QED self energies from lattice QCD without power-law finite-volume errors
QQED self energies from lattice QCD without power-law finite-volume errors
Xu Feng
1, 2, 3, 4 and Luchang Jin
5, 6 School of Physics, Peking University, Beijing 100871, China Collaborative Innovation Center of Quantum Matter, Beijing 100871, China Center for High Energy Physics, Peking University, Beijing 100871, China State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China Physics Department, University of Connecticut, Storrs, Connecticut 06269-3046, USA RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA
Using the infinite-volume photon propagator, we developed a method which allows us to calcu-late electromagnetic corrections to stable hadron masses with only exponentially suppressed finite-volume effects. The key idea is that the infinite volume hadronic current-current correlation functionwith large time separation between the two currents can be reconstructed by its value at modest timeseparation, which can be evaluated in finite volume with only exponentially suppressed errors. Thisapproach can be extended to other possible applications such as QED corrections to (semi-)leptonicdecays and some rare decays.
I. INTRODUCTION
Electromagnetic and strong interactions are two fun-damental interactions known to exist in nature. Theyare described by the first-principle theories of quantumelectrodynamics (QED) and quantum chromodynamics(QCD), respectively. In some physical processes, QEDand QCD are both present and both play indispensableroles. A typical example is the neutron proton mass dif-ference, which is attributed to both electromagnetic andstrong isospin-breaking effects. Although this mass dif-ference is only 2.53 times the electron mass, it deter-mines the neutron-proton abundance ratio in the earlyUniverse, which is an important initial condition for BigBang nucleosynthesis. This quantity attracts a lot of in-terest and has motivated a series of lattice QCD studieson the isospin breaking effects in hadron spectra [1–7].Generally speaking, QED effects are small due to thesuppression of a factor of the fine-structure constant α QED ≈ / f + (0) and the leptonic decay constant ratio f K /f π havereached the precision of (cid:46) .
3% [8]. At this precisionthe isospin symmetry breakings cannot be neglected. Pi-oneering works [9–11] have been carried out to includeQED corrections to leptonic decay rates.The conventional approach to include QED in latticeQCD calculations is to introduce an infrared regulatorfor QED. One popular choice is QED L , first introducedin Ref. [12], which removes all the spatial zero modes ofthe photon field. There are also some other methods:QED TL [6], massive photon [13], C ∗ boundary condi-tion [14]. In general, by including the long-range electro-magnetic interaction on a finite-volume lattice, all thesetreatments introduce power-law suppressed finite-volumeerrors. This is different from typical pure QCD latticecalculations where finite-volume errors are suppressed ex- ponentially by the physical size of the lattice. Ref. [15]provides an up-to-date systematic analysis of the finite-volume errors for the hadron masses in the presence ofQED corrections.Another approach to incorporate QED with QCD is toevaluate the QED part in infinite volume analytically andcompletely eliminate the power-law suppressed finite vol-ume errors. Such an approach, called QED ∞ , has beenused in the calculation of hadronic vacuum polarization(HVP) and the hadronic light-by-light (HLbL) contribu-tion to muon g − ∞ , while its ab-solute size is still exponentially suppressed by the sizeof the system, is only power-law suppressed comparedwith the correction functions. In this paper, we proposea method to solve this problem. We show that the QEDself-energy diagram can be calculated on a finite volumelattice with only exponentially suppressed finite-volumeeffects. , ν x, µ FIG. 1. Self-energy diagrams a r X i v : . [ h e p - l a t ] D ec II. MASTER FORMULA
We first consider the self-energy calculation in an infi-nite space-time volume. For the case of a stable hadronicstate N , the self-energy diagram shown in Fig. 1 can becalculated in Euclidean space from the integral:∆ M = I = 12 (cid:90) d x H µ,ν ( x ) S γµ,ν ( x ) , (1)where hadronic part H µ,ν ( x ) = H µ,ν ( t, (cid:126)x ) is given by H µ,ν ( x ) = 12 M (cid:104) N ( (cid:126) | T [ J µ ( x ) J ν (0)] | N ( (cid:126) (cid:105) , (2)where J µ = 2 e ¯ uγ µ u/ − e ¯ dγ µ d/ − e ¯ sγ µ s/ | N ( (cid:126)p ) (cid:105) indicates a hadronic state N with the mass M and spatial momentum (cid:126)p , and S γµ,ν isthe photon propagator whose form is analytically known.The states | N ( (cid:126)p ) (cid:105) obey the normalization convention (cid:104) N ( (cid:126)p (cid:48) ) | N ( (cid:126)p ) (cid:105) = (2 π ) E (cid:126)p δ ( (cid:126)p (cid:48) − (cid:126)p ). The current oper-ator J µ ( t, (cid:126)x ) is a standard Euclidean Heisenberg-pictureoperator J µ ( t, (cid:126)x ) = e Ht J µ (0 , (cid:126)x ) e − Ht . A possible shortdistance divergence of the integral can be removed byrenormalizing the quark mass.If we examine an L finite-volume system, the mainfeature of the conventional methods such as QED L isto design a finite-volume form for photon propagator, S γ,Lµ,ν , and calculate the hadronic correlation function in afinite volume in the presence of finite-volume QED using S γ,Lµ,ν . Unfortunately, it results in power-law suppressedfinite-volume effects in the mass extracted from the finite-volume hadronic correlation function. For the QED ∞ approach, one may begin with the infinite volume formulain Eq. (1) to extract the QED self energy, but then limitthe range of the integral and replace H µ,ν ( x ) by a finitevolume version. However, as we will explain later, theresult still suffers from power-law finite-volume effects.To completely solve the problem, we develop a methodas follows. We choose a time t s ( t s (cid:46) L ), that is suffi-ciently large that the intermediate hadronic states be-tween the two currents are dominated by single hadronstates since all the other states (resonance states, multi-hadron states, etc) are exponentially suppressed by t s . I = I ( s ) + I ( l ) , I ( s ) = 12 (cid:90) t s − t s dt (cid:90) d (cid:126)x H µ,ν ( x ) S γµ,ν ( x ) , I ( l ) = (cid:90) ∞ t s dt (cid:90) d (cid:126)x H µ,ν ( x ) S γµ,ν ( x ) . (3)We propose to approximate I ( s ) and I ( l ) using thelattice-QCD calculable expressions I ( s,L ) and I ( l,L ) ,which are defined as I ( s,L ) = 12 (cid:90) t s − t s dt (cid:90) L/ − L/ d (cid:126)x H Lµ,ν ( x ) S γµ,ν ( x ) , I ( l,L ) = (cid:90) L/ − L/ d (cid:126)x H Lµ,ν ( t s , (cid:126)x ) L µ,ν ( t s , (cid:126)x ) , (4) where L µν ( t s , (cid:126)x ) is a QED weighting function, defined as: L µ,ν ( t s , (cid:126)x ) = (cid:90) d p (2 π ) e i(cid:126)p · (cid:126)x (cid:90) ∞ t s dt e − ( E (cid:126)p − M )( t − t s ) × (cid:90) d (cid:126)x (cid:48) e − i(cid:126)p · (cid:126)x (cid:48) S γµ,ν ( t, (cid:126)x (cid:48) ) . (5)Here the energy E (cid:126)p is given by the dispersion relation E (cid:126)p = (cid:112) M + (cid:126)p . The integral in L µ,ν ( t s , (cid:126)x ) can becalculated in infinite volume (semi-)analytically. In sec-tion IV, detailed expressions for L µ,ν ( t s , (cid:126)x ) are given forboth Feynman- and Coulomb-gauge photon propagators.The finite-volume hadronic part H Lµ,ν ( x ) is definedthrough finite-volume lattice correlators (assuming t ≥ H Lµ,ν ( t, (cid:126)x ) = L (cid:104) N ( t + ∆ T ) J µ ( t, (cid:126)x ) J ν (0) ¯ N ( − ∆ T ) (cid:105) L (cid:104) N ( t + ∆ T ) ¯ N ( − ∆ T ) (cid:105) L , (6)where ¯ N ( t )/ N ( t ) is an interpolating operator which cre-ates/annihilates the zero momentum hadron state N attime t , ∆ T is the separation between the source and cur-rent operators, which only needs to be large enough tosuppress the excited states effects.We will demonstrate below the quantities I ( s,L ) and I ( l,L ) defined in the master formula (4) only differ from I ( s ) and I ( l ) by exponentially suppressed finite-volumeeffects. III. PATH TO THE MASTER FORMULAA. Comparison between I ( s ) and I ( s,L ) We adopt the conventional expectation (which can bedemonstrated in perturbation theory using the Poissonsummation formula [20]) that for a theory such as QCDwith a mass gap a matrix element such as H Lµ,ν ( t, (cid:126)x ),evaluated in a finite space-time volume L × T with pe-riodic boundary conditions, will differ from the corre-sponding matrix element H µ,ν ( t, (cid:126)x ) in infinite volume byterms that are exponentially suppressed in the spatialand temporal extents of the volume. In addition, thevalue of the infinite-volume H µ,ν ( t, (cid:126)x ), when | (cid:126)x | (cid:38) t , isexponentially suppressed in | (cid:126)x | .These considerations suggest that the integral for I ( s ) is dominated by the region inside the finite-volume lat-tice and well approximated by the finite-volume integral I ( s,L ) . We therefore conclude that I ( s,L ) differs fromits infinite-volume version I ( s ) by an exponentially sup-pressed finite-volume effect. B. Comparison between I ( l ) and I ( l,L ) We remind the reader that the value of H µ,ν ( x ) is notalways exponentially suppressed at large | x | . In fact, forlarge | t | , we shall have: H µ,ν ( t, (cid:126)x ) ∼ e − M ( √ t + (cid:126)x − t ) ∼ e − M (cid:126)x t ∼ O (1) . (7)Therefore if we limit the range of the integral for I inEq. (1), it will contain an O (1 /L ) power-law finite volumeeffect even if the infinite-volume photon propagator S γµ,ν is used instead of S γ,Lµ,ν . This is one of the reasons why thetraditional QED ∞ method, which works for the cases ofHVP and HLbL, does not work for the QED self-energydiagram. As both ends of the photon propagator coupleto the quark current, one can only perform the intergralover a finite time window. Even if we could create aninfinite time-extent lattice and use the integral (cid:90) ∞−∞ dt (cid:90) L/ − L/ d (cid:126)x H Lµ,ν ( t, (cid:126)x ) S γµ,ν ( t, (cid:126)x ) , (8)the result would still carry an O (1 /L ) finite-volume ef-fect, due to the fact that H Lµ,ν ( t, (cid:126)x ) − H µ,ν ( t, (cid:126)x ) is notexponentially suppressed at large | t | .Instead of using H Lµ,ν ( t, (cid:126)x ) at large | t | directly, westudy the t -dependence of the infinite-volume H µ,ν ( t, (cid:126)x )for | t | > t s . By inserting a complete set of intermediatestates, we can rewrite H µ,ν ( t, (cid:126)x ) as H µ,ν ( t, (cid:126)x ) = (cid:88) n (cid:90) d (cid:126)p (2 π ) E n,(cid:126)p e i(cid:126)p · (cid:126)x e − ( E n,(cid:126)p − M ) t (9) × M (cid:104) N ( (cid:126) | J µ (0) | n ( (cid:126)p ) (cid:105)(cid:104) n ( (cid:126)p ) | J ν (0) | N ( (cid:126) (cid:105) . Without losing generality, positive t is assumed in theabove equation. In Euclidean space with large t , the con-tribution from excited states si exponentially suppressed.The following approximation, where only the lowest en-ergy states’ contributions are kept, is then valid for t > t s : H µ,ν ( t, (cid:126)x ) ≈ (cid:90) d (cid:126)p (2 π ) E (cid:126)p e i(cid:126)p · (cid:126)x e − ( E (cid:126)p − M ) t (10) × M (cid:104) N ( (cid:126) | J µ (0) | N ( (cid:126)p ) (cid:105)(cid:104) N ( (cid:126)p ) | J ν (0) | N ( (cid:126) (cid:105) . where E (cid:126)p = (cid:112) M + (cid:126)p . On one hand, Eq. (10) sug-gests that we can calculate H µ,ν ( t, (cid:126)x ), for large t , via thematrix element (cid:104) M ( (cid:126)p ) | J µ (0) | M (cid:105) . On the other hand, itindicates that the Fourier transformation of H µ,ν ( t, (cid:126)x ) atfixed t = t s gives the relevant matrix element: (cid:90) d (cid:126)x H µ,ν ( t s , (cid:126)x ) e − i(cid:126)p · (cid:126)x = 12 E (cid:126)p e − ( E (cid:126)p − M ) t s (11) × M (cid:104) N ( (cid:126) | J µ (0) | N ( (cid:126)p ) (cid:105)(cid:104) N ( (cid:126)p ) | J ν (0) | N ( (cid:126) (cid:105) . Putting Eq. (11) into Eq. (10), we are able to reconstructthe needed infinite volume hadronic matrix element atlarge t from its value at modest t s : H µ,ν ( t, (cid:126)x (cid:48) ) ≈ (cid:90) d (cid:126)x H µ,ν ( t s , (cid:126)x ) (12) × (cid:90) d (cid:126)p (2 π ) e i(cid:126)p · (cid:126)x e − ( E (cid:126)p − M )( t − t s ) e − i(cid:126)p · (cid:126)x (cid:48) . We will refer this relation, which is the crucial step inthe derivation, as the “infinite volume reconstructionmethod”. Here the ≈ symbol reminds us that the excited-state contributions in H µ,ν ( t, (cid:126)x ) and H µ,ν ( t s , (cid:126)x ) are ex-ponentially suppressed and have been neglected.In the previous section, we have confirmed that H µ,ν ( t s , (cid:126)x ) is equal to H Lµ,ν ( t s , (cid:126)x ) up to some exponen-tially suppressed corrections if (cid:126)x ∈ [ − L/ , L/ I ( l ) can be well approximated by I ( l,L ) through I ( l ) = (cid:90) ∞ t s dt (cid:90) d (cid:126)x H µ,ν ( t, (cid:126)x ) S γµ,ν ( t, (cid:126)x ) ≈ (cid:90) d (cid:126)x H µ,ν ( t s , (cid:126)x ) L µ,ν ( t s , (cid:126)x ) ≈ (cid:90) L/ − L/ d (cid:126)x H Lµ,ν ( t s , (cid:126)x ) L µ,ν ( t s , (cid:126)x )= I ( l,L ) (13)where the weighting function L µ,ν ( t s , (cid:126)x ) has been givenin Eq. (5) and will be discussed in the following section. IV. QED WEIGHTING FUNCTION L µ,ν ( t s , (cid:126)x ) Detailed expressions for the QED weighting func-tion L µ,ν ( t s , (cid:126)x ) defined in Eq. (5) can be evaluated forFeynman- and Coulomb- gauge photon propagators: • Feynman gauge S γµ,ν ( x ) = δ µ,ν π x = δ µ,ν (cid:90) d p (2 π ) e ipx p . (14) L µ,ν ( t s , (cid:126)x ) = δ µ,ν π (cid:90) ∞ dp sin( p | (cid:126)x | )2( p + E p − M ) | (cid:126)x | e − pt s . (15) • Coulomb gauge S γµ,ν ( t, (cid:126)x ) (16)= π | (cid:126)x | δ ( t ) µ = ν = 0 (cid:82) d (cid:126)p (2 π ) | (cid:126)p | (cid:16) δ i,j − p i p j (cid:126)p (cid:17) e −| (cid:126)p | t + i(cid:126)p · (cid:126)x µ = i, ν = j .L i,j ( t s , (cid:126)x ) (17)= ( δ i,j − x i x j (cid:126)x ) 1(2 π ) (cid:90) ∞ dp sin( p | (cid:126)x | )2( p + E p − M ) | (cid:126)x | e − pt s +( δ i,j − x i x j (cid:126)x ) 1(2 π ) (cid:90) ∞ dp p | (cid:126)x | cos( p | (cid:126)x | ) − sin( p | (cid:126)x | )2( p + E p − M ) | (cid:126)x | e − pt s . Only the spatial polarization components are needed forthe large time expression in Coulomb gauge. All othercomponents of L are zero. V. EXTENDED DISCUSSIONS
Eq. (12) tells us that the large time hadronic matrixelements H µ,ν ( t, (cid:126)x ) can be determined using H µ,ν ( t s , (cid:126)x ),while H µ,ν ( t s , (cid:126)x ) can be calculated using lattice. Be-fore reaching Eq. (12), we explored other methods todetermine (cid:104) N ( (cid:126) | J µ (0) | N ( (cid:126) (cid:105) . We recognized that by us-ing the Coulomb-gauge photon propagator and assuming | N ( (cid:126) (cid:105) is a spin-0 charged particle, the corresponding ma-trix element can be determined easily. Here follows ourdiscussion.The infinite volume photon propagator in Coulombgauge is given in Eq. (16). This implies, for I ( l ) , only S γi,j is relevant. For a spin-0 charged particle, we have (cid:104) N ( (cid:126)p ) | J µ (0) | N ( (cid:126)p ) (cid:105) = ( p + p ) µ F ( q ) , (18)where the matrix element is expressed in terms of theform factor F ( q ) with q = p − p . If the initial or thefinal state has zero momentum, as is the case in Eq. (10),we have (cid:104) N ( (cid:126) | J i (0) | N ( (cid:126)p ) (cid:105) = p i F ( p ) . (19)Therefore, we can obtain that I ( l ) = 0 simply because ofthe Coulomb-gauge condition. Thus I ≈ I ( s,L ) = 12 (cid:90) t s − t s dt (cid:90) L/ − L/ d (cid:126)x H Lµ,ν ( t, (cid:126)x ) S γµ,ν ( t, (cid:126)x )(20)for a spin-0 charged particle and a Coulomb-gauge pho-ton propagator, and all the finite volume errors are ex-ponentially suppressed by the lattice size, L , or the inte-gration range in the time direction, t s . Note that t s (cid:46) L is required for the above statement to be valid. VI. CONCLUSION
We have demonstrated that the QED self-energy for astable hadron can be calculated on a finite volume latticewith only exponentially suppressed finite volume effects.The power-law finite volume effects, which are commonin QCD+QED calculations, are completely eliminated.This is achieved with the following three ideas: 1. QED ∞ : We start with an integral I , where theQED part in the integrand can be calculated in in-finite volume analytically, and the hadronic part ispurely a QCD matrix element, and enjoys an expo-nential suppressed long distance behavior becauseof the mass gap, as is familiar from pure QCD lat-tice calculations;2. Window method: We introduce a cut in the timeextent of the integral, t s , to separate the integralinto the short-distance part, which can be calcu-lated within finite volume directly, and the remain-ing long-distance part.3. Infinite volume reconstruction method: We usethe fact that the long-distance hadronic function isdominated by the lowest isolated pole (the hadronwhose QED mass shift is under study) in the spec-tral representation to express the infinite-volumehadronic function at large t in terms of its value atmodest t s , which can be evaluated in finite volume.The first idea, QED ∞ , has already been employed insome QED+QCD calculations, e.g. HVP [16], HLbL[17, 18] and the QED correction to HVP [19]. For thesecalculations, this idea by itself is able to remove all thepower-law suppressed finite-volume errors. The secondidea used in this work, the window method, is relativelynew. The name of the method comes from Ref. [19],where the integrand is also divided into parts, and dif-ferent treatments are applied to different parts. Thethird idea, the infinite volume reconstruction method,combined with the window method, is the essential partof our framework. It should be emphasized that, it isthe infinite-volume hadronic function, H µ,ν ( t, (cid:126)x ), at large t , being expressed in terms of H µ,ν ( t s , (cid:126)x ) at modest t s ,which helps eliminate the power-law finite-volume errors.In additional to QED self-energy, the framework de-veloped here can also be applied to other QED+QCDproblems. One example is the QED corrections to (semi-)leptonic decays, which can be used to determine someimportant CKM matrix elements like V ud and V us [9–11].Another example is the rare kaon decays [21–25], wherethe light electron propagator can be treated in a similarway as the photon discussed in this paper to reduce thefinite-volume error. ACKNOWLEDGMENTS
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