Point Particle Effective Actions
1612.07334 and 1706.01063 with Cliff Burgess, Peter Hayman, Matt Williams and Laszlo Zalavari
Markus Rummel PASCOS 2017
Point Particle Effective Actions Markus Rummel 1/ 13 What is it
• An effective field theory (PPEFT) to describe a system of a heavy compact object and a light particle (bulk field) • first-quantized language for the heavy compact object (localized at x = 0) • second-quantized language for the light particle
Point Particle Effective Actions Markus Rummel 2 / 13 What is it good for Lots of examples in nature: atoms, solar system, … • [Goldberger, Rothstein 2006] • Only a few properties of the compact object actually matter: observables as expansion in R/a
• Atoms • Contact interactions (Pionic Atoms, Branes) "R ✏ a • Condensed Matter systems • Monopole catalysis, Black holes Point Particle Effective Actions Markus Rummel 3 / 13 Figure 1. A schematic of the scales arising when relating near-source boundary conditions to the 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 satisfies " ✏ a. We require " ✏ in ⌧ ⌧ ⌧ order to have the first few multipole moments (in our example only the first is considered) dominate the field 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 flow describes how the e↵ective couplings within the PPEFT action must change for di↵erent choices of ✏ in order to keep physical quantities unchanged.
that follow from these questions). Our treatment here broadly agrees with what is done in the litera- ture, di↵ering mainly in emphasis. We argue though that PPEFTs bring the following two conceptual benefits. A first benefit 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 delta-function 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 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 benefit. In particular, most treatments of the system agree there is a basic ambiguity to do with how to choose the boundary condition on the wave-function at r = 0. This is often phrased as a failure of the Hamiltonian to be self-adjoint [4, 5],
1See also [7], whose point of view on this is close to our own.
–2– What is new
• Explicit connection between PPEFT and near- source boundary condition for bulk fields • Recipe to calculate finite size corrections including back reaction • Easier than full 2-quantized framework such as NRQED/ NRQCD in certain limits (bound states) [Caswell, Lepage 1986] • By-product: Understanding of divergent potentials ( 1 /r 2 ) in Quantum mechanics via classical renormalization [Holstein 1996, Essin, Griffith 2006]
Point Particle Effective Actions Markus Rummel 4 / 13 charge distributions are also considered in the Appendix, and are compared with the more general EFT estimates. charge distributions are also considered in the Appendix, and are compared with the more general EFT estimates. charge distributions are also considered in the Appendix, and are compared with the more general EFT2 estimates. Nonrelativistic mixed Coulomb and inverse-square potentials 2 Nonrelativistic mixed Coulomb and inverse-square potentials 2Much Nonrelativistic of the physics mixed needed Coulomb for the relativistic and inverse-square case hinges potentials on the competition between the inverse- Much of the physicssquare needed and for Coulomb the relativistic potentials, case hinges so we on start the competition our discussion between with the the inverse- Schr¨odinger system involving these Much of the physics needed for the relativistic case hinges on the competition between the inverse- square and Coulombtwo potentials, potentials. so we Our start treatment our discussion follows with closely the Schr¨odinger that of system [1], which involving examines these the classical renormalizations square and Coulomb potentials,where soindicates we start theour discussion integration with is theover Schr¨odinger the world-line, systemyµ involving(⌧), of the these source. In the final equality two potentials. Ourtwoassociated treatment potentials.Action: follows Our with treatment the closely inverse-square W thatSchrödinger follows of [ closely1], which that potential, examines of [1], which though the field classical examines we renormalizations extend the classical this renormalizations analysis here by adding also a associated with the inverse-square potential,B and L thoughp are both we extend regarded this as analysis being functions here by of adding the bulk also fields a evaluated at an arbitrary spacetime associatedCoulomb with potential. the inverse-squareL µ potential, though we extend this analysis here by adding alsoµ a Coulomb potential. point, x . Lp is also a function of the ‘brane-localized’ position field, y (⌧). Coulomb potential. Stot = SBulk + Sbrane 2.1 Schr¨odinger action Potential 2.1 Schr¨odinger2.1 action Schr¨odinger action2.1 Action and field equations bulk particle mass We take, therefore,We our take, action therefore, to be STaking= S our+ the actionS bulkwhere to dynamicsS beisS the= to Schr¨odingerSB be+ QEDSb where with ‘bulk’ aS fermion actionB is the of Schr¨odinger charge e, the ‘bulk’ bulk action action becomes We take, therefore, our actionB tob be S = SBB+ Sb where SB is the Schr¨odinger ‘bulk’ action
3 i 3 i 3 i 1 2 14 2 1 µ⌫1 2 SB = dt d x SB = S⇤@d t=d x d@t d ⇤x⇤@t ⇤ @⇤t@ ⇤ SB =+ ⇤V@(x )d⇤ x ,+ FV⇤µ(⌫xF) +(2.1), (D/++Vm(x) )(2.1) , , (2.1) (2.2) 2 Bt 2t 2mt r t 2m r 4 ⇢ 2 2m r Z ⇢ ⇣ Z Z ⇣ ⌘ ⇢ ⇣ ⌘ Z ⌘ where m is the particle mass and S describes a microscopic contact interaction between the Schr¨odinger where m is the particlewhere massm andis theSb describes particlewith D a massµ microscopic b=( and@µ S+ contactieAdescribesµ) . interaction This a microscopic should between be considered thecontact Schr¨odinger interaction in the spirit between of a Wilson the Schr¨odinger action, and so in field and the point source localized at the originb r = 0: field and the point source localized atprinciple the3 origin3 alsor = includes 0: µ an infinite series˜ of subdominant local terms involving more powers of the fields field andSb = the pointdt d x source (x) localizedM QAµy˙ at+ theh origin h r =E 0:+ ... and their derivatives (whose e↵ects are · not important in what follows). 3 3 Sb = dt b[ (x = 0), ⇤(x =3 0)] = dt d3x b( , ⇤) (x) , (2.2) Sb = dt b[ (x = 0),The ⇤L( point-particlex = 0)] = dt actiond x b is( similarly, ⇤) (x givenL) , by an expansion(2.2) in these fields, for which (for a spinless, L Z L Z 3 3 Z parity-preservingSb = dt b[ source) Z (x = the 0), leading⇤(x = parity-even 0)] = dt termsd x areb(2 , ⇤) (x) , (2.2) with L L with source mass Z s sourceg world ContactZ s gV (x)= and b = h ⇤ (2.3) with V (x)= source andcharge r b =r2 lineh ⇤ L interactionsµ (2.3) µ ˜ r r2 Sp L= ds⌧ Mg QAµy˙ + cs + icv µ y˙ h E + , (2.3) used, for coupling constants s, g and h, when an explicit form is required. r · ··· used, for coupling constants s, g and h, when an explicitV ( formx)=Z isW required.h 2 and b = h ⇤ i (2.3) The field equation found by varying then is ther Schr¨odingerr equation,L Point Particle Effective Actionswhere the over-dot⇤ denotes di↵erentiationMarkus with Rummel respect5 / to13 proper time, the coe cients c , c and h˜ all The field equationused, found for bycoupling varying constants ⇤ then iss the, g Schr¨odingerand h, when equation, an explicit form is required. s v have dimension length-squared1 and the@ b ellipses indicate terms suppressed by more than two powers of i@ = 2 + V (x) + L 3(x) , (2.4) The field equation1 2 foundt by varying@ b 3 ⇤ then is the Schr¨odinger equation, i@t = length. Notice + V ( thatx2)m termsr+ L involving (x) , more@ ⇤ than two powers(2.4) of first arise suppressed by a coupling with 2m r @ ⇤ dimension (length)5, andiEt so are nominally subdominant to several terms involving only two powers of which for energy eigenstates, (x,t)= (x) e ,1 and with the choice (2.3@) becomes iEt 2 b 3 which for energy eigenstates, (x,t)= but(x) moree , derivatives andi@t with = the than choice those ( 2.3 written+)V becomes(x above.) + L (x) , (2.4) s g 2m r @ Specializing2 +2m to the+ rest frameh 3(x) for a= motionless2 , source,⇤ y ˙µ(⌧)= µ(2.5), with charge Q = Ze the bulk 2 s rg 3 r r2 2 0 +2m + h (x) = , iEt (2.5) which for energyfield eigenstates, equations2 become(xh,t)= (x) e i , and with the choice (2.3) becomes 2 r r r with = 2mE. For boundh states — wheni E 0— is real, but when discussing scattering 2 2 µ⌫ ⌫ ⌫ with = 2mE—. where For boundE 0 states — we — switch when toE = ik0—2with(D/ realis+ real,mk) sgiven but+ g by when=k 0=+2 discussing3 andmE. Expanding scattering@µF2 ie in spherical + j =0, (2.4) +2m2 +J h (x) = , (2.5) — where E 0harmonics, — we switchY`` to(✓, ),= impliesik with the real radialk given equation by k is given=+2mE by2 . Expanding in spherical z r r r harmonics, Y`` (✓, ), implies the radialwhere equation is given by h i z 2 1 d d `(` + 1) with = 2mE. For bound2 ``z states — when E@Lp 0—2 is real,0 but3 when discussing scattering 2 r 2 + U:=(r) ``z == c s ``+z ic, v (x)+ (2.6), (2.5) 1 d d `` r dr `(` +d 1)r r J @ 2 ··· — where E 2 0z — we✓ switch◆ to = ik with2 real k given by k =+2mE. Expanding in spherical 2 r 2 + U(r) ``z = ``z , ⇣ ⌘(2.6) r dr dr 3 r whereharmonics,U =2m✓[VY+ h( and✓(,◆x )]), while implies` =0 the, 1, 2 radial, and equation`z = `, is` +1 given, , by` 1, ` are the usual angular ``z ··· ··· 2 momentum3 quantum numbers. ⌫ @Lp rp 2 3 ⌫ where U =2m[V + h (x)] while ` =0, 1, 2, and `z = `, ` +1, j :=, ` 1, ` are= Ze the usual1+ angular (x) 0 . (2.6) ··· ··· @A⌫ 6 r momentum quantum numbers. 1 d 2 d ``z `(` + 1) !2 2.2 Source action and boundary2 conditionsr 2 + U(r) ``z = ``z , (2.6) r dr dr r ˜ 2 2 This last equality✓ trades the◆ parameter h for the mean-square charge radius: rp = r of the source 2.2 Source actionThe source and boundary action, Sb, appears conditions here only through the delta-function contribution to U and the only ˜ 1 2 h i e↵ect of this is to determinecharge the3 distribution boundary using conditionh = satisfied6 Zerp. by at r = 0. This can be obtained as The source action,whereS , appearsU =2 herem[V only+ h through (x)] whilethe delta-function` =0, 1, 2, contributionand `z = to U`,and` +1 the, only , ` 1, ` are the usual angular describedb in [1] by integrating (2.5) over an infinitesimal··· sphere, , of radius 0 r ···✏ around x =0 e↵ect of this is to determinemomentum the quantum boundary2.2 condition Bulk numbers. solutions satisfied by at r = 0. ThisS can be obtained as described in [1] by integrating (2.5) over an infinitesimal sphere, , of radius 0 r ✏ around x =0 2.2 Source actionWe seek and solutions boundary to the bulkconditionsS equations with a motionless point charge situated at the origin. The Maxwell equation is straightforwardly solved for the given source by choosing A = 0 and electrostatic The source action,potentialSb, appears here only–7– through the delta-function contribution to U and the only 2 e↵ect of this is to determine the boundary condition0 satisfied1 byr p at3 r = 0. This can be obtained as –7– A = Ze (x) . (2.7) described in [1] by integrating (2.5) over an infinitesimal" sphere,4⇡r 6 , of radius# 0 r ✏ around x =0 S 2Our metric is mostly plus and our Dirac conventions in rectangular and polar coordinates are given in Appendix A.
–7–
–4– 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 ! @ @ Boundary(0) = d3x 2 = Conditiond2x n = d2⌦ r2 =4⇡✏2 , (2.7) r @ · r @r r=✏ @r r=✏ ZS Z S Z ✓ ◆ ✓ ◆ 0-th order in R (Gauss law): 1-st order in R: where := 2mh while n dx =dr is the outward-pointing radial unit vector, d2⌦ =sin✓ d✓ d is · 2 the volume2 0 element3 on the surface of the angular 2 2-spherem [V (x) andE the] last equality assumes a spherically A = Q (x) 0 symmetric source so that is also spherically symmetric=2mh 3 to(x good) approximation for ✏ su ciently small. and using continuity ofBecause there to solutions see that only(r the) vary integral like of a power the secondrp as derivativer 0, contributes the boundary from condition given above `m ! the left-hand side ofbecomes (2.5) asIntegrate singular✏ 0. This as ✏over leads0. Gaussian to This the is result dealt pillbox with by of renormalizing size : — 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. 3 2 2 2 2 @ 2 @ (0) =Withd x this in= mindd —x andn defining = d ⌦(0)r := (r = ✏=4) —⇡✏ our problem, is to(2.7) solve the radial equation, r @ · r @r r=✏ @r r=✏ (Z2.6S ), subject toZ theS boundary conditionZ ✓ ◆ ✓ ◆ where := 2mh while n dx =dr is the outward-pointing radial unit vector, d2⌦ =sin✓ d✓ d is · @ the volume element on the4 r surface2 A of the angular= Q 2-sphere and4⇡r the2 lastln equality = assumes ,=2mh a spherically (2.8) r 0 r= @r symmetric source so that is also spherically symmetric to good approximation r=✏ for ✏ su ciently small. p Because solutions `m(r) vary like a power r as r 0, the boundary condition given above at the regulated0 radiusQ r = ✏. ! becomes singular as ✏ 0.A This= is dealt2 with by renormalizing Boundary— i.e. by associating condition an implicit ✏- !As mentioned 4 in [1], this boundary condition can be regarded as a specific choice of self-adjoint dependence to inextension such a way [18 as, to19 ensure] of the that inverse-square the precise value Hamiltonian. of ✏ drops out The of inverse-square physical predictions. potential requires such an With this in mindPointextension — Particle and defining Effective because Actions (0) its := wave-functions (r = ✏) — our are problem su ciently isMarkus to bunched solve Rummel the at radial6 the/ 13 origin equation, that physical quantities (2.6), subject to the boundary condition actually care about the nature of the physics encapsulated by the source action, Sb.Writingthe extension in this way usefully@ casts its ambiguities in terms of a physical action describing the physics 4⇡r2 ln = , (2.8) that can act as a potential@ sinkr (or not) of probability at r = 0. As might be expected, this extension r=✏ is self-adjoint provided that the source action is real and involves no new degrees of freedom. In the at the regulated radius r = ✏. present instance this can be seen from the radial probability flux, As mentioned in [1], this boundary condition can be regarded as a specific choice of self-adjoint 2 extension [18, 19] of the inverse-square Hamiltonian. The2 inverse-square2⇡r potential requires such an J =4⇡r n J = @ ⇤ ⇤@ , (2.9) extension because its wave-functions are su ciently bunched· at them originr that physicalr quantities ⇣ ⌘ actually care aboutemerging the nature from of the the source physics through encapsulated the surface by the at sourcer = ✏. Evaluating action, Sb.Writingthe with energy eigenstates gives extension in this way usefully casts its ambiguities in terms of a physical action describing the physics 2 that can act as a potential sink (or not) of probability2⇡✏ at r = 0. As might be expected, this extension J(✏)= (✏)@r ⇤(✏) ⇤(✏)@r (✏) =(h⇤ h) ⇤ (✏) , (2.10) is self-adjoint provided that the source action ism real and involves no new degrees of freedom. In the h i present instance thiswhich can beshows seen no from probability the radial flows probability into or outflux, of the source when its action is real (ie h⇤ = h). 2 2 2⇡r 2.3 SolutionsJ =4⇡r n J = @ ⇤ ⇤@ , (2.9) · m r r The radial equation (2.6) to be⇣ solved is ⌘ emerging from the source through the surface at r = ✏. Evaluating with energy eigenstates gives 2 2 d d 2⇡✏ r2 +2r + wr + v 2r2 =0, (2.11) J(✏)= (✏)@ ⇤(✏) ⇤(✏)@2 (✏) =(h⇤ h) ⇤ (✏) , (2.10) m r drr dr where w =2msh and v =2mg `(`+1). Thisi can be written in confluent hypergeometric form through which shows no probability flows into or out of the source when its action is real (ie h⇤ = h). l z/2 12 the transformation (r)=z e u(z), for z =2r where l(l + 1) + v = 0 so that 2.3 Solutions 1 1 1 1 2 The radial equation (2.6) to be solved isl = ( 1+p1 4v)= ( 1+⇣)= + ` + ⇠ , (2.12) 2 2 2 2 s✓ ◆ d2 d 12 2 2 2 Choosingr the other2 +2 rootr for+p justwr + exchangesv r the roles=0 of, the two independent solutions(2.11) encountered below, so does not introduce anydr new alternatives.dr where w =2ms and v =2mg `(`+1). This can be written in confluent hypergeometric form through l z/2 12 the transformation (r)=z e u(z), for z =2r where l(l + 1) + v = 0 so that
1 1 1 1 2 l = ( 1+p1 4v)= ( 1+⇣)= + `–8–+ ⇠ , (2.12) 2 2 2 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.
–8– 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 ! @ @ (0) = d3x 2 = d2x n = d2⌦ r2 =4⇡✏2 , (2.7) r @ · r @r r=✏ @r r=✏ ZS Z S Z ✓ ◆ ✓ ◆ 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 assumes a spherically symmetric source so that is also spherically symmetric to good approximation for ✏ su ciently small. Because solutions (r) vary like a power rp as r 0, the boundary condition given above `m ! 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 defining (0) := (r = ✏) — our problem is to solve the radial equation, (2.6), subject to the boundary conditionBoundary condition @ 4⇡r2 ln = ,=2mh (2.8) @r r=✏ at the regulated radius r = ✏.• depends on integration constants of bulk As mentioned in [1], this boundaryequations condition can be regarded as a specific choice of self-adjoint extension [18, 19] of the inverse-square• The BC relates Hamiltonian. these with The the inverse-square effective coupling potential requires such an extension because its wave-functionsconstants are like su hciently bunched at the origin that physical quantities actually care about the nature of the physics encapsulated by the source action, Sb.Writingthe extension in this way usefully casts its ambiguities in terms of a physical action describing the physics Physical observables depend on h 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 bePoint seen Particle from Effective theActions radial probability flux, Markus Rummel 7 / 13
2 2 2⇡r J =4⇡r n J = @ ⇤ ⇤@ , (2.9) · m r r ⇣ ⌘ emerging from the source through the surface at r = ✏. Evaluating with energy eigenstates gives 2⇡✏2 J(✏)= (✏)@ ⇤(✏) ⇤(✏)@ (✏) =(h⇤ h) ⇤ (✏) , (2.10) m r r h i which shows no probability flows 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
d2 d r2 +2r + wr + v 2r2 =0, (2.11) dr2 dr where w =2ms and v =2mg `(`+1). This can be written in confluent hypergeometric form through l z/2 12 the transformation (r)=z e u(z), for z =2r where l(l + 1) + v = 0 so that
1 1 1 1 2 l = ( 1+p1 4v)= ( 1+⇣)= + ` + ⇠ , (2.12) 2 2 2 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.
–8– 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 where we define for later notational simplicity ⇠ := 2mg and
Muchwhere of we the define physics for later needed notational for simplicity the⇣ relativistic:=⇠p:=1 2mg4v =and case1+4` hinges(` + 1) on4⇠ = the(2` competition+ 1)2 4⇠ . between(2.13) the inverse- where we define for later notational simplicity ⇠ := 2mg and square and Coulomb⇣ potentials,:= p1 4v = so1+4 we start`(` + 1) our4⇠ discussionp= (2` + 1)2 with4⇠ . thep Schr¨odinger(2.13) system involving these The two linearly independent radial profiles therefore are 2 two potentials. Our treatment⇣ := p1 4v follows=p 1+4 closely`(` + 1) that4⇠ ofp= [1(2],` which+ 1) examines4⇠ . the classical(2.13) renormalizations The two linearly independent radial profiles therefore are1 ( 1 ⇣) r 1 w (r)=(2r) 2 ± e +1 ⇣ , 1 ⇣;2r , (2.14) associatedThe two linearly with the independent inverse-square radial profilesp± potential, therefore are thoughpM we2 extend ± this analysis± here by adding also a 1 ( 1 ⇣) r 1 w ⇣ ⌘ (r)=(2r) 2 ± e +1 ⇣ , 1 ⇣;2r , (2.14) Coulomb potential. ±where (a, b; z)=1+(az/b)+ is the confluent hypergeometric function regular at z = 0. We 1 M 2 1 w ± ± M 2 ( 1 ⇣) r ··· therefore(r)=(2 taker) our general± e radial⇣ solution+1 to have⌘⇣ the, 1 form⇣;2 =r C,+ + + C . (2.14) where (a, b; z)=1+(± az/b)+ is the confluentM 2 hypergeometric ± function± regular at z = 0. We We next impose the boundary condition at r = 0 to determine the ratio C /C+. Regularizing for 2.1 Schr¨odingerM action ··· ⇣ ⌘ thereforewhere take(a, b; ourz)=1+( generalsmallaz/b radialr =)+✏ the solution solutionsis theto have confluent(r the) behave form hypergeometric as= C+ + + C function . regular at z = 0. We ± WeM next impose the boundary condition··· at r = 0 to determine the ratio C /C+. Regularizing for Wetherefore take, therefore, take our general our action radial solution to be toS have= SB the+ formSb where = C+ S+B+isC the . Schr¨odinger ‘bulk’ action 1 ( 1 ⇣) w ✏ 2 small r = ✏ the solutions (r) behave as (✏)=(2✏) 2 ± 1 + (✏ ) , (2.15) We next impose the boundary± condition at r±= 0 to determine the ratio1 ⇣C /CO +. Regularizing for ± small r = ✏ the solutions (r) behavei as1 w✏ 1 3 2 ( 1 ⇣) l l 1 2 2 SB =which has±d t thed(✏)=(2x familiar✏) form ⇤±@ oft r 1or r @,witht + ⇤ (l✏as) defined , ⇤ in (2.12). This+ V shows((2.15)x) that for some, choices (2.1) ± 2 1 ⇣ O 2m r of ⇠ neither of is bounded1 ( 1 at⇣) the origin.±w✏ This implies 2 that boundedness at the origin cannot be the ⇢ 2 Z l (✏)=(2±l 1 ✏⇣) ± 1 +⌘ (✏ ) , (2.15) which has the familiarright form physical of r ±or r criterion ,with there,l as at defined least 1 in in the (2.12⇣ presence).O This there shows of that a physical for some source. choices This is not really a ± whereof ⇠mneitheris theof particleis boundedsurprise mass atsince andthe fields origin.Sb genericallydescribes This implies diverge a thatmicroscopic at the boundedness presence contact of at a the source, origin interaction such cannot as does be the between the Coulomb the potential Schr¨odinger ± l l 1 fieldrightwhich and physical has the the point criterion familiar sourceitself. form there, of localized atr leastor r in the at,with presence thel as origin defined therer of in= a (2.12 physical 0: ). This source. shows This that is for not some really choices a of ⇠ neither of is boundedWe do at demand the origin. solutions This be implies normalizable, that boundedness however, and at the the convergence origin cannot of the be integral the d3x 2 surprise since fields± generically diverge at the presence of a source, such as does the Coulomb potential 3/2 | | right physical criterionas r there,0 implies at least incannot the presence diverge faster there than of ar physical as r source.0. For This isthis not implies really 2 a ⇣ > 0. For itself. ! !3 ± 3 ± R surpriseWe do since demand fields solutionsS genericallyconcreteness’b = bed diverge normalizable,t sakeb[ in at(x what the= however, presence 0) follows, ⇤ we( and ofx follow a= the source, 0)] convergence [1 =] and such specialized ast of doesd thex the to integralb( the Coulomb , case ⇤d) where3x potential (x2 the) , inverse-square (2.2) 3 5 The potentialpotential satisfies Land< ⇠ bulk< ,3/ because2 solutions this captures all of the examplesL of most| interest| and has the asitself.r 0 implies cannotZ diverge faster 4 than r 4 as r 0. For Z this implies 2 ⇣ > 0. For ! property that is not normalizable at!r = 0 for± any ` = 0. This± ensuresR that3 that2 the boundary concreteness’We do demand sake in solutions what follows be normalizable, we follow [1] however, and specialize and the to convergence the case where6 of the the integral inverse-squared x with 3 condition5 at the origin implies C3/2= 0 and so + for ` = 0. | | potentialas r 0 satisfies implies cannot< ⇠ < diverge, because faster this than capturesrs allasg of ther examples0. For/ of mostthis6 impliesinterest 2and⇣ has> the0. For 4 4 ± R !Consider potentialIt is only forV (`x=)= 0 that the contact!and interactionb is= neededh to⇤ determine± C /C+, and for such (2.3) propertyconcreteness’• that sakeis in not what normalizable follows we at followr = 0 [ for1] and any specialize`2= 0. This to ensures the case that where that the the inverse-square boundary1 s-wave states we have ⇣(` = 0)r = ⇣s r:=6 p1 4⇠ andL so 0 ⇣ < 1 for 0 ⇠ , and so both solutions conditionpotential atsatisfies the origin3 < implies⇠ < 5C, because= 0 and this so captures for all` of= the 0. examples of most interest and4 has the 4diverge but4 are normalizable at+ the origin.13 If 1 < ⇠ 5 then ⇣ becomes imaginary, in which case used, for couplingFor small constants r the solutions s, g and areh, when/ = C an + 6 explicit + + 4 C form 4 is required.s property•It is only that for `is=both not 0 that normalizable the2 and contact at2 interactiondiverger = 0 for near any isr needed=` 0= while 0. to This remaining determine ensures normalizable.C that/C+ that, and the In for this boundary such case eq. (2.8)isthe + | | | | 6 1 sconditionThe-wave field states at theequation we have origin⇣( implies` found= 0) =C⇣ bys1 :== varying 0p and1 4 so⇠ and so then 0 for⇣ ` is<=1 the 0. for 0 Schr¨odinger⇠ , and so equation, both solutions condition that fixes( 1 C /C) +, evaluating⇤ + the derivative using4 the small-r form for leads to with (r) (2 r) 2 ± 13s 1 / 5 6 ± divergeIt is but only are for normalizable` = 0 that at the the contact origin. interactionIf < ⇠ isthen needed⇣s becomes to determine imaginary,C /C in+, which and for case such ± 4 4 1 1 2 2 ( 3+⇣s) ( 3 ⇣s) both and diverge near2 r@= 0 while remaining2 C normalizable.+ ( 1+⇣s)(2✏ In) 2 this 1 case+ eq.C ((2.81)isthe⇣s)(2✏) 2 s-wave+ states we have ⇣(` = 0) = ⇣s := p1 4⇠1and so2 0 ⇣ < 1 for 0 @⇠ b , and3 so both solutions | | | | =4⇡✏ ln =2⇡✏ 1 4 1 i@@tr = 1 +5 V (x) + 2L(1+ ⇣s) (x) , 2 ( 1 ⇣s) (2.4) conditiondiverge with but that are fixes normalizable = C /C2 mE + , evaluating at and the origin. thes = derivative13r=✏If1 <8 using⇠mg" thethen small-⇣C+becomesr(2form✏) for imaginary, +leadsC (2 to in✏) which case # ✓ ◆ 2m4 r 4 s @ ⇤ ± 2 2 both + and diverge near r = 0 while remaining1 normalizable.R 1 In this case1 eq. (2.8)isthe = 2⇡✏ 21+( 3+⇣ ⇣s) , 2 ( 3 ⇣s) (2.16) | | 2 @| | 2 C+ ( 1+⇣s)(2✏) s + C ( 1 ⇣s)(2✏) iEt R +1 whichcondition for=4 energy⇡✏ that fixes eigenstates,ln C /C+=2, evaluating⇡✏ (x,t the)= derivative (x) e using1 , the and✓ small- with◆ r form the 1 for choice leads (2.3 to) becomes @r 2 ( 1+⇣s) 2 ( 1 ⇣s) ✓ ◆ r=✏ " C+(2✏) + C (2✏) ± # where 1 1 2 ( 3+⇣s) 2 ( 3 ⇣s) @ C+ ( R1+1⇣ss)(2✏g) C + C ( 1⇣ ⇣s)(2✏) 2 2 2 3 s 2 =4⇡✏ ln ==22⇡✏⇡✏ 1+⇣s , R1 := (2 ✏ ) 1, (2.16) (2.17) +2m + ( 1+h⇣Cs ) (x) =(1 ⇣s), (2.5) @r r=✏ " R +1C+(2✏)22 ++ C (2✏) 2 # ✓ ◆ r ✓ ◆ r r ✓ ◆ where and so, in particular, R =0whenR h 1 C = 0. i 2 = 2⇡✏ 1+⇣ , (2.16) To use this equations itC is useful to rewrite⇣ it as with = 2mE. For bound states —R +1 when Es 0— is real, but when discussing scattering R := ✓ (2◆ ✏) , (2.17) C+ 1 R2 — wherePoint ParticleE Effective0 —Actions we switch to =✓ ik ◆with ˆ real:=Markusk+1= givenRummel⇣ by8 /k 13 =+2, mE. Expanding(2.18) in spherical andwhere so, in particular, R =0whenC = 0. 2⇡✏ s 1+R C ⇣ ✓ ◆ harmonics, Y``z (✓, ), implies the radial equation iss given by To use this equation13 it is useful to rewriteR := it as (2✏) , (2.17) The only exception to this isC the+ case ⇠ =0forwhichl = ` 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 deficiency and so, in particular, R =0whenC = 0. 1 R of most1 treatmentsd ˆ of:=2 thed Coulomb``+1=z potential,⇣s `( ` a+ point 1), to which we return below. 2 (2.18) r 2⇡✏ 1+R + U(r) `` = `` , (2.6) To use this equation itr2 isd usefulr to rewritedr it as ✓ r2 ◆ z z 13 ✓ ◆ The only exception to this is the case ⇠ =0forwhich l = ` and so1 +Ris bounded. However once having discarded boundedness as a valid criterion at the origin,ˆ it cannot be revived in this special case. In our view this is a deficiency 3 := +1=⇣s , (2.18) whereof mostU treatments=2m[V of+ theh Coulomb (x)] potential, while a` point=02⇡✏ to, 1 which, 2, we return1+andR below.`z = `, ` +1, , ` 1, ` are the usual angular ···✓ ◆ –9– ··· 13 momentumThe only quantum exception to this numbers. is the case ⇠ =0forwhichl = ` 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 deficiency 2.2of most Source treatments action of the Coulomb and boundary potential, a point conditions to which we return below. –9– 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 satisfied by at r = 0. This can be obtained as –9– described in [1] by integrating (2.5) over an infinitesimal sphere, , of radius 0 r ✏ around x =0 S
–7– Running d Physical observables do not depend on ! d =0 ˆ for a general ` = 0 solutionDifferentiating to the Klein-Gordon the boundary equation, condition then yields:= ( /2⇡✏) + 1 satisfies the RG equation • 2 d ˆ ⇣ ˆ ✏ = s 1 (3.19) d✏ ⇣s ! 2 2 ⇣s ! 3 4 5 for a general ` = 0 solution to thefor Klein-Gordon✏ small enough equation, to use where the then small- ˆ :=r asymptotic ( /2⇡✏) + solution 1 satisfies for the(r). RG Here ⇣ := 1 4(Z↵)2. As s equation Z↵ 0 it follows as defined in (3.18) again satisfies the RG equation (3.15). ! Fixed points: ˆ = = 1 8mg p These considerations• show2 that when Z↵ vanishes,s if we define the quantity d ˆ ⇣ ˆ ± ± ✏ = s 1 Trivial fixed points =0 only if g =0 (3.19) (no 1 /r 2 • g d✏ ⇣s ! 2 2 + :=⇣s (m!+3!)(c + c )=(m + !)4⇡✏2 + =4⇡✏2 0 , (3.20) D potential)s v f ✓ + ◆ ✓ ◆ 4 5 2 for ✏ small enough to use the small-r asymptotic solutionPoint1 Particle for Effective (r Actions).ˆ+ Here+⇣s := 1 4(Z↵)Markus. As Rummel 9 / 13 for parity-even j = 2 states, then D := ( D /2⇡✏) + 1 satisfies the same RG equation, eq. (3.15), as Z↵ 0 it follows as defined in (3.18does) again ˆ in the satisfies Klein-Gordon the RG case. equation Notice (3.15 that). in the nonrelativistic limit we have + 2m(c + c ) ! p D ! s v These considerations show that whenin agreementZ↵ vanishes, with the ifZ we↵ define0 limit the of ( quantity3.14). !
+ Parity-odd case 2 g+ 2 0 D := (m + !)(cs + cv)=(m + !)4⇡✏ =4⇡✏ , (3.20) f+ 1 A similar argument goes✓ through◆ for the parity-odd✓ ◆ j = 2 states. Parity-odd states satisfy the radial equations (2.15) and so when Z↵ =0wehave 1 ˆ+ + for parity-even j = 2 states, then D := ( D /2⇡✏) + 1 satisfies the same RG equation, eq. (3.15), as does ˆ in the Klein-Gordon case. Notice that in the nonrelativistic limit we have g 0 + 2m(c + c ) f = D !. s v (3.21) in agreement with the Z↵ 0 limit of (3.14). m ! ! Repeating the arguments of the parity-odd case then shows that g = satisfies the Klein-Gordon ˆ Parity-odd case equation and so implies that D =( D /2⇡✏) + 1 satisfies (for small ✏) the same RG equation, (3.15) as do the parity-even and1 Klein-Gordon cases, provided we define A similar argument goes through for the parity-odd j = 2 states. Parity-odd states satisfy the radial equations (2.15) and so when Z↵ =0wehave 2 f 2 0 := (m !)(c c )=(m !)4⇡✏ =4⇡✏ . (3.22) D s v g ✓ ◆ ✓ ◆ g0 Flow patternsf = . (3.21) m ! The flow obtained by integrating (3.15) is given (for ✏ small enough that f and g are dominated by Repeating the arguments of the parity-odd case then shows that g = satisfies the Klein-Gordon their near-source asymptotic forms) by equation and so implies that ˆ =( /2⇡✏) + 1 satisfies (for small ✏) the same RG equation, (3.15) D D ˆ ⌘ 0±(✏ + ✏0 )+(✏ ✏0 ) ✏ + ✏? ± as do the parity-even and Klein-Gordon cases, provided we ˆ define±(✏)= ± ± = ± , (3.23) D ˆ ✏ ✏ (✏ + ✏0 )+ 0±(✏ ✏0 ) ? ± ± ✓ ± ◆ 2 f 2 0 D := (m !)(csa flowcv)=( thatm is shown!)4 in⇡✏ Fig. 1 . In=4 the⇡✏ first equality. the integration constant(3.22) is chosen using the initial g ˆ condition D±(✏0 )= 0±✓, while ◆ in the second✓ ◆ equality ⌘ = sign( D± 1) and the RG-invariant ± ± | | quantities ✏? are defined as the scales where the ˆ± approach zero (or diverge). Which of these one Flow patterns ± D uses depends on whether the RG trajectory of interest has ˆ± greater than or smaller than 1. In | D | The flow obtained by integrating (3.15either) is case given✏? (foris given✏ small explicitly enough by inverting that f and the firstg are equality dominated of (3.23 by): their near-source asymptotic forms) by ± ˆ ˆ ˆ ˆ ˆ ✏? D± 0± 1 ( D± 0±) 0± 1 ± =lim ⌘ = ⌘ . (3.24) ˆ ✏ 0 ˆ ˆ ± ˆ ˆ ± ˆ 0±(✏ + ✏0 )+(✏ ✏0 ) 0 ✏ D±+ ✏? D± 0± 1+( D± 0±) 0± +1! ˆ±(✏)= ± ± =± ! 1 ± , (3.23) D ˆ ✏ ✏ (✏ + ✏0 )+ 0±(✏ ✏0 ) ? As shown± in detail in± [3, 4✓], the ✏-independence± ◆ of physical quantities implies they depend only on ±(✏) and ✏ through RG-invariant quantities like ✏? . a flow that is shown in Fig. 1. In the firstD equality the integration constant is chosen± using the initial ˆ condition D±(✏0 )= 0±, while in the second equality ⌘ = sign( D± 1) and the RG-invariant ± ± | | quantities ✏? are defined as the scales where the ˆ± approach zero (or diverge). Which of these one ± D uses depends on whether the RG trajectory of interest has ˆ± greater than or smaller than 1. In | D | either case ✏? is given explicitly by inverting the first equality of (3.23): – 10 – ± ˆ ˆ ˆ ˆ ˆ ✏? D± 0± 1 ( D± 0±) 0± 1 ± =lim = ⌘ . (3.24) ✏ 0 ˆ ˆ ˆ ˆ ± ˆ 0 D± D± 0± 1+( D± 0±) 0± +1! ± ! 1 As shown in detail in [3, 4], the ✏-independence of physical quantities implies they depend only on ±(✏) and ✏ through RG-invariant quantities like ✏? . D ±
– 10 – Running �
� �
� λ
-�
-� -�� -� � � �� �� ϵ/ϵ* Klein-Gordon equation can be brought into Figure 1.PlotoftheRGflowof ˆ± (as defined in the main text) vs ln ✏/✏ when Z↵ = 0. A representative • D ? 2 of each of the two RG-invariant classes of flows is shown, and ✏? is chosen as the2 placeZ where ˆ(Z=0or ) ˆ , Schrödinger form with V (r)= 2 !1 depending on which class of flows is of interest. r r Relativistic scalars always run non-trivially! For•✏ ✏? (though ✏ not so large as to invalidate the small-r expansion of the mode functions ± at r = ✏) the flow approaches the fixed point at ˆ± = +1, with ˆ± 1 ✏? /✏. Because ˆ± 1 D D / ± D / Point(c Particlec )/✏ this Effective implies Actionsc and c simply become independent of ✏ in this limit.Markus Rummel 10 / 13 s ± v s v For small ✏ the flow emerges from the repulsive fixed point at ˆ± = 1 with ˆ± +1 2⌘ (✏/✏? ) D D ' ± ± with (as before) ⌘ = sign( ˆ± 1). Consequently for small ✏ the couplings cs and cv evolve linearly ± | D | with ✏ (as opposed to the naive quadratic behaviour expected on dimensional grounds): 1 + 4⇡m✏ c (✏)= D + D = + (✏2) s 2 m + ! m ! m2 !2 O ✓ ◆ 1 + 4⇡!✏ and c (✏)= D D = + (✏2) . (3.25) v 2 m + ! m ! m2 !2 O ✓ ◆ The flow describes the transition between these two asymptotic states, and clearly no source coupling (cs = cv = 0) is an RG-invariant fixed point, and it is also RG-invariant to have cv =0whilecs runs (corresponding to ✏?+ = ✏? ). As a concrete example, suppose matching to a UV completion were to give the predictions
2 2 cv = gvR and cs = gsR at ✏ = R, (3.26) < for a microscopic scale 1/R ! m and dimensionless constants gv , gs (1) . Then D±(R)= 2 | | | | ⇠ O (m !)(gs gv)R while the signs ⌘ = sign( ˆ± 1) are ⌘+ = sign(gs + gv) and ⌘ = sign (gv gs). ± ± ± D Then the RG-invariant scales are ✏? /R = ⌘ ( ˆ± 1)/( ˆ± + 1) and so ± ± D D ✏? (m !)(gs gv)R/4⇡ ± = ⌘ ± ± , (3.27) R ± 1+(m !)(g g )R/4⇡ ± s ± v and so ✏? R requires (gs gv)R 4⇡/(m !). Unlike for the nonrelativistic case there is always ± ± ' ± an ! for which this can be satisfied, but because !R 1 this is only possible in the e↵ective theory ⌧ if g g is su ciently large and has the right sign. s ± v
– 11 – What about fermions?
µ ¯ ¯ µ 2 2 Sp = d M QAµy˙ + cs + icv µ y˙ 3 Z rp E + W · ··· E ( Z ) 4 r 2 is IR fixed point of the Dirac PPEFT • p nothing new on the Proton radius problem
2 where theEfficient e↵ective coupling parametrizationh± is given order by order of incorrections: (Z↵) and (mRZ↵)by: limited sense of ‘model-independence’ than• we use here, since the model-independencee↵ of the predic- tions of the e↵ective action apply not just to static charge distributions, but essentially to any kind 4 3 of source physics that is su ciently localized. (This+ model-independence(Z ) m 22 of EFT methods2 for atomic2mRZ↵ Eh ⇡Z↵RR 2(1 + 2ˆg1) 1 (Z↵) ln + Hn+1 + e↵ 'n3 n measurements are emphasized within the 2nd-quantized framework⇢ in [17, 21].) ✓ ◆ 2 We verify in Appendix B that for a general static charge distribution, ⇢(x), the quantity2 g ˆ1 that 12n n 9 2 + 4ˆg3 +5+8ˆg1 2 2ˆg1 +(1+2ˆ2 g1) 2 (Z↵) (6.4) dominates how source physics appears in g+/f+ is related to the rms charge density, rp = r ,by 2n (n + 1) moments e.g. h i 2 1 2 3 2 rp + 2ˆg2 1+8n 1+ gˆ1(ˆg1 + 2) (mRZ↵)+... , (1 + 2ˆg1)R = , 3n2 (5.3)2 3 ✓ ◆ which implies that the leading energyand shift given by (4.18) becomes Checked with literature for toy charge distributions •+ 2⇡ 2 ⇡(n2 1) 2 fˆ 5 h Z↵ r , 2 3 (5.4) 1 [Friar 1979, Nickel 2013] e↵ h p (Z↵) mR fˆ 1+ (mRZ↵) ' 3 e↵ ' n2 1 3 2 3 Point Particle Effective Actions (✓ ◆ " Markus! Rummel# 11 / 13 as required for consistency with (5.1). On the other hand, the boundary condition (3.8)showshow 1 the parameterg ˆ is also interchangeable with one combination of c and c through + fˆ (mRZ↵)+... , (6.5) 1 s v tot 2 9 ✓ ◆ 2⇡ 2 2 g+ 2 cs + cv tot = cs + cv + Z↵ rp =4⇡R =4⇡gˆ1ˆZ↵ R . (5.5) 2 ✏=R These3 expressions applyf for general fi andg ˆi out to subdominant order mRZ↵ and (Z↵) , and ✓ + ◆r=R ⇣ ⌘ so su ce for modern comparisons with precision measurements. As such they provide a model- This implies independent description2 of source e↵ects, allowing source e↵ects to be e ciently parameterized when cs + cv = 2⇡Z↵ R , (5.6) comparing modern measurements [28] with other precisions corrections, such as those of QED. i.e. the infrared fixed point found in (3.42Finally,). Note the we di have↵erence verified from explicitly the Schr¨odinger that these running expressions where reproduce those in the literature when we found that h = 0 is a fixed point thatspecialized parametrizes to the a trivial special boundary case where condition. the source is modelled as an explicit charge distribution, and The subdominant (mRZ↵) contribution also provides a relation betweeng ˆ2 and the higher moment for comparison purposes give expressions for the leading values of fˆi andg ˆi for several simple models r3 . Comparing (4.21)with(5.1) and using (5.3)shows h i(2) considered elsewhere. 3 1 r3 6R3 2ˆg 1+8n2 1+ gˆ (ˆg + 2) , (5.7) h i(2) ' 2 2 1 1 3n2 ⇢ Acknowledgements ✓ ◆ Although we do not have a general proof of this result, we can verify it for specific charge distributions. These higher terms can be related toWe higher-dimension thank Paddy interactions Fox, Richard — Hill, such Bob as those Holdom, of (3.44 Marko)—in Horbatsch, Friederike Metz, Bernie Nickel, Sasha Penin, Ryan Plestid, Maxim Pospelov, Ira Rothstein, Andrew Tolley and Michael Trott for Sp, using matching conditions similar to (5.5), although we do not pursue this here. helpful discussions and Ross Diener, Leo van Nierop, Claudia de Rham and Matt Williams for their 5.1.2 Specific charge distributionshelp in understanding singular fields and classical renormalization. This research was supported in The detailed calculations done for specificpart charge by funds distributions from the [12 Natural, 19, 20 Sciences] provide useful and Engineering checks on the Research Council (NSERC) of Canada and higher-order terms, since these must agreeby a on postdoctoral the series coe fellowship cients for from the specific the National charge distributions Science Foundation of Belgium (FWO). Research at studied. To provide this check we computethe Perimeter the couplings Instituteg ˆi and isfˆ supportedi for various in charge part by distributions the Government in of Canada through Industry Canada, Appendix B, and we here use these inand the above by the expressions Province for of Ontariohe↵ to verify through agreement the Ministry where overlap of Research and Information (MRI). is possible.
Spherical charged shell A Gamma-matrix conventions
The simplest such example is that of aWhen charged necessary shell, for we which use the following representation for the tangent-frame gamma matrices: Ze ⇢ = (r R)= (r R)0 I (5.8) 0 4⇡R2 0 = i = i , = i k , (A.1) I 0 k 0 which is convenient since the interior solution can be solved exactly in closed form.✓ (We have◆ checked ✓ k ◆ that our numerical results for this case agree with those of [20].) For this distribution the rms charge where k are the Pauli matrices, radius is r2 = R2 and r3 = 16R3/5. p h i(2) 01 0 i 10 = , = and = , (A.2) 1 10 2 i 0 3 0 1 ✓ ◆ ✓ ◆ ✓ ◆
– 24 –
– 29 – More applications
Scattering for contact interactions (Deser formula) • [Deser, Goldberger, Baumann, Thirring 1954] • Vacuum polarization (Ueling potential) [Ueling 1935] • (Mesonic) Atoms with e.g. ,K , p ¯ orbiting [e.g. Tucker-Smith, Yavin 2011, Barger, Chiang, Keung, Exotic interactions Marfatia 2011, Batell, McKeen, Pospelov 2011, • Carroll, Thomas, Rafelski, Miller 2011]
Point Particle Effective Actions Markus Rummel 12 / 13 Conclusions & Outlook
• PPEFT is efficient way to calculate finite size effects when source particle is approximately at rest (e.g. finite size corrections of nucleus in atoms) • Manifests itself via boundary condition and classical renormalization • Works in Schrödinger, Klein-Gordon and Dirac context • Only the beginning: More applications on the way
Point Particle Effective Actions Markus Rummel 13 / 13