Point-particle effective field theory I: classical renormalization and the inverse-square potential

Journal of High Energy Physics, Apr 2017

Singular potentials (the inverse-square potential, for example) arise in many situations and their quantum treatment leads to well-known ambiguities in choosing boundary conditions for the wave-function at the position of the potential’s singularity. These ambiguities are usually resolved by developing a self-adjoint extension of the original prob-lem; a non-unique procedure that leaves undetermined which extension should apply in specific physical systems. We take the guesswork out of this picture by using techniques of effective field theory to derive the required boundary conditions at the origin in terms of the effective point-particle 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 self-adjoint 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 effective-field picture provides a simple framework for understanding well-known renormalization effects that arise in these systems, including how renormalization-group techniques can resum non-perturbative interactions that often arise, particularly for non-relativistic applications. In particular we argue why the low-energy effective theory tends to produce a universal RG flow 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 baryon-number violation by scattering from magnetic monopoles.

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Point-particle effective field theory I: classical renormalization and the inverse-square potential

Received: January eld theory I: classical renormalization and the inverse-square 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 , B-3001 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 inverse-square potential, for example) arise in many situations and their quantum treatment leads to well-known ambiguities in choosing boundary conditions for the wave-function at the position of the potential's singularity. These ambiguities are usually resolved by developing a self-adjoint extension of the original problem; a non-unique 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 point-particle 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 self-adjoint 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 well-known renormalization e ects that arise in these systems, including how renormalization-group techniques can resum non-perturbative interactions that often arise, particularly for nonrelativistic applications. In particular we argue why the low-energy 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 baryon-number violation by scattering from magnetic monopoles. ArXiv ePrint: 1612.07313 ective; E ective eld theories; Nonperturbative E ects; Renormalization Group - Near-source boundary condition Source action vs self-adjoint extension Bound states Scattering and catalysis Scattering when d = 3 Nonrelativistic s-wave fermion-monopole scattering A The boundary action and the meaning of the classical RG B Scattering in 1 dimension 1 Introduction & discussion: point-particle e ective eld theories Standard formulations Classical renormalization and boundary conditions The Schrodinger inverse-square potential in d dimensions Introduction & discussion: point-particle 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 ground-work 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 well-known 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 lowest-dimension 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 scale-invariance 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 low-energy point-particle 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 inverse-square potential in isolation. Rather, one is obliged also to include a delta-function coupling whose strength runs | in the renormalization-group (RG) sense, though purely within the classical approximation | in a way that depends on the inverse-square coupling g. Many otherwise puzzling features of the inverse-square system become mundane once this inevitable presence of the delta-function potential is recognized. For instance, the inverse-square 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 delta-function, 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 self-adjoint [4{6], and is usually dealt with by constructing a self-adjoint 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 inverse-square 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 point-particle 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 centre-of-mass 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 `point-particle' action of the centre-of-mass coordinates, y , of For instance, the interactions in this point-particle action are often taken to be2 Sb = ds Lb = which s is an arbitrary parameter. Here = g y_ y_ denotes the induced world-line metric determinant, and both g (x) and A (x) are evaluated along the world-line of the source particle of interest: x main applications) represent the point-particle 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 stress-energy 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 in-spiralling binary systems [56{64]). In order to couple the source to the bulk eld equations it is useful to re-express 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 near-source 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 brane-bulk 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 stress-energy 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 right-hand 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 long-distance 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 source-bulk couplings (couplings like q or c1 in the above example). It is source-bulk couplings (rather than couplings internal to the bulk, say) that renormalize these divergences because the divergences themselves are localized at the source position. Standard power-counting 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 well-known phenomenon of renormalization for delta-function 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 renormalization-group (RG) running it implies for the source-bulk 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 inverse-square 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 source-bulk 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 RG-invariant scales that characterize the ow. (This is reminiscent of how the basic scale of the strong interactions is given by the RG-invariant 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 RG-invariant 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 source-bulk interactions can appear to be surprisingly large. We argue here (and in more detail in [1, 2]) that this fact may partly underlie otherwise-puzzling phenomena like monopole catalysis [79, 80, 82, 83] of baryon-number 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 non-relativistic quantum mechanics of a bulk Schrodinger eld interacting with the source through an inverse-square potential in d spatial dimensions. These calculations closely parallel other treatments of inverse-square potentials in the (quite extensive) literature (for reviews of the quantum mechanics of the inverse-square 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 point-particle 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 RG-invariant scales are much larger than the microscopic size of the source, and apply these results to the special case of non-relativistic s-wave scattering of a charged particle (like a proton or electron) from a magnetic monopole. Whether this mechanism actually arises once the low-energy theory is matched to the underlying monopole is a more detailed question that goes beyond the scope of the present paper. The Schrodinger inverse-square 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 inverse-square potential, V (r) = and for which the appearance of classical renormalization is well-known. 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 low-energy theory must be renormalized to cancel the regularization dependence from physical quantities). We do this by interpreting the well-known divergences and dimensional-transmutation that arise for the inverse-square potential in terms of the renormalization of the action of a source situated at the origin, that in the non-relativistic case studied boils down to a delta-function 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 inverse-square potential itself. nal reason for re-exploring 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 baryon-number 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 delta-function contribution to the potential. For bound states E 0 =) is real; for scattering states E 0 =) k is real. In spherical-polar coordinates, the metric for d-dimensional 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 sometimes-articulated alternative picture where what is being renormalized is the value of the cut-o potential, V ( ), in the far-UV 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 d-dimensional 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($ Near-source 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 non-zero eigenvalue of the angular laplacian,7 as this is the region s-wave states (e.g. in three dimensions, this is the range is insu cient in itself in this case to determine the boundary condition for s-wave 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 inverse-square potential draws the s-wave 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 self-adjoint, and this is the way the need to choose boundary conditions is usually framed. Although usually not incorrect,8 demanding boundary conditions ensure self-adjointness (i.e. nding a self-adjoint 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 non-uniqueness 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 delta-function 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 n-sphere, 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 delta-function interactions in the absence of inverse-square 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 self-adjoint extension above encodes how the source back-reacts onto its environment over distances much larger than the size of the source itself. Does this boundary condition provide a self-adjoint 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 non-unitary boundary condition be desired. Bound states We next seek bound-state solutions to the bulk equations, using their small-r 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 $. For bound states we seek solutions normalizable at in nity, and the large-z asymptotic expansion of hypergeometric functions shows this leads to the following combination of solutions (with arbitrary normalization constant C) 1(r) = C which shows that integer can be problematic (and so is obtained by a limiting procedure On the other hand the solutions (r) for small r = is only normalizable at r = 0 when 2 > 0. Adopting the convention 0 when real, we see that + is always normalizable at r = 0 but normalizable there when < 2. Keeping in mind that (in agreement with the discussion above) that = p(d 2)2 + 4($ 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 small-r 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 delta-function 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 limit-cycle 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 inverse-square potential is not deep enough to support a bound state and so the lone bound state is the one supported by the delta-function 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 delta-function itself, and this is what allows the bound state to have low-enough energy to be reliably described purely within the low-energy 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 Euler-Mascheroni 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 bound-state 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 low-energy 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 inverse-square potential does not support as we now show. In this regime, the bound-state 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 delta-function 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 xed-point coalescence can lead to universal features. Focusing on the case j^j > s, the bound-state 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 left-hand 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 delta-function 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 bound-state condition, the size of this state satis es ? the state does not lie believably within the low-energy 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 bound-state 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 RG-invariant 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 (non-relativistic s-wave 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 RG-invariant sign that determines which de nition of ? is to be used. large-r 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 delta-function coupling only through the RG-invariant 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 delta-function potential. = 0 in the above formulae reduces them to a useful special case: where the scattering is purely from the delta-function part of the interaction. In this limit (3.13) and (3.14) give the phase shift for scattering from the 3D delta-function potential when evaluated with = 2` + 1. In this limit the ow, at which point there s-wave 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 k-independent. This comparison gives an independent measure of the RG-invariant scale ?, and using this to trade y ? for as in the earlier predictions for bound-state 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 energy-level shifts due to nuclear forces to nuclear scattering lengths. Nonrelativistic s-wave fermion-monopole scattering The delta-function scattering result just described has an immediate application to s-wave elastic scattering of non-relativistic 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 RG-invariant scale de ned by the contact interaction.13 The utility of something as simple as delta-function scattering to something as complicated as fermion-monopole scattering arises because of the great simplicity of s-wave 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 s-wave 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 non-relativistic s-wave 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. GUT-scale) 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 point-like 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 spin-half 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 angular-momentum barrier in the radial equation. This is borne out by detailed calculations, for which the Schrodinger-Pauli equation acting on the 2-component 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 delta-function coupling coming from any contact interaction, such as Lb = at the position of the source monopole. This is the mode believed to participate in monopole-catalyzed baryon violation [79, 80], for which the absence of an angularmomentum barrier allows an incoming s-wave 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 inverse-square potential and keeping only the delta-function interaction. Consequently its size is controlled by the RG-invariant 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 point-particle 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 inverse-square 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 renormalization-group techniques for resumming the breakdown of perturbation theory that often arises in the near-source 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 renormalization-group 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 delta-function story. Recall rst the argument establishing the boundary conditions used in the main text. This uses the naive reasoning appropriate for delta-function interactions in the absence of singularities in the rest of the scalar potential. In it one starts with the time-independent 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 small-distance 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 boundary-value 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 world-volume, IB is always integrated over a codimension-1 world-volume that B sweeps out as time evolves. For instance for point particles in 3 spatial dimensions Sb comes to us as a one-dimensional integral over the particle world-line, while IB is a 3-dimensional integral over time and the two angular directions of a 2-sphere 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 inverse-square potential in the bulk, and this is the underlying reason why the singularity of the inverse-square 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 bound-state calculations automatically does the same for other observables, and shows how the RG-invariant 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 right-moving wave, whilst H(2)(kx) similarly asymptotes at large x to a left-moving 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 small-k 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 bound-state 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 RG-invariance 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 delta-function 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 Euler-Mascheroni constant. Consequently ) = What is noteworthy about these expressions is that the re ection and transmission amplitudes are controlled by the RG-invariant 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 delta-function potential that drives the scattering. It is in strong contrast to what one might have expected if treating a delta-function 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 baryon-number 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. 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C.P. Burgess, Peter Hayman, M. Williams, László Zalavári. Point-particle effective field theory I: classical renormalization and the inverse-square potential, Journal of High Energy Physics, 2017, 106, DOI: 10.1007/JHEP04(2017)106