On the application of Effective Field Theory to finite-volume effects in a HVP μ
Christopher Aubin, Thomas Blum, Maarten Golterman, Santiago Peris
OOn the application of Effective Field Theory tofinite-volume effects in a HVP µ Christopher Aubin, a Thomas Blum, b Maarten Golterman, c Santiago Peris da Department of Physics and Engineering Physics,Fordham University, Bronx, New York, NY 10458, USA b Physics Department,University of Connecticut, Storrs, CT 06269, USA c Department of Physics and Astronomy, San Francisco State University,San Francisco, CA 94132, USA d Department of Physics and BIST, Universitat Aut`onoma de Barcelona,E-08193 Bellaterra, Barcelona, Spain
ABSTRACTOne of the more important systematic effects affecting lattice computationsof the hadronic vacuum polarization contribution to the anomalous magneticmoment of the muon, a HVP µ , is the distortion due to a finite spatial volume. Inorder to reach sub-percent precision, these effects need to be reliably estimatedand corrected for, and one of the methods that has been employed for doing thisis finite-volume chiral perturbation theory. In this paper, we argue that finite-volume corrections to a HVP µ can, in principle, be calculated at any given order inchiral perturbation theory. More precisely, once all low-energy constants neededto define the Effective Field Theory representation of a HVP µ in infinite volumeare known to a given order, also the finite-volume corrections can be predictedto that order in the chiral expansion. I. INTRODUCTION
Recent years have seen renewed efforts to obtain a more reliable and more preciseStandard-Model estimate of the muon anomalous magnetic moment. The discrepancy be-tween the best Standard-Model estimates and the experimental value from the Brookhavenexperiment [1] has not only persisted, but also has become more acute, as many of thesystematic errors associated with the Standard-Model estimate have become more con-trolled. According to a recent review of the Standard-Model calculation [2] the best averageStandard-Model value is estimated to be 3 . σ smaller than the value reported in Ref. [1].The renewed efforts to improve the Standard-Model estimate are driven by new experi-mental programs at both Fermilab [3] and J-PARC [4], which aim to improve on the precisionof the measurement of the magnetic moment, in the case of Fermilab, by a factor four. Itis thus important to improve the precision of the Standard-Model estimate to a level com-mensurate with the experimental goal. 1 a r X i v : . [ h e p - l a t ] A ug he bulk of the error of the Standard-Model estimate originates from the hadronic con-tributions to the anomalous magnetic moment, which cannot be computed in perturbationtheory. The hadronic contribution consists of two parts: the hadronic vacuum polarization(HVP) contribution and the hadronic light-by-light contribution; in this paper, the focuswill be the HVP contribution. The most precise estimate has been based on the dispersive,data-driven approach, but more recently lattice QCD computations have started to becomemore precise, and are expected to become competitive in the near future. It is thus im-portant to gain a thorough understanding of the various systematic errors afflicting latticecomputations of the HVP contribution to the muon anomalous magnetic moment, a HVP µ .Lattice computations of the HVP are necessarily done in a finite physical spatial volume,typically a cubic volume with periodic boundary conditions. As a HVP µ is dominated bymomenta at a scale set by the muon mass, which is roughly equal to the pion mass, itturns out that finite-volume (FV) effects constitute one of the more important systematicerrors in state-of-the-art lattice computations of the HVP. These effects are large enoughthat lattice results for a HVP µ need to be corrected, and it is thus important to computethis correction, as well as the systematic errors associated with such a correction, reliably.Various methods have been used to provide reliable estimates: an adaptation [5] of theGounaris–Sakurai model [6] for the low-momentum HVP to finite volume using the methodof Ref. [7]; chiral perturbation theory (ChPT) [8–11] in finite volume [12]; a model for low-energy pions including the ρ and ω resonances [13]; a systematic estimate of the leadingFV effects in terms of the forward Compton amplitude of the pion [14, 15], based on themethods of Ref. [16]; and, finally, by varying the lattice volume directly in the numericalcomputations of a HVP µ [11, 17].Because the relevant scale is so low, the proper systematic Effective Field Theory (EFT)to analyze these FV effects is Chiral Perturbation Theory (ChPT). In Ref. [10], we com-puted FV corrections to NNLO, i.e. , to two loops in ChPT. Since the lowest-order pioniccontribution to a HVP µ already involves a pion loop, we will follow the convention of referringto the lowest order contribution as NLO, the contribution that involves two loops in ChPTas NNLO, etc. Our motivation here is not to push this to higher orders, but to consider whether, as amatter of principle, FV corrections for a HVP µ can be computed to arbitrary orders in ChPT.Even if orders beyond NNLO may never be pursued in practice, it is important to establishthat ChPT allows, in principle, a systematic approach to FV effects which is well-defined ateach order in the chiral expansion. Our main motivation is to illustrate how the propertiesof an EFT guarantee this to happen through simple examples.The HVP contribution to the muon anomalous magnetic moment to lowest order in α isgiven by [18] (see also Ref. [19]) a HVP µ = 4 α (cid:90) ∞ dq f ( q ) ˆΠ( q ) , (1.1)where ˆΠ( q ) = Π(0) − Π( q ) , (1.2) The euclidean time extent of the lattice is usually significantly larger than the linear spatial dimension.We will not consider effects of the finite extent of the lattice in euclidean time in this paper. A small mistake was corrected in Ref. [11]; for the computation of Ref. [10] the numerical effect of thismistake is negligibly small.
2s the subtracted HVP, obtained from (cid:0) q δ µν − q µ q ν (cid:1) Π( q ) = (cid:90) d x e iqx (cid:10) j EM µ ( x ) j EM ν (0) (cid:11) , (1.3)with j EM µ ( x ) the hadronic part of the electromagnetic current and α the fine-structure con-stant. Here the momentum q is euclidean, and throughout this paper, we will work ineuclidean space. The weight f ( q ) in Eq. (1.1) depends on the muon mass m µ , and is givenby f ( q ) = m µ q Z ( q )(1 − q Z ( q ))1 + m µ q Z ( q ) , (1.4) Z ( q ) = (cid:113) m µ /q − m µ . The integral in Eq. (1.1) is finite in QCD, as can be seen from the operator product expansionof j EM µ ( x ) j EM ν (0), which governs the behavior of ˆΠ( q ) at large q . However, ChPT, which isdesigned to parametrize the small- q behavior of ˆΠ( q ), does not get the large- q behaviorright, and therefore, when we insert a ChPT representation of ˆΠ( q ) into Eq. (1.1), theintegral over q does not converge beyond some order. In fact, at k -loop order in ChPT,ˆΠ( q ) ∼ ( q ) k − , k = 1 , , . . . , (1.5)modulo logarithmic corrections. Because f ( q ) ∼ m µ /q for large q , this means that theChPT result for a HVP µ is finite only up to NNLO, i.e. , k = 2, while at N LO and higherthe integral in Eq. (1.1) is UV divergent, and new counter terms need to be introduced torender the result finite. Such counter terms introduce new low-energy constants (LECs), andthus ChPT cannot be used to estimate a HVP µ quantitatively beyond two-loop order, unless itwould be possible to estimate the value of the (renormalized) LECs from some other physicalprocesses.A key point is that these new counter terms arise because of the integral over q inEq. (1.1), and they are thus not part of the ChPT lagrangian used to calculate Π( q ) to anygiven order. They will necessarily be constructed not only from the pion fields of ChPT butalso from the muon and photon fields. As the photon is massless, and the muon is lighterthan a pion, this raises the additional question whether it is consistent to consider only FVeffects associated with pions, while muons and photons are kept in an infinite volume.In Ref. [10] we claimed that, nevertheless, pionic FV corrections can be computed inChPT to all orders, based on the notion that FV effects refer to the IR behavior of thetheory, while counter terms fix the UV behavior. As we will see, the way this separationof scales works is subtle, and the claim is perhaps not obvious. Also in a finite volumeˆΠ( q ) takes the form of an expansion in powers of q in ChPT, and one might fear thatthus also the FV part of a HVP µ , defined as in Eq. (1.1) with ˆΠ( q ) replaced by its FV part,diverges beyond NNLO in ChPT. This concern, that the computation of higher-order FVcorrections to a HVP µ in ChPT might break down, was raised in Ref. [15]. We will argue that, in fact, the FV part of Eq. (1.1) starts diverging at N LO.
3n this paper, we will argue that, nevertheless, FV corrections to a HVP µ can be computed atany order in ChPT. More precisely, our claim is that, once all counter terms needed to make a HVP µ finite in a ChPT calculation in infinite volume have been introduced, also the finite-volume corrections to a HVP µ will be UV finite, with no need to introduce any further counterterms. In addition, this also holds when only the hadronic part of Eq. (1.1) is consideredin a finite volume, while the muons and photons (the QED part of Eq. (1.1)) are kept ininfinite volume. This is actually the situation encountered in lattice QCD computations of a HVP µ , in which only ˆΠ( q ) is calculated on the lattice, and thus in a finite volume.It is beyond the scope of this paper to give an all-order proof, or even to carry out explicitChPT calculations beyond NNLO. Instead, we will present general arguments supporting ourclaim, and discuss the form of the new counter terms introduced to absorb UV divergenceswhich arise in a ChPT calculation of the q integral in Eq. (1.1) in more detail. This isdone in Sec. II. Then, in Sec. III, we invent a toy model that allows us to demonstrate howour claim works already at two-loop order, thus illustrating the mechanism underlying ourclaim in a simple example. We end with our conclusions.There are two appendices. In the first appendix, we discuss the relation between counterterms in the momentum representation for a HVP µ , Eq. (1.1), which is exclusively used inthe main text, and the “time-momentum” representation [20] often employed in latticecomputations of a HVP µ . The other appendix contains a number of technical details neededfor Secs. II and III. II. COUNTER TERMS FOR a HVP µ In Sec. II A, we begin with a qualitative discussion of the problem, based on a diagram-matic picture, explaining the necessary introduction of counter terms not present in thelow-energy pion effective theory. Then, in Sec. II B, we discuss the explicit form of thesecounter terms in more detail, and in Sec. II C we present illustrative examples of the role ofthese counter terms. We will have a first look at FV effects in Sec. II D, to argue that theinterplay between UV divergences and FV effects only becomes non-trivial at N LO. WhileFV effects will also be qualitatively discussed in Sec. II A below, most of the finite-volumediscussion will be postponed to Sec. III; the bulk of this section will concentrate on thecounter-term structure for a HVP µ in infinite volume. A. Qualitative discussion
In Fig. 1, diagram N0 depicts the standard diagrammatic picture of the HVP contributionto the muon anomalous magnetic moment. According to Refs. [14, 15], the HVP contributionin Eq. (1.1) may, in more detail, be thought of in terms of a forward pion Compton scatteringsubdiagram. This is schematically depicted in the panel N1 of Fig. 1 where the internalCompton amplitude is obtained by cutting open the pion loop. In this diagram the fat linedepicts any number of pions as well as any other heavier hadrons, such as e.g. the ρ , ω , etc. resonance contributions or a proton loop, which are denoted in the diagram by H . In thispicture, ChPT is the result of “integrating out” all heavier states denoted by H , giving rise4o the low-energy Effective Field Theory (EFT) which is ChPT. Although it is not knownhow to make this quantitative in the real world, it presents a clear picture which will provehelpful in what follows.Diagram N1
EFT schematically depicts the contribution of the low-energy degrees of free-dom, i.e. , the pions, while diagrams N2-N5 originate from integrating out all heavier hadroniccontributions, collectively denoted as H in diagram N1. As the H states are considered in-finitely heavier than the pions in the setting of the EFT, all H propagators get shrunk to apoint, as seen, e.g. , in diagram N2. Only pion loops remain, with one of them schematicallyindicated in N2. The sum of N1 EFT and N2 has a different UV behavior than diagramN1, which must be compensated by the necessary low-energy constants which play the roleof counter terms to subtract the UV divergences which are produced by the “incorrect”large-momentum behavior of the sum of N1
EFT and N2. These counter-term contributionscorrespond to the diagrams denoted by N3-N5. Both the loop containing the muon lineas well as pion loops become more divergent as a consequence of contracting heavy-hadronpropagators to a point. Diagram N3 renormalizes the UV sub-divergence from the muon-photon loop in N1
EFT and N2, while N4 renormalizes the UV sub-divergence from thepion loop in N1 EFT and N2. Diagram N5 is needed to renormalize the product of thesedivergences. Each of these new “vertices” (denoted by solid squares in diagrams N2-N5)corresponds to a set of new higher-dimension operators suppressed by the right power ofthe scale M H characterizing the heavy states H . In this sense, ChPT is nothing but anexpansion in inverse powers of M H . As is clear from these diagrams, one should expect notonly an LEC associated with the HVP subdiagram (N4) but also a mixed one involvingpions, muons and photons (N3), as well as a counter term without pions (N5). Integratingout H , therefore, yields an EFT which is not just ChPT but ChPT enlarged by the pres-ence of muons and photons; in other words, what we need is the EFT for low-energy QCDplus QED. Diagram N4 corresponds to a counter term in standard ChPT, but both N3 andN5 correspond to Pauli-like counter terms coupling to the muon and the photon, with N3coupling to two pions as well.So far, our discussion has been restricted to infinite volume, and we now turn to theimplications for the EFT in a finite volume.While the analysis in this paper will be carried out in momentum space, it is useful tothink about the origin of FV effects in position space. Pion induced FV corrections to theHVP are obtained by having (at least) one pion wrap (at least once) around the “periodicworld,” thus producing a factor e − nm π L , where L is the linear size of the periodic spatialvolume [12]. Here n is a number relating to how many times in which directions the pionwraps around the world: if the pion wraps once in one direction, n = 1; if it wraps once intwo different directions, n = √ etc. Since a pion wrapping at least once around the worldtravels a long distance, it will not contribute to the degree of divergence of the diagram.However, the remainder of the diagram may still be UV divergent, and one may thus endup with a FV correction multiplied by a UV divergence. This appears to be in conflict withthe claim that FV corrections can be systematically computed in an EFT framework. Aswe will see explicitly in Sec. III, in momentum space, forcing a pion to wrap around theperiodic world corresponds to cutting the corresponding pion loop, i.e. , putting that pion While the diagrams show only one pion loop explicitly, there can be more than one pion loop. While the muon-photon(-pion) loop in N1
EF T and N2 looks convergent, at sufficiently high order in ChPTthere will be derivatives at the photon-pion vertex making this loop divergent. IG. 1: a HVP µ in terms of hadronic Feynman diagrams. on shell.Within ChPT, there will be a counter term corresponding to the divergent subdiagram.At orders beyond the order at which the counter term first appears, it will contribute to loopdiagrams as well, and again cutting pion lines on these loops will also lead to FV correctionsof the same form, multiplied by the LEC associated with the counter term. Since this LEC6s UV divergent, this will renormalize the coefficient of the FV correction, and the completeFV correction will turn out to be finite, thus exhibiting the expected separation of UV andIR effects in the EFT. FV effects in ˆΠ( q ) are thus predicted in terms of the LECs of ChPT.An explicit example of this appears in Ref. [10] at NNLO, where the O ( p ) counter term (cid:96) appears inside a one-loop diagram. The pion line in this diagram can wrap around the world,and thus the diagram contributes to FV effects, yielding a FV contribution proportional to (cid:96) . But, FV effects also appear in the two-loop diagrams that appear at this order. Takingthe pion on one of the loops around the world still leaves a one-loop subdivergence. Thisone-loop subdivergence is renormalized by (cid:96) , and the sum of the FV effects coming fromthe two-loop diagrams and the one-loop diagrams containing (cid:96) is UV finite. The vertexcorresponding to the counter term with LEC (cid:96) is an example of a diagram of type N4 inFig. 1.In the case of a HVP µ , there appear also counter terms corresponding to the black squares indiagrams N3 and N5. The reason for this is as follows. a HVP µ is proportional to the projectionon the Dirac structure associated with the muon form factor F ( q ) at low momentum ofa correlation function in QCD coupled to muons and photons. This enlarged theory isdescribed at low energy by a theory of pions coupled to muons and photons (which, bothbeing lighter than pions, have to be kept as explicit degrees of freedom in the EFT). Inthis extended EFT, new counter terms can appear which contain muon and photon fieldscombining into a Pauli-like operator. While in the renormalizable UV-complete theoryconsisting of QCD plus QED such a counter term cannot (and does not) appear since ithas at least dimension five, in the EFT there is no restriction on the dimension of possiblecounter terms, because an arbitrary inverse power of the heavy scale M H that has beenintegrated out can appear multiplying these counter terms. Pauli-like counter terms arethus expected to appear. In the next subsection, we will consider the explicit form of suchcounter terms. B. Pauli-like counter terms
In this section, we consider the explicit form of counter terms corresponding to diagramsN3 and N5 in Fig. 1. Counter terms corresponding to diagram N4 are those already appearingin the EFT for pions only. At NNLO, examples of the LECs associated with N4-type counterterms are the O ( p ) LEC (cid:96) (for its role in the ChPT approach to a HVP µ see Ref. [10]) andthe O ( p ) LEC c [22] (see App. A); their role is to renormalize the HVP subdiagram ofdiagram N1 EFT in Fig. 1. Here, instead, we will construct the simplest example of a countertem in the EFT which includes also muons and photons.At N LO, ˆΠ( q ) ∼ q modulo logarithmic corrections, and, since f ( q ) ∼ m µ /q , theintegral in Eq. (1.1) diverges at this order, requiring an N5-type counter term. Of course,the counter term that is needed must be proportional to ψ µ σ κλ ψ µ F κλ , (2.1)where ψ µ is the muon field, and F κλ is the electromagnetic field strength. However, wewill also need N3-type counter terms, and in general, in the theory with pions, muons andphotons, we need to analyze the general structure N3- and N5-type counter terms can have.The standard method for carrying out this analysis is through the use of spurions.We start with coupling massless, two-flavor QCD to [ SU (2) × U (1)] L × [ SU (2) × U (1)] R vector sources (cid:96) κ and r κ , which we will eventually set equal to the photon field A κ by7hoosing (cid:96) κ = −
12 (1 − τ ) A κ , r κ = −
12 (1 − τ ) A κ , (2.2)where τ i are the Pauli matrices. We also introduce a muon doublet, ψ = ( ψ ν , ψ µ ) T , whichcouples to the spurions (cid:96) κ and r κ through¯ ψγ κ ( ∂ κ + i(cid:96) κ P L + ir κ P R ) ψ , (2.3)where P R and P L are right- and left-handed projectors. The non-linear pion field U =exp[2 iπ/f π ] couples to the spurions through the covariant derivative D κ U = ∂ κ U + i(cid:96) κ U − iU r κ . (2.4)Since, for large momenta, the weight f ( q ) is proportional to m µ , the N3- and N5-typecounter terms have to contain the third power of the muon mass, and we thus need tointroduce a spurion χ ( µ ) for the muon mass as well. Finally, since a HVP µ contains internalphoton lines, we need the charge matrix spurions Q L and Q R through which the photoncouples to the left-handed and right-handed quarks, respectively. Of course, Q L = Q R = Q = diag( , − ), but the spurions Q L and Q R transform differently, under [ SU (2) × U (1)] L and [ SU (2) × U (1)] R ], respectively. QCD coupled to the muon doublet ψ and the spurions (cid:96) κ , r κ and χ ( µ ) , and thus our EFT, is invariant under U → LU R † , (2.5) ψ → ( LP L + RP R ) ψ ,(cid:96) κ → L(cid:96) k L † − iL∂ κ L † ,r κ → Rr k R † − iR∂ κ R † ,χ ( µ ) → Lχ ( µ ) R † ,Q L → LQ L L † ,Q R → RQ R R † , where L ∈ [ SU (2) × U (1)] L and R ∈ [ SU (2) × U (1)] R . For a complete construction of theEFT also a spurion χ ( π ) for the quark mass transforming in the same way as χ ( µ ) would beneeded, but we will not need it for the counter terms discussed below.Ignoring the pions for the moment, the simplest counter term leading to the Pauli struc-ture (2.1) is¯ ψ L σ κλ χ ( µ ) r κλ ψ R + ¯ ψ R σ κλ r κλ χ ( µ ) † ψ L + ¯ ψ R σ κλ χ ( µ ) † (cid:96) κλ ψ L + ¯ ψ L σ κλ (cid:96) κλ χ ( µ ) ψ R . (2.6)At least one power of χ ( µ ) is needed, consistent with the fact that one factor m µ has toappear because of the helicity flip of the muon associated with its magnetic moment. Sincewe need the third power of the muon mass, two more spurion factors χ ( µ ) or χ ( µ ) † need tobe inserted. This can be done in various ways consistent with the symmetry (2.5), but theyall collapse to the same factor m µ once we set χ ( µ ) = m µ . One factor m µ in f ( q ) comes from the definition of a HVP µ , and not from diagram N0. We ignore the U (1) axial anomaly, because the corresponding source will be set equal to zero. U istr ( Q L U Q R U † ) , (2.7)where Q L and Q R appear because of the internal photon lines in diagram N0. MultiplyingEq. (2.6) with two more insertions of the spurion χ ( µ ) with Eq. (2.7), setting χ ( µ ) = m µ , (cid:96) κ and r κ equal to the values in Eq. (2.2) and Q L = Q R = Q , we obtain the counter term α m µ (4 πf π ) F κλ ψ µ σ κλ ψ µ tr (cid:0) Q L U Q R U † (cid:1) = α m µ (4 πf π ) F κλ ψ µ σ κλ ψ µ (cid:18) − f π π + π − + O ( π ) (cid:19) . (2.8)We multiplied with the factor 1 / (4 πf π ) to make this a dimension-four operator, with thescale 4 πf π standing in for the hadronic scale M H of the “heavy” hadrons that have beenintegrated out. The two powers of α reflect the presence of the two internal photon lines.The fact that the pion fields are traced over corresponds to the fact that they appear in aloop; if there were n pairs of charged pions, their contribution to a HVP µ would be n timeslarger. Furthermore, it is clear that the square of the quark charge matrix has to appearfrom the quark picture of the HVP.Setting the pion field π = 0 in Eq. (2.8), this counter term is of the form (2.1), andthus is the simplest example of an N5-type counter term, with its coefficient renormalizingthe overall divergence that can appear in the EFT calculation of Eq. (1.1). Since such adivergence appears for the first time at N LO, we expect this counter term to be of order1 / (4 πf π ) . Indeed, dimensional analysis leads to the appearance of this factor in Eq. (2.8),and the counter term thus takes a natural form. Of course, divergences also appear beyondN LO, and corresponding counter terms proportional to powers of 1 / (4 πf π ) larger than fourwill be needed as well. Such counter terms are easily constructed by inserting (covariant)derivatives and/or powers of the pion mass.The counter term (2.8) also produces a counter term of order 1 / (4 πf π ) with a photon,two muon and two pion external lines, and such a counter term leads to diagrams of typeN3 in Fig. 1. The power of 1 / (4 πf π ) thus suggests that this counter term will only beneeded at N LO. Moreover, only N3-type (and not N5-type) counter terms will contributeto pion-induced FV effects. This is consistent if the first UV-divergent FV effects requiringan N3-type counter term only appear at N LO, so that they are renormalized by the N3-type counter term in Eq. (2.8). As we will see in Sec. III, in order to produce a FV effect,a pion line needs to be cut. This makes it plausible that no UV-divergent FV effects occurat three-loop order, because cutting one pion line reduces the degree of divergence. While afull three-loop calculation is beyond the scope of this paper, in the next subsection we givea more quantitative argument supporting this conjecture.
C. An explicit example
A very simplified model for the “three-loop” HVP is given byΠ ( q ) = q f π (cid:90) d p p + m π p − q ) + m π . (2.9) Indeed, other charged mesons, such as the K + K − pair, do contribute, but their contribution is suppressedbecause of the larger mass of these mesons. O ( q log q ) and is divergent, and theseare the essential ingredients we need for our discussion. Regulating the integral as d p → dp p ( p /µ ) (cid:15) , thus introducing the unphysical scale µ ,and ignoring numerical factors from angle integrations, etc. , the result isΠ ( q ) = q f π (cid:90) dx (cid:18) − − (cid:15) − log x (1 − x ) q + m π µ (cid:19) = q f π (cid:32) − (cid:15) + 1 − log m π µ + (cid:112) m π + q q log (cid:112) m π + q − q (cid:112) m π + q + q (cid:33) . (2.10)Adding the “ChPT” counter termΠ CT ( q ) = q f π (cid:18) (cid:15) + (cid:96) ( µ ) − (cid:19) , (2.11)in order to subtract the divergence in Eq. (2.10), and allowing for an additional finite renor-malization q f π ( (cid:96) ( µ ) − ( q ) ≡ Π ( q ) + Π CT ( q ) (2.12)= q f π (cid:18) (cid:96) ( µ ) − (cid:90) dx log x (1 − x ) q + m π µ (cid:19) = q f π (cid:32) (cid:96) ( µ ) − log m π µ + (cid:112) m π + q q log (cid:112) m π + q − q (cid:112) m π + q + q (cid:33) . The renormalized quantity ˆΠ ( q ) is a physical quantity, and thus should not depend on theunphysical scale µ . Therefore, in this simple example the running of the renormalized LEC (cid:96) ( µ ) satisfies µ ddµ (cid:96) ( µ ) = − . (2.13)Since the model of Eq. (2.9) is not UV complete, we cannot determine the dependence of (cid:96) ( µ ) on the UV physics that has been integrated out, and the running of (cid:96) ( µ ) in (2.13) isall we can know. In Sec. III we will study a simple model which is UV complete.As a model for the contribution to the muon anomaly from ˆΠ ( q ) we will choose themodel “anomaly” a to be given by the integral ( m ≡ m µ ) a = m (cid:90) d n qq + m q ˆΠ ( q ) (cid:18) M q + M (cid:19) , (2.14)where 1 /q represents the combination of photon propagators and other kinematical factorsand 1 / ( q + m ) regulates the IR divergence ( i.e. , it gives the dependence on the muon massin our simplified representation). The factor M / ( q + M ) has been inserted to make the Of course, at three loops more complicated logarithmic corrections to the q behavior can appear, but webelieve this is not essential to the point we wish to make. integral in Eq. (2.14) finite, and “stands in” for the non-pionic hadron physics of QCD.One might think of it as the insertion of a fake ρ propagator, but this is not essential: theonly job of this factor is to regulate the UV divergence of the q integral. As we will see, itwill allow us to determine the form of the counter term needed to make a finite without thisfactor.As it stands, the integral (2.14) is finite and well behaved for d = 4. The “ChPT” versionof a is obtained by setting M / ( q + M ) → i.e. , sending M → ∞ , but of course, thatreintroduces the UV divergence of the q integral.In order to proceed, we split M q + M = 1 − q q + M , (2.15)and using the expression for ˆΠ ( q ) in Eq. (2.12) one may split the contributions to a as a = a EFT + a CT = m f π (cid:90) ∞ dq q + m (cid:18) q µ (cid:19) (cid:15) (cid:18) (cid:96) ( µ ) − (cid:90) dx log q x (1 − x ) + m π µ (cid:19) (2.16a) − m f π (cid:90) ∞ dq q + m (cid:124)(cid:123)(cid:122)(cid:125) neglect (cid:18) q µ (cid:19) (cid:15) q q + M (cid:18) (cid:96) ( µ ) − (cid:90) dx log q x (1 − x ) + m π µ (cid:19) , (2.16b)where, as indicated, the muon mass m may be neglected in the second integral, a CT , onaccount of the extra q in the numerator. As we will see, this also allows the limit m π → a CT independent of the IR scales m and m π , as expected for a counterterm.The integrals yield cumbersome expressions for m (cid:54) = m π . This is why we will simplifyour example by taking m = m π as a common low-energy scale, m = m π (cid:28) M , in the restof this section. This simplification only serves to simplify the math and is not essential, ofcourse.Setting m π = m , the result for a EFT can be obtained by evaluating the integrals inEq. (2.16a), a EFT = m f π (cid:20) (cid:96) ( µ ) (cid:18) − (cid:15) − log m µ (cid:19) − (cid:15) − (cid:15) − m µ + 12 log m µ + 227 π + C (cid:21) , (2.17)where the first (second) line corresponds to the first (second) term in the integrand ofEq. (2.16a), and C is a constant given by C = − ψ (cid:48) (cid:18) (cid:19) − ψ (cid:48) (cid:18) (cid:19) + 19 ψ (cid:48) (cid:18) (cid:19) + 118 ψ (cid:48) (cid:18) (cid:19) , (2.18)where ψ (cid:48) ( x ) = dψ ( x ) dx with ψ ( x ) is the digamma function ψ ( x ) = Γ (cid:48) ( x )Γ( x ) .11e next turn to a CT in Eq. (2.16b). The result of the integral is a CT = − m f π (cid:20) (cid:96) ( µ ) (cid:18) − (cid:15) − log M µ (cid:19) − (cid:15) − (cid:15) − M µ + 12 log M µ + π O (cid:18) m π M log m π M (cid:19)(cid:21) , (2.19)where again the first (second) line corresponds to the first (second) term in the inte-grand of Eq. (2.16b). We emphasize that for the second line, i.e. , the term with thelog ( q x (1 − x ) + m π ), the limit m π → m π . In the third line of Eq. (2.19) we have kept m π explicit for illustration, but thiswhole term is O (cid:16) m π M (cid:17) and therefore is to be neglected at the order we consider. In App. Bwe show how to calculate it from Eq. (2.16b).Adding a EFT and a CT in Eqs. (2.17) and (2.19) we finally obtain the result a = m f π (cid:18) (cid:96) ( µ ) log M m + 2 log M m + 12 log M m log µ m M − π + C (cid:19) , (2.20)where C is given in Eq. (2.18). Note how the condition (2.13) for the running of (cid:96) ( µ ) makesour result independent of µ .The result for a CT in Eq. (2.19) admits an expansion in powers of m π /M . To lowestorder in this expansion, there are no logarithms such as log m π /M , which could only comefrom a pion loop. Therefore, the conclusion is that the leading order is given by an N5-typePauli-like operator, without any pion loops. Thus at leading order (which, recall, in the realworld corresponds to three-loop order), only an N5-type counter term is needed. It is onlyat the next order in the chiral expansion, i.e. , O ( m π /M × m µ /f π ), that these logarithmsappear: this is when tadpole diagrams of type N3 will start to contribute. D. The example in finite volume
Let us now extend the discussion of our example to have a first look at FV effects comingfrom the pions. We thus want to consider the integral D ( q ) = (cid:90) ∞−∞ dp π (cid:90) d p (2 π ) p + p + m π p − q ) + p + m π (2.21)in a finite spatial volume V = L of linear dimension L , with periodic boundary conditions.Here we took q = (0 , , , q ) to point in the 4-direction, without loss of generality, as thiswill be convenient in our explicit calculations. In finite volume, the integral over (cid:126)p is replacedby a sum, and in finite volume D becomes D FV ( q ) = (cid:90) ∞−∞ dp π L (cid:88) (cid:126)p =2 π(cid:126)n/L p + p + m π p − q ) + p + m π , (2.22) Dependence on m π would be provided by higher-order counter terms. (cid:126)n has integer components. In order to isolate the FV effects, we will use Poissonresummation,1 L (cid:88) (cid:126)p f ( (cid:126)p ) = (cid:88) (cid:126)n (cid:90) d (cid:126)p (2 π ) e i(cid:126)n · (cid:126)p L f ( (cid:126)p ) = (cid:88) (cid:126)n (cid:18) iπ nL (cid:19) (cid:90) ∞−∞ dp p f ( p ) e inpL , (2.23)in which n = | (cid:126)n | . In position space, the vector (cid:126)n represents the number of times thepion wraps around the periodic spatial volume in each direction. Thus, the term with n = 0 corresponds to the infinite-volume part, and the n > D FV ( q ) = (cid:88) (cid:126)n D ( n ) ( q ) , (2.24) D ( n ) ( q ) = (cid:90) ∞−∞ dp π (cid:90) ∞−∞ dp π p p + p + m π p − q ) + p + m π e inpL inL = 18 π q nL (cid:90) ∞ dp p (cid:16) e − nL √ ( p − q / + m π − e − nL √ ( p + q / + m π (cid:17) , where we used contour integration to evaluate the integral over p , and we shifted p → p + q /
2. For large m π L , the integral over p is (exponentially) dominated by the region( p ± q / (cid:28) m π , allowing us to expand the square roots for large m π . For large m π L and n > D ( n ) ( q ) ≈ π q nL e − m π nL (cid:90) ∞ dp p (cid:18) e − nLmπ p − q / m π − ( q → − q ) (cid:19) = 18 π q nL e − m π nL π e − q nL mπ Erfi (cid:115) q nL m π , (2.25)where Erfi( x ) is the imaginary error function erf( ix ) /i , which is real for real x . Using the asymptotic expansionErfi( x ) ≈ e x (cid:18) √ πx + 12 √ πx + 34 √ πx + ... (cid:19) , ( x → ∞ ) , (2.26)we obtain, for n > D ( n ) ( q ) ≈ √ π π e − m π nL ( q nL ) (cid:112) m π nL (cid:18) O (cid:18) m π nL ; m π nL ( q nL ) (cid:19)(cid:19) . (2.27)We recall that Π ( q ) = q f π D ( q ), cf. Eq. (2.9). Consequently, we see here an exampleof what we anticipated earlier in this section about the N LO contribution to a HVP µ . Theinsertion of ˆΠ ( q ) into the a integral, as in Eq. (2.14), leads to a divergence in infinite volumewhich is renormalized by the corresponding counter term, cf. Eqs. (2.16a), (2.16b). On the In order to carry out the integral in Eq. (2.25), we regulated the pole at p = 0 by introducing a factor p (cid:15) in each term, then combined both terms and only at the end took (cid:15) → IG. 2: Feynman diagram for the quantity a defined in Eq. (3.2). other hand, the finite-volume contribution, Eq. (2.27), when inserted into the integral (2.14)for a is UV finite even for M → ∞ . Intuitively, this can be understood as follows. To obtain the leading FV correction, wetake the one pion loop in any diagram to wrap “around the world.” Such a pion can be seenas an on-shell pion (as we will show explicitly in Sec. III). Effectively, one thus “removes”a loop and a pion propagator, decreasing the degree of divergence by (at least) two.However, as we have discussed above, we expect that FV effects in the real world willlead to a divergent a HVP µ integral at N LO. Therefore, in the next section, we will study aneven simpler example which, while keeping integrals at an elementary level, does lead todivergent FV effects and allows us to illustrate the interplay of these FV effects with thecorresponding UV counter terms which appear at infinite volume.
III. A TOY MODEL
Our goal is to investigate the interplay between UV divergences, renormalization, and FVeffects in more detail. As we have argued, this interplay shows up only in full force at threeand four loops in the case of a HVP µ . Therefore, in this section, we will study this interplayin a very simple model, in which we do not have to go beyond two loops in order to seethis interplay at work, and in which details have been kept simple enough to make explicitcalculations feasible.We will define the model in Sec. III A, where we explain how keeping things simple ledus to consider this model. Then, in Sec. III B, we will essentially repeat the analysis ofSec. II for the model, in infinite volume. In Sec. III C we will demonstrate how, once theinfinite-volume counter terms have been identified, FV corrections due to the “pions” in ourmodel are always UV finite, and thus well defined. Equation (2.27) is only valid for large q L . This is sufficient for our argument, as it is the large- q regionof the integral over q in Eq. (2.14) that leadsto a UV divergence. . Definition of the model We wish to construct a simple toy model with pions, muons, photons, and some “heavystrong-interaction” physics in which a quantity a analogous to a µ can be defined. While wereally just need an integral like the one in Eq. (1.1), it is instructive to cast the model interms of a lagrangian and Feynman rules obtained from the lagrangian. This will allow fora diagrammatic analysis analogous to that based on Fig. 1.The lagrangian for our model is L = 12 ( ∂ κ π ) + 12 m π π + 12 ( ∂ κ σ ) + 12 m σ σ (3.1)+ 12 ( ∂ κ ψ µ ) + 12 m µ ψ µ + 12 ( ∂ κ A ) + 12 e Aπ + 12 e Aψ µ + e Aσπ + 12 gψ µ σπ . All fields are scalars, but we can intuitively think of the massless scalar A as a “photon,”the scalar ψ µ as a “muon,” while of course π is our “pion.” The strong-interaction physicsis represented by the massive scalar σ , and we will thus always think of m σ as much largerthan m π and m µ .In this model, we define a “muon anomaly” by a = e e e g (cid:90) ∞ dq m µ m µ + q Π( q ) , (3.2)where, as in Eq. (1.1), q > q , and a possible symmetry factor. In Secs. III B and III Cwe will omit the factor e e e g as well. The function Π( q ) is given byΠ( q ) = (cid:90) d p (2 π ) p + m σ ) 1[( p − q ) + m π ] , (3.3)which can be seen by traversing around the hadronic loop in the diagram depicted in Fig. 2.We emphasize that this model for Π( q ) has been chosen to be UV finite without any furthersubtraction, unlike Eq. (1.2). In particular, this means that also Π(0) is finite in the model.We could have chosen a model with Π( q ) defined by Eq. (3.3) without the square on the pionpropagator, in which case Π( q ) would have been logarithmically divergent. In that case,we would have considered the UV-finite difference Π( q ) − Π(0), as in Eq. (1.1). However,this would make the mathematical treatment of the model more cumbersome, and it is notessential, as we will see next. It is thus important to keep in mind, for the rest of this section,that Π( q ) itself is finite, and does not need to be subtracted, unlike in the real world.The diagram for the “anomaly” a is shown in Fig. 2. In detail, it looks rather differentfrom diagram N0 in Fig. 1. But, it shares the following essential properties with a HVP µ . All similarity with the linear sigma model is purely coincidental. All couplings in Eq. (3.1) except g have mass dimension one. In fact, it bears some similarity to the hadronic light-by-light contribution to a µ , with the external photonline attached to the pion. IG. 3: a HVP µ for the toy model. First, a is a UV-finite quantity, just like a HVP µ , and it is an integral over a weight functiontimes a hadronic loop, Π( q ), which itself is finite, just like ˆΠ( q ) in Eq. (1.1). However,as can be seen from Fig. 2, if we “integrate out” the σ , i.e. , we contract its propagatorto a point by replacing 1 / ( p + m σ ) → /m σ in Eq. (3.3), both the pion loop and themuon-photon loop become logarithmically divergent. This can be seen in diagram T2 ofFig. 3. This implies that in an EFT containing only the pions, photons and muons, whilethe σ has been integrated out, counter terms will need to be introduced to renormalize thesedivergences. Counter terms will be needed for the pion-loop subdivergence (diagram T4),for the muon-photon-loop subdivergence (diagram T3), and the overall two-loop divergence(diagram T5). The precise form of these counter terms will be derived in Sec. III B below.The example of the quantity a in the toy model thus mimics the situation that arisesat N LO in the case of a HVP µ . At N LO in ChPT not only are counter terms needed torenormalize UV divergences in the HVP, but also the integral over q (the momentumthrough the muon and photon lines in Eq. (1.1)) which becomes divergent and leads tothe counter terms discussed in Sec. II. Because of the construction of our model, and16he definition of the quantity a , the same phenomena happen here already at the lowestpossible number of loops. Since we want to study in our model both “strong interaction”counter terms of type N4 and “electromagnetic” counter terms of type N3 and N5, we needa subdivergence from a pion loop, and a subdivergence from a loop containing a muon, andboth are present in diagram T2 of Fig. 3, which represents the EFT version of the diagramin Fig. 2 obtained by contracting the σ propagator to a point. The detailed form of Π( q )makes the example somewhat contrived, and leads to the detailed form of the diagram inFig. 2 and thus the diagrams in Fig. 3 to be different from the diagrams in Fig. 1. Weemphasize that this is not important; what is important is the fact that the model generatesboth types of counter terms already at two loops. In the next subsection, we will carry outthe integrals, transition to the EFT, and fully elucidate the counter-term structure in thetoy model.We end this subsection with a comment. In the model (3.1), there are other contributionsto the quantity a , i.e. , other diagrams with an external photon and two external muon lines.In particular, a contribution exists with one µ σπ and one Aσπ vertex, proportional to e g ,and this contribution is UV divergent. But this will not affect our discussion of the diagramin Fig. 2. While the EFT version of the simpler one-loop contribution can be analyzedwith the same method as we will use below to analyze the integral in Eq. (3.2), it does notexhibit the parallel behavior with a HVP µ we are after. We will thus ignore this contribution,and define a for the rest of this section by the integral in Eq. (3.2). B. The toy model in infinite volume
We begin with the calculation of a in the complete theory, Eq. (3.1). Using Feynmanparametrization of the integral in (3.3), we find (omitting, from now on, the couplings e , , and g ) Π( q ) = 116 π (cid:90) dx xq x (1 − x ) + m σ (1 − x ) + m π x (3.4)= 116 π (cid:18) m σ + q log m σ m π + m σ − q q ( m σ + q ) log m σ m σ + q (cid:19) + O (cid:18) m π m σ (cid:19) . We emphasize again that Π(0) is finite, as can be seen by setting q = 0 in Eq. (3.4). Nosubtraction is needed. In fact, we note that, to leading order in an expansion in 1 /m σ ,Π( q ) = Π(0). As indicated in Eq. (3.4), we will work to leading order in m π /m σ , as thiswill be sufficient for our purposes. Inserting this into the integral defining a in Eq. (3.2), wefind for a the explicit result a = m µ π m σ (cid:20)(cid:18) − log m σ m π (cid:19) log m µ m σ − π O (cid:18) m µ , m π m σ (cid:19)(cid:21) . (3.5)This is the result that an EFT analysis of our model is expected to reproduce, and thisanalysis is what we will turn to next. One way to do the calculation is to first do perform the integral over q using the Feynman-integralrepresentation for Π( q ) of Eq. (3.4) and then carry out the integral over x . σ field, which amounts toreplacing every σ propagator in the theory by 1 /m σ , thus making the exchange of a σ meson into a point vertex. This corresponds to splitting the σ propagator in Eq. (3.3) as1 p + m σ = 1 m σ − p m σ ( p + m σ ) . (3.6)The EFT contribution corresponding to the first term on the right-hand side is depicted indiagram T2 of Fig. 3. To begin with, this replacement makes Π( q ) divergent, and a counterterm renormalizing this divergence will thus have to be introduced in the EFT. This counterterm should of course reproduce the contribution from the second term on the right-handside of Eq. (3.6), in an expansion in inverse powers of m σ . The replacement also gives riseto the divergent muon-photon loop in diagram T2; we will return to this divergence below.As a side comment, one could study also this model at higher orders in m π /m σ and m µ /m σ by further expanding the second term on the right-hand side of Eq. (3.6), using − p / ( m σ ( p + m σ )) = − p /m σ + O ( p /m σ ). But since our goal is to keep things mathematically as simpleas possible, we will only consider the expansion in inverse powers of m σ to leading order, inthe rest of this paper.Replacing d p → µ (cid:15) d d p with d = 4 − (cid:15) in Eq. (3.3) allows us to calculate the EFT partof Π( q ), and we findΠ EFT ( q ) = 116 π m σ (cid:18) (cid:15) − γ E − log m π πµ (cid:19) = Π EFT (0) . (3.7)We note that this result reproduces the − log m π / (16 π m σ ) of Eq. (3.4). The toy model isthe UV completion of our EFT, and thus the second term in Eq. (3.6) should lead us to theform of the counter term. At q = 0 we thus find the counter-term contribution by replacingthe σ propagator in Eq. (3.3) by this second term:Π CT (0) = − π m σ (cid:18) (cid:15) − γ E − log m σ πµ + 1 + O (cid:18) m π m σ (cid:19)(cid:19) . (3.8)The sum Π EFT (0) + Π CT (0) reproduces Π(0):Π(0) = 116 π m σ (cid:18) log m σ m π − O (cid:18) m π m σ (cid:19)(cid:19) . (3.9)Of course, if we do not know the underlying UV completion, we cannot calculate thecontribution (3.8), as the precise form of the second term in Eq. (3.6) is not known if we onlyhave the EFT. But, the divergent and logarithmic terms in Π(0) CT can be inferred from thoseof Π(0) EFT and dimensional analysis, as the sum of the two has to be finite and independentof µ . This is precisely how a counter term is introduced in the EFT, in order to absorb thedivergence and the scale dependence that shows up in the calculation of Eq. (3.7). Thisdoes not determine the remaining finite part (here the term ( − γ E − log 4 π ) / (16 π m σ ))of the counter term. If only the EFT is known, the finite part gets replaced by an unknownfinite constant of the right dimension: the value of the renormalized LEC in a particularregularization scheme (here, dimensional regularization with minimal subtraction). Thecounter term discussed here takes the form T C ψ µ A ( µ )4 ψ µ A , (3.10)18nd corresponds to the black square in diagram T4 in Fig. 3.We now return to the EFT calculation of a in Eq. (3.2). Diagram T2 has more divergencesfor which counter terms need to be introduced, and we aim to identify those from splitting a into a low-energy part a and a counter-term part a , following the same reasoning as abovefor Π( q ): a = (cid:90) ∞ dq (cid:18) q µ (cid:19) (cid:15) m µ q + m µ (cid:8) Π EFT (0) + Π CT (0) + (cid:2) Π( q ) − Π(0) (cid:3)(cid:9) (3.11)= a + a = m µ (cid:90) ∞ dq q + m µ (cid:18) q µ (cid:19) (cid:15) { Π EFT (0) + Π CT (0) } (cid:124) (cid:123)(cid:122) (cid:125) a + m µ (cid:90) ∞ dq q + m µ (cid:124)(cid:123)(cid:122)(cid:125) neglect (cid:18) q µ (cid:19) (cid:15) (cid:8) Π( q ) − Π(0) (cid:9)(cid:124) (cid:123)(cid:122) (cid:125) a . Since we want to split a into two divergent parts, a and a , we introduced a regulatingfactor ( q /µ ) (cid:15) , necessary to define each of the terms a and a separately; of course, a itself is finite in the limit (cid:15) →
0. We emphasize that this regulator is independent of theregulator used in the calculation of Π
EFT (0) and Π CT (0) in Eqs. (3.7) and (3.8) (the sumof which is finite); the use of dimensional regularization again here is just for calculationalsimplicity. The m µ in the denominator of the integral defining a may be neglected becauseΠ( q ) − Π(0) ∼ q as q →
0. Indeed, while a will depend on m µ logarithmically, a isanalytic in m µ to O ( m µ ), which is the order to which we have constructed the EFT. Thecontribution a is what one would obtain using the EFT calculation of Π( q ) (which in thisexample we carried out to lowest order, i.e. , we obtained an EFT representation of Π(0),including the counter term contribution Π CT (0)). However, the integral over q defining a is itself divergent, and we thus will need to add new counter terms, which are only definedin the theory containing not only the pion, but also the photon and muon as dynamicalfields, with the integral over q representing the photon-muon loop. Of course, since in ourmodel we have the exact expression (3.2), we know what a is exactly, but we will see howit corresponds to counter terms in the pion-muon-photon EFT below.A direct calculation of the first term in Eq. (3.11) gives a = m µ { Π EFT (0) + Π CT (0) } (cid:18) − (cid:15) − log m µ µ (cid:19) , (3.12)showing explicitly that this contribution is divergent; this divergence comes from the muon-photon loop in diagram T2. A direct calculation of the second term yields a = m µ { Π EF T (0) + Π CT (0) } (cid:18) (cid:15) + log m σ µ (cid:19) + (cid:18) m µ π m σ (cid:19) (cid:18) − π (cid:19) , (3.13) It can be shown that counter terms are analytic in the low-energy scales order by order in the EFTexpansion. One way to do the calculation is to start from the first line of Eq. (3.4) for Π( q ), subtract Π(0), carryout the integral over q in Eq. (3.11), and finally the integral over x in Eq. (3.4).
19o that the total result is a = a + a = m µ Π(0) log m σ m µ + (cid:18) m µ π m σ (cid:19) (cid:18) − π (cid:19) . (3.14)Using Eq. (3.9) this equals the complete result, Eq. (3.5).Equation (3.13) is the result in the underlying theory, which we do not know if do notknow the UV completion of the EFT. However, the divergence in a demands that we addcounter terms to the EFT that absorbs this divergence, yielding the contribution shown asthe first term in Eq. (3.13). Again, the divergent and logarithmic terms in Eq. (3.13) can beinferred from those in Eq. (3.12) and dimensional analysis as the sum has to be finite andindependent of µ . The second term in Eq. (3.13) again corresponds to a finite contribution,which is unknown if we only have access to the EFT, and will thus be represented in theEFT by renormalized LECs.In order to disentangle the complete counter-term structure, we split the different con-tributions as: a = m µ Π EFT (0) (cid:18) − (cid:15) − log m µ µ (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) (1) (3.15)+ m µ Π CT (0) (cid:18) − (cid:15) − log m µ µ (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) (2) + m µ Π EFT (0) (cid:18) (cid:15) + log m σ µ (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) (3) + m µ Π CT (0) (cid:18) (cid:15) + log m σ µ (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) (4) + m µ π m σ (cid:18) − π (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) (5) , a = m µ Π EFT (0) (cid:18) − (cid:15) − log m µ µ (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) (1) (3.16)+ m µ C ψ µ A ( µ ) (cid:18) − (cid:15) − log m µ µ (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) (2) + m µ C ψ µ π ( µ ) (cid:18) − (cid:15) + γ E + log m π πµ (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) (3) + m µ C ψ µ A ( µ ) (cid:124) (cid:123)(cid:122) (cid:125) (4)+(5) , (3.17)where C ψ µ A ( µ ) = − π m σ (cid:18) (cid:15) − γ E − log m σ πµ + 1 (cid:19) , (3.18a) C ψ µ π ( µ ) = − π m σ (cid:18) (cid:15) + log m σ µ (cid:19) , (3.18b) C ψ µ A ( µ ) = − π m σ (cid:18) (cid:15) − γ E − log m σ πµ + 1 (cid:19) (cid:18) (cid:15) + log m σ µ (cid:19) (3.18c)+ 116 π m σ (cid:18) − π (cid:19) , Note the presence of terms proportional to 1 /(cid:15) , as expected at two loops. where the T4-typeLEC C µ A ( µ ) already appeared in Eq. (3.10), and we encounter the new counter terms T C ψ µ π ( µ )4 ψ µ π , (3.18d) T C ψ µ A ( µ )2 ψ µ A .a (1) corresponds to the diagram T2 in Fig. 3. This diagram has two subdivergences, eachneeding a counter term, one for the pion loop and one for the muon-photon loop, leading tocounter term vertices ψ µ A and ψ µ π , respectively. The first vertex, ψ µ A , is obtained bycontracting the pion loop to a point, and was already encountered in the EFT calculationof Π( q ); it corresponds to diagram T4, and leads to a (2) . The second vertex, ψ µ π , is newand is obtained by contracting the muon-photon loop to a point; note that this counterterm involves both the pions of the “strong interactions” and the muons of “QED.” Thisnew counter term corresponds to diagram T3, and leads to a (3) . An overall two-loop diver-gence corresponding to the 1 /(cid:15) pole requires a new counter term proportional to ψ µ A . It In a minimal subtraction scheme, one drops the combinations 1 /(cid:15) + finite constant to obtain the renor-malized LECs. a (4) ; this counter termcan also have (and does have, in the UV completion provided by our model) a finite part, a (5) . This latter counter term is to be compared to the Pauli term in the case of a HVP µ . Theappearance of 1 /(cid:15) poles is consistent with Eq. (3.2) being a two-loop integral. Again, thespecific result for a (5) is only known because in this example we know the underlying theory;if we only had access to the EFT, the factor − π / m π /m σ and m µ /m σ . C. The toy model in finite volume
We now turn to the analysis of the interplay between FV effects and UV divergences,our primary reason for introducing the toy model (3.1) in the first place. We begin withcalculating FV effects in the model itself, and then calculate and compare them with thecalculation in the EFT developed in the previous subsection.It is convenient to use the Schwinger parametrization for Π( q ) given byΠ( q ) = (cid:90) d p (2 π ) (cid:90) ∞ dα α (cid:90) ∞ dα e − α ( p + m σ ) e − α ( ( p − q ) + m π ) . (3.19)Making the change of variablesˆ x = α α + α , x = α + α ←→ α = x ˆ x , α = x (1 − ˆ x ) , (3.20)one can rewrite Π( q ) in a finite spatial volume with linear dimension L and periodic bound-ary conditions as:Π FV ( q ) = (cid:88) (cid:126)n Π ( n ) ( q ) (3.21)Π ( n ) ( q ) = (cid:90) d ˆ x ˆ x (cid:90) ∞ dx x e − xN (cid:90) ∞−∞ dp π p (cid:90) ∞−∞ dp π e − x ( p + p ) e inpL inL , where N = q ˆ x (1 − ˆ x ) + N , (3.22) N = m σ (1 − ˆ x ) + m π ˆ x , and we made use of the Poisson resummation (2.23). As we are interested here in the FVcontributions, we will take n >
0. We took (cid:126)q = 0, as in Sec. II D and shifted p − xq → p ,and q = q >
0. Carrying out the integrals over p and p yieldsΠ ( n ) ( q ) = 116 π (cid:90) d ˆ x ˆ x (cid:90) ∞ dx e − x q ˆ x (1 − ˆ x ) e − x N e − n L / (4 x ) . (3.23)We can also write a FV = (cid:88) (cid:126)n a ( n ) , (3.24)22o that, performing the integrals over q in Eq. (3.2) and x in Eq. (3.23), we find a ( n ) = − m µ π (cid:90) d ˆ x ˆ xN (cid:18) K ( N nL ) + N nLK ( N nL ) (cid:18) γ E + log m µ n L ˆ x (1 − ˆ x )2 N nL (cid:19)(cid:19) , (3.25)where K ν ( z ) is the modified Bessel function of order ν . Carrying out the integral over ˆ x (see App. B for details), we find for our final result: a ( n ) = m µ π m σ K ( m π nL ) (cid:124) (cid:123)(cid:122) (cid:125) Π ( n )EFT (0) log m σ m µ . (3.26)Here we dropped terms ∼ e − m σ nL and terms that are suppressed by additional powersof m π /m σ or 1 / ( m σ L ). In Eq. (3.26), we already anticipate the result to be derived inEq. (3.27) below, that the prefactor of the logarithm is nothing but m µ Π ( n )EFT (0) in finitevolume. This result conforms with the intuition that FV effects associated with the pionsare infrared effects due to the low-energy degrees of freedom contained in the EFT. As the σ is not part of the EFT, its FV effects cannot be obtained from the EFT.We now turn to the calculation of FV effects in the EFT version of our toy model,beginning with Π EFT (0). Defining Π ( n )EFT analogous to the definition of D ( n ) in Eq. (2.24),we obtain for n > ( n )EFT (0) = 1 m σ (cid:90) ∞−∞ dp π (cid:90) ∞−∞ dp π p ( p + p + m π ) e ipnL inL . (3.27)= 18 π m σ K ( m π nL )= 18 π m σ (cid:114) π e − m π nL √ m π nL (cid:18) O (cid:18) m π nL (cid:19)(cid:19) , where we evaluated the integral over p using the residue theorem, and then the integral over p to yield the modified Bessel function.There is another instructive way to obtain the result in Eq. (3.27). This is by noting thatthe case n > ( n )EFT (0) = (cid:18) − im σ (cid:19) (cid:90) ∞−∞ dp π (cid:90) ∞−∞ dp π p ( p − p − m π + i(cid:15) ) e inpL inL . (3.28)Putting the pion in the loop on shell amounts to the following replacement1( p − p − m π + i(cid:15) ) = 12 p ddp (cid:18) p − p − m π + i(cid:15) (cid:19) → − iπ p ddp δ ( − p + p + m π ) , (3.29)so that Eq. (3.28) becomesΠ ( n )EFT (0) = − m σ (cid:18) π (cid:19) (cid:90) ∞−∞ dp (cid:90) ∞−∞ dp ddp (cid:2) δ ( p − p − m π ) (cid:3) e inpL inL = 1 m σ π (cid:90) ∞ m π dp (cid:112) p − m π cos (cid:18) nL (cid:113) p − m π (cid:19) . (3.30)23 change of variable y = (cid:112) p − m π finally yieldsΠ ( n )EFT (0) = 1 m σ π (cid:90) ∞−∞ dy (cid:112) y + m π e inLy , (3.31)which equals Eq. (3.27). We note that the result for Π ( n )EFT (0) for n > CT (0) was only needed in the infinite-volume theory. It does not involve a pion loop,so there is no contribution to the FV part of Π(0) in the EFT. In other words, Π CT (0)renormalizes Π EFT (0) in infinite volume, while FV corrections to Π
EFT (0) are finite.The implication of this is that we expect the FV part of Π CT (0) to vanish. Since wecalculated Π ( n )EFT (0) to order 1 /m σ we thus expect that if we calculate Π ( n )CT (0) for n > CT (0) from the model in Eq. (3.8) it will turn out not tohave a term of order 1 /m σ , implying that the FV corrections to Eq. (3.8) vanish. This iswhat we will demonstrate next.Replacing the σ propagator in Eq. (3.3) by the second term of Eq. (3.6), and again goingthrough the steps to arrive at an expression for the FV correction term Π ( n )CT (0) for n > ( n )CT (0) = − m σ (cid:90) ∞−∞ dp π (cid:90) ∞−∞ dp π p ( p + p + m π ) (cid:20) p + p p + p + m σ (cid:21) e ipnL inL . (3.32)In addition to the double poles at p = ± i (cid:112) p + m π also present in Eq. (3.27), the integrandnow also has simple poles at p = ± i (cid:112) p + m σ . The pole at p = i (cid:112) p + m σ gives a con-tribution ∼ e − L √ p + m σ which, however, is negligible when m σ (cid:29) m π , in comparison to thecontribution ∼ e − L √ p + m π from the pole at p = i (cid:112) p + m π , and may thus be discarded. Furthermore, we can immediately see why the remaining part of Π ( n )CT (0) is suppressed rela-tive to Π ( n )EFT (0) by an extra factor m π /m σ . The suppression by this factor follows from thepresence of the term in square brackets in Eq. (3.32) and dimensional analysis. The factor e − L √ p + m π present after integration over p makes the integral over p sufficiently convergentthat the rest of the integrand can be expanded in inverse powers of m σ . We conclude that,to the order we are working, indeed, no counter-term FV correction is produced by theunderlying model. We note that, in this argument, we always assume that m π L (cid:29) i.e. ,that we consider FV effects in the p -regime.To summarize, to leading order in an expansion in m π /m σ , we conclude that, for n > ( n )CT (0) = 0 ⇒ Π ( n ) (0) = Π ( n )EFT (0) . (3.33)No counter term is needed for the FV corrections to Π(0) calculated using the EFT.Now, let us return to our “anomaly,” a defined in Eq. (3.2). First, substituting Eq. (3.27) This reflects the fact that light particles dominate the FV effects in the EFT for our toy model. a in the EFT, a ( n )EFT = (cid:90) ∞ dq (cid:18) q µ (cid:19) (cid:15) m µ q + m µ Π ( n )EFT (0)= m µ Π ( n )EFT (0) (cid:18) − (cid:15) − log m µ (cid:19) = m µ Π ( n ) (0) (cid:18) − (cid:15) − log m µ µ (cid:19) . (3.34)This corresponds to the FV part of a defined in Eq. (3.11): it is the FV contribution fromthe pion loop in diagram T2 in Fig. 3. Clearly, a counter-term contribution will be needed tomake this FV contribution finite. The counter term needed here is the same infinite-volumecounter term used in diagram T3 of Fig. 3, with the pion loop in that diagram producingthe FV part. Diagrams T4 and T5 do not contribute to the pion-induced FV corrections(T4 would only yield FV contributions due to a muon wrapping around the world). Weemphasize how it is consistent to keep the q integral in Eq. (3.11) in infinite volume, eventhough Π(0) itself is replaced by its FV part. In other words, we can treat the “strong-interaction” ( i.e. , π, σ ) physics in finite volume, while keeping the “electromagnetic” ( i.e. , A, µ ) physics in infinite volume.Finally, as before, we need to consider a given by ( cf. Eq. (3.11)) a = m µ (cid:90) ∞ dq q + m µ (cid:124)(cid:123)(cid:122)(cid:125) neglect (cid:18) q µ (cid:19) (cid:15) (cid:8) Π( q ) − Π(0) (cid:9) , (3.35)but now replacing the pion-physics part, Π( Q ) − Π(0), by its FV correction. As before, the m µ in the denominator can be neglected because Π( q ) − Π(0) ∼ q as q →
0. The termproportional to Π(0) in Eq. (3.35) then vanishes in dimensional regularization, so the netresult becomes a ( n )2 = m µ (cid:90) ∞ dq q (cid:18) q µ (cid:19) (cid:15) Π ( n ) ( q ) . (3.36)Inserting Eq. (3.23) and using the method of App. B that was used to evaluate Eq. (3.25), we obtain a ( n )2 = m µ Π ( n ) (0) (cid:18) (cid:15) + log m σ µ (cid:19) . (3.37)An important remark here is that, even though the end result (3.37) is proportional toΠ ( n ) (0), the q dependence of Π ( n ) ( q ) in the integrand in Eq. (3.36) is crucial for obtainingthis result. The factor 1 /(cid:15) + log( m σ /µ ) is the same as that appearing in a (3) in infinitevolume, on the third and fourth line of Eq. (3.15). Thus, adding Eqs. (3.34) to (3.37), onearrives at a ( n ) = a ( n )1 + a ( n )2 = m µ Π ( n ) (0) log m σ m µ , (3.38)where Π ( n ) (0) = Π ( n )EFT (0) and Π ( n )EFT (0) is given in Eq. (3.27). This is equal to the completeresult (3.26) calculated in the underlying toy model. In summary, the volume dependence For the 1 /(cid:15) part, all that is needed is K ( y ) = − dK ( y ) /dy .
25s contained in the contribution from the EFT, i.e. , Π ( n ) (0), but the infinite-volume counterterm with coefficient C ψ µ π ( µ ) given in Eq. (3.18b) is needed to render the FV contributionfinite. The contributions a (4) and a (5) in Eq. (3.15) do not contribute pion-induced FVeffects, as the corresponding diagrams T4 and T5 in Fig. 3 do not contain pion loops. IV. CONCLUSION
In this paper, we considered the effective field theory approach to the hadronic vacuumpolarization contribution to the muon anomalous magnetic moment, a HVP µ . Our primaryaim was a better understanding of how the evaluation of finite-volume effects works in anEFT framework, but this led us to a discussion of the counter-term structure needed for acomplete EFT representation of a HVP µ in infinite volume. The specific motivation for ourinterest in finite-volume effects is to contribute to a deeper understanding of the systematicerror caused by these effects in lattice computations of a HVP µ .Finite-volume effects in lattice computations of a HVP µ are dominated by pions, and thusthe natural EFT candidate is chiral perturbation theory. As we showed, for a completelow-energy description of a HVP µ , also muons and photons have to be included in the EFT. Inparticular, since a HVP µ also contains a loop with muons and photons, counter terms in ourEFT can contain not only pion fields, but also muon and photon fields. Just like counterterms are needed to regulate the UV behavior of pion loops in ChPT, additional counterterms are also needed to regulate the UV behavior of the muon-photon loop. These counterterms lead to the presence of new low-energy constants in the EFT, in addition to thosealready present in ChPT. The values of these LECs can, of course, only be fixed by matchingwith the underlying UV-complete theory, QCD plus QED.Once these LECs are taken into account, ChPT, augmented with muon and photon fields,gives a complete representation of a HVP µ . In addition, finite-volume effects at any given orderare completely fixed in terms of the LECs that appear to one order less in the EFT. In thissense, finite-volume effects can be predicted, at any given order, in terms of the LECs of theinfinite-volume EFT. This is not surprising, as a HVP µ is nothing else than (the projection tozero external photon momentum and onto the Pauli spin structure of) a correlation functionin QCD plus QED.In the case of a HVP µ , we explained that the need for new counter terms, not present inChPT, the EFT for pions alone, arises for the first time in the EFT expansion at N LO,and that their role in the study of finite-volume effects starts at N LO. We constructedthe new counter terms at the lowest order at which they appear, and showed that indeeddimensional arguments imply that they become relevant at N LO. While the constructionof these counter terms is relatively straightforward, it would require three- and four-loopcalculations to demonstrate how it all works in the case of a HVP µ . Therefore, instead, wedemonstrated our observations by working out a toy model in which the effects appearalready at two loops, which is the minimum number. While it is unlikely that the fullN LO analysis will ever be worked out in practice, it is important to establish the validityof the EFT framework for the study of a HVP µ in order to be assured that even the NNLOanalysis of finite-volume effects carried out in Ref. [10] has a solid EFT basis. We believethat our discussion in this paper illustrates why indeed this is the case. We expect thesame separation between the UV physics represented by the counter terms and IR physicsof finite-volume effects will also take place in the hadronic light-by-light contribution when26he QED part is taken in infinite volume [21]. Acknowledgments
We thank Max Hansen for discussions. TBs and MGs work is supported by the U.S.Department of Energy, Office of Science, Office of High Energy Physics, under Award Num-bers DE-SC0010339, DE-SC0013682, respectively. S.P. is supported by CICYTFEDER-FPA2017-86989-P and by Grant No. 2017 SGR 1069.
Appendix A: Counter terms in the time-momentum representation
In this appendix, we revisit the time-momentum representation for the HVP [20]. Inlattice computations, a HVP µ is often obtained from C ( t ) = 13 (cid:88) i (cid:90) d x (cid:104) j i (0 , j i ( (cid:126)x, t ) (cid:105) , (A1)where j µ is the hadronic part of the electromagnetic current. We can, of course, express C ( t ) in terms of the HVP Π( q ): C ( t ) = 13 (cid:88) i (cid:90) d x (cid:90) d q (2 π ) e iqx (cid:0) δ ii q − q i q i (cid:1) Π( q ) (A2)= (cid:90) dq π e iq t q Π( q )= − (cid:90) dq π e iq t q ˆΠ( q ) − Π(0) δ (cid:48)(cid:48) ( t ) , where ˆΠ( q ) = Π(0) − Π( q ) . (A3)Equation (A2) contains the divergent counter term − Π(0) δ (cid:48)(cid:48) ( t ), and we note that, in gen-eral, counter terms in C ( t ) live at t = 0, and thus appear in the form of the Dirac deltafunction δ ( t ) and its derivatives. In particular, in ChPT, higher derivatives than δ (cid:48)(cid:48) ( t ) canappear, corresponding to the fact that counter terms are polynomials in q in the momentumrepresentation.Inverting Eq. (A2), we find q Π( q ) = (cid:90) ∞−∞ dt (cid:0) e − iq t − (cid:1) C ( t ) = − (cid:90) ∞ dt sin ( q t/ C ( t ) , (A4)where we inserted the − q = 0, and we used that C ( t ) is even in t in the second step. It follows thatˆΠ( q ) = (cid:90) ∞ dt (cid:18) ( q t/ q − t (cid:19) C ( t ) , (A5)and thus a HVP µ = (cid:90) ∞ dq f ( q ) ˆΠ( q ) (A6)= (cid:90) ∞ dq f ( q ) (cid:90) ∞ dt (cid:18) ( q t/ q − t (cid:19) C ( t ) .
27n Ref. [10] we made the incorrect observation that at NNLO a HVP µ is UV divergent inChPT, because at NNLO C ( t ) ∼ /t for t →
0, which makes the integral over t in Eq. (A6)divergent, as the weight function in the integral over t behaves like t for t →
0. If true, thiswould be a paradox, because, as we have seen in Sec. I, the expression for a HVP µ in terms ofˆΠ( q ) is UV finite at NNLO in ChPT. The problem, in fact, has nothing to do with a HVP µ ,as can be seen from Eq. (A5). While ˆΠ( q ) is finite by construction, the integral on theright-hand side of this equation is UV divergent if indeed C ( t ) ∼ /t for small t at NNLO.As we will show here, this apparent paradox arises if one is not careful with the treatmentof counter terms in the time-momentum representation. The correct expression connecting (cid:98) Π( q ) and C ( t ) in ChPT isˆΠ( q ) = lim (cid:15) → (cid:90) ∞ − dt (cid:18) ( qt/ w − t (cid:19) (cid:18) C ChPT ( t ; (cid:15), µ ) − δ IV ( t ) C ( (cid:15), µ ) (cid:19) , (A7)where δ IV ( t ) is the 4th derivative of the Dirac delta and C ( (cid:15), µ ) is a counter term directlyrelated to the LECs of ChPT. As usual, in order to define a counter term, a regulator needsto be introduced, and the parameter (cid:15) represents this regulator, as we will see below. Therest of this appendix will prove this result. While we keep our discussion concrete by usinga simple example to make the argument, the end result carries over to the case of ChPT.In the following, it will be useful to express C ( t ) in terms of the spectral function ρ ( s ).From the subtracted dispersion relationˆΠ( q ) = q (cid:90) ∞ ds ρ ( s ) s ( s + q ) , (A8)it follows from Eq. (A2) that C ( t ) = − (cid:90) ∞ ds ρ ( s ) (cid:90) ∞−∞ dq π e iq t q s ( s + q ) − Π(0) δ (cid:48)(cid:48) ( t ) (A9)= − (cid:90) ∞ ds ρ ( s ) (cid:18) − δ ( t ) − s δ (cid:48)(cid:48) ( t ) + 12 √ s e −√ s | t | (cid:19) − Π(0) δ (cid:48)(cid:48) ( t )= − (cid:90) ∞ ds ρ ( s ) (cid:18) − δ ( t ) + 12 √ s e −√ s | t | (cid:19) , where in the last step we used the unsubtracted dispersion relation.To illustrate the claim and understand what is going on, let us consider the simple exampleof a spectral function given by ρ ( s ) = (cid:18) ss + M (cid:19) θ ( s − m ) , (A10)where M is like the ρ -meson mass, for example, and we are interested in a “ChPT” resultwhere M (cid:29) m ∼ q ≡ m L ( L for light). With this spectral function, one may calculatethe once-subtracted vacuum polarization, ˆΠ( Q ) asˆΠ( q ) = q (cid:90) ∞ m dss ( s + q ) ρ ( s ) (A11)= log m + q m + q M − q log m + M m + q ≈ log m + q m + q M log M m + q + O (cid:18) m L M (cid:19) .
28n this world, the identification of the counter term can be made by splitting the spectralfunction as ρ ( s ) = 1 + sM (cid:124) (cid:123)(cid:122) (cid:125) ρ ChPT ( s ) − s M ( s + M ) (cid:124) (cid:123)(cid:122) (cid:125) ∆ ρ ( s ) (A12)in terms of the “ChPT” spectral function, ρ ChPT ( s ), and the “UV completion,” ∆ ρ ( s ). Withthis split, also the dispersion relation (A11) splits into two parts:ˆΠ( q ) = q (cid:90) ∞ m dss ( s + q ) (cid:16) sM (cid:17) (cid:18) sµ (cid:19) (cid:15) (A13)+ q (cid:90) ∞ m (cid:124)(cid:123)(cid:122)(cid:125) dss ( s + q (cid:124)(cid:123)(cid:122)(cid:125) ) (cid:18) − s M ( s + M ) (cid:19) (cid:18) sµ (cid:19) (cid:15) ≡ ˆΠ ChPT ( q , (cid:15), µ ) + q C ( (cid:15), µ ) , where ˆΠ ChPT ( q , (cid:15), µ ) = q (cid:90) ∞ m dss ( s + q ) (cid:16) sM (cid:17) (cid:18) sµ (cid:19) (cid:15) (A14)and C ( (cid:15), µ ) = (cid:90) ∞ m (cid:124)(cid:123)(cid:122)(cid:125) dss ( s + q (cid:124)(cid:123)(cid:122)(cid:125) ) (cid:18) − s M ( s + M ) (cid:19) (cid:18) sµ (cid:19) (cid:15) . (A15)We have added a regulating factor ( q /µ ) (cid:15) because both integrals are now separately UVdivergent. In the second integral, Eq. (A15), we may take the limit q , m → ρ ( s ), making C ( (cid:15), µ ) depend only on the UV scale M . Both integrals may be evaluated to give:ˆΠ( q ) = (cid:18) − q M (cid:19) log (cid:18) q m (cid:19) − q M (cid:18) (cid:15) + log m µ (cid:19) + q C ( (cid:15), µ ) , (A16)and C ( (cid:15), µ ) = 1 M (cid:18) (cid:15) + log M µ (cid:19) . (A17) C ( (cid:15), µ ) is the necessary counter term, and encodes the UV completion of the theory. Onecan check that the combined result (A16) agrees with the last line of Eq. (A11), as it should.Let us now discuss consider Eq. (A7), using our example. Using Eqs. (A9) and (A12): C ChPT ( t ) = − (cid:90) ∞ m ds ρ ChPT ( s ) (cid:18) √ s e − t √ s − δ ( t ) (cid:19) ( t > − ) (A18)= e − mt t (cid:20) t ( − − mt (2 + mt )) + − − mt (24 + mt (12 + mt (4 + mt ))) M (cid:21) ≈ − t − M t ( t → + ) . The term proportional to δ ( t ) will not contribute to (cid:98) Π( q ) in Eq. (A7), since the kernel (cid:16) ( wt/ w − t (cid:17) ∼ t , and we will thus neglect this term in what follows. The result in29q. (A18) shows that our example (A10) reproduces the 1 /t behavior for t → (cid:98) Π( q ) in Eq. (A5) diverge. Inview of Eq. (A11), this result is clearly incorrect.First, using for instance Eq. (A14), we need to regulate this divergence. The result ofEq. (A14) may be expressed as:ˆΠ
ChPT ( q , (cid:15), µ ) = (cid:90) ∞ dt (cid:18) ( qt/ q − t (cid:19) C ChPT ( t ; (cid:15), µ ) , (A19)with C ChPT ( t ; (cid:15), µ ) = − (cid:90) ∞ m ds (cid:16) sM (cid:17) √ s e − t √ s (cid:18) sµ (cid:19) (cid:15) . (A20)Then, using the identity q = (cid:90) ∞ − dt (cid:18) ( qt/ q − t (cid:19) (cid:18) − δ IV ( t ) (cid:19) , (A21)we can express the result in Eq. (A16) asˆΠ( q ) = (cid:90) ∞ − dt (cid:18) ( qt/ q − t (cid:19) (cid:18) C ChPT ( t ; (cid:15), µ ) − δ IV ( t ) C ( (cid:15), µ ) (cid:19) , (A22)which is precisely Eq. (A7), as promised. To verify this, it is easiest to do the t integralfirst, then express the integral over s in terms of the Beta-function, and finally, to use theidentities ( x > (cid:15) < − (cid:15) B ( − x, − (cid:15),
0) = 1 (cid:15) + log 1 + xx + O ( (cid:15) ) , (A23)( − (cid:15) B ( − x, − (cid:15),
0) = log(1 + x ) + O ( (cid:15) ) . Our example generalizes to ChPT, because it shares the 1 /t behavior at NNLO. In orderto properly treat the divergence arising in Eq. (A7), C ChPT ( t ) needs to be properly regulated,and thus a counter term needs to be introduced to define the integral when the regulatoris removed. In momentum space, the counter term is proportional to q , as follows fromEq. (A21), and it corresponds to the LEC c in two-flavor ChPT [22]. Appendix B: Integrals
In this appendix, we provide the details of some calculations in the main text. Webegin with a derivation of the explicit form of the third line of Eq. (2.19). Starting fromEq. (2.16b), the additional term containing m π can be written as δa CT = m f π (cid:90) dx (cid:90) ∞ dq q + M ( q ) (cid:15) log q x (1 − x ) + m π q x (1 − x ) , (B1) Alternatively one could also introduce a t η for η → C ( (cid:15), µ ) in Eq. (A17) but the final result for ˆΠ( q ) willbe the same. (cid:15) → q + M = 1 M ∞ (cid:88) n =0 ( − n (cid:18) q M (cid:19) n (B2)= 1 M πi (cid:90) γ ds (cid:18) q M (cid:19) − s π csc ( πs ) , < Re s < , where γ is a line parallel to the imaginary s -axis with 0 < Re s <
1. (One can see this byusing π csc( πs ) = ∞ (cid:88) n = −∞ ( − n n + s , (B3)and then closing the contour in the left half plane counter clockwise.) Then, δa CT = m f π M πi (cid:90) γ ds π csc ( πs ) (cid:90) dx (cid:90) ∞ dq (cid:18) q M (cid:19) − s log q x (1 − x ) + m q x (1 − x )= m f π πi (cid:90) γ ds (cid:18) m M (cid:19) − s Φ( s ) , (B4)where Φ( s ) = π csc ( πs ) Γ( s ) (1 − s )Γ(2 s ) (cid:16) s + 2 s + 6 + π s + O ( s ) , s → . (B5)The expansion after the symbol “ (cid:16) ” is the singular expansion of the function Φ( s ). Accord-ing to the Converse Mapping Theorem [23], the expansion for δa CT in powers of m /M isthen δa CT = m f π m M (cid:18) log m M − m M + 6 + π (cid:19) . (B6)The function Φ( s ) has poles not only at s = 0, but also at all the negative integers in theleft half plane, but those poles lead to higher powers of m /M . For example, near s = − s ) = π csc ( πs ) Γ( s ) (1 − s )Γ(2 s ) (cid:16) s + 1) − s + 1) + −
15 + 2 π s + 1 + O (( s + 1) ) , (B7)which leads to a contribution of order ( m /f π )( m /M ) times logarithms to δa CT .Next, we give some details about the derivation of Eq. (3.26) from Eq. (3.25). Forsimplicity, we set n = 1; the dependence on n can be restored in the final result by replacing L → nL . A change of variables y = m σ L (1 − ˆ x ) + m π L ˆ x ↔ ˆ x = y − m σ L m π L − m σ L , (B8)allows us to rewrite Eq. (3.26) as a (1) = − m µ π m σ − m π ) L (cid:90) m σ Lm π L dy ( m σ L − y ) (B9) × (cid:18) K ( y ) y + K ( y ) (cid:18) γ E + log m µ ( m σ L − y )( y − m π L )2( m σ − m π ) L y (cid:19)(cid:19) . m π /m σ or e − m σ L relativeto the dominant contribution. Therefore, we can replace the upper limit m σ L of the integralby ∞ and we may neglect y in comparison with m σ L in the combinations m σ L − y appearing in Eq. (B9), because the functions K , ( y ) fall off exponentially like e − y at large y and thus suppress such contributions by a factor e − m σ L . This simplifies Eq. (B9) to a (1) ≈ − π m µ m σ (cid:90) ∞ m π L dyy (cid:20) K ( y ) + y K ( y ) (cid:18) γ E + log m µ ( y − m π L )2 m σ y (cid:19)(cid:21) , (B10)and, using the representation K ν ( y ) = y ν (cid:90) ∞−∞ dz e iz ( z + y ) ν + ( ν = 0 , , (B11)one may carry out the y integral and obtain a (1) ≈ − π m µ m σ (cid:90) ∞−∞ dz e iz ( z + m π L ) (cid:20) γ E + log 2 m µ ( z + m π L ) m σ m π L (cid:21) = 18 π m µ m σ K ( m π L ) log m σ m µ . (B12)Finally, the replacement L → nL yields the result (3.26), as promised. [1] G. W. Bennett et al. [Muon g-2], Final Report of the Muon E821 Anomalous Magnetic MomentMeasurement at BNL,
Phys. Rev. D , 072003 (2006) [arXiv:hep-ex/0602035 [hep-ex]].[2] T. Aoyama et al. , The anomalous magnetic moment of the muon in the Standard Model, [arXiv:2006.04822 [hep-ph]].[3] J. Grange et al. [Muon g-2],
Muon (g-2) Technical Design Report, [arXiv:1501.06858[physics.ins-det]].[4] M. Abe, et al. , A New Approach for Measuring the Muon Anomalous Magnetic Moment andElectric Dipole Moment,
PTEP , no.5, 053C02 (2019) [arXiv:1901.03047 [physics.ins-det]].[5] A. Francis, B. Jaeger, H. B. Meyer and H. Wittig,
A new representation of the Adler functionfor lattice QCD,
Phys. Rev. D , 054502 (2013) [arXiv:1306.2532 [hep-lat]].[6] G. Gounaris and J. Sakurai, Finite width corrections to the vector meson dominance predictionfor ρ → e + e − , Phys. Rev. Lett. , 244-247 (1968).[7] M. L¨uscher, Signatures of unstable particles in finite volume,’
Nucl. Phys. B , 237-251(1991); L. Lellouch and M. L¨uscher,
Weak transition matrix elements from finite volumecorrelation functions,
Commun. Math. Phys. , 31-44 (2001) [arXiv:hep-lat/0003023 [hep-lat]].[8] C. Aubin, T. Blum, P. Chau, M. Golterman, S. Peris and C. Tu,
Finite-volume effects inthe muon anomalous magnetic moment on the lattice,
Phys. Rev. D , no.5, 054508 (2016)[arXiv:1512.07555 [hep-lat]].[9] J. Bijnens and J. Relefors, Vector two-point functions in finite volume using partially quenchedchiral perturbation theory at two loops,
JHEP , 114 (2017) [arXiv:1710.04479 [hep-lat]].
10] C. Aubin, T. Blum, C. Tu, M. Golterman, C. Jung and S. Peris,
Light quark vacuum polariza-tion at the physical point and contribution to the muon g − , Phys. Rev. D , no.1, 014503(2020) [arXiv:1905.09307 [hep-lat]].[11] S. Borsanyi, Z. Fodor, J. Guenther, C. Hoelbling, S. Katz, L. Lellouch, T. Lippert, K. Miura,L. Parato, K. Szabo, F. Stokes, B. Toth, C. Torok and L. Varnhorst,
Leading-orderhadronic vacuum polarization contribution to the muon magnetic moment from lattice QCD, [arXiv:2002.12347 [hep-lat]].[12] J. Gasser and H. Leutwyler,
Spontaneously Broken Symmetries: Effective Lagrangians atFinite Volume,
Nucl. Phys. B , 763-778 (1988).[13] B. Chakraborty, C. Davies, P. de Oliviera, J. Koponen, G. Lepage and R. Van de Water,
Thehadronic vacuum polarization contribution to a µ from full lattice QCD, Phys. Rev. D , no.3,034516 (2017) [arXiv:1601.03071 [hep-lat]].[14] M. T. Hansen and A. Patella, Finite-volume effects in ( g − HVP , LO µ , Phys. Rev. Lett. ,172001 (2019) [arXiv:1904.10010 [hep-lat]].[15] M. T. Hansen and A. Patella,
Finite-volume and thermal effects in the leading-HVP contri-bution to muonic ( g − , [arXiv:2004.03935 [hep-lat]].[16] M. L¨uscher, Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories.1. Stable Particle States,
Commun. Math. Phys. , 177 (1986).[17] E. Shintani et al. [PACS],
Hadronic vacuum polarization contribution to the muon g − with2+1 flavor lattice QCD on a larger than (10 fm ) lattice at the physical point, Phys. Rev. D , no.3, 034517 (2019) [arXiv:1902.00885 [hep-lat]].[18] T. Blum,
Lattice calculation of the lowest order hadronic contribution to the muon anomalousmagnetic moment,
Phys. Rev. Lett. , 052001 (2003) [arXiv:hep-lat/0212018 [hep-lat]].[19] B. E. Lautrup, A. Peterman and E. de Rafael, Recent developments in the comparison betweentheory and experiments in quantum electrodynamics,
Phys. Rept. , 193-259 (1972)[20] D. Bernecker and H. B. Meyer, Vector Correlators in Lattice QCD: Methods and applications,
Eur. Phys. J. A , 148 (2011) [arXiv:1107.4388 [hep-lat]].[21] N. Asmussen, J. Green, H. B. Meyer and A. Nyffeler, Position-space approach to hadroniclight-by-light scattering in the muon g − on the lattice, PoS
LATTICE2016 , 164 (2016)[arXiv:1609.08454 [hep-lat]]; T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin, C. Jungand C. Lehner,
Using infinite volume, continuum QED and lattice QCD for the hadronic light-by-light contribution to the muon anomalous magnetic moment,
Phys. Rev. D , no.3, 034515(2017) [arXiv:1705.01067 [hep-lat]]; N. Asmussen, A. G´erardin, H. B. Meyer and A. Nyffeler, Exploratory studies for the position-space approach to hadronic light-by-light scattering in themuon g − , EPJ Web Conf. , 06023 (2018) [arXiv:1711.02466 [hep-lat]].[22] J. Bijnens, G. Colangelo and G. Ecker,
The Mesonic chiral Lagrangian of order p , JHEP ,020 (1999) [arXiv:hep-ph/9902437].[23] P. Flajolet, X. Gourdon and P. Dumas, Mellin transforms and asymptotic harmonic sums,
Theoretical Computer Science , 3 (1995)., 3 (1995).