Point-particle effective field theory II: relativistic effects and Coulomb/inverse-square competition

Journal of High Energy Physics, Jul 2017

We apply point-particle effective field theory (PPEFT) to compute the leading shifts due to finite-sized source effects in the Coulomb bound energy levels of a relativistic spinless charged particle. This is the analogue for spinless electrons of calculating the contribution of the charge-radius of the source to these levels, and our calculation disagrees with standard calculations in several ways. Most notably we find there are two effective interactions with the same dimension that contribute to leading order in the nuclear size, one of which captures the standard charge-radius contribution. The other effective operator is a contact interaction whose leading contribution to δE arises linearly (rather than quadratically) in the small length scale, ϵ, characterizing the finite-size effects, and is suppressed by (Zα)5. We argue that standard calculations miss the contributions of this second operator because they err in their choice of boundary conditions at the source for the wave-function of the orbiting particle. PPEFT predicts how this boundary condition depends on the source’s charge radius, as well as on the orbiting particle’s mass. Its contribution turns out to be crucial if the charge radius satisfies ϵ ≲ (Zα)2 a B , where a B is the Bohr radius, because then relativistic effects become important for the boundary condition. We show how the problem is equivalent to solving the Schrödinger equation with competing Coulomb, inverse-square and delta-function potentials, which we solve explicitly. A similar enhancement is not predicted for the hyperfine structure, due to its spin-dependence. We show how the charge-radius effectively runs due to classical renormalization effects, and why the resulting RG flow is central to predicting the size of the energy shifts (and is responsible for its being linear in the source size). We discuss how this flow is relevant to systems having much larger-than-geometric cross sections, such as those with large scattering lengths and perhaps also catalysis of reactions through scattering with monopoles. Experimental observation of these effects would require more precise measurement of energy levels for mesonic atoms than are now possible.

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Point-particle effective field theory II: relativistic effects and Coulomb/inverse-square competition

HJE Point-particle e ective eld theory II: relativistic e ects and Coulomb/inverse-square competition C.P. Burgess 0 1 2 3 5 Peter Hayman 0 1 2 3 5 Markus Rummel 0 1 2 3 5 Matt Williams 0 1 2 4 Laszlo Zalavari 0 1 2 3 5 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 - Abstract: We apply point-particle e ective eld theory (PPEFT) to compute the leading shifts due to nite-sized source e ects in the Coulomb bound energy levels of a relativistic spinless charged particle. This is the analogue for spinless electrons of calculating the contribution of the charge-radius of the source to these levels, and our calculation disagrees with standard calculations in several ways. Most notably we nd there are two e ective interactions with the same dimension that contribute to leading order in the nuclear size, one of which captures the standard charge-radius contribution. The other e ective operator is a contact interaction whose leading contribution to E arises linearly (rather than quadratically) in the small length scale, , characterizing the nite-size e ects, and is suppressed by (Z )5. We argue that standard calculations miss the contributions of this second operator because they err in their choice of boundary conditions at the source for the wave-function of the orbiting particle. PPEFT predicts how this boundary condition depends on the source's charge radius, as well as on the orbiting particle's mass. Its contribution turns out to be crucial if the charge radius satis es . (Z )2aB, where aB is the Bohr radius, because then relativistic e ects become important for the boundary condition. We show how the problem is equivalent to solving the Schrodinger equation with competing Coulomb, inverse-square and delta-function potentials, which we solve explicitly. A similar enhancement is not predicted for the hyper ne structure, due to its spin-dependence. We show how the charge-radius e ectively runs due to classical renormalization e ects, and why the resulting RG ow is central to predicting the size of the energy shifts (and is responsible for its being linear in the source size). We discuss how this ow is relevant to systems having much larger-than-geometric cross sections, such as those with large scattering lengths and perhaps also catalysis of reactions through scattering with monopoles. Experimental observation of these e ects would require more precise measurement of energy levels for mesonic atoms than are now possible. ArXiv ePrint: 1612.07334 HJEP07(21) 1 Introduction 2 Nonrelativistic mixed Coulomb and inverse-square potentials 3 Applications 2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2 3.3 3.4 Schrodinger action Source action and boundary conditions Solutions RG evolution Bound states Scattering Pionic atoms and the Deser formula RG scales and reaction catalysis Mixing through contact interactions Klein-Gordon Coulomb problem A Matching to a simplistic nuclear model A.1 Spherical surface-charge distribution A.2 Constant charge distribution B Gamma function approximations when s . 1 or time) is to be directly probed. We here extend results from a companion paper [ 1 ] to some issues arising from the application of EFTs to a particular system of this type: the problem of a spinless point charge | which we henceforth generically call a `meson' or `spinless-electron' interacting electromagnetically with another charged particle that might contain some substructure (like a proton or a nucleus | henceforth called the `nucleus' or `source'). In this case the hierarchy of interest is the ratio between the large size, a, of the meson a we expect an appropriate e ective description to lead e ciently to a series expansion of observables in powers of "=a. E ective theories capture this expansion by writing an e ective action for a source with structure, which includes all possible interactions involving the `bulk' elds of interest (like the EM eld A or meson eld , say) consistent with the symmetries of the problem. In practice one organizes these interactions with increasing (mass-) dimension1 in the expectation that dimensional analysis then requires the couplings of higher-dimension interactions to be suppressed by additional powers of " relative to couplings of lower-dimensional interactions. Of course, simply writing down a point-particle action is not new in itself. The new part | and what we mean by `point-particle e ective eld theory' (or PPEFT) | is the explicit connection that is made between this action and the near-source boundary conditions for the various `bulk' elds to which it couples (this connection is laid out more formally in [ 1 ], building on the earlier construction of [2, 3]). It is through these boundary conditions that the e ective couplings of the source action can in uence the integration constants arising when solving bulk eld equations, and thereby express how the source back-reacts onto its surrounding environment. Concretely, for a rotationally invariant nucleus coupled to photons and spinless electrons (respectively described by the bulk elds A and ), such an e ective action might have lowest-dimension interactions of the form Sb = Z d h M Q A y_ + h ~ h r E + i ; (1.1) where the integral is along the world-line, y ( ), of the nucleus for which is the proper time and y_ := dy =d . where N = Ze 1 + 16 rp r 2 2 + The constants M , Q represent the nuclear mass and charge (we take Q = Ze), while the couplings h and h~ are the rst of a succession of possible e ective couplings having dimensions that are a positive power of length. The rest of these terms are collectively denoted by the ellipses in (1.1), and include all possible local interactions involving A and and their derivatives, and it turns out that all of those not written are negligible for the present purposes because they are suppressed by more powers of the small scale " than are those explicitly written.2 In what follows we keep only the above three terms, dropping all other e ective interactions with higher mass dimensions than these. The coupling h~ describes the traditional charge-radius of the nucleus. It is related to the root-mean-square charge radius, rp2, by h~ = 16 Ze rp2, as might be measured by scattering photons from the nucleus. For observables not involving photons the electromagnetic eld may be integrated out, which amounts in this case to using Maxwell's equations to rewrite r E in terms of the total charge density, which for a Schrodinger eld is = e + N , 3(r) is the rest-frame nuclear charge density obtained by varying Sb with respect to A0. The term quadratic in can then be absorbed into h, 1We use fundamental units for which ~ = c = 1. 2The dimension of the interaction depends on the canonical dimension of , which is mass for a KleinGordon eld but mass3=2 for a Schrodinger eld. In most of what follows it is the Schrodinger eld that is leading to an e ective interaction of the form with htot ' h + 1 6 htot contributes to physical observables comes from recognizing that it is equivalent to a delta-function potential of the form indeed using htot = 16 Ze2rp2 in the perturbative formula E = htotj (0)j2 using standard Coulomb wave-functions reproduces the leading expression for the nuclear-radius contriV = htot 3(r), and bution to atomic energy shifts [5]. If had been a Klein-Gordon scalar then h = hKG would have dimension length (rather than length-squared) and so naively might be expected to contribute to observables linearly in the small scale ". The main point of this paper is to argue that this is basically true for both Schrodinger and Klein-Gordon elds. (If is a Schrodinger eld, it turns out only the combination hKG = 2mh contributes to observables and it is this combination that scales linearly in ".) It is also more subtle than it looks even for Klein-Gordon elds. Two surprises turn out to be buried within the statement that h scales linearly with ": Reaction `catalysis': although the leading in uence of h on physical observables is linear in microscopic scales, it turns out that the scale involved need not strictly be " and in some cases can be much larger. In particular, because we nd that the coupling h must be renormalized | even at the classical level | it runs with scale according to a renormalization-group (RG) evolution. It therefore contributes to observables proportional to the RG-invariant scale, ?, associated with this running, which can (but need not) be much larger than the underlying physical scale ". When ? " physical processes like scattering can be strongly enhanced, in a way that resembles how scattering from magnetic monopoles can catalyze [6{8] the violation of baryon number in grand-uni ed theories.3 Larger than expected shifts in atomic energy levels: even when ? ' " we argue that h shift energy levels (and a ects scattering) in surprising ways. First, because (for the Schrodinger eld) only the combination hKG = 2mh appears in physical quantities, when the orbiting particle is relativistic at nuclear radii then matching to a nucleus leads to the expectation h = B=m where B is of nuclear size and independent of the mass. (The same need not be true when it is nonrelativistic at the nuclear surface.) This leads to unexpected shifts in the energy levels of spinless particles that are of order E / h / "=m. Beyond this, the classical renormalization adds additional m-dependence in the in uence of h on observables. In particular, although h = 0 is an RG xed point for a non-relativistic particle experiencing only a Coulomb potential, it is not a xed point for a relativistic particle in a Coulomb potential or for a non-relativistic particle experiencing a superposition of both Coulomb and inverse-square potentials. Because zero coupling is not in this case a xed point, contact interactions become compulsory rather than optional: h = 0 can at best only 3We argue this running is a part of the mechanism for understanding monopole catalysed events within the PPEFT. { 3 { be chosen at a particular scale (perhaps at the UV scale "). If so then h runs to become nonzero at larger scales and where it contributes to observables linearly in ?. In particular, for hydrogen-like states both of these e ects imply s-wave states are shifted in energy by amounts that depend di erently on mass than does the normal (Z )4rp2m3 charge-radius term (where is the usual ne-structure constant). Unfortunately4 these e ects seem not to be shared by spin-half pariticles [4], and so their experimental veri cation requires more precise measurements for the energies of - or K-mesic atoms than are presently possible. Contact interactions, boundary conditions and classical renormalization. Although we ll in the details explicitly in the bulk of the paper, because the results are so (1.3) derivative satis es6 surprising we rst provide here a brief sketch of the logic of the argument. The crucial role is played by the coupling h, of the lagrangian (1.1), which from the point of view of the eld equation appears as would a `contact' interaction (i.e. a deltafunction contribution to the inter-particle interaction potential).5 As might be expected, even in the absence of Coulomb interactions, the presence of a delta-function potential necessarily modi es the boundary condition that satis es at the origin, as can be seen by integrating the eld equations over an in nitesimal Gaussian pillbox that encircles the source nucleus (see gure 1 and the discussion in [ 1 ]). This implies that as r ! 0 the radial 4 r where = hKG = 2mh. It is , rather than h, that is approximately independent of m for sources small enough that orbiting particles are relativistic in their vicinity, a feature we further motivate in the appendix using several toy models of the nuclear charge distribution. Recall that for free particles (or for particles interacting through the Coulomb interaction) the two independent solutions to the radial equation behave for small r like and r ` 1, for angular-momentum quantum number `. When h = 0 eq. (1.3) reduces + r ` to the usual condition that the overlap with must vanish. More generally the solution satisfying (1.3) involves both + and . But because requires it to be regulated and evaluated at in nitesimal r = rather than strictly at zero. diverges7 as r ! 0 the use of (1.3) Once this is done (1.3) makes sense, but also seems to require that physical quantities must depend in detail on the value of the regularization scale , which seems odd given we at this point only needed to choose to be small and not precisely equal to the physical 4`Unfortunately' because if shared by spin-half particles such e ects have the right size and sign to have accounted for the experimental `proton-radius' discrepancy [5, 9{16] without the need for exotic new interactions [17{20]. 5Such contact interactions sometimes arise in the Coulomb problem, such as to describe strong mesonnucleus interactions in mesonic atoms where a negative pion or kaon orbits the nucleus. 6If one resists imposing this boundary condition, such as by perturbing in the interaction h, one nds that graphs involving repeated meson interactions with the nucleus are not small and their resummation [21{25] simply imposes (1.3). 7Regularity at the origin is not in itself a good boundary condition for two reasons. First, it is generic that bulk elds diverge at the position of a source | as is clearest for the Coulomb potential, A0 / q=(4 r). Second, for many simple potentials (such as the inverse-square: V / 1=r2) it can happen that both radial solutions diverge at r = 0, so boundedness cannot distinguish them. { 4 { " a HJEP07(21) by " the actual UV physics scale associated with the underlying size of the source (e.g., the size of the proton), which by assumption 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 require 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. UV scale ". What really happens though is that physical quantities are -independent because the explicit 's in (1.3) can be renormalized into h [26, 27]. That is, the explicit -dependence of (1.3) can cancel against an implicit -dependence buried in h( ), which turns out to require h = f (h0= 0; = 0) where f (x; y) is a nontrivial dimensionless function (given explicitly below) and 0 is a scale where h = h0. This required -dependence of h is what we call its renormalization-group (RG) evolution. An RG invariant scale ? can then be de ned, such as by specifying the scale where h0 = 0 or h0 ! 1. In the end it is only ? on which physical quantities typically do depend, and this is ultimately the origin of the rst bullet point given above. A reality check on this running is that it has a xed point at h = 0, in the sense that f (0; y) = 0 for all y. This means (for delta-functions plus Coulomb potentials, at least) one can always choose not to have a contact interaction if that is what one wants. That is, once h = 0 at any scale, then its running ensures it remains zero for all scales. And if h is nonzero at some scales it turns out that its ow is towards the xed point at zero in the far infrared (IR), as for nonzero and positive g. The starting observation is that such an inverse-square potential can compete with the centrifugal barrier and so modify the asymptotic form of solutions as r ! 0 to become more singular there.8 As many have observed [28{40] because of this the Schrodinger Hamiltonian can fail to be self-adjoint, depending the boundary conditions that hold at r = . Selecting a choice of boundary condition to secure its self-adjointness | not a unique construction | is known as constructing its self-adjoint extension [41{48]. We here follow closely the treatment of inverse-square potentials given in [ 1 ]. Physically, what is happening is that the inverse-square potential concentrates more at the origin and so increases the probability of interacting with whatever the physics is that describes the source there. In particular the resulting time-evolution could be non-unitary if these interactions excite other degrees of freedom besides (or otherwise remove particles for whatever reason). On the other hand interactions with the source might preserve probability if there are no other degrees of freedom and the action describing the source is real. In this language the freedom inherent in choosing self-adjoint extensions is precisely the freedom in choosing the form of the source action. In particular, given an action like (1.1) the boundary condition (1.3) speci es a speci c self-adjoint extension provided h is real. But casting the extension in terms of the source action has the advantage that it gives a criterion for how to choose it; in particular it shows why the lowest-dimension interactions should dominate at low energies, and so why (1.1) should commonly apply at low-energies.9 In detail, the way this connection between inverse-square potentials and contact interactions arises is through the contribution the inverse-square potential makes to the RG evolution of h. (This observation is also not in itself new, since it has long been known to be an example of renormalization and dimensional transmutation within a quantum mechanical setting.10) Naively, perturbing in h leads to formulae like E = h j (0)j2, however the inverse-square potential causes (0) to diverge and this turns the expansion in h into a more dangerous expansion in h= that breaks down as dependence on h= that the RG e ciently resums. ! 0. It is this nontrivial What is important, however, is that when g 6= 0 the IR xed point of the RG evolution for h gets driven away from h = 0 towards a nonzero value. As a result the presence of a contact interaction becomes compulsory, rather than merely being an option. At best h = 0 can only hold at a speci c scale, 0 say, after which RG evolution requires it not to vanish anywhere else. This is equally true if a Coulomb interaction is also present or not. Contributions of the contact interaction to bound-state energy levels and scattering amplitudes turn out to be linear in ? or 0, and so contribute negligibly if these scales should be vanishingly small, as would be true if the value of h were set at a vanishingly small length scale (as appropriate if the nucleus were a point particle like a muon). 8When the boundary condition is imposed at nite r = rather than zero, the presence of an inversesquare interaction competing with the Coulomb potential can be expected to be important if < g=s. 9Because quadratic actions often dominate, the RG evolution described here is likely universal for many systems. 10When cast in terms of a self-adjoint extension it has not always been clear | see however [40] | that it is usually the strength of a delta-function contact potential that is being renormalized. { 6 { HJEP07(21) Most crucially, an inverse-square potential is always present for the relativistic Coulomb problem. We consider here the Klein-Gordon/Coulomb system, for which the square of the Coulomb potential appears within the second time derivative,11 Dt2 . The radial part of the Klein-Gordon/Coulomb equation is precisely the same as in (1.4) with s ' 2mZ while g = (Z )2. From the above discussion, the presence of the relativistic inverse-square potential ensures h = 0 is not a xed point, provided boundary conditions are required at distances as small as < g=s ' Z =m. When this is so h must instead be driven away from zero in the far infrared to be of order 2mh ? . The presence of such a contact interaction then turns out to shift s-wave energy levels by an amount E (Z )3 ?m2, that is linear in ? as claimed in the second bullet point above. Linearity in the microscopic scale ? is unlike standard contributions to energy shifts due to nuclear nite-size e ects [51{53]. It is intriguing that for plausible nuclear values of ? this energy shift is similar to what is seen experimentally in comparisons between energy levels for electrons and muons bound to nuclei, though (alas) similar-sized contributions do not also arise in the Dirac equation appropriate for spin-half particles [4]. A road map. The rest of this paper is organized as follows. The next section, section 2, sets up and solves the Schrodinger equation in the presence of the potential (1.4). Both bound-state energies and scattering amplitudes are computed explicitly, as is the detailed RG ows for the contact coupling h for several regimes that di er according to the size of the inverse-square coupling g. Section 3 then follows this with several applications of these results, designed as checks or illustrations of the two bullet points given above. A reality check rst uses the results of section 2 to derive the Deser formula [59] for mesonic atoms, that relates the energy-level shift of s-wave states and the low-energy scattering length due to the short-range meson-nuclear force. This is followed by a short discussion identifying under what circumstances the RG evolution of the contact interaction can enhance scattering cross sections, as is reminiscent of monopole-catalyzation of exotic GUT-scale reactions. Next is a detailed treatment of the Klein-Gordon/Coulomb system, and the estimates of the size of the energy-level shifts that are implied by the running of the contact interaction. Some toy models checking these results against speci c nuclear charge distributions are also considered in the appendix, and are compared with the more general EFT estimates. 2 Nonrelativistic mixed Coulomb and inverse-square potentials eliminate Dt2 spatial derivatives like r 4 in the action. Much of the physics needed for the relativistic case hinges on the competition between the inverse-square and Coulomb potentials, so we start our discussion with the Schrodinger system involving these two potentials. Our treatment follows closely that of [ 1 ], which examines the classical renormalizations associated with the inverse-square potential, though we extend this analysis here by adding also a Coulomb potential. 11Field rede nitions allow e ects to be moved around within an EFT (these are standard arguments, and we follow here the formulation made in [49, 50]), and if the lowest-order Schrodinger equation is used to the self-adjointness problems remain, being attributable now to the appearance of higher { 7 { with used, for coupling constants s, g and h, when an explicit form is required. The eld equation found by varying then is the Schrodinger equation, V (x) = s r g r2 and Lb = h 1 2m r 2 r 2 + 2m + h s r g r2 h 3(x) i = 2 ; which for energy eigenstates, (x; t) = (x) e iEt, and with the choice (2.3) becomes with 2 = 2mE. For bound states | when E 0 | is real, but when discussing scattering | where E 0 | we switch to = ik with real k given by k Expanding in spherical harmonics, Y``z ( ; ), implies the radial equation is given by 2 = +2mE. 1 d r2 dr r 2 d ``z dr `(` + 1) r2 + U (r) ``z = 2 ``z ; We take, therefore, our action to be S = SB + Sb where SB is the Schrodinger `bulk' action Z SB = dt d3x i 2 (2.1) where m is the particle mass and Sb describes a microscopic contact interaction between the Schrodinger eld and the point source localized at the origin r = 0: Z Z Sb = dt Lb[ (x = 0); (x = 0)] = dt d3x Lb( ; ) 3(x) ; where U = 2m[V + h 3(x)] while ` = 0; 1; 2; usual angular momentum quantum numbers. 2.2 Source action and boundary conditions and `z = `; ` + 1; ; ` 1; ` are the The source action, Sb, appears here only through the delta-function contribution to U and the only e ect of this is to determine the boundary condition satis ed by at r = 0. This can be obtained as described in [ 1 ] by integrating (2.5) over an in nitesimal sphere, S, of radius 0 r around x = 0 and using continuity of there to see that only the integral of the second derivative contributes from the left-hand side of (2.5) as ! 0. This leads to the result Z S (0) = d3x r 2 = Z = Z d 2 r ; (2.7) where := 2mh while n dx = dr is the outward-pointing radial unit vector, d2 = sin d d is the volume element on the surface of the angular 2-sphere and the last equality { 8 { (2.2) (2.3) (2.4) (2.5) (2.6) assumes a spherically symmetric source so that is also spherically symmetric to good approximation for su ciently small. Because solutions `m(r) vary like a power rp as r ! 0, the boundary condition given above becomes singular as ! 0. This is dealt with by renormalizing | i.e. by associating an implicit -dependence to in such a way as to ensure that the precise value of drops out of physical predictions. With this in mind | and de ning (0) := (r = ) | our problem is to solve the radial equation, (2.6), subject to the boundary condition (2.8) HJEP07(21) at the regulated radius r = . As mentioned in [ 1 ], this boundary condition can be regarded as a speci c choice of selfadjoint extension [41{48] of the inverse-square Hamiltonian. The inverse-square potential requires such an extension because its wave-functions are su ciently bunched at the origin that physical quantities actually care about the nature of the physics encapsulated by the source action, Sb. Writing the extension in this way usefully casts its ambiguities in terms of a physical action describing the physics that can act as a potential sink (or not) of probability at r = 0. As might be expected, this extension is self-adjoint provided that the source action is real and involves no new degrees of freedom. In the present instance this can be seen from the radial probability ux, 2 r 2 m J = 4 r2 n J = (2.9) emerging from the source through the surface at r = . Evaluating with energy eigenstates gives J ( ) = 2 m 2 ( ) i h) ( ) ; (2.10) which shows no probability ows into or out of the source when its action is real (ie h = h). 2.3 Solutions The radial equation (2.6) to be solved is r 2 d 2 dr2 + 2r d dr + wr + v `(` + 1). This can be written in con uent hypergeometric form through the transformation (r) = zl e z=2u(z), for z = 2 r where l(l + 1) + v = 0 so that12 1 2 l = ( 1 + p 1 1 2 4v) = ( 1 + ) = 1 2 + s ` + 12Choosing the other root for p just exchanges the roles of the two independent solutions encountered below, so does not introduce any new alternatives. { 9 { where we de ne for later notational simplicity := 2mg and The two linearly independent radial pro les therefore are := p 1 4v = p1 + 4`(` + 1) is the con uent hypergeometric function regular at z = 0. We therefore take our general radial solution to have the form . We next impose the boundary condition at r = 0 to determine the ratio C =C+. Regularizing for small r = the solutions (r) behave as which has the familiar form of rl or r l 1, with l as de ned in (2.12). This shows that for some choices of neither of is bounded at the origin. This implies that boundedness at the origin cannot be the right physical criterion there, at least in the presence there of a physical source. This is not really a surprise since elds generically diverge at the presence of a source, such as does the Coulomb potential itself. We do demand solutions be normalizable, however, and the convergence of the integral R d3x j j2 as r ! 0 implies cannot diverge faster than r 3=2 as r ! 0. For 2 > 0. For concreteness' sake in what follows we follow [ 1 ] and specialize to the case this implies where the inverse-square potential satis es < 54 , because this captures all of the examples of most interest and has the property that is not normalizable at r = 0 for any ` 6= 0. This ensures that that the boundary condition at the origin implies C = 0 and so / + for ` 6= 0. for such s-wave states we have (` = 0) = s := p 1 4 and so 0 < 1 for 0 It is only for ` = 0 that the contact interaction is needed to determine C =C+, and and so both solutions diverge but are normalizable at the origin.13 If 14 < becomes imaginary, in which case both j +j2 and j j2 diverge near r = 0 while remaining normalizable. In this case eq. (2.8) is the condition that xes C =C+, evaluating the 54 then s derivative using the small-r form for r= = 2 2" C+ ( 1 + s) (2 ) 21 ( 3+ s) + C ( 1 s) (2 ) 21 ( 3 s) # C+(2 ) 21 ( 1+ s) + C (2 ) 21 ( 1 s) = 2 1 + s R R + 1 1 ; (2.16) (2.13) (2.14) (2.15) 13The only exception to this is the case = 0 for which l = ` and so + is bounded. However once having discarded boundedness as a valid criterion at the origin, it cannot be revived in this special case. In our view this is a de ciency of most treatments of the Coulomb potential, a point to which we return below. where and so, in particular, R = 0 when C To use this equation it is useful to rewrite it as C+ R := where the rst equality de nes14 the dimensionless coupling ^. This shows that physical quantities depend only on the ratio ^= s. Solving for C =C+ leads to s and negative otherwise. For scattering calculations we take = ik and then (2.19) xes (r) up to normalization, thereby allowing scattering phases to be read o by examining the large-r limit. Alternatively, for bound states it is the compatibility of (2.19) with the value C =C+ obtained by the normalization condition at in nity that picks out the quantized value for (and so also E = 2=2m). Two points about this boundary condition are noteworthy: Even though s need not always be real (2.18) always amounts to a single real condition on C =C+ or , because R is either real (when s is real, and so v < 14 ) or R is a pure phase (when s is pure imaginary, and so when v > 14 ). Our main interest is in small v, so in what follows we restrict attention to real s. Although (2.18) seems to imply depends on , this naive dependence is cancelled by the -dependence implicit in the renormalization of . The required -dependence is worked out below separately for the two cases where s is real or imaginary. 2.4 RG evolution The -dependence of required to make physical quantities like independent of can be found by di erentiating the quantization condition (2.18) or (2.19), being careful to hold physical quantities like or C =C+ xed. We focus here on real s, though the imaginary case goes through along the lines found in [ 1 ] since the RG discussion does not depend on the Coulomb interaction. When s is real then so is R and it is convenient to write R = e for a real parameter . The sign is chosen because it turns out below that C =C+ is negative once normalizability is imposed at in nity. In this case (2.18) becomes 14Notice that vanishing coupling, = 0, corresponds to ^ = 1, and so attractive (repulsive) -potentials chosen so that s is real. A representative of each of the two RG-invariant classes of ows is shown, and ? is chosen as the place where ^ = 0 or ^ ! 1, depending on which class of ows is of interest. nding ^( ) is to demand its dependence cancel the explicit dependence that is hidden within R (or ) in (2.20). Di erentiating this expression with respect to using the -independence of and C =C+ in (2.17) to infer d =d = s leads to the RG equation d d ^ ! s = 1 sinh2( =2) d 2 d = 2 s 1 coth2 2 = 2 s 2 41 ^ !23 s ow clearly has xed points at ^ = s and integrates to give ^( ) s = (^0= s) + tanh 12 s ln( = 0) : (2.21) (2.22) This shows how ^ ows with increasing (i.e. from the UV to the IR) from the xed point at s when ! 0 up to + s as ! 1, passing through the value 0 when = 0. Notice this depends only on the inverse-square coupling through s, but remains nontrivial even when this coupling vanishes (i.e. when 2mg = = 0 and so s = 1). Of particular later interest is the observation that zero coupling (that is, = 0 and so ^ = 1) is only a xed point when 2mg = = 0. Notice also that there are two distinct classes of ows | as illustrated in gure 2 | that di er in the RG-invariant criterion of whether j^= sj is larger than or smaller than unity. Of later interest is the asymptotic form for this running as ^( ) nears the xed points at ^ = s . Using the asymptotic expression for tanh z for large positive or negative z ^( ) s for ^. Bound states are found by imposing normalizability of = C+ + +C at large r, which can be written (with arbitrary normalization constant C) as 1(r) = C " 1 2 ( w + 1 ) +(r) + 1 2 ( ) can be problematic in this expression and so is obtained by a limiting procedure. Clearly this xes the ratio C =C+ to be C C+ = ( ( ) ) 1 2 1 2 w + 1 ; (2.25) and so demanding this be consistent with the condition (2.19) gives the quantization conditions for . For all but the s-wave we have seen (at least for < 54 ) that normalizability at r = 0 requires C = 0, so consistency with (2.25) is not possible at all in the absence of a Coulomb potential (i.e. when w = 0), indicating the absence of a bound state in this case. On the other hand, when w 6= 0 consistency requires must sit at a pole of the denominator, which ensures = w 2N + 1 + ; (2.26) ) 3 4 . This is also the solution for s-wave states if ^ = s, since R = 0 in this for N = 0; 1; 2; case too. For the Schrodinger Coulomb problem (with no inverse-square potential) we have g = 0 and w = 2mZ while = c := 2`+1 where ` = 1; 2; : : : is the angular momentum quantum number. For all ` 6= 0 states (2.26) then returns the usual Schrodinger eigenvalues: E = 2=(2m) = m(Z )2=(2n2), where the principal quantum number is n = N +1+` `+1. Eq. (2.26) also captures the Klein-Gordon energy levels once we include also the inversesquare term in the potential. In this case we nd = 2l + 1, with the non-integer l now de ned by (2.12), which gives the standard result when inserted into !2 = 2 + m2. Perturbing of s-wave energies when ^ 6= s. Consider next the more general s-wave case, in the case where s = p 1 that determines j^j > s the solution is found by solving for 4 is real. In this case the quantization condition (2.20) has no solutions for RG trajectories satisfying j^j < s and for ows with C C+ = " ^ # s s + ^ (2 ) s = 1 2 1 2 s w + 1 + s : (2.27) in ( s) ( s) As mentioned above, this reduces to the standard Coulomb energy level when the left-hand side vanishes, as it would if either C = 0 or ^ = s . An extreme limit occurs when w = 0 (so where there is no 1=r component to the potential), in which case the solution reduces to the result found in [ 1 ]: ' ( s) ( s) < 14 this bound state is dominantly supported by the deltafunction potential furnished by the contact interaction whose strength is governed by . When w 6= 0 a useful formula for how energy levels are perturbed from their Coulomb (or Klein-Gordon) limit when ^ function near its pole by (z ( )N h i s is not too large is found by approximating the gammaN ) ' N! z 1 + O(z) , where z is near zero. Using this when is near a zero of C we nd (2.27) takes the approximate form w 2 + 1 ( s + N + 1) (2 ) s ' N ! ( s) ( s + 1) ^ ! s s + ^ ' (n ( s + n) 1)! ( s) ( s + 1) 0 s s ^ 0 s + ^0 ; (2.29) (2.30) for = N + 1 + l and l = 12 (1 s) as above, with N = n 1 = 0; 1; 2; the principal quantum number n of the Coulomb limit, as above. The second line assumes ^( ) is speci ed by giving its value ^0 = ^( 0) at some microscopic scale 0, and uses the asymptotic expression (2.23). Notice the cancellation of the explicit -dependence in this formula. The solution perturbatively close to the zeroth order solution of the Coulomb/inversesquare problem is = + = (w=2 ) + with Of course, the mere existence of a solution for does not su ce to ensure the presence of a physical bound state. In order to be trusted the bound state must be much larger than the UV scale that characterizes the structure of the source, and which provides a lower limit to the length scales for which an analysis purely within the point-particle EFT can be valid. For the Coulomb-like solutions the size of the bound state is given as usual by the `Bohr radius', or r w 1 where w = 2ms(= 2mZ ). Believability of the bound state requires 1 where is a UV scale. For bound states where the contact interaction plays an important role demanding the bound state be much larger than UV scales imposes a condition on , and this is how we see why the delta-function potential must be attractive and su ciently strong. To see how this works we must identify the scale of the bound state determined by (2.28), and this is most simply identi ed by exploiting the -independence of equations like (2.28) to express the result in terms of an RG-invariant scale. Since j^j > RG-invariant scale to be the scale ? where j^( ?)j = 1, leading to s it is natural to choose this HJEP07(21) ' 1 ? ( s) ( s) For generic s this shows the bound state is of order ? in size. To be trusted for any UV on the RG ow we must ask ^( ) to be such that ? . Taking ^0 ! 1 in the ^( ) s couplings in the UV that a macroscopic bound state of the form (2.28) can be trusted. (2.31) (2.32) boundary condition ensures C which states Scattering calculations go through in a very similar way, and for later purposes we collect results here for the scattering amplitude, restricted to the case real. Our treatment here follows that of [ 1 ] fairly closely. 3 4 14 for which s is The scattering result also shows how renormalization makes the contribution to scattering of the contact interaction, , depend only on RG-invariant scales like ?, rather than being set directly by the microscopic scale where ( ) is matched to the UV completion of the source. This can make scattering e ects surprisingly large in those circumstances where ? As before the starting point is the radial solution in the form C =C+ set by the boundary condition as r ! 0. For the range of = 0 for all ` 6= 0, while the s-wave state satis es (2.19), C C+ = R(^= s)(2ik ) s = " ^ # s s + ^ (2ik ) s ; (2.33) which also writes = ik, as appropriate for a state with E = k2=2m > 0. Unlike for bound states the ratio C =C+ is not independently set by normalizability at large r. Notice it is again the di erence between ^ and its IR xed point value that drives C =C+ away from what would be found in the absence of a contact interaction with the source (i.e. drives it away from C = 0). or ^( ?) = 0 (if j^0j < s). Here y = sign[j^j which of these de nitions of ? is to be used. wavefunction as and de ne the phase shift by [60] e2i ` = hypergeometric function leads to Evaluating asymptotically close to the IR xed point at ^ = s using (2.23) and inserting into (2.33) we see the expected cancellation of powers of leaving C C+ ' (2ik 0) s s where the last equality uses the RG-invariant scale ?, de ned by ^( ?) = 1 (if j^0j > s) s] is the RG-invariant sign that determines To match C =C+ to the scattering amplitude we write the large-r behaviour of our A`=B`. Taking the large-r limit of the con uent (2.35) (2.36) (2.37) (2.38) which permits reading o the phase shift. For large r we drop oscillating factors like (2kr) w=2ik = e i(w=2k) ln(2kr) that are subdominant to the exponentials e ikr, leading for ` 6= 0 (and for ` = 0 when ^ = s) to the phase shift e2i ` = 1 2 ikw + 1 + 21 ikw + 1 + ei (` l) ; which uses = 2l + 1. Notice that in the absence of an inverse-square potential ( = 0 and so l = `) this expression reduces to the usual one for Rutherford scattering [60] e2i ` = (` + 1 iw=2k) (` + 1 + iw=2k) (Rutherford limit) : On the other hand, for s-wave scattering in general C =C+ is given by (2.34), leading to Of later interest is the case where the Coulomb contribution is turned o , and so for which w = 0. In this case | as shown in more detail in [ 1 ] | the scattering phase shift simpli es to become where with A := y k ? 2 s " 1 1 + 12 s 12 s # : A nal limit is the case of scattering from a delta-function, obtained by turning o the inverse-square potential and taking = 0 and s = 1. In this limit we have This agrees with standard calculations [26, 27] and in particular gives tan 0 = low energies the scattering length, as, is given by k cot 0 ' low-energy cross section is = 4 as2). When the -function dominates in the scattering we 1=as + O(k2) (so that the therefore nd as directly xes the RG-invariant scale through the relation 3 Applications We now turn to several practical applications to the developments of the previous section. These include the reproduction and clari cation of some well-known results (such as the Deser formula relating the energy-level shift and scattering length of pion-nucleon interactions in pionic hydrogen states); a brief recap of the argument of [ 1 ] as to why classical renormalization provides a simple and intuitive low-energy description for how scattering from small objects like magnetic monopoles can catalyze reactions; a treatment of mixing induced by contact interactions; and a discussion of how the interplay of relativistic e ects with the classical renormalization of contact interactions can amplify the size of contact interactions within mesonic atoms. Although (as shown in a companion paper [4]) some of these features also carry over to a Dirac-equation treatment including spin, this is not so for the spectacular energy level shift of order ?=m. 3.1 Pionic atoms and the Deser formula For our rst application we consider mesonic (pionic and kaonic) atoms, in which a relatively long-lived and negatively charged meson orbits a nucleus (or proton, in the simplest case). Such `atoms' are of interest because the mesons live long enough to be captured by the nucleus once a beam is brought to impinge on a target material. Once captured, the meson cascades down to the ground state and detection of the X-rays emitted in this process allows the measurement of the bound-state energies. The binding is electromagnetic because the mesonic Bohr radius is much larger than the range of nuclear forces (that are set by the pion Compton wavelength) and because of this v2=c2 is small enough to be well within the non-relativistic regime. Additionally, since the meson mass, m, is at least 300 times larger than the electron mass its orbital radius is at least 300 times smaller, bringing the mesonic orbit well inside the various electronic ones. In this case the in uence of meson-nuclear strong forces can be modelled by a contact interaction in an e ective theory that does not resolve the nuclear size, making the formalism of this paper appropriate. Measurements of the energy-level shift induced by the meson-nucleon strong interaction probe the detailed nature of meson-nucleon interactions [55{58]. In this section we use the previously presented formalism to derive the Deser formula [59] relating the strong-interaction shift to the mesonic bound state energy to the meson-nucleus scattering length. This formula usually is derived using a model for the nuclear potential acting over short distances, in which the need to go beyond Born approximation is often emphasized. Our presentation here shows how the discussion naturally ts within the framework of a point-particle EFT and how the need for contributions beyond Born approximation are captured in a controlled way by the RG evolution of the nuclear contact interaction. The starting point is the Schrodinger action (2.1) coupled to a contact interaction, (2.2), meant to represent the short-range strong meson-nucleon interactions. We parameterize this interaction here in terms of the coupling h, as above, though a more systematic exploration of the kinds of contact interactions possible might also be warranted.15 In this case the results of section 2 can be taken over in whole cloth, and neglecting very small relativistic e ects (more about which below) we can take the Coulomb potential to have strength s = Z and the inverse-square potential to vanish: g = 0. The quantization condition that sets the binding energy of the hydrogen-like mesonic state is then given by (2.27), in which we use w = 2ms = 2mZ and = 2mh while the condition = 2mg = 0 ensures s = 1. Because we set g = 0 (and so have no inverse-square potential) it is RGinvariant to choose h = 0, although in this case we do not do so because its value captures a physical e ect: the strength of the short-range meson-nucleon force. In the regime of interest the bound-state condition is solved by a relatively small change from the Schrodinger Coulomb solution as in (2.30), leading to ' 2 0 1 1 + ^0 ^ ! 0 ? naB = 2 y (if ` = 0) : (3.1) y = sign[j^0j Here n is the principal quantum number and aB = (mZ ) 1 is the mesonic Bohr radius, while h0 ' 02 is a typical nuclear scale when speci ed at nuclear distances, 0 ' 1 fm. This ensures ^0 ' O(1) and so also that ? | de ned as the scale where ^ diverges (if j^0j > 1) or where ^ = 0 (if j^0j < 1) is also a typical nuclear size ? ' 0. (As in previous sections 1] is the RG-invariant sign that distinguishes the two types of RG ow.) This leads to the following shift in the mesonic bound state energy, En = 2 2m = m = 2 y ? mn3a3B 15Much thought has been put into the meson-nucleon e ective interaction within chiral perturbation theory, in which the dominant term is momentum-dependent though smaller momentum-independent Yukawastyle interactions are also possible [11{14]. For the purposes of illustration we restrict ourselves here to a simple Yukawa interaction, though a more sophisticated and systematic treatment is clearly possible. As usual the size of the in uence of the contact interaction on physical quantities is set by the RG-invariant scale ? found from its coupling . In the present instance this is generically similar in size to the nuclear scale, 0, at which matching to the UV completion describing the nucleus occurs. But in the end, both 0 and ? are just parameters, and a real prediction comes only once they are traded for another observable. One such observable is the scattering length, as, of mesons from nucleons, which if governed at low-energies by the same contact interaction is given by (2.44), or y ? = as. Using this in (3.2) leads to the following relationship between the fractional strong-interaction shift in the s-wave energy levels of mesonic atoms to the low-energy elastic scattering length for mesons scattering from the 4as naB (3.3) For the ground state n = 1 this reproduces the Deser formula [59] for mesonic atoms. As is usual for an EFT analysis, corrections to this expression should arise from higherdimension interactions localized at the source, and because of their higher dimension would be expected to be suppressed by further powers of ?=aB. We see that for mesonic atoms it is well-known that energy shifts can receive contributions linear in a microscopic UV scale. RG scales and reaction catalysis For completeness we brie y reiterate here a point made in [ 1 ] concerning reaction catalysis. In some problems the scattering of interest between a particle and a point source is dominated by the -function contact interaction h 3(x), rather than the longer-range Coulomb or inverse-square potentials. When this is true, (2.42) and (2.44) show that the low-energy cross section is ' 4 as2 where the scattering length is of order the RG-invariant scale ? set by the classical running of h. The value of ? is in turn predictable from the RG evolution in terms of any initial condition h( 0) = h0 that might x h at a UV scale 0, perhaps where the low-energy point-particle EFT is matched to whatever UV completion describes the source's internal structure. Now comes the main point. Although it is often the case that ? is of order the geometrical size 0 suggested by such a matching (such as was found for mesonic atoms in the previous example), it can also happen that ? di ers considerably, with ? 0 when ^0 is very close to the UV xed point (at ^ = s) or with ? 0 when ^0 is close to the IR xed point (at ^ = + s). In particular, if the UV theory happens to match to the e ective theory at also guarantees that ? = 0 with h ' ( =m)(1 + s) then because this ensures ^ ' s it 0. In such a case the low-energy scattering cross section can be much larger than the geometrical one suggested by the UV scale 0 . As discussed in [ 1 ] a concrete case where we believe these observations to apply is to s-wave scattering of charged particles from magnetic monopoles [8]. The radial equation studied here applies to the non-relativistic limit (and | see below | to the relativistic case for spinless particles), though in general such scattering also involves an inversesquare potential because the magnetic monopole alters the particle angular momentum. In particular, for spinless particles the angular part of the problem alters the angularmomentum quantum number away from a non-negative integer to ` = ; + 1; where = eg=4 = n^=2 with g the monopole's magnetic charge and e the electric charge of the scattering particle (and the relation to an integer n^ is as required by the Dirac quantization condition). In terms of these quantities the dimensionless coe cient, v, of the inversesquare potential is l(l + 1) = `(` + 1) 2 and so = 2 These expressions show that when = 12 (say) there is no value of angular quantum numbers for which l(l + 1) vanishes, so the inverse-square coupling always plays a role. But the same exercise shows that for spin- 12 particles there is an s-wave combination for which the spin combines with = 12 to allow v = 0, in which case the scattering is purely governed by the -function component. As we see above (and is argued in [ 1 ]), this opens the possibility for cross sections being much larger than geometric in size provided the matching in the UV provides a coupling h0 in the right range. This leaves open (see, however, [4]) why the standard arguments associated with monopole catalysis of baryon-number violation [6, 7] provide the microscopic UV boundary conditions required to enhance scattering cross sections, thereby allowing classical RG evolution to provide a simple explanation for the unexpectedly large size of these cross sections. enhanced. Pauli eld, Mixing through contact interactions Since we have seen that renormalization can cause contact interactions to cause surprisingly large e ects, one might ask whether this renormalization oats all boats and ampli es all possible contact interactions. This section explores this issue by considering the RG evolution of contact interactions for two species of particles and shows why for some contact interactions zero coupling remains a xed point even in the presence of an inverse-square potential. The interactions that are not ampli ed do not share the same selection rules as does the inverse-square potential itself, and this is what decides which interactions become To explore this further imagine extending the Schrodinger eld to a 2-component (3.4) on which internal SU(2) ` avour' rotations are represented by the usual Pauli matrices. We take the bulk description to be SU(2)-invariant but imagine this symmetry to be broken by the source action, which is taken to be Sb = Z d h h0 y + h3 y 3 i = Z d h(h0 + h3) 1 1 + (h0 i h3) 2 2 : (3.5) As usual we de ne 0 = 2mh0 and 3 = 2mh3. boundary condition as before: Repeating the argument given above for each of 1 and 2 returns precisely the same = 1 2 ! ; = ( 0 + 3) 1( ) and 3) 2( ) ; (3.6) d d ^0 := ^ ! 0 s 2 0 = and ^3 := 3 ; s 2 1 (^0 + 1)2 + ^23 # ; d d ^ ! 3 s = " ^3(^0 + 1) # s : Notice that ^3 = 0 is a xed point of this last equation, indicating that it is RG-invariant equation for ^3 integrates to give ^3( ) = ^3( 0)( 0= ), which states that 3 = 2 for this coupling to vanish, even if ^0 6= 0 and s 6= 1. In particular, when ^0 = 0 the RG ^3 is The general solutions to the RG equation are given by the same ows as found earlier, ^ ( ) s [ ^ ( 0)= s] + tanh 12 s ln( = 0) 1 + [^ ( 0)= s] tanh 12 s ln( = 0) where the second equality specializes the reference point to ?, for which lim 0! ? where and and -independent. for ^ : 1. Consequently ^3( ) ^+( ) ^ ( ) s s 2 s 2 s s 2 2 s (3.7) (3.8) (3.9) (3.10) (3.11) ; (3.12) ^ ( 0) = ; (3.13) ; (3.14) (3.15) and because @r ln ( ) is a function of s and that depends only on the bulk eld equations and how their radial solutions approach the origin, the RG equation found by di erentiating the above with respect to is also the same as found in earlier sections: d d ^ ! ^ = = 2 2 41 0 2 Notice that when s = 1 the xed point for these ows occurs at 0 = case where Sb projects out either 1 or 2 . HJEP07(21) Suppose we de ne so that ^ = ^0 ^3 + 1. Then the evolution for these two new variables is given by coth 2 s ln 1 2 1 2 = coth 2 s ln ?+ ?+ + coth 2 s ln coth 2 s ln ? ? Since coth x ! 1 for x ! 1 these enjoy the IR xed points !1 lim ^0 + 1 = s and !1 lim ^3 = 0 ; coth x ' 1 + 2e 2x + showing that it is only ^0 that is driven away from zero when s 6= 1. For we use to infer the following approach to the IR xed points: and s ' 1 + ' ?+ s + ; ?+ s ? s + : where !2 m2 = 2 and for bound state solutions (for which ! < m) we take real. This has the same form as (2.5) | i.e. r given by 2 U = 2 | with potential U (x) U (r) = 2!eA0 (eA0)2 = 2!Z r (Z ) 2 r2 ; The upshot is this: because zero coupling remains a xed point for 3 even in the presence of an inverse-square potential, it need not be driven to run as dramatically as . They di er in this way because 0 shares the selection rules of the inverse-square potential while 3 does not. As the above arguments show, rather than implying a complete absence of evolution the RG e ects are instead suppressed to enter at higher order in Z . Klein-Gordon Coulomb problem (3.16) (3.17) (3.18) (3.19) to be (3.20) We now argue why the non-relativistic Schrodinger analysis given above also carries over directly to a relativistic spinless particle moving in the presence of a Coulomb potential. (We discuss the case of spin- 12 relativistic particles in [4].) In particular, the interaction of relativistic particles from point sources turns out to provide a practical example of competing Coulomb and inverse-square potentials, with the Coulomb potential arising with coe cient of order Z and the inverse-square potential arising due to relativistic e ects with a coe cient of order (Z )2. The signi cance of having both Coulomb and inverse-square potentials in this case is that this ensures that ' (Z )2 6= 0 and so s 6= 1. As a result zero-coupling, h = 0, is not a xed point of the RG evolution of the contact interaction, with the consequence that such a contact interaction must be nonzero for all scales except perhaps for a speci c scale, 0, at which point ^( 0) = 1. This makes the presence of a contact interaction mandatory, rather than optional, in relativistic Coulomb problems. Relativistic eld equation. The Klein-Gordon equation for a Coulomb potential is given by (D D m2) = 0 potential to be eA0(x) = becomes where D 1 particle). Assuming the only nonzero gauge Z =r and choosing a stationary state, (x; t) = '(x) e i!t, this 0 = h 2 m2i = h 2 r 2!eA0 + (eA0)2 2i' ; and so the parameters v and w are which gives = (Z )2 and w = 2!Z and v = (Z ) 2 `(` + 1) ; We see the radial part of the KG equation has the form considered earlier, specialized to these choices for v and w. In particular, for s-wave states we have s = p1 4(Z )2 ' 1 2(Z )2 : Boundary conditions. Because a canonically normalized Klein-Gordon eld has dimensions of mass, a contact interaction like Lb = dimension length. Following the steps of [ 1 ] and integrating over a small Gaussian pillbox 3(r) has coupling hKG with to obtain the boundary condition implied for this interaction gives 4 r 2 ; by = p 2m , and so Lb = 3(r) with hKG = 2mh. with = hKG. This is consistent with the result = 2mh found for the Schrodinger case because canonical normalization of the Schrodinger eld, , requires it to be related to The renormalization described earlier goes through as before for , and (as also noted earlier) because s < 1 it is inconsistent to choose h = 0 for all scales. Should we happen to know h0 = 0 at some UV scale 0 then the ow towards the IR xed point is given by hKG = 2mh = + (for 0) : (3.25) Energy shifts in mesonic atoms. Any departure of ^ from s implies a deviation from the standard energy-eigenvalue predictions, at least for s-wave states, and the surprise is that this is also true in particular if h0 = 0 at some scale. To second order in , the mode functions (2.14) specialized to the Klein-Gordon Coulomb problem take the form + ' (2 ) 21 ( 1+ s) 1 ' (2 ) 21 ( 1 s) 1 using 1 s ' 2(Z )2 and in appendix B we nd Z m + m Z + (m ) = p(m 2n2 + 1 6n2 2 2n2 + 1 6n2 (Z m )2 + O((Z m )3); Z (m )3 + O((Z )2(m )4) ; !)(m + !) ' Z m=n, where n is the principal quantum number. Combining this with the higher order pole approximation (B.6) derived ' 2m 0Z n s 0 Z m 0 (2 + s ^0) # (3.21) (3.22) (3.24) (3.23) (3.26) (3.27) for Z m 0 (3.27), gives found perturbatively near the IR xed point using (3.25). To leading order in ' 2m 0Z s where we drop all subdominant powers of Z and as before ^ = 1+mh= = 1+hKG=(2 ). The fractional energy shift of the s-wave states (using non-relativistic kinematics, as appropriate for the leading order e ect) is then En En 4m 0Z s ; and so using En ' (Z )2m=(2n2) we have the main result: En ' 3 3 : For instance, if h0 = 0 at Here the last equality specializes to the case ^0 ! 0 (if y = 1) or to ^0 ! 1 (if y = +1). = 0, then y = +1 and ^0 = +1 leading to s ^ 0 ' En ' +2 (Z ) 5 n3 0m2 : What is noteworthy about these expressions is that they are linear in the UV scale 0, precisely as was the Deser formula, above. This linearity di ers from the usual assessment of nite-size e ects, such as for the e ects in atoms of the nite size of the nucleus, which arise quadratically in the charge-radius of the nucleus. The Deser formula is also of practical value since trading ? (or 0) for the contact-interaction scattering length, as, again leads to (3.3).16 It is useful to quote these results in a more transparent way. For these purposes recall that a potential of the form V = he 3(x) naively shifts atomic energy levels by an amount This corresponds to an operator En = he j n(c)(0)j2 ' he Z m n 3 : he = m s using (3.27). A given charge distribution of the nucleus parametrizes the boundary condi^0 = ^(0) + ^(1)(k 0)2 + O(k 0) 4 where k is the momentum inside the nucleus. Generically, in the ultra-relativistic limit Z the rst term in (3.33) will dominate while the second term or a combination of 16In the appendix we examine a toy model of nuclear charge, to develop intuition as to why the boundary conditions should care about the mass of the particle orbiting the nucleus. (3.28) (3.29) HJEP07(21) ϵ appropriate for a spherical surface charge distribution discussed in appendix A.1. The dispersion relation for k is given in (A.11). the two terms dominates in the non-relativistic limit m 0 Hence, interpreting (3.33) as predictions for an `e ective' charge radius as a function of orbiting particle mass, m, the value of he strongly depends on how m 0 compares to Z and is not simply given by 23 Z rp2. We have demonstrated this point in gure 3 below. Z and yields he = 23 Z rp2. Acknowledgments We thank Brian Batell, Richard Hill, Ted Jacobson, Friederike Metz, Sasha Penin, Maxim Pospelov, Michael Trott and Itay Yavin for helpful discussions and Ross Diener, Leo van Nierop and Claudia de Rham for their help in understanding singular elds and classical renormalization. We are grateful to Marko Horbatsch and Henry Lamm for their careful reading of the manuscript, including the catching of several errors. This research was supported in part by funds from the Natural Sciences and Engineering Research Council (NSERC) of Canada and by a postdoctoral fellowship from the National Science Foundation of Belgium (FWO). 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). A Matching to a simplistic nuclear model In this appendix we describe several simple toy models of a nuclear charge distribution, with the goal of making more explicit how = 2mh should be expected to depend on m. We examine two distributions: one where all of the nuclear charge is located at the nuclear surface, r = R, and one where the charge is uniformly distributed throughout the nucleus, r R. We show both predict / m (and so h to be roughly m-independent) when computed within the non-relativistic Schrodinger regime, but both also predict to be m-independent (and so h / 1=m) when examined in the regime where the orbiting particle would be relativistic at the nuclear surface. We always demand R to be much smaller than the Bohr radius, which implies R 1=(Z m). In the Schrodinger analysis we also demand m 1=R and so its range of validity is for the window 1 Z mR 1 ; (A.1) which is non-empty because Z 1. The relativistic analysis requires only the rst of these inequalities and so assumes only mRZ Spherical surface-charge distribution The simplest (but least realistic) distribution assumes that the charge is concentrated in an in nitely thin sphere at r = R: where the constant charge per unit area, 0, is related to the total charge by 0 = Ze=(4 R2). In this case, the electrostatic potential is = 0 (r R) A0 = ( Ze=(4 R) for r Ze=(4 r) for r > R R ; which is chosen to be continuous at r = R with the external Coulomb potential. Schrodinger formulation. Let us discuss the s-wave solution with this potential. Outside the charged sphere (r > R) it is the Schrodinger solution, out(r), for the Coulomb problem, though without imposing regularity at the origin. We denote the energy of the state by E and determine this by matching the solution to the one found for r Inside the charge sphere (r < R) the wavefunction is that of a free particle, for which we choose regularity at the origin (because there is no source located there). This leads to the interior solution with k given in terms of E by in(r) = Cin sin(kr) r ; k2 = 2m E + eA0 = 2m E + Z R ' 2mZ R ; where the last, approximate, equality uses the condition R 1=(Z m) to infer Z =R jEj, since in the ground state jEj ' 21 (Z )2m. The wave function and its derivative must be continuous across r = R, and matching in(R) = out(R) relates the overall normalization constants of in(r) and out(r). For the (A.2) (A.3) (A.4) (A.5) present purposes it is the matching of the derivatives that is more interesting, which can be written as o0ut(R) out(R) = = k cot(kR) 1 R ; the main text at small r. function out satis es showing a possible underlying origin of the nontrivial boundary condition entertained in On the other hand, recall that outside the nucleus for su ciently small the wave(A.6) (A.7) 2R 1 : ' h(R) = 2m ' 3 For our toy model we nd in this way 2 o0ut(R) (R) = 4 R = 4 kR2 cot(kR) where we use (A.5) to infer (kR)2 ' 2mRZ from (A.1). We see this model predicts 1 kR ' 8 3 mR2Z ; (A.9) 1, with this last inequality following o0ut( ) out( ) = ( ) ^( ) =(2 ) + 1. Applying this to function of the interior in(R) e ectively ! R, the logarithmic derivative of the wavexes the function ^(R). The m-dependence and other properties of ^(R) in the external theory can be directly related to the properties of the source through this matching condition ! + Z R 2 k2 = m2 : (kR)2 ' (Z ) 2 1 : which is the same for any particle (independent of their mass) at the matching scale R. Klein-Gordon formulation. We can describe the same distribution using the KG equation, in order to treat the regime where m 1=R. We still require R to be much smaller than the Bohr radius, and so continue to require mRZ the matrix element, (x) = h0j (x)jni, where is the KG 1. To do so we compute eld and jni is an atomic meson state. For s-wave solutions with energy ! this function (r) solves the KG equation, with solutions still given by (A.4) but dispersion relation giving k now being This reduces to the non-relativistic Schrodinger dispersion relation for ! = m + E with, as before, E ' 12 (Z )2m for the ground state. In the regime Z =R ! ' m this dispersion relation can be approximated as Again expanding (A.9) for small kR we get = 4 RhkR cot(kR) i 1 ' in this regime is independent of m (as must also be the KG source coupling hKG = ). The equivalent Schrodinger coupling therefore becomes h = hKG = 2m 2m = 2 (Z )2R 3m ; which varies inversely with m. A.2 Constant charge distribution A slightly more realistic choice is a constant charge distribution: (A.13) (A.14) (A.15) (A.17) (A.18) (A.19) (A.20) (A.21) where the constant 0 is related to the total charge by 0 = 3Ze=(4 R3). In this case the electrostatic potential ' = A0 satis es A0 = + Ze R 2 = Ze 3 ; (A.16) where the integration constants ensure A0 is nonsingular at r = 0 and is continuous with the external Coulomb potential at r = R. Schrodinger formulation. s-wave solutions to the Schrodinger equation with this potential satisfy where Eq. (A.26) has as its general solution 1 E + eA0 V0 V2 r2 ; 3Z 2R V0 = 2m E + and V2 = 1 x (x) = e x2=2hC+ +(x) + C mZ R3 : i (x) ; where (x) are a pair of basis solutions that can be written in terms of con uent hypergeometric functions and the dimensionless coordinate is x = r where Since (A.17) is invariant under r ! C+ = 0 is required for regularity at x = 0, and x2 = 2r2, given explicitly by r we may choose ( r) = (r), in which case (x) is ultimately a series in powers of 4 = V2 = mZ R3 : (x) = x 2( x 3 3! 1) ; where V0 2 2 2 ER + 3Z 2 r mR Z 1 2 ' 2 1 p 3 mRZ 1 ' 1 2 ; which simpli es using (A.1). Because ( r)2 ( R)2 = mRZ eq. (A.1) also says that the regime of interest is small x for which (r) ' C C2( ) = 1 3 + 12 ' 0. Therefore @r ln in(r = R) ' C4 4R3, and so 1 + 12 C2x2 + 14 C4x4 + / m and so h = =2m independent of m. Klein-Gordon formulation. The KG equation to be solved with this potential is in this case 2 1 where we de ne the constants = hm2 ! + eA0 2i = V0 + V2 r2 + V4r4 ; (A.24) V0 = ! + V2 = V4 = 3Z 2R 2 m2 ' 4 9 Z R ! + Z 2R3 3Z 2R 2 1 4R4 Z R3 ' Z R 2R2 2 3 regime again predicts = hKG to be independent of m, and so h / R=m. and we focus on the regime Z =R when ! = m + E and we take m 1=R and E ' (Z )2m as before. ! ' m. The above reduces to the Schrodinger result Although not simply solvable, its dependence on scales is made explicit by changing coordinates to z = r=R and multiplying the equation through by R2, giving 00 + 2 0 + A + Bz2 + Cz4 = 0 ; 9 A = R2V0 ' 4 (Z )2 ; B = R4V2 ' 2 3 (Z ) 2 and C = R6V4 ' 4 1 (Z )2 : Our interest is r < R and so z < 1. To evaluate we can then approximate in(z) ' f [z2; (Z )2] with f going over to spherical Bessel functions as Z ! 0. Therefore @r ln in(r = R) becomes with (A.23) (A.25) (A.26) (A.27) (A.28) ) of this function close to the origin z = 0. The black solid line is f (z) = (z a)= (z)= ( a), the blue dashed line f (z) = a + 1)= ( a) (double pole approximation) and the red dashed line f (z) = z (single pole approximation). The chosen numerical value is a = 0:9. B Gamma function approximations when s . 1 The ratio C =C+ is given by a ratio of Gamma functions (2.25) We can rewrite this expression as where C C+ ( s) s) 1 2 1 2 w + 1 C C+ ( s) (z + 1 s (N + 1)) s) (z N ) 1 2 w + 1 + s) + N 1 ; : : since is close to the Bohr energy (2.26). If j s 1 j 1, as for instance in the Klein Gordon Coulomb problem, both -functions depending on z are in the vicinity of a pole, see gure 4. This makes it necessary to approximate both -functions by their respective poles whereas (2.29) is su cient if s is not close to one. (B.1) (B.2) (B.3) HJEP07(21) N ) ' ( 1)N =(N ! z) for z 1 we can then make the approximations (z + 1 (N + 1)) ' (N + 1)! 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C. P. Burgess, Peter Hayman, Markus Rummel, Matt Williams, László Zalavári. Point-particle effective field theory II: relativistic effects and Coulomb/inverse-square competition, Journal of High Energy Physics, 2017, 72, DOI: 10.1007/JHEP07(2017)072