K l3 form factors in N f =2+1 QCD at physical point on large volume
Junpei Kakazu, Ken-ichi Ishikawa, Naruhito Ishizuka, Yoshinobu Kuramashi, Yoshifumi Nakamura, Yusuke Namekawa, Yusuke Taniguchi, Naoya Ukita, Takeshi Yamazaki, Tomoteru Yoshié
aa r X i v : . [ h e p - l a t ] F e b K l form factors in N f = + QCD at physical pointon large volume
J. Kakazu ∗ , K.-I. Ishikawa , , N. Ishizuka , , Y. Kuramashi , , Y. Nakamura ,Y. Namekawa , Y. Taniguchi , , N. Ukita , T. Yamazaki , , and T. Yoshie , (PACS Collaboration) Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki305-8571, Japan Graduate School of Science, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526,Japan Core of Research for the Energetic Universe, Hiroshima University, Higashi-Hiroshima739-8526, Japan Center for Computational Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan RIKEN Center for Computational Science, Kobe, Hyogo 650-0047, Japan Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization(KEK), Tsukuba 305-0801, JapanE-mail: [email protected]
We present our results of the K l form factors on the volume whose spatial extent is more than L =
10 fm, with the physical pion and kaon masses using the stout-smearing clover N f = + a − ≈ . K l form factor at zero momentumtransfer is obtained from fit based on the next-to-leading (NLO) formula in SU(3) chiral perturba-tion theory. We estimate systematic errors of the form factor, mainly coming from the finite latticespacing effect. We also determine the value of | V us | by combining our result with the experimentand check the consistency with the standard model prediction. The result is compared with theprevious lattice calculations. ∗ Speaker. 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/ l form factors in N f = + QCD at physical point on large volume
J. Kakazu
1. Introduction
Semileptonic decay of kaon into pion and lepton-neutrino pair, so-called K l decay plays animportant role to determine V us , which is one of the Cabbibo-Kobayashi-Maskawa (CKM) ma-trix [1] elements to explain mixture between up and strange quarks. | V us | is necessary to examinethe existence of physics beyond the standard model (BSM). In the standard model, the unitary con-dition of the up quark part in the CKM matrix leads to vanishment of ∆ u ≡ | V ud | + | V us | + | V ub | −
1. Since | V ud | has been obtained accurately and | V ub | is tiny, high precision determination of | V us | is required for a check of ∆ u = | V us | is the combination of experimental resultsand lattice calculations [2, 3]. However, there are some uncertainties, for instance, chiral extrapo-lations to the physical pion and kaon masses and finite size effect. To recognize the BSM signa-ture, these uncertainties should be reduced. Hence we determine the K l form factor in dynamical N f = + . The detailed analyses in this study are presented in Ref. [4].
2. Calculation of form factors
The K l form factors f + ( q ) and f − ( q ) are defined by the matrix element of the weak vectorcurrent as, h K ( ~ p ′ ) (cid:12)(cid:12) V µ (cid:12)(cid:12) π ( ~ p ) i = ( p ′ + p ) µ f + ( q ) + ( p ′ − p ) µ f − ( q ) , (2.1)where V µ is weak vector current and q = p ′ − p is the momentum transfer. The scalar form factor f ( q ) is defined by combination of vector form factors, f ( q ) = f + ( q ) + − q ( m K − m π ) f − ( q ) = f + ( q ) (cid:18) + − q ( m K − m π ) ξ ( q ) (cid:19) , (2.2)where ξ ( q ) = f − ( q ) / f + ( q ) . At q =
0, the two form factors, f + ( q ) and f ( q ) give the samevalue, f + ( ) = f ( ) .To obtain the form factors, we calculate the meson 3-point function with the weak vectorcurrent C π K µ ( ~ p ,~ p ′ , t ) , which is given by C π K µ ( ~ p ,~ p ′ , t ) = h | O K ( ~ p ′ , t f ) V µ ( ~ q , t ) O † π ( ~ p , t i ) | i (2.3) = Z K ( ~ p ′ ) Z π ( ~ p ) E K ( ~ p ′ ) E π ( ~ p ) Z V h K ( ~ p ′ ) (cid:12)(cid:12) V µ (cid:12)(cid:12) π ( ~ p ) i e − E K ( ~ p ′ )( t f − t ) e − E π ( ~ p )( t − t i ) + · · · , (2.4)where Z V is the renormalization factor of the vector current, and t i < t < t f . E π and E K denotethe energy of pion and kaon, respectively. Their energies are determined by the equation E X = q m X + ( π L ~ n ) using the fitted mass m X with the label X assigned to π or K , where L is the spatialextent and ~ n is integer vector which represents the direction of meson’s momentum. m X and Z X ( ~ ) are evaluated from the meson 2-point functions given by C X ( ~ , t ) = h | O X ( ~ , t ) O † X ( ~ , t i ) | i = | Z X ( ~ ) | m X ( e − m X | t − t i | + e − m X ( T −| t − t i | ) ) + · · · , (2.5)1 l form factors in N f = + QCD at physical point on large volume
J. Kakazu where T is the temporal extent. The periodicity in the temporal direction is effectively doubledthanks to averaging the 2-point functions with the temporal periodic and anti-periodic conditions.The terms of the dots ( · · · ) denote the contributions from excited states.For construction of the form factors, we define the quantity R µ ( ~ p ,~ p ′ , t ; t sep ) which consists of2- and 3-point functions in t sep ≡ | t f − t i | , R µ ( ~ p ,~ p ′ , t ; t sep ) = C π K µ ( ~ p ,~ p ′ , t ) C π ( ~ p , t ) C K ( ~ p ′ , t ) t i ≪ t ≪ t f −−−−→ A µ ( ~ p ,~ p ′ ) + B π , µ ( ~ p , t ) + B K , µ ( ~ p ′ , t ) + · · · . (2.6)The term A µ is proportional to matrix element h K ( ~ p ′ ) (cid:12)(cid:12) V µ (cid:12)(cid:12) π ( ~ p ) i , and the terms B X , µ are the firstexcited state contributions of meson X and ( · · · ) denotes the other excited state contributions. Afterextracting A µ , we could obtain the vector form factor f + ( q ) and f − ( q ) by the combination withthe relation Eq. (2.1).
3. Simulation setup
We use the configurations which were generated at the physical point, m π = .
135 GeV, onthe large volume corresponding to La = Ta = . L = T = N f = + ρ = .
1) and theimprovement coefficient c SW = .
11, and the Iwasaki gauge action [7] at β = .
82 corresponding to a − = . ( ) GeV. The hopping parameters of two degenerate light quarks and strange quarkare ( κ ud , κ s ) = ( . , . ) , respectively.In the calculation of the form factors, we use 20 configurations in total. We adopt 8 sourcesin time with 16 time separation per configuration, and 4 choices of the temporal axis thanks tothe hypercube lattice. In the calculation of 2- or 3-point functions, we use Z ( ) ⊗ Z ( ) randomwall source which is spread in the spatial sites, and also color and spin spaces [8]. We choosethe three temporal separations t sep = , ,
48 ( ≈ . , . , .
05 fm) to dominate A µ ( ~ p ,~ p ′ ) forsufficiently large t sep . One random source is used in the calculations of t sep =
36 and two sourcesfor the others. The 3-point function is calculated using the sequential source technique at the sinktime slice t f , where the meson momentum is fixed to zero. We calculate C π K µ ( ~ p ,~ p ′ , t ) and C X ( ~ p , t ) with the momentum ~ p = ( π / L ) ~ n of n ≡ | ~ n | ≤
6. For suppression of the wrapping around effectof the 3-point functions, which is similar to that in Ref. [9], we average the 3-point functions withthe periodic and anti-periodic boundary conditions in the temporal direction.
4. Result
We extract A µ by fitting R µ at each momentum transfer, q n = − ( m K + m π − m K q m π + n ( π L ) ) ,with the fit form considering first excited state contributions R µ ( ~ p ,~ p ′ , t ; t sep ) = A µ + B µ exp ( − t ( ∆ E π )) + C µ exp ( − ( t f − t )( ∆ m K )) , (4.1) ∆ E π = ( E π ∗ − E π ) , ∆ m K = ( m K ∗ − m K ) , where E π ∗ = q m π ∗ + ( π L ~ n ) is the energy of the first excited state pion. We apply the experimentalvalues of first excited state pion and kaon m π ∗ = . m K ∗ = . l form factors in N f = + QCD at physical point on large volume
J. Kakazu
Figure 1 indicates the fit result at ~ p = π / L ( q = − ( m K + m π − m K q m π + ( π L ) ) ) combinedall the t sep with the fit form in Eq. (4.1). We use the fit range of 7 ≤ t ≤ ( t sep − ) in R (the leftpanel in Fig.1), and use the range of 6 ≤ t ≤ ( t sep − ) in R i (the right panel in Fig.1). (a) Temporal component A t R A t sep =36t sep =42t sep =48 (b) Spatial component A i t R i A i t sep =36t sep =42t sep =48 Figure 1: Extraction of A µ from the quantity R µ when | ~ p | = π / L . Black circle, red squareand green diamond symbols in both panels represent the quantities R µ ( ~ p ,~ , t ; 36 ) , R µ ( ~ p ,~ , t ; 42 ) , R µ ( ~ p ,~ , t ; 48 ) , respectively. Bold fit curves are drawn by Eq. (4.1). Cyan bands represent A µ withthe statistical error.After constructing the form factors from the extracted A µ , we employ formulae based on theNLO SU(3) ChPT [10] for interpolation to q = f + ( q ) = − q F L ( µ ) + H K π ( − q ) + H K η ( − q ) + c + + c + ( − q ) , (4.2) f ( q ) = − q F L ( µ ) − π F (cid:18) q + Σ π K − ∆ π K q (cid:19) ¯ J π K ( − q ) − π F (cid:18) q + Σ π K − ∆ π K q (cid:19) ¯ J K η ( − q ) − q ∆ π K ( µ π − µ K − µ η )+ c + c ( − q ) , (4.3)where Σ XY = m X + m Y , ∆ XY = m X − m Y , µ X = m X π F ln (cid:16) m X µ (cid:17) , and L ( µ ) , L ( µ ) are the lowenergy constants of the NLO SU(3) ChPT. H XY , ¯ J XY , ( X , Y = π , K , η ) represent one-loop integralsin Ref. [10]. µ = .
77 GeV is the renormalization scale. F = .
079 GeV is the decay constant inthe SU(3) chiral limit. The NLO ChPT formulae have only trivial terms at q =
0, thus we add thenontrivial term c + , . We also add the c + , ( − q ) as the NNLO analytic term has − q dependence.3 l form factors in N f = + QCD at physical point on large volume
J. Kakazu (a) q dependence -0.15 -0.1 -0.05 0 0.05 q [GeV ] f ( q ) , f + ( q ) f (q max2 )f (q )f + (q ) (b) near q = -0.005 0 0.005 0.01 0.015 0.02 0.025 q [GeV ] f ( q ) , f + ( q ) f (q )f + (q )f + (0)=f (0) Figure 2: (a) q dependence of K l form factors. The square and circle symbols denote f + ( q ) and f ( q ) , respectively. The dashed and dot-dashed curves represent the simultaneous fit resultof f + ( q ) and f ( q ) , respectively, with the ChPT forms in Eqs.(4.2) and (4.3) added the analyticterms. The orange triangle at the far left in Fig. (a) represents f ( q min ) , where q min = − ( m K − m π ) .The diamond symbol denotes the fit result of f + ( ) = f ( ) . Fig. (b) is the magnification of theleft panel near q = f + ( q ) and f ( q ) using the ChPT forms inEqs. (4.2) and (4.3). The datum of f ( q min ) is not included in the fit. We also check that theinclusion of the datum in the fit does not change the fit result qualitatively. The ChPT forms welldescribe our data, and χ / d . o . f . ≈ .
06 in the uncorrelated fit.We investigate systematic errors in f + , ( ) . The largest error is from the discretization. Thediscrepancy of Z V is regarded as a systematic error coming from the finite lattice spacing effect,and it must vanish in the continuum limit. We obtain Z V of weak current using two methods. One isdetermination of Z V using electromagnetic current conservation. We obtain the quantities from thecombination of 3-point functions of t sep =
36 with electromagnetic current and 2-point functionsof pion and kaon, and estimate by geometric mean of these. This yields Z V = . ( ) . Theother is Schrödinger functional method, resulting in Z V = . ( ) [12]. The difference of theform factor by choice of Z V is 0.45%. The systematic error from finite size effects could be ignoredbecause of O ( e − m π L ) ≈ . f + , ( ) = . ( )( ) , (4.4)where the first error is statistical error and the second is the discrepancy of Z V .After interpolating to q = | V us | by combining the value | V us | f + ( ) = . ( ) derived from the K l decay rate [18] as | V us | = . ( )( )( ) , (4.5)where the first error is statistical error, and the second comes from the choice of Z V and the thirdcomes from experiment. Figure 3 shows the comparison of our results from the ChPT with the4 l form factors in N f = + QCD at physical point on large volume
J. Kakazu
PDG’s estimations [2], other lattice results [8, 13–17], and the one from the unitarity condition ∆ u =
0. Our result with total error covers the SM prediction which is estimated by the combination ∆ u = | V ud | from Ref. [19] (ignoring | V ub | because of too small effect). The largest deviationfrom the previous lattice calculation of continuum limit is 1 . σ . |V us | PDG result (FF)ETM Nf=2+1+1FNAL/MILC Nf=2+1+1RBC/UKQCD Nf=2+1FNAL/MILC Nf=2+1JLQCD Nf=2+1ETM Nf=2PACS Nf=2+1 (decay const.)PDG result (decay const.)
This work
Form factorDecay constant ratio continuum limitsingle lattice spacing
Form factor
Figure 3: Comparison of | V us | . The filled square symbol is our result from the K l form factor withthe ChPT fit. The filled diamond symbol is estimated by the decay constant ratio calculated withthe same configuration [5]. The star and cross symbols express the PDG’s results [2] from the K l form factor and the ratio of the decay constant, respectively. Brown band is the SM prediction from ∆ u = K l form factor [8, 13–17].We also confirm the consistency on the slope of the form factors between our results andexperimental results. We estimate the slopes of the form factors λ + , defined by λ s = m π ± phys f s ( ) d f s ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = − q = , (4.6)where label s is assigned to + or 0. Our results from the ChPT fit are λ + = . ( ) × − , λ = . ( ) × − . (4.7)They are consistent with the experimental results λ + = . ( ) × − and λ = . ( ) × − [18].
5. Summary
We present the results of the K l form factors f + ( q ) and f ( q ) in N f = + m π ≈ .
135 GeV) on the large volume lattice with (10 fm) . Since we calculate The result of the latest determination of | V ud | = . ( ) [20], the | V us | estimation from ∆ u = | V us | = . ( ) . The difference from our result is 1.7 σ . l form factors in N f = + QCD at physical point on large volume
J. Kakazu the form factors in close to zero momentum transfer, we perform an interpolation to q = q = | V us | is estimated by combining the form factor with the K l decay rate. Our result is consistentwith the SM prediction and is slightly larger than previous lattice results in the continuum limit.The choice of Z V , which is regarded as the systematic error from the finite lattice spacing effect,is the largest systematic error. Thus, an important future work is to evaluate form factors at one ormore finer lattice spacings for taking the continuum limit. Acknowledgement
Numerical calculations in this work were performed on the Oakforest-PACS at Joint Center forAdvanced High Performance Computing under Multidisciplinary Cooperative Research Programof Center for Computational Sciences, University of Tsukuba. A part of the calculation employedOpenQCD system This work is supported in part by Grants-in-Aid for Scientific Research fromthe Ministry of Education, Culture, Sports, Science and Technology (MEXT) (Nos.16H06002,18K03638, 19H01892).
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