Pointparticle effective field theory I: classical renormalization and the inversesquare potential
Received: January
eld theory I: classical renormalization and the inversesquare potential
C.P. Burgess 0 1 2 3 5 6 7
Peter Hayman 0 1 2 3 5 6 7
M. Williams 0 1 2 4 6 7
Laszlo Zalavari 0 1 2 3 5 6 7
0 Celestijnenlaan 200D , B3001 Leuven , Belgium
1 Waterloo , Ontario N2L 2Y5 , Canada
2 Hamilton , ON, L8S 4M1 , Canada
3 Perimeter Institute for Theoretical Physics
4 Instituut voor Theoretische Fysica, KU Leuven
5 Physics & Astronomy, McMaster University
6 Open Access , c The Authors
7 produce a universal RG
Singular potentials (the inversesquare potential, for example) arise in many situations and their quantum treatment leads to wellknown ambiguities in choosing boundary conditions for the wavefunction at the position of the potential's singularity. These ambiguities are usually resolved by developing a selfadjoint extension of the original problem; a nonunique procedure that leaves undetermined which extension should apply in speci c physical systems. We take the guesswork out of this picture by using techniques of e ective eld theory to derive the required boundary conditions at the origin in terms of the e ective pointparticle action describing the physics of the source. In this picture ambiguities in boundary conditions boil down to the allowed choices for the source action, but casting them in terms of an action provides a physical criterion for their determination. The resulting extension is selfadjoint if the source action is real (and involves no new degrees of freedom), and not otherwise (as can also happen for reasonable systems). We show how this e ective eld picture provides a simple framework for understanding wellknown renormalization e ects that arise in these systems, including how renormalizationgroup techniques can resum nonperturbative interactions that often arise, particularly for nonrelativistic applications. In particular we argue why the lowenergy e ective theory tends to ow of this type and describe how this can lead to the phenomenon of reaction catalysis, in which physical quantities (like scattering cross sections) can sometimes be surprisingly large compared to the underlying scales of the source in question. We comment in passing on the possible relevance of these observations to the phenomenon of the catalysis of baryonnumber violation by scattering from magnetic monopoles. ArXiv ePrint: 1612.07313
ective; E ective eld theories; Nonperturbative E ects; Renormalization Group

Nearsource boundary condition
Source action vs selfadjoint extension
Bound states
Scattering and catalysis
Scattering when d = 3
Nonrelativistic swave fermionmonopole scattering
A The boundary action and the meaning of the classical RG
B Scattering in 1 dimension
1 Introduction & discussion: pointparticle e ective eld theories Standard formulations Classical renormalization and boundary conditions
The Schrodinger inversesquare potential in d dimensions
Introduction & discussion: pointparticle e ective
eld theories
E ective eld theory (EFT) provides an e cient way of exploiting hierarchies of scale when
extracting the predictions of physical systems, and the adoption of its methods ever more
widely throughout physics has brought the reach of practical calculation to ever more
problems over time. It is therefore useful to extend its applications to areas where its
e ciency of description is not yet completely exploited. In this  and several companion
papers [1{3]  we develop EFT tools for use with common problems involving the
backreaction of small `point' sources (PPEFTs), in which the hierarchy of scales to be exploited
is the small ratio of the linear size, ", of the source relative to the size, a, of a macroscopic
probe. For instance, when applied to atoms (which are more the focus of [1{3], rather
than this paper) " might be the size of the nucleus, regarded as the point source of the
nuclear electrostatic potential, while a might be the size of an atomic electron orbit, or the
wavelength of a particle that is scattered from the source's gauge or gravitational elds.
Our interest is in the limit where "=a
1 and so for which we can be largely ignorant
about the details of the source's underlying structure (see gure 1).
We here lay out the groundwork for the later papers in a simple (and widely
studied [4{54]) system: the quantum mechanics of a particle interacting with a point source
g=r2. This problem is wellknown
source action. We denote by " the actual UV scale associated with the size of the source (e.g., the
size of the proton), which we assume is very small compared to the scale a of physical interest (e.g.,
the size of an atom). The PPEFT uses the action of the point source to set up boundary conditions
on the surface of a Gaussian pillbox of radius . The precise size of this pillbox is arbitrary, so long
as it satis es "
a. We require "
in order to have the rst few multipole moments (in
our example only the rst is considered) dominate the eld on the surface of the pillbox, and we
a in order to be able to truncate the e ective action at the few lowestdimension terms.
The classical RG
ow describes how the e ective couplings within the PPEFT action must change
for di erent choices of in order to keep physical quantities unchanged.
to be very rich and involve many subtleties to do with its scaleinvariance and how this
larization/renormalization issues and dimensional transmutation that follow from these
questions). Our treatment here broadly agrees with what is done in the literature, di
ering mainly in emphasis. We argue though that PPEFTs bring the following two
conceptual bene ts.
A rst bene t of PPEFTs is the clarity they bring to which quantity is being
renormalized in these systems. The coupling being renormalized is usually an e ective contact
coupling within the lowenergy pointparticle e ective action (more about which below). In
the simplest situations this coupling turns out to contribute to physics precisely like a
deltafunction potential would,1 showing that one never really has an inversesquare potential in
isolation. Rather, one is obliged also to include a deltafunction coupling whose strength
runs  in the renormalizationgroup (RG) sense, though purely within the classical
approximation  in a way that depends on the inversesquare coupling g. Many otherwise
puzzling features of the inversesquare system become mundane once this inevitable
presence of the deltafunction potential is recognized. For instance, the inversesquare potential
is known sometimes to support a single bound state, even when g is small (and sometimes
1See also [15], whose point of view on this is close to our own.
even when it is negative, so the potential is repulsive). It turns out this state is simply the
bound state supported by the deltafunction, which can remain attractive for the range of
g for which the bound state exists.
Using the EFT lens also has a second conceptual bene t. In particular, most
treatments of the system agree there is a basic ambiguity to do with how to choose the boundary
tonian to be selfadjoint [4{6], and is usually dealt with by constructing a selfadjoint
extension [20{39]; a process that is not unique (nor appropriate, if the physics of the
source should not be unitary, as in [55]). The main bene t of thinking in terms of PPEFTs
is that it provides an explicit algorithm that directly relates the boundary conditions at
guesswork from the problem, and because it is cast in terms of an action, standard EFT
reasoning xes which boundary conditions should be most important in speci c situations
(such as at low energies).
Before exploring the inversesquare potential in detail (and some applications) in the
next sections, we rst pause to esh out these two claims and to be more explicit about
what we mean by a pointparticle EFT.
Standard formulations
There is a regime for which the EFT for point particles is very well understood. When
studying the motion of objects like planets or stars within external elds it is common
to neglect their internal structure and instead focus completely on their centreofmass
describes the action of any `bulk' elds  such as the Maxwell action of electromagnetism
for the bulk vector eld A (x), or the Einstein action of gravity for the bulk tensor eld
g (x)  while Sb is the `pointparticle' action of the centreofmass coordinates, y , of
For instance, the interactions in this pointparticle action are often taken to be2
Sb =
ds Lb =
which s is an arbitrary parameter. Here
= g y_ y_
denotes the induced worldline
metric determinant, and both g (x) and A (x) are evaluated along the worldline of the
source particle of interest: x
main applications) represent the pointparticle mass and charge. Variation of y in this
action gives the equations of motion of the particle within a given bulk eld con guration,
while variation of g
and A in this action gives the contributions of the particle to the
stressenergy and electromagnetic current that respectively source the gravitational and
electromagnetic elds.
PPEFT adds to this framework the e ects of particle substructure, which can be
incorporated by additional interactions within Sb that involve more elds, more derivatives of
2We use units for which ~ = 1.
the elds, and/or more degrees of freedom (such as spin), in what is essentially a generalized
multipole expansion for the sources. Such an expansion has been developed systematically
in some situations (such as for slowly moving gravitating objects when computing the orbits
and radiation elds of inspiralling binary systems [56{64]).
In order to couple the source to the bulk eld equations it is useful to reexpress Sb as
an integral over all of spacetime using a function representation of the form
Sb =
where (x) generically denotes any bulk
When Lb is linear in the bulk elds its
variation provides the usual inhomogeneous source term for the bulk eld eld equations.
For instance variation of A , which appears linearly in (1.1), leads to a Maxwell equation
of the form
= q
relationship between electric eld and the source charge:
E = q 3(x) :
The importance of this expression is that its integral over a small spherical gaussian
pillbox, S
that expresses how the integration constants in the bulk
elds  such as the coe cient,
q appearing in the source action:
q =
d3x q 3(x) =
E = 4
2 er E(r = ) = 4 k :
The main result for a more general PPEFT is to derive and use the analogue of this
expression, which relates directly the nearsource boundary conditions in terms of the
action describing the properties of the source.
Classical renormalization and boundary conditions
So far so standard. However two related subtleties arise in this approach once terms are
examined in Sb that are not simply linear in the bulk elds. One of these is the need that
then arises to regularize and renormalize the branebulk couplings even at the classical
level. The second is the need to include in this action also the Schrodinger eld,
the related change to the boundary conditions of bulk elds that such terms imply. We
next brie y summarize both of these issues in turn.
3Although it is tempting to treat the function as independent of the bulk elds this is not in general
possible for the metric, since by construction the localizing function is designed to discriminate points
according to whether or not they are far enough away to be inside or outside of the source. Assuming it to
be metric independent can sometimes lead to inconsistencies with the balance of stressenergy within any
microscopic source (for a recent instance of this issue see [65]).
rst, terms in Sb that are not linear in bulk
elds, such as Lb =
)2 where c1 is a coupling constant. The subtlety is that use of such
form c1r [(E
terms in the above argument involves evaluating the bulk
eld near the source, and this
generically diverges.4 For instance, a term like c1(E
E)2 in Lb produces a term of the
E)E] 3(x) on the righthand side of (1.4), leading to a similar term in (1.5)
to be related to the physical scales of the problem as sketched in gure 1.
There are several reasons to entertain terms in SB that are nonlinear in the elds.
For the problems studied in this paper we do so because the bulk
eld we wish to follow
is the Schrodinger eld, , (rather than the electromagnetic eld, say) and this eld rst
appears in Sb at quadratic order. But even if we were to focus exclusively on electromagnetic
interactions, nonlinear terms are not only allowed, they are usually obligatory for actions
describing realistic objects. They arise because they express the dominant implications for
longdistance physics of any substructure of the source.
The divergences that appear in nonlinear terms when bulk elds are evaluated at
classical solutions (i.e. divergences at small ) can be dealt with by appropriately renormalizing
sourcebulk couplings (couplings like q or c1 in the above example). It is sourcebulk
couplings (rather than couplings internal to the bulk, say) that renormalize these
divergences because the divergences themselves are localized at the source position. Standard
powercounting arguments then ensure that renormalization is possible provided all
possible interactions are included in the source action that are allowed by the eld content
and symmetries. This renormalization program has been worked out most explicitly for
branes coupled to bulk elds, both for scalars and for gravitational elds5 [66{72], and our
presentation here is essentially an adaptation of that developed in [71, 72]. The instance
where Sb is quadratic in the elds contains as a special case the wellknown phenomenon
of renormalization for deltafunction potentials [6, 76, 77].
So far as our companion papers are concerned, the most useful result of this paper
follows directly from this renormalization story: the renormalizationgroup (RG) running
it implies for the sourcebulk couplings. As we shall see, this running turns out to be fairly
universal for source actions that stop at quadratic order in the elds (as often dominates
at low energies). In some circumstances the running of the source couplings can lead
to surprising consequences. In particular, in the presence of an inversesquare potential,
V =
scales. When this happens they can at best be set to zero at a single scale, say at high
energies in the deep ultraviolet (UV), but once this has been done they are free to run,
sometimes with unexpected physical consequences at lower energies.
4A notable exception is the special case of only one dimension transverse to the source.
5In the gravitational case, some papers [73{75] regularize these divergences, such as by replacing the
source by a `thick' brane, but without following through with their renormalization. As argued in [71], such
regularization arguments often work in practice  basically because physics far from the sources depends only
on a few multipole moments. However, because they do not renormalize they can indicate a dependence on
the microphysical scale
that is misleading.
This RG running of sourcebulk couplings sometimes contains surprises. RG invariance
generally ensures that physical quantities do not simply depend on the values of the
couplings speci ed at UV scales. Dimensional transmutation instead ensures they are xed by
RGinvariant scales that characterize the ow. (This is reminiscent of how the basic scale
of the strong interactions is given by the RGinvariant QCD scale,
QCD, rather than the
more microscopic scale, , at which point the value of the QCD coupling, g( ), might be
speci ed.) What is important about these RGinvariant scales is that they can sometimes
be much larger than the size of the source being described by the e ective theory, and
when this is true sourcebulk interactions can appear to be surprisingly large. We argue
here (and in more detail in [1, 2]) that this fact may partly underlie otherwisepuzzling
phenomena like monopole catalysis [79, 80, 82, 83] of baryonnumber violation in Grand
Uni ed Theories (GUTs).
A road map.
Our presentation of these arguments is organized as follows. Section 2
examines in detail the running of source couplings that are driven by the nonrelativistic
quantum mechanics of a bulk Schrodinger
eld interacting with the source through an
inversesquare potential in d spatial dimensions. These calculations closely parallel other
treatments of inversesquare potentials in the (quite extensive) literature (for reviews of the
quantum mechanics of the inversesquare potential see [4{15]), for which renormalization
e ects have been widely studied. Our main new ingredient here is our construction of the
boundary condition in terms of the pointparticle action.
spatial dimensions, with one eye on deriving results of later utility. Along the way we clarify
how catalysis (unusually large scattering cross sections) can arise when the RGinvariant
scales are much larger than the microscopic size of the source, and apply these results to
the special case of nonrelativistic swave scattering of a charged particle (like a proton
or electron) from a magnetic monopole. Whether this mechanism actually arises once the
lowenergy theory is matched to the underlying monopole is a more detailed question that
goes beyond the scope of the present paper.
The Schrodinger inversesquare potential in d dimensions
To make the above discussion concrete we now turn to a hoary old saw: the quantum
mechanics of a particle interacting with a point source through an inversesquare potential,
V (r) =
and for which the appearance of classical renormalization is wellknown.
The main di erence in our presentation is mostly one of emphasis rather than
computational di erence, with the important exception that we argue there is a systematic
way to determine the boundary conditions in terms of the action of the point source. In
particular we try to cleanly separate regularization issues (the need to cut o the
inversesquare potential in the deep UV regime where r <
) from renormalization issues (the
identi cation of which couplings in the lowenergy theory must be renormalized to cancel
the regularization dependence from physical quantities). We do this by interpreting the
wellknown divergences and dimensionaltransmutation that arise for the inversesquare
potential in terms of the renormalization of the action of a source situated at the origin,
that in the nonrelativistic case studied boils down to a deltafunction potential.6 Because
renormalization implies the source coupling runs, it must be present. We argue that its
presence helps simplify the understanding of many physical features, such as why bound
states sometimes exist in regimes where they are hard to understand purely in terms of the
inversesquare potential itself.
nal reason for reexploring this system in some detail is that it captures in the
simplest context several properties that also arise in more complicated applications, in
particular catalysis of baryonnumber violation and novel relativistic e ects in Coulomb
systems  that are treated in several companion papers [1{3].
Schrodinger `bulk'
with the `bulk' described by the Schrodinger action, SB, and where Sb describes microscopic
SB =
where m is the particle mass and
Sb =
dtLb[ (r = 0);
(r = 0)] =
V (r) =
Lb =
used when an explicit form is required.
eigenstates, for which
(x; t) =
(x) e iEt, becomes
The eld equation found by varying
, with the choice (2.3) and specializing to energy
U = 2mV (r) + 2mh (d)(r) ;
and 2 =
2mE =
k2. Eqs. (2.4) and (2.5) show the equivalence of the quadratic source
action with a deltafunction contribution to the potential. For bound states E
0 =)
is real; for scattering states E
0 =) k is real.
In sphericalpolar coordinates, the metric for ddimensional Euclidean space can be
sphere, parameterized entirely by the periodic coordinates
1; : : : ; d 1. In terms of this
6This is to be contrasted with a sometimesarticulated alternative picture where what is being
renormalized is the value of the cuto potential, V ( ), in the farUV regime r < .
metric, the Schrodinger equation is separable into a radial piece and an angular piece:
is the Laplacian for g^mn on the
Clearly the equation is separable and we can write
(r; 1; 2; : : : ; d 1) =
where ! is one or more parameters associated with the ddimensional spherical harmonics
de ne the eigenvalue of the (d
1)dimensional spherical laplacian by
$ (e.g., this is the
`(` + 1) in three dimensions, while it is
`2 in two dimensions, and of course 0 in
one dimension), then the Schrodinger equation reduces to the following radial equation for
r 6= 0:
! =
functions of order l + (d
2)=2, where l is de ned by 2l + d
2 =
2)2 + 4($
Nearsource boundary condition
For small r the radial solutions asymptote to become proportional to rl and r l (d 2). For
(indeed, the two functions j
2)=4). This shows
2)2=4
that boundedness of the solutions as r ! 0 cannot be the right boundary condition to use
at the origin.
It is perhaps not a big surprise that boundedness is not the right criterion because
there are many examples where
elds diverge at the positions of point sources, such as
does the Coulomb potential at the position of a point charge. Once this point is conceded
one must recognize that boundedness also cannot be the right criterion to use in general,
and in particular it might not be appropriate to discard solutions in situations for which one
What boundary condition should be imposed instead? A weaker criterion at the origin
imposes the normalizability of
r ! 0. This condition excludes solutions that diverge faster than
4)=4 + $, we nd
> 2, and so
1) =
1, hence the solution that goes as r l (d 2) is excluded.
, which asks R dr r(d 1)
j j2 to converge in the regime
r d=2.
1) =
The case of most interest in what follows is when d(d
where $1 denotes the rst nonzero eigenvalue of the angular laplacian,7 as this is the region
swave states (e.g. in three dimensions, this is the range
is insu cient in itself in this case to determine the boundary condition for swave states.
It is for these that we argue the correct condition instead is given by the properties of the
source action, Sb, as indicated below. Physically this occurs because the inversesquare
potential draws the swave wavefunction su ciently towards the origin that the net ow of
probability there cannot be determined without knowing more precisely how the particle
the centrifugal barrier is
strong enough to keep this from happening for any nonzero $.
The choice of boundary conditions as r ! 0 also bears on whether or not the
inversesquare Hamiltonian is selfadjoint, and this is the way the need to choose boundary
conditions is usually framed. Although usually not incorrect,8 demanding boundary conditions
ensure selfadjointness (i.e. nding a selfadjoint extension) typically does not determine
them uniquely. The advantage of casting the boundary condition directly in terms of the
source action is that it makes explicit the connection between any nonuniqueness and the
choices available for the physics of the source. When the source action is real the resulting
boundary conditions ensure no loss of probability, as we show in a speci c example below.
The implications of the source action, Sb, for the boundary condition is obtained9 by
integrating (2.4) over an in nitesimal sphere, S, of radius 0
around r = 0 (and
considered here the result is the same as expected for a deltafunction potential where
implies the result gets only contributions from the delta function and from
the integral of the second derivative, leading to the result
(0) =
@r r=
@r r=
= 2mh, n
g^ d 1d 2 : : : d n is the volume element on the surface of the unit nsphere, and
is the corresponding volume. The last equality assumes a spherically symmetric source
and that is small enough that
is also spherically symmetric to good approximation.
7Of course in one dimension, $ does not strictly exist, since there is no angular Laplacian. In that case,
we de ne $1 as 1, since the lack of higher angular momentum state means there is no upper bound on .
8Unless the physics of the source does allow it to be a sink  or source  of probability, such as in
situations like those described in [55], for example.
9Strictly speaking the reasoning presented here is only true for deltafunction interactions in the absence
of inversesquare potentials, because the singularity of the
g=r2 potential at r = 0 undermines the
argument that only the derivatives and delta function can contribute when integrating the equations of motion
over a small pillbox. A better derivation that also applies when inverse potentials are present is given in
appendix A, with the bonus that its formulation also provides a clearer picture of what the RG equations
physically mean.
The required boundary condition at the origin then is
which uses the de nition (0) :=
values, and so the integral (2.9) evaluates to
= ;
(d = 1):
As we see later, this boundary condition implies physical quantities depend on ,
and (at face value) also on . The main point in what follows is that the dependence
of all physical quantities on
can be absorbed into an appropriate renormalization of the
parameter . After this is done, all explicit dependence becomes cancelled by the implicit
dependence in
implied by the boundary condition (2.10) or (2.11). Before pursuing this
further we pause brie y to ask whether or not this boundary condition is unitary.
Source action vs selfadjoint extension
above encodes how the source backreacts onto its environment over distances much larger
than the size of the source itself.
Does this boundary condition provide a selfadjoint
extension [20{39] in the sense of conserving probability at the source?
To see how this works we use the boundary condition to compute whether the region
r < j j is a source or sink of probability using the radial probability ux,
J = d 1rd 1 n J =
Evaluating with energy eigenstates gives
J ( ) =
= (h
(d > 1) ;
J ( ) = J ( ) + (h
(d = 1) :
This states something reasonable: probability is conserved at the source provided either
a nonunitary boundary condition be desired.
Bound states
We next seek boundstate solutions to the bulk equations, using their smallr form to
de
runs with . (Clearly this running also could be determined using scattering
solutions rather than bound states. As we see explicitly below, the results are the same.)
Although these solutions can be written in terms of modi ed Bessel functions, we here
instead analyze them in terms of con uent hypergeometric functions since this generalizes
more easily to other applications (such as those in [1, 2]). To this end we write the radial
equation (2.8) in the form
= 0 ;
where v =
2) + v = 0, and so10 given by
d + ). In this case the two linearly independent radial pro les can be written
where M(a; b; z) = 1 + (a=b)z +
is the usual con uent hypergeometric function and
= (2 r) 21 (2 d
:= p(d
2)2 + 4($
) = 2l + d
l = 12 (2
if it arises).
nonzero $.
For bound states we seek solutions normalizable at in nity, and the largez asymptotic
expansion of hypergeometric functions shows this leads to the following combination of
solutions (with arbitrary normalization constant C)
1(r) = C
which shows that integer
can be problematic (and so is obtained by a limiting procedure
On the other hand the solutions
(r) for small r =
is only normalizable at r = 0 when 2
> 0. Adopting the convention
0 when real, we see that
+ is always normalizable at r = 0 but
normalizable there when
< 2. Keeping in mind that
(in agreement with the discussion above) that
= p(d
2)2 + 4($
can be discarded for all $ 6= 0 whenever
choose a value for
in (2.18) to ensure this implies no bound states exist in this case for
4)=4 + $1, either
< 2 or is imaginary when $ = 0, and
so both solutions are normalizable. In this case we instead use the boundary condition at
the origin given in (2.10) which, when using the smallr expansion,
10Choosing the other root for l just exchanges the roles of the two independent solutions encountered
below, so does not introduce any new alternatives.
d + ) + R (2
R :=
C+(2 ) 21 (2 d+ ) + C (2 ) 21 (2 d )
choices of $.
Once values for
C =C+ and so implies
Notice R = 0 when C
= 0, which for su ciently small
applies for all nonzero
To use this equation we rewrite it as
^ :=
2 =
where the rst equality de nes ^. Notice that the deltafunction potential is repulsive when
2, while attractive potentials (
< 0) imply ^ < d
and ^ are given, eq. (2.23) is used by solving it for R in terms of the
R(^) =
and negative otherwise. Equating this to (2.22) either allows the
R =
which uses Euler's duplication formula. Since
is a square root, it will either be entirely
With this, we can write (2.23) cleanly as
= ln
^ =
It should be noted that when
= i is purely imaginary,
= i is also purely imaginary,
and the hyperbolic cotangent becomes a regular cotangent. That is,
^ =
Notice that (2.21) either diverges or vanishes as
! 0, and this is where the
renormalization story comes in. We must choose
also to diverge or vanish as
way as to ensure what remains is a nite, sensible, independent expression for
also for the bound state energy E).
Renormalization
Again focusing on the case d(d
4)=4 + $1, we rst consider the case
2)2=4
4)=4 + $1. To
determine in this case how
must depend on
in order to renormalize any divergences
! 0, we use that the energy
@ =@ =
cannot depend on
and di erentiate (2.28) using
2 sin2( =2)
0( ) = (2n + 1) = arg
En =
e2[ ( ) (2n+1) ]= ;
This can be integrated to nd ^( ) giving
in terms of 0 by
Solving then gives
where ei :=
(i s=2)= ( i s=2).
This indicates an in nite number of states
tan h 2 ln( = 0)
+ ^0 tan h 2 ln( = 0)
Inserting this into (2.28) gives
in a manifestly independent way, with a simple
calculation showing that
is again given by (2.28), but with ^ ! ^0 and
! 0 := ( 0):
^0 =
is given explicitly
parameter ? is chosen such that ^( ?) = 0.
2)2=4 <
grows, and the
where n = 0; 1; 2;
up until n is so large that jEnj is greater than the UV cuto
of the EFT in which the source size is not resolved (allowing it to contribute simply a
delta function). As is often observed, dimensional transmutation is at work here with the
appearance in En of the scale 0
. The above discussion shows that it is the value of the
dimensionless coupling ^ that is traded for the scale 0 (as opposed to the dimensionless
coupling, , say).
This is the RG limitcycle regime that has been extensively studied, in particular within
the context of the E mov e ect [40{49].
The limit
2)2=4.
The case where
2)2=4 (and so s = i s ! 0) is
instructive for two reasons. First, the tower of bound states collapses to a single bound
state, and the reason for the one remaining state is clear: in this limit the inversesquare
potential is not deep enough to support a bound state and so the lone bound state is
the one supported by the deltafunction potential. Second, dimensional transmutation for
this state can allow an exponential suppression of the binding energy of this state relative
to the typical (microscopic) scale set by the deltafunction itself, and this is what allows
the bound state to have lowenough energy to be reliably described purely within the
lowenergy theory.
we have ( s) =
s + O( s2) as s ! 0. Therefore
Recalling the small form
i s=2) = 1
i s=2 + O( s2), and so ei ( s) =
the EulerMascheroni constant,
0 =
and so in this limit (2.31) becomes
^0 =
+ ln( 0=2)
which integrates to
Again this is precisely what is required to make
determined from (2.37) independent of .
A convenient way to write this is in terms of the `QCD' scale, ?, de ned as the scale
where ^ ! 1 becomes strong. In terms of ? (2.37) becomes
and so the boundstate condition becomes
As s ! 0 the running of is most easily found by returning to the RG equation (2.29)
and integrating it from scratch, leading to
2 sinh2( =2)
2=^0i :
which shows that it is ? that sets the scale of the lone bound state in this limit. This last
result is only really interesting if ?
0, since this allows a hierarchy between the bound
state and the scale of the `brane' source. We see that ^ ! +1 for
! ? satisfying
These last two equations show that the bound state scale ? is only much greater
than the microscopic scale 0 when ^0 is negative and very small. When this is not true
the bound state is not macroscopic and so its existence cannot reliably be inferred purely
within the lowenergy e ective theory. It is in this way that we see why the existence of the
The case
bound states when
2)2=4.
Although the inversesquare potential does not support
as we now show. In this regime, the boundstate condition is given by (2.27), which we
repeat here
^ =
This admits no solutions if ^ < s and admits one solution when ^ > s
The running of ^ is again found by di erentiating the quantization condition with
respect to , holding
xed. When s is real we use d =d =
s and d =d = d =d = 0,
4)=4 <
of ows is of interest.
This integrates to
It is important to note that this RG
ow now has xed points11 at
(^0= s) + tanh 12 s ln( = 0)
1 + (^0= s) tanh 12 s ln( = 0)
^ := ^[ln( = 0) !
1] =
with the ow towards the infrared (large ) going from ^
to ^+. As shown in gure 3,
this evolution supports the following two disjoint classes of ow, according to the size of
j^j= s throughout the ow:
cally from ^
ows within this class climb
monotoniincreases, ensuring that there always exists a scale, 0, for
s systems that ow along these
trajectories never have deltafunction couplings that can support any bound states.
The second class of ows satis es j^j
initially drop monotonically through negative values of ^ from ^
ows within this class
! ? from below. For
> ? the coupling ^ falls from ^ ! +1 as
above, and falls monotonically with increasing
! 1. Systems with these RG
until eventually approaching ^
s and so can support a bound
state. We shall see that the bound state arises with characteristic scale ?, and this
is only macroscopic when ^ is negative and very small, just as was found above for
= (d
2)2=4.
11Ref. [15] uses the disappearance of these xed points as
! 41 as an archetype of how xedpoint
coalescence can lead to universal features.
Focusing on the case j^j > s, the boundstate condition is e ciently given in terms of
! 0, implying R =
1 and so
where ? is given in terms of and ^( ) by
For s positive and ?
requires ^= s =
tanh x = (1
(1 + ) with 0 <
e 2x)=(1 + e 2x) ' 1
the lefthand side is just a hair smaller than +1 and this
1. Using in this regime the asymptotic expression
2e 2x for large x then leads to the approximate
These show how the formulae for a bound state with macroscopic size become extended
into the regime where
2)2=4 for ^ su ciently close to
s. In this regime the
underlying deltafunction potential (whose strength is ) is su ciently attractive to give
rise to this bound state, and this is its physical origin.
The RG running of
can sometimes have surprising implications. In particular,
because it runs it cannot be set to zero for all scales at once. When
for at most a single scale, 0
. This is because
not a xed point of the RG
ow unless s = d
= 0 corresponds to ^ = d
2, which is
2 (and so
= 0). If there should exist a
2. Because s < d
2 this satis es12
passing through ^ = d
= ?
6= 0 this can be done
Of later interest is the asymptotic form for this running as ^( ) nears the xed points
. Using the asymptotic expression for tanh z for large positive or negative z
^ = + s < d
at ^ =
(^0= s) + tanh 12 s ln( = 0)
1 + (^0= s) tanh 12 s ln( = 0)
12Although this naively satis es the boundstate condition, the size of this state satis es ?
the state does not lie believably within the lowenergy theory.
which reveals how the quantity d
s = d
4 acts as an `anomalous
to large is given by
0 when ^0 = d
Scattering and catalysis
We next describe several applications of the above boundary conditions to scattering
problems. Doing so illustrates several separate points. First, it shows that the same
renormalizations required to make sense of the boundstate problem also do so automatically for
scattering problems. Second, it shows how scattering cross sections need not depend on
the coupling
in the naive way expected from the Born approximation,
might be expected to vanish  or be suppressed by powers of the UV scale (k )
the limit where k
! 0. Instead we nd them to be controlled by the RGinvariant scale
k ? or k 0, which can be much larger than the UV scale
associated with the underlying
structure of the source.
For concreteness we restrict ourselves here to scattering in three dimensions, and
specialize at the end to a particularly simple example (nonrelativistic swave scattering of
fermions from a magnetic monopole). As a reality check, in an appendix we also present
results for scattering in one dimension since this allows us to connect our framework to
standard results for that particularly simple case.
Scattering when d = 3
For scattering problems the radial part of the Schrodinger equation in three dimensions
= 0 ;
where v =
radial pro les that solve this are given in terms of con uent hypergeometric functions by
(r) = (2ikr) 21 ( 1 ) e ikr
4v = p(2` + 1)2
= 2l + 1.
Writing the general solution as
1(r) = C+ +(r) + C
above, to 0
< 54 ) and to the expression
we may in this case use the boundary condition (2.22) through (2.24) at the origin to
determine the ratio C =C+, leading to C
= 0 for ` 6= 0 (for simplicity we restrict, as
= (2ik ) s s
= 0 if ^ should lie at the IR
point ^ = s
It is the running of
and it is the di erence between ^ and its IR
xed point (rather than, say, having
= 0)
that drives its value away from the value found in the absence of a source. It is also this
(that can be surprisingly di erent than the size of the underlying source).
To see how this works it is convenient to exploit the independence of the result to
evaluate it asymptotically close to the IR
into (3.4) this shows how the explicit powers of cancel to leave
xed point at ^ = s using (2.48). Inserted
= (2ik ) s s
0 =
y = sign[j^j
is the RGinvariant sign that determines which de nition of ? is to be used.
larger behaviour of our wavefunction to the form
from which we determine the phase shift by [84]
M(a; b; z)
To use this we write the asymptotic behaviour of the M functions for large imaginary
ei(kr ` =2)
e i(kr ` =2)
e2i ` =
ei `=2; and
e(1+ `)i =2 :
This gives ?
e2i 0 =
1 + A e i s=2
e(1 s)i =2 ;
A :=
Notice that this depends on the deltafunction coupling only through the RGinvariant
quantity y ?. It is this feature in particular that opens the door to the possibility of
catalysis: contact interactions can contribute to observables (such as scattering rates) by
an amount much larger than the UV size naively associated with the microscopic source.
Such catalysis occurs in situations where RG evolution predicts values for ? to be much
larger than the values appearing in ^( ) and :
? =
when ^( ) is close to the UV
s(1 + ), where
Special case: the deltafunction potential.
= 0 in the above formulae
reduces them to a useful special case: where the scattering is purely from the deltafunction
part of the interaction. In this limit (3.13) and (3.14) give the phase shift for scattering
from the 3D deltafunction potential when evaluated with
= 2` + 1. In this limit the
ow, at which point there
swave scattering is nontrivial.
For any ` 6= 0 this simpli es using C
` =
(for ` 6= 0) :
p , instead gives the phase shift
0, Euler's duplication formula in the form
where (with y as de ned in (3.6))
e2i 0 =
A =
tan 0 = =(1
in agreement with standard calculations [77]. The energy dependence of this result is
k1 tan 0 = y ? is kindependent. This
comparison gives an independent measure of the RGinvariant scale ?, and using this to
trade y ? for as in the earlier predictions for boundstate energy shifts provides them as
functions of as; thereby directly relating physical observables to one another. As applied to
mesic atoms (such as a
orbiting a nucleus) [1] similar reasoning leads to the Deser
formula [78] relating energylevel shifts due to nuclear forces to nuclear scattering lengths.
Nonrelativistic swave fermionmonopole scattering
The deltafunction scattering result just described has an immediate application to swave
elastic scattering of nonrelativistic fermions from a magnetic monopole, and shows how
these processes might also exhibit catalysis. In this section we develop this connection a bit
more explicitly, with the goal of showing that this type of scattering can also be controlled
by the RGinvariant scale de ned by the contact interaction.13
The utility of something as simple as deltafunction scattering to something as
complicated as fermionmonopole scattering arises because of the great simplicity of swave
scattering for these systems; in particular the possibility of there being no angular
momentum barrier. For scattering from a magnetic monopole the absence of such a barrier for
swave scattering is a bit more subtle than it looks, and would not be possible for monopole
scattering of a spinless boson.
Finally we apply the above insights to nonrelativistic swave scattering of a charged
particle from a magnetic monopole in the Pauli approximation of spin, where the running
provides a simple understanding of why such scattering need not be suppressed by
microscopic (i.e. GUTscale) lengths, as would have been naively expected in Born
approximation. This bears more than passing resemblance to earlier discussions of monopole
catalysis of baryon number violation, as we explore in more detail in a companion paper [2]
dedicated to the full relativistic treatment.
The electromagnetic potential of a pointlike monopole with magnetic charge g sitting
at the origin is given in spherical polar coordinates by
A =
where e is the unit vector in the azimuthal direction and the Dirac quantization condition
monopole eld. As discussed in many references  see for example the reviews [82, 83]
 the magnetic eld contributes to the angular momentum for motion in this potential,
The eigenvalues of L2 for this modi ed angular momentum remain `(` + 1) but now with
` = ;
:= eg=4
= n=2.
magnetic monopole. When
12 it is possible for spinhalf particles to have zero total
angular momentum, however, and so it is for this speci c combination that one might hope
13Whether or not this leads to catalysis depends on whether or not matching to the underlying monopole
actually does give couplings ( ) for small that lie su ciently close to the UV xed point.
to nd no angularmomentum barrier in the radial equation. This is borne out by detailed
calculations, for which the SchrodingerPauli equation
acting on the 2component Pauli spinor, , leads to the following radial equation
= 0 ;
= 0
for energy eigenstates
e iEt with momentum k = p
2mE as above.
er term here has its origin in the
B interaction with the fermion's magnetic
moment. For ` =
precisely one of the spinor harmonics satis es
coe cient of 1=r2 becomes proportional to L2
= ( + 1)
, so that the
= 0, as claimed.
In this case the bulk potential interaction vanishes, leaving only the deltafunction coupling
coming from any contact interaction, such as
Lb =
at the position of the source monopole.
This is the mode believed to participate
in monopolecatalyzed baryon violation [79, 80], for which the absence of an
angularmomentum barrier allows an incoming swave state to penetrate right down to the
monopole position.
For the present purposes what is important is that the scattering story for this mode
is told precisely as above, dropping the inversesquare potential and keeping only the
deltafunction interaction. Consequently its size is controlled by the RGinvariant scale ?
raising the possibility that monopole catalysis might be regarded as a special case of the
more general catalysis phenomenon discussed above.
We extend the above discussion to a more detailed examination of the implications of
PPEFTs to monopole scattering and to the problem of scattering in the Coulomb potential
in several companion papers [1{3].
Summary. In short, we propose the systematic use of the e ective pointparticle
action for deriving the boundary conditions appropriate near any source, and show why this
point of view helps understand the subtleties of the quantum mechanics of singular
potentials like the inversesquare potential. The presence of an action makes transparent the
process of regularization and renormalization of elds that are classically singular at the
source, and allows use of renormalizationgroup techniques for resumming the breakdown
of perturbation theory that often arises in the nearsource limit due to these singularities.
The great virtue of this approach is its constructive nature, since the linking of
boundary conditions to source properties removes the guesswork associated with their
determination. Furthermore, because only a few types of actions typically dominate at low energies
(for instance a choice quadratic in elds often does, as we describe here) it is likely that a
very broad variety of systems will fall onto a universal class of renormalizationgroup ows.
Acknowledgments
We thank Horacio Camblong, Friederike Metz, Duncan O'Dell, Carlos Ordon~ez, Ryan
Plestid, Markus Rummel and Kai Zuber for useful discussions and Ross Diener, Leo van
Nierop and Claudia de Rham for their help in understanding singular elds and classical
renormalization. CP3 Odense and the Neils Bohr Institute kindly hosted CB while part
of this work was done. This research was supported in part by funds from the Natural
Sciences and Engineering Research Council (NSERC) of Canada, by a postdoctoral felloship
from the National Science Foundation of Belgium (FWO) and by the Ontario Trillium
Fellowship. Research at the Perimeter Institute is supported in part by the Government of
Canada through Industry Canada, and by the Province of Ontario through the Ministry
of Research and Information (MRI).
The boundary action and the meaning of the classical RG
This appendix provides a more systematic derivation of the boundary conditions used,
following closely the reasoning used in [71]. More careful reasoning is required because the
standard argument becomes suspicious in the presence of a su ciently singular potential.
The argument of [71] has the added value of providing a simple interpretation for what the
classical RG equations physically mean.
The deltafunction story.
Recall rst the argument establishing the boundary
conditions used in the main text. This uses the naive reasoning appropriate for deltafunction
interactions in the absence of singularities in the rest of the scalar potential. In it one
starts with the timeindependent Schrodinger equation (say),
+ U (r) +
= k
and integrates over the small pillbox, S, with radius
main point: if U (r) is smooth within S, then one expects the integral of [U (r)
S to vanish in the limit of vanishing pillbox radius,
! 0. When this is true the remaining
centred at r = 0. Now comes the
= h (0) ;
leading to the boundary condition used in the main text. However if U (r) is su ciently
! 0, undermining
faith in its validity.
A better argument.
To arrive at a better argument it helps to refer again to gure 1,
which identi es three scales subject to the hierarchy "
a, where " and a are physical
scales respectively associated with the size of the underlying source and the size of the
physics that is of interest far from the source. The scale
is not part of the original
physical problem, however, but rather is introduced purely as a calculational crutch in
order to conceptually separate the calculation into two parts. One imagines drawing a
, of radius
around the source and then separately thinking about the
largedistance story relevant for r
outside this sphere and about a smalldistance story
relevant to r
well inside this sphere.
The two steps of the problem are then to derive the elds and their derivatives at the
surface of S given the properties of the source (typically as speci ed by its action, Sb).
The advantage is that this can be done once and for all, without specifying precisely which
observables are of interest outside S
. These enter only in the second step, where their
properties are computed using only the boundary data on the surface of S.
Although in principle
could be arbitrary, in practice we choose
" in order to
pro t from a multipole expansion (or its generalization) when computing the boundary
information on the surface of S, since successive multipoles are suppressed by powers of
"= . We similarly choose a
so that the observables of interest are su ciently distant
from S that they are not inordinately sensitive to boundary e ects there.
The boundary action. Literally specifying
elds and their radial derivatives at the
boundaryvalue problem exterior to S
. This is because in practice the actual values taken
elds on B also depend somewhat on the positions of other source and boundaries
elsewhere in the problem, and although this dependence is weaker the further away they
are (hence the condition
a), it is there and the boundary information at B must be
encoded in a way that leaves the elds free to adjust as required to meet its demands.
A simple and e cient way to do so is to specify the boundary data at B in terms of
a boundary action, IB. This action is related to, but not the same as, the original source
action, Sb. Whereas Sb is an integral over the source's worldvolume, IB is always integrated
over a codimension1 worldvolume that B sweeps out as time evolves. For instance for
point particles in 3 spatial dimensions Sb comes to us as a onedimensional integral over
the particle worldline, while IB is a 3dimensional integral over time and the two angular
directions of a 2sphere surrounding the particle. For instance, for the type of interaction
studied in the main text we have (using adapted coordinates in the particle rest frame)
Sb =
and IB =
and so whereas h has dimension length2  for the 3D Schrodinger eld, for which
length 3  the coupling h~ is dimensionless.
Once IB is speci ed the surface B can be regarded as a boundary of the exterior region,
with its in uence on physics exterior to S obtained in the usual way.14
What is needed is a way to construct I
B given Sb. In principle this is a matching
calculation: if N e ective couplings are of interest on Sb then we compute any N convenient
quantities exterior to S from which they could be determined. We then compute these same
N quantities in terms of the most relevant interactions on IB and equate the results in order
to infer the couplings of IB in terms of those of Sb.
14That is, the source physics contributes to the path integral a phase eiIB , or phrased another way, classical
physics is determined by requiring the total action, SB + IB, be stationary with respect to variations of the
elds throughout the bulk and on the boundary surface B.
For the simple cases considered in the main text our interest is largely restricted to a
single term in the source action, like h
, and so this matching process is particularly
easy. In practice in such situations the relation between Sb and IB boils down to something
very simple: Sb is the dimensional reduction of I
B (as would be appropriate if S were
examined with insu cient resolution to see that it has a nite radius). For instance the
couplings of (A.3) would be related by
Finally, given the boundary action I
B of (A.3) the boundary condition is simple to
derive by varying the combined bulk+boundary action SB + IB with respect to variations
of the elds on the boundary B
. In the present instance only the spatial derivatives of
SB play any role, due to the integration by parts required when computing
SB=
generalizing the scalar boundary condition (A.2) to
= 0 ;
where the rst equality assumes SB =
], where none of the terms
rep. In situations
where Sb = 4
IB this takes the form found in [71, 72]:
@r r=
This is the boundary condition used in the main text, and shows how it would be generalized
to situations beyond the simplest
What is clear is that this construction goes through equally well with and without
there being an inversesquare potential in the bulk, and this is the underlying reason why
the singularity of the inversesquare potential does not alter the boundary condition used
in the main text.
Physical interpretation of the RG.
The boundary action provides a simple physical
interpretation of what the RG evolution described in the text. What it expresses is that
there is nothing unique about the choice of S, whose radius, , could be anything subject
only to the condition "
a. The RG condition expresses this fact.
In detail, when computing the RG evolution by di erentiating with respect to
always hold
xed all physical quantities. What we are doing is adjusting the couplings
in IB in such a way as to not change the physical bulk eld pro le. This relates the RG
evolution of the couplings in IB to the classical bulk eld equations.16 The couplings of Sb
that naturally arose in the RG derivation given in the main text.
, inasmuch as
16This resembles what occurs in AdS/CFT [85].
they appear in the r !
limit of the bulk eld equation.
= 4 2.
scattering observables.
Scattering in 1 dimension
This appendix computes the transmission and re ection coe cients implied by the main
Besides providing a
comparison with standard results, it also illustrates (as in the 3D case) that the source
renormalization required to make sense of boundstate calculations automatically does
the same for other observables, and shows how the RGinvariant scale can be related to
For scattering we take E > 0 and so
= ik is imaginary with k2 = 2mE
0. Eq. (2.4)
z u(z) with z = kx implies
u(z) satis es Bessel's equation,
z2 u00 + z u0 + (z2
2)u = 0 ;
with 2 := 14
0 eq. (B.2) is most usefully solved by the Hankel functions H(1)(z) and
H(2)(z), which in our conventions (see next appendix) asymptote for large real z to
Since H(1)(kx) e ikt asymptotes at large x to be proportional to e ik(t x) it represents a
rightmoving wave, whilst H(2)(kx) similarly asymptotes at large x to a leftmoving wave.
Similar expressions hold for negative x, as may be found using the re ection properties of
the Hankel functions given in the appendix.
For a particle that initially approaches from x ! +1 we therefore take the following
solutions for x >
+(x) = N
(x) = N
where N is an irrelevant constant and the re ection and transmission coe cients, R and
Demanding +( ) =
) and using the re ection properties of the appendix implies
H(2)(k ) + R H(1)(k ) = iei
and so using the smallk limit of the Hankel functions then gives,
X :=
gives a second relation between
Inserting this into (B.9) gives
= X e
= @x ln +( )
The divergence here as
! 0 is, as before, cancelled by the dependence implicit in , and
can in principle be determined by di erentiating with respect to
with k held
xed. But
because this boundary condition is precisely the same one that was used to determine this
dependence in the boundstate problem this argument is guaranteed to return the same
evolution equation for
that was found earlier.
For the sake of concreteness we specialize now to
> 0 (and so
To write a more explicit form for R and T it is useful to exploit the RGinvariance of the
above conditions for T and R by evaluating them using a convenient value for , which
we choose as
?. This choice allows two simpli cations to be used. First, we can use
1 to expand in powers of X because X is proportional to a positive power of the
small quantity k
1, leading to
Second we can use the RG evolution to infer that ^ ! ^
asymptotic small expression (2.48) for the approach of ^ to this limit:
! 0, and so use the
from which the dependence completely drops out, as it must.
The re ection coe cient, R, found by solving this last equation then is
R =
T = ie i (1
R) =
of deltafunction scattering in the limit
These last expressions use
^ =
1 asymptotes to ^
2= ?, corresponding to using
= 12 in (B.10) to learn
itself satis es
in the small regime.
X? :=
The special case where
! 0 is also a delicate one since in this limit
+ O( 2) =: 1 + A? + O( 2) ; (B.16)
1 = A? + O( 2), leading to
where the last equality de nes A? and
1 = (A?
is the EulerMascheroni constant. Consequently
) =
What is noteworthy about these expressions is that the re ection and transmission
amplitudes are controlled by the RGinvariant combination k ? rather than by the value of
( ) as measured at a microscopic scale . This is particularly striking in the limit
for which it is only the deltafunction potential that drives the scattering. It is in strong
contrast to what one might have expected if treating a deltafunction potential in Born
approximation, since then would have been simply depended on
rather than the
. This carries the seeds of the explanation of why microscopic scales can drop
out of scattering cross sections  such as is known to be the case for monopole catalysis
of baryonnumber violation [79, 80]  and of a more systematic e ective description of
such processes.
Bessel function properties.
This appendix gathers useful properties of Bessel
functions, de ned as the series solutions to
of the form
z2 u00 + z u0 + (z2
2)u = 0 ;
J (z) =
It is also often useful to use the explicit form of the linearly independent solution
N (z) =
J (z) cos(
For large real z these enjoy the asymptotic forms
N (z) =
J (z) =
H(1)(z) = J (z) + iN (z) =
H(2)(z) = J (z)
iN (z) =
+ O(1=z)
+ O(1=z) :
For scattering problems of more use are the Hankel functions, de ned by
since these satisfy (for large real z)
H(1)(e i z) =
; (B.24)
and so the direction of the phase rotation makes a di erence. In particular, of use in the
main text is
H(1)(ei z) =
H(2)(z) and similarly
H(2)(e i z) =
though the formulae are more complicated if we rotate in the other direction by .
For small z we have the expansions
H(1)(z) =
H(2)(z) =
: (B.26)
Open Access.
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