Calculating the Two-photon Contribution to π 0 → e + e − Decay Amplitude
CCalculating the Two-photon Contribution to π → e + e − Decay Amplitude
Norman H. Christ
Columbia UniversityE-mail: [email protected]
Xu Feng
Peking UniversityE-mail: [email protected]
Luchang Jin
University of ConnecticutE-mail: [email protected]
Cheng Tu
University of ConnecticutE-mail: [email protected]
Yidi Zhao ∗† Columbia UniversityE-mail: [email protected]
We develop a new method that allows us to deal with two-photon intermediate states in a latticeQCD calculation. We apply this method to perform a first-principles calculation of the π → e + e − decay amplitude. Both the real and imaginary parts of amplitude are calculated. The imaginarypart is compared with the prediction of optical theorem to demonstrate the effectiveness of thismethod. Our result for the real part of decay amplitude is 19 . ( )( . ) eV, where the firsterror is statistical and the second is systematic. ∗ Speaker. † This work is a part of the USQCD QCD+QED project and is partially supported by US DOE grant c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] J a n alculating the Two-photon Contribution to π → e + e − Decay Amplitude
Yidi Zhao
1. Introduction
In the standard model, there is a class of decays involving a combination of QED and QCDprocesses that are dominated by long-distances contributions. One example that is of interest andcan provide important tests of standard model is the rare Kaon decay K L → µ + µ − . This decayis well established experimentally [1] but includes a large long-distance contribution from two-photon intermediate states which migh be computed using lattice QCD. One difficulty of carryingout this lattice computation is the presence of intermediate two-photon states which can have alower energy than the initial Kaon state. In this article, we propose a method to tackle this problemand apply this method to perform a lattice computation on a simpler process, the π → e + e − decay.The π → e + e − decay process is described by the Feynman diagram in Fig. 1. The decayamplitude can be separated into two parts: the matrix element (cid:104) | T (cid:8) J µ ( u ) J ν ( v ) (cid:9) | π (cid:105) containingcontribution from hadronic interaction, and a regular Feynman integral stemming from the internaltwo-photon and electron loop. The decay amplitude consists of both a real and an imaginary part.The latter is easy to calculate using optical theorem, and gives the well-known unitary bound forthe π → e + e − branching ratio. Calculation of the real part of the decay amplitude requires anon-perturbative approach and is the primary goal of this work. π J µ ( v ) e + ( ~k + ) e − ( ~k − ) J µ ( u ) p − P p + P p + P − k + P Figure 1: The Feynman diagram for π → e + e − decay.
2. Analytic Continuation Approach
To calculate π → e + e − decay on the lattice, we need to solve the problem created by thetwo-photon intermediate state, whose energy can be lower than initial π state which implies thata direct Euclidean-space calculation can result in terms which grow exponentially in the time sep-aration. We start with the Minkowski-space expression for the π → e + e − decay amplitude andwrite down the matrix element in position space with Minkowski-space time dependence: A = (cid:90) d w (cid:104) | T (cid:8) J µ (cid:0) w (cid:1) J ν (cid:0) − w (cid:1)(cid:9) | π (cid:105) (2.1) (cid:90) d p e − ip · w (cid:34) g µµ (cid:48) ( p + P ) − i ε (cid:35) (cid:34) g νν (cid:48) ( p − P ) − i ε (cid:35) u ( k − ) γ µ (cid:48) (cid:34) γ · ( p + P − k − ) + m e ( p + P − k − ) + m e − i ε (cid:35) γ ν (cid:48) v ( k + ) . To convert the matrix element to a Euclidean-space quantity that is calculable on lattice, werotate the time coordinate w → − iw and meanwhile perform the opposite rotation on p , i.e. p → ip . However, because intermediate two-photon state can have less energy than the initialpion state, we have poles in the p complex plane that prevent a direct rotation of p integration1 alculating the Two-photon Contribution to π → e + e − Decay Amplitude
Yidi Zhao contour from the real axis to the imaginary axis. Thus, as shown in Fig. 2, we deform the contour atthe position of the two poles which could cross over imaginary axis when energy of photon is small.It can be shown that the exponential growth of leptonic factor introduced by the deformed contouris overcome by the hadronic matrix matrix which drops faster. More details on this approach canbe found in the companion proceeding [2].
Re( p )Im( p ) − m π − | ~p | − r ( ~p − ~k + ) + m e m π − | ~p |− m π + | ~p | r ( ~p − ~k + ) + m e m π + | ~p | Contour C Figure 2: A diagram of the complex p plane showing integration contours before and after therotation of the p contour.The new integral can be written as: A = (cid:90) d w L µν ( w ) H µν ( w ) (2.2) L µν ( w ) = (cid:90) d p (cid:90) C d p e − i (cid:126) p · (cid:126) w e + p w (cid:34) (cid:101) g µµ (cid:48) ( p + P ) − i ε (cid:35) (cid:34) (cid:101) g νν (cid:48) ( p − P ) − i ε (cid:35) u ( k − ) γ µ (cid:48) (cid:34) γ · ( p + P − k − ) + m e ( p + P − k − ) + m e − i ε (cid:35) γ ν (cid:48) v ( k + ) (2.3) H µν ( w ) = (cid:104) | T (cid:8) J µ (cid:0) w (cid:1) J ν (cid:0) − w (cid:1)(cid:9) | π (cid:105) E . (2.4)We express the result as the space-time integral of the product of leptonic and hadronic factors.The subscript E on the hadronic matrix element indicates that it is evaluated using Euclidean timedependence and conventions. The diagonal metric tensor (cid:101) g µµ (cid:48) with elements ( , , , i ) has beenintroduced to correctly connect the Minkowski conventions for the E&M currents in the leptonicfactor with the Euclidean conventions used in the hadronic matrix element.
3. Computational Method
Due to Lorentz invariance, when the initial pion state is stationary, the hadronic matrix elementcan be written as: H µν ( w ) = ε µνρ w ρ h ( w ) , (3.1)where h ( w ) is a scalar factor. In our lattice computation, the hadronic factor H µν can be extractedfrom the lattice three-point function through the following relationship: (cid:104) | T (cid:8) J µ ( x ) J ν ( ) (cid:9) | π (cid:105) = Z V m π N π lim t →− ∞ e m π | t | (cid:104) | T (cid:8) J µ ( x ) J ν ( ) π ( t ) (cid:9) | (cid:105) , (3.2)2 alculating the Two-photon Contribution to π → e + e − Decay Amplitude
Yidi Zhao where 2 m π comes from the normalization of pion state, and N π is the normalization factor for pionground state, i.e. N π = (cid:104) π | π ( ) | (cid:105) . The J µ and J ν operators on the right hand side of the equationare non-conserved local lattice currents and must be multiplied by a renormalization factor Z V . J µ ( u ) J ν ( v ) (a) Connected diagram J µ ( u ) J ν ( v ) (b) Disconnected diagram Figure 3: Feynman diagrams for the contractions in the calculation of hadronic factor. The dashedline on the left represents the location of the pion wall source.As shown in Fig. 3, there are two types of diagrams involved in calculating the three pointfunction: connected and disconnected diagrams. The connected diagram is made up of two wallsource propagators and one point source propagator. To make sure that the E & M current is sepa-rated far enough from pion operator but not too far such that it crosses the periodic boundary andgoes to the other side of pion, we always keep the time difference from the pion wall source at t to the closer current fixed to be a constant ∆ t . That means, for every lattice site x in [ − L , L ] , wealways choose a t such that t = { min ( x , ) − ∆ t } mod L . The values of ∆ t for each ensembles canbe found in Table 1.The disconnected diagram is more difficult to calculate as it involves large noise and wouldrequire much more statistics to be calculated accurately. In this work, we make use of the EMloops Tr (cid:2) D − ( x , x ) γ µ (cid:3) generated with random grid sources from the hadronic vacuum polarizationcalculation carried out by the RBC/UKQCD collaboration [3] [4]. As shown in Section 4, thisallows us to determine the amplitude of disconnected diagram up to an error of about 60%. The leptonic factor L µν ( w ) is evaluated by performing the p integral using Cauchy’s theorem,which leads us to a remaining three-dimensional integral over (cid:126) p . Note that imaginary part ofamplitude is well preserved in this expression. To get the imaginary part, we replace the pole | (cid:126) p |− M π / by a delta function i πδ ( | (cid:126) p | − M π / ) . The real part of decay amplitude is obtained bytaking the principal value.To further simplify the leptonic factor integral, we make use of the fact that this integral isindependent of the direction of the momentum of the outgoing electron (cid:126) k − . This enables us tointegrate over the angular direction of (cid:126) k − and divide it by 4 π . We present the integration result for3 alculating the Two-photon Contribution to π → e + e − Decay Amplitude
Yidi Zhao spatial components here. For the imaginary and real part: L im i j ( w , | (cid:126) w | ) = ε i jk w k | (cid:126) w | m e M π πα β ln (cid:18) + β − β (cid:19) M π | (cid:126) w | (cid:34) cos ( M π | (cid:126) w | ) − sin ( M π | (cid:126) w | ) M π | (cid:126) w | (cid:35) (3.3) L re i j ( w , | (cid:126) w | ) = ε i jk w k m e α π | (cid:126) w | (cid:18) − e M π | w | π M π p − ln (cid:18) + β − β (cid:19) | (cid:126) w | (3.4) (cid:90) ∞ d p e − p | w | M π − p (cid:20) cos ( p | (cid:126) w | ) − sin ( p | (cid:126) w | ) p | (cid:126) w | (cid:21) + e − M π | w | π M π p − ln (cid:18) + β − β (cid:19) | (cid:126) w | (cid:90) ∞ d p e − p | w | M π + p (cid:20) cos ( p | (cid:126) w | ) − sin ( p | (cid:126) w | ) p | (cid:126) w | (cid:21) + π | (cid:126) w | (cid:90) ∞ d p d cos θ e − E pe | w | E pe ( − M π + p − cos θ )( M π + p − cos θ ) (cid:20) cos ( p | (cid:126) w | ) − sin ( p | (cid:126) w | ) p | (cid:126) w | (cid:21)(cid:33) , where E pe is energy of internal electron. The leptonic factor for the real part is now a two-dimensional integral and can be evaluated numerically. Note that only spatial components areshown in this equation, namely, i , j = ( x , y , z ) . The time components do not contribute to thisamplitude because of the tensor structure of Eq. (3.1). The numerical integration error is easilycontrolled to be no more than 0 . w and | (cid:126) w | . Their values on the lattice sites are obtained by linear interpolation.
4. Results and Analysis
We have performed the lattice computation on four different ensembles, whose parameters arelisted in Table 1. All ensembles use the Iwasaki gauge action and Möbius domain wall fermions.For all ensembles except 48I, the dislocation-suppressing-determinant-ratio (DSDR) is also used toreduce the chiral symmetry breaking effects. For each configuration, we have 1024 or 2048 pointsource propagators whose sources are randomly distributed, and Coulomb gauge-fixed wall sourcepropagators with sources on every time slice. The number of point source propagators for eachensemble is listed in Table 1. a − (GeV) 1.015 1.015 1.37 1.73 m π (MeV) 140 140 143 139Configuration separation 10 10 10 20Configurations 47 47 61 31point sources 1024 2048 1024 1024 ∆ t
10 10 14 16
Table 1: Table of lattice ensembles used in this work. All ensembles are generated by theRBC/UKQCD collaborations [5]. Here, ∆ t is the time difference from the pion wall source at t to the closer current, as explained in SectionThe results for the real and imaginary parts of the decay amplitude calculated on these fourensembles are listed in Table 2. Note that only the contribution from the connected diagram isincluded. For the 24ID ensemble, the amplitude from the disconnected diagram is calculated andlisted in Table 3. A plot of amplitude is shown in Fig. 4.4 alculating the Two-photon Contribution to π → e + e − Decay Amplitude
Yidi ZhaoSource Im A (eV) Re A (eV)24ID 38.58(54) 23.06(40)32ID 39.80(36) 23.88(29)32IDF 36.17(47) 21.48(33)48I 35.26(57) 19.68(52)Experiment 35.07(37) 23.88(1.99) Table 2: Table for comparison among the lattice results and experimental results. The error inparenthesis is statistical. Experimental branching ratio of this decay after radiative correctionsis presented in [6]. The experimental value for imaginary part is obtained by combining opticaltheorem and the experimental pion life time. The experimental real part is calculated by subtractingthe imaginary part contribution from the total experimental decay rate.
Diagram Im A (eV) Re A (eV)24ID 38.58(54) 23.06(40)Disconnected -1.11(55) -0.62(40) Table 3: Contribution to amplitude from connected and disconnected diagrams for the 24ID en-semble. The error in parenthesis is statistical. I m ( e V ) Experimen al resul Experimen al error24ID a −1 = 1GeV32ID a −1 = 1GeV32IDF a −1 = 1.37GeV48I a −1 = 1.73GeV (a) Imaginary Part of Amplitude R e ( e V ) Experimen al resul Experimen al error24ID a −1 = 1GeV32ID a −1 = 1GeV32IDF a −1 = 1.37GeV48I a −1 = 1.73GeV (b) Real Part of Amplitude Figure 4: Plot of lattice results for decay amplitude v.s. cutoff in time direction.We choose the results obtained from the 48I ensemble because it has the smallest lattice spac-ing and use our other results to estimate the systematic error presented in Table 4. First of all, wedid not include the contribution from disconnected diagram. This error can be estimated from theamplitude of disconnected diagram from 24ID ensemble shown in Table 3. The second systematicerror is the finite volume error. We have two ensembles with the same lattice spacing but differ-ent spatial volume: the 24ID and 32ID ensembles. Assuming that finite volume error behaves as e − m π L , we can use the difference of 24ID and 32ID results to estimate the finite volume error for48ID. The third systematic error is finite lattice spacing error. We estimate this error to be 3% byan order O (( a Λ QCD ) ) counting with Λ QCD =
300 MeV. Another systematic error arises becausethe pion masses on the four ensembles are slightly larger than physical π mass. Using ChPT onecan show that the leading order of the hadronic factor is independent of pion mass. We ignore the5 alculating the Two-photon Contribution to π → e + e − Decay Amplitude
Yidi Zhao effects of pion mass on hadronic factor. The imaginary part of the leptonic factor has been workedout analytically in Eq. (3.3). The major contribution comes from the region where w is small. WithTaylor expansion, one can show that the leptonic factor is independent of pion mass in the limit M π w (cid:28)
1. Therefore, we ignore the impact of slight pion mass deviation on imaginary part. Thereal part of the leptonic factor is not analytic. Through numerical experiments, by comparing thedifference in the real part of the leptonic part after changing the pion mass, we estimate the errorin the leptonic factor in real part to be no more than 1%. Additionally, the measurement of renor-malization factor in Ref. [5], Z V = . ( ) also brings an error. Finally, the errors in leptonicfactor numeral integrals are easy to control and are ignored.We report our final result Im A = . ( )( . ) eV (4.1)Re A = . ( )( . ) eV , (4.2)where the first error is statistical and the second is systematic. Sources Im A (eV) Re A (eV)Finite volume 1.09 0.74Finite lattice spacing 1.06 0.59Disconnected diagram 1.01 0.53Pion mass 0 0.20 Z V Table 4: Sources of systematic error
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