron as the WGC
particle. If the hierarchy problem is then tied to the WGC, as applied to the Standard
Model in this way, then the quantitative formulation would predict a specific relation to high
precision between the gauge and Yukawa couplings that could be experimentally tested.
In this note we proposed a new way to think about separation of scales and UV/IR
mixing. While we focused on a possible application to the gauge hierarchy problem, it would be
very interesting to establish if it also has implications for the cosmological constant problem.
We have discussed how quantum gravity may ensure the absence of towers of bound
states that are protected from decay by their charge. If the U(
1
) is broken then generically
we would expect that the states would not be stable. However, if the particle forming the
tower is the lightest in the theory by far, then the states would still not decay until a very
large number. In the Standard Model the neutrinos behave in such a way, they are charged
under hypercharge and are far lighter than the other particles. Therefore neutrinos could
form a tower of stable bound states. At long distances the only force that neutrinos feel
is gravity. Therefore, in empty space they would indeed form bound states. However, this
is not the case in the presence of a cosmological constant. In the weak gravity regime the
cosmological constant can be modelled as a repulsive linear force (see for example [50]) so
that the total gravitational force acting on two neutrinos is
FGravity = mν
− mr2ν +
Λr
3
,
(6.1)
HJEP02(
18
)4
where mν denotes the lightest neutrino mass. We can apply this approximately up to the
scale of the neutrino mass r ∼ mν−1. Therefore, for neutrinos to not form stable bound
states we require
Λ > mν4 .
(6.2)
This bound can be viewed either as a bound on how small the cosmological constant
could be or as a bound on how heavy neutrinos could be. In the latter case, this can be
translated to a bound on the scale of electroweak symmetry breaking, and therefore the
mass of the Higgs. We therefore find the striking result that the mass of the Higgs could
not be much higher else, for the given value of the cosmological constant, neutrinos would
form a tower of stable bound states. This could lead to a species problem and in that
sense be inconsistent with quantum gravity. It would be interesting to explore a possible
connection to the results of [32, 34, 35].
Acknowledgments
We would like to thank Gia Dvali, Inaki GarciaEtxebarria, Arthur Hebecker, Luis Ibanez,
Daniel Junghans and Irene Valenzuela for useful discussions. The work of DL is supported
by the ERC Advanced Grant “Strings and Gravity” (Grant No. 320040).
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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