Calculation of PCAC mass with Wilson fermion using gradient flow
Atsushi Baba, Shinji Ejiri, Kazuyuki Kanaya, Masakiyo Kitazawa, Asobu Suzuki, Hiroshi Suzuki, Yusuke Taniguchi, Takashi Umeda
CCalculation of PCAC mass with Wilson fermionusing gradient flow
UTHEP-745, UTCCS-P-130, J-PARC-TH-0212, KYUSHU-HET-204
Atsushi Baba ∗ , Asobu Suzuki Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki,305-8571, JapanE-mail: [email protected]
Shinji Ejiri
Department of Physics, Niigata University, Niigata 950-2181, Japan
Kazuyuki Kanaya
Tomonaga Center for the History of the Universe, University of Tsukuba, Tsukuba, Ibaraki,305-8571, Japan
Masakiyo Kitazawa
Department of Physics, Osaka University, Osaka 560-0043, JapanJ-PARC Branch, KEK Theory Center, Institute of Particle and Nuclear Studies, KEK, 203-1,Shirakata, Tokai, Ibaraki, 319-1106, Japan
Hiroshi Suzuki
Department of Physics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan
Yusuke Taniguchi
Center for Computational Science (CCS), University of Tsukuba, Tsukuba, Ibaraki, 305-8571,Japan
Takashi Umeda
Graduate School of Education, Hiroshima University, Higashihiroshima, Hiroshima 739-8524,Japan
We calculate the PCAC mass for ( + ) flavor full QCD with Wilson-type quarks. We adoptthe S mall F low- t ime e X pansion ( SF t X ) method based on the gradient flow which provides us ageneral way to compute correctly renormalized observables even if the relevant symmetries forthe observable are broken explicitly due to the lattice regularization, such as the Poincáre andchiral symmetries. Our calculation is performed on heavy u , d quarks mass ( m π / m ρ (cid:39) .
63) andapproximately physical s quark mass with fine lattice a (cid:39) .
07 fm. The results are compared withthose computed with the Schrödinger functional method. ∗ 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/ a r X i v : . [ h e p - l a t ] J a n alculation of PCAC mass with Wilson fermion using gradient flow Atsushi Baba
1. Introduction
Though the quark masses are fundamental parameters of QCD, they cannot be measured ex-perimentally since quarks are confined in hadrons. Here, non-perturbative calculation by latticeQCD plays an important role to determine the quark masses. When we calculate the renormalizedquark mass, the PCAC mass is often used. The PCAC mass is the quark mass parameter appearingin the PCAC relation, which is a chiral Ward-Takahashi identity given by (cid:104) | (cid:8) ∂ µ A a µ ( x ) + m f P a ( x ) (cid:9) O ( y ) | (cid:105) + (cid:104) | δ ax O ( y ) | (cid:105) = , (1.1)where m f is the PCAC mass and δ ax means the infinitesimal chiral transformation. Axial-vectorcurrent A a µ ( x ) and pseudo-scalar density P a ( x ) are defined by A a µ ( x ) = ¯ ψ f ( x ) γ γ µ T a ψ f ( x ) , (1.2) P a ( x ) = ¯ ψ f ( x ) γ T a ψ f ( x ) , (1.3)and O ( x ) is an operator which we can set arbitrarily. When we set O ( y ) = P a ( y ) and integrate overthe spacial coordinates, we obtain a PCAC relation (cid:10) ∂ A a µ ( x ) P a ( ) (cid:11) = − m f (cid:104) P a ( x ) P a ( ) (cid:105) , (1.4)with which we can calculate the PCAC mass by m f = − (cid:10) ∂ A a ( x ) P a ( ) (cid:11) (cid:104) P a ( x ) P a ( ) (cid:105) . (1.5)Recently, a new use of the gradient flow method [1–4] was proposed to calculate correctlyrenormalized observables [5, 6]. The new method is called Small Flow- t ime eXpansion (SF t X)method . Making use of the finiteness of flowed operators, non-perturbative estimates of observ-ables are extracted by taking a vanishing flow-time extrapolation. The SF t X method was firstapplied to evaluate the energy-momentum tensor for which the explicit violation of the Poincaréinvariance on the lattice has been a hard obstacle in obtaining a non-perturbative estimate [7, 8].Because the SF t X method is generally applicable to any observables including chiral observ-ables [6], we are applying it to QCD with dynamical quarks [8, 9]. In this paper, we study thePCAC mass by the SF t X method in QCD with ( + ) -flavors of improved Wilson quarks.
2. SF t X method
In this study, we adopt the simplest gradient flow for the gauge field [2]: ∂ t B µ ( t , x ) = D ν G νµ ( t , x ) , B µ ( , x ) = A µ ( x ) , (2.1)where the field strength G νµ and the covariant derivative D ν are defined in terms of the flowedgauge field B µ . The flow equations for quarks are given by [4]: ∂ t χ f ( t , x ) = D χ f ( t , x ) , χ f ( , x ) = ψ f ( x ) , (2.2) ∂ t ¯ χ f ( t , x ) = ¯ χ f ( t , x ) (cid:126) D , ¯ χ f ( , x ) = ¯ ψ f ( x ) , (2.3)1 alculation of PCAC mass with Wilson fermion using gradient flow Atsushi Baba with D µ χ f ( t , x ) = (cid:0) ∂ µ + B µ ( t , x ) (cid:1) χ f and ¯ χ f ( t , x ) (cid:126) D µ = ¯ χ f ( t , x ) (cid:16) (cid:126) ∂ µ − B µ ( t , x ) (cid:17) .In terms of the flowed fields, the correctly renormalized axial-vector current and pseudo-scalardensity in the MS scheme at µ = A a µ ( x ) = lim t → A a µ ( t , x ) = lim t → c A ( t ) ϕ f ( t ) ¯ χ f ( t , x ) γ γ µ T a χ f ( t , x ) , (2.4) P a ( x ) = lim t → P a ( t , x ) = lim t → c S ( t ) ϕ f ( t ) ¯ χ f ( t , x ) γ T a χ f ( t , x ) , (2.5)where the matching coefficients c A ( t ) , c S ( t ) and fermion wave function renormalization factor ϕ ( t ) are c A ( t ) = (cid:26) + ¯ g ( µ (cid:48) )( π ) (cid:20) − +
43 ln 432 (cid:21)(cid:27) , (2.6) c S ( t ) = (cid:26) + ¯ g ( µ (cid:48) )( π ) (cid:20) (cid:0) ln (cid:0) t µ (cid:48) (cid:1) + γ E (cid:1) + +
43 ln 432 (cid:21)(cid:27) ¯ m f ( µ (cid:48) ) ¯ m f ( ) , (2.7) ϕ f ( t ) = − ( π ) t (cid:68) ¯ χ ( t , x ) ↔ / D χ ( t , x ) (cid:69) , (2.8)where ¯ g ( µ (cid:48) ) and ¯ m ( µ (cid:48) ) are running coupling and running mass in the MS scheme at the renor-malization scale µ (cid:48) ( t ) , and γ E is the Euler-Mascheroni constant. Then, the PCAC mass is givenby m f = lim t → m f ( t ) = − lim t → c A ( t ) c S ( t ) ϕ f ( t ) (cid:10) ∂ A a ( t , x ) P a ( t , ) (cid:11) c S ( t ) ϕ f ( t ) (cid:104) P a ( t , x ) P a ( t , ) (cid:105) . (2.9)Final results of m f should be independent of the scale µ (cid:48) ( t ) as far as it is O ( / √ t ) to preservethe quality of the perturbation theory. A conventional choice is µ (cid:48) = µ d ( t ) ≡ / √ t , which isa natural scale of flowed observables because the gradient flow smears the fields over a physicalextent of ∼ √ t [2]. Recently, a new choice was proposed by Harlander et al. as µ (cid:48) = µ ( t ) ≡ / √ e γ E t [10]. Because µ (cid:39) . µ d , we expect that, in asymptotically free theories, the range of t in which the perturbative expansion is well applicable is extended towards larger t with the µ -scalethan the µ d -scale. In a study of Ref. [11] on the energy momentum tensor and chiral observables infinite-temperature QCD, we found that the wider range of t with the µ -scale is helpful in reducingsystematic uncertainties from the t → µ -scale also in this study.We evaluate Eq. (2.9) non-perturbatively by performing lattice simulations. The original pro-cedure of the SF t X method is to take the continuum limit a → t correction to the flowed PCAC mass will be m f ( t ) = m f + tA + O ( t ) , where A is the contamina-tion from dimension-five operators. In Refs. [8, 9], an alternative procedure to take t → O ( a ) we will have additionalcontaminations like m f ( t , a ) = m f ( t ) + O ( a / t , a T , a m , a Λ ) . (2.10)Among the O ( a ) terms, the term O ( a / t ) is singular in the t → t , “linear window”, a range of t in which terms like O ( a / t ) and2 alculation of PCAC mass with Wilson fermion using gradient flow Atsushi Baba
60 65 70 75 80 0 10 20 30 40 50 P C A C m a ss [ M e V ] x flowtime=0.5flowtime=1.0flowtime=1.5flowtime=2.0 100 105 110 115 120 125 130 135 0 10 20 30 40 50 P C A C m a ss [ M e V ] x flowtime=0.5flowtime=1.0flowtime=1.5flowtime=2.0 Figure 1:
PCAC mass of u quark ( left ) and s quark ( right ) as function of the Euclidean time x , at flow-time t / a = . . .
5( cyan) and 2 . µ -scale was adopted. Vertical linesindicate the range of constant fit. Fit range is the same for all flow-times. Errors are statistical only, estimatedby the jackknife method. O ( t ) are not dominating, and taking a t → a → O ( a T , a m , a Λ ) lattice artifacts. We may check the validity of the linear windows byperforming non-linear fits including O ( a / t ) and O ( t ) terms. The difference between the linearand non-linear fits gives an estimate of the systematic error due to the fit ansatz. See Ref. [8] formore details.
3. Lattice Setup
We study ( + ) -flavor QCD adopting a non-perturbatively O ( a ) -improved Wilson quark ac-tion and the RG-improved Iwasaki gauge action. We choose a set of CP-PACS+JLQCD configura-tions generated at β = .
05 corresponding to a (cid:39) .
07 fm, degenerate u , d quark mass correspond-ing to m π / m ρ (cid:39) .
63, and almost physical s quark mass corresponding to m η ss / m φ (cid:39) .
74 on a28 ×
56 lattice [12]. At this simulation point, the O ( a ) -improvement of axial-vector current c A isculculated by CP-PACS/JLQCD and ALPHA Collaborations as c A = − . ( ) [13]. We em-ploy the 5-loop order β -function [14] and anomalous dimension [15] to calculate running couplingand running mass in matching coefficients c A ( t ) and c S ( t ) .In the study of Ref. [12], PCAC masses at each simulation points have been calculated by theSchrödinger functional method. The bare PCAC quark masses at the simulation point of this studyusing the same configurations are am ud = . ( ) and am s = . ( ) , which correspondto m SFu = . ± . , m SFs = . ± . , (3.1)in MeV unit.
4. Numerical results
In Fig. 1, we show the PCAC mass for u and d quarks computed with the µ -scale as function3 alculation of PCAC mass with Wilson fermion using gradient flow Atsushi Baba
40 45 50 55 60 65 70 75 80 0 0.5 1 1.5 2 P C A C m a ss [ M e V ] t/a u quarkScale µ d Scale µ
50 60 70 80 90 100 110 120 130 140 0 0.5 1 1.5 2 P C A C m a ss [ M e V ] t/a s quarkScale µ d Scale µ Figure 2:
PCAC mass of u quark ( left ) and s quark ( right ) as function of the flow-time time t / a . Reddiamonds and black circles are the results of µ d - and µ -scales, respectively.
65 70 75 80 85 90 0 0.5 1 1.5 2 P C A C m a ss [ M e V ] t/a u quark PCAC masslinear fitnonlinear fitSF scheme
110 115 120 125 130 135 140 145 150 0 0.5 1 1.5 2 P C A C m a ss [ M e V ] t/a s quark PCAC masslinear fitnonlinear fitSF scheme
Figure 3:
PCAC mass of u quark ( left ) and s quark ( right ) as function of flow-time. The µ -scale isadopted. Vertical lines indicate the linear window we adopt. We take t → t / a ∼ of Euclidean time at four different flow-times. We perform constant fits at each flow-time withinthe range indicated by the vertical lines. We use the same fit range for all flow-times. The resultsof the fits are shown by colored bands. The errors of the constant fits are statistical only, estimatedusing the jackknife method.In Fig. 2, we show the PCAC mass as function of flow-time. Red diamonds and black circlesare the results with µ d - and µ -scales, respectively. We see that, with the conventional µ d -scale, itis not well unambiguous to identify a linear window due to the bend at large t , and thus the t → µ -scale, wesee a linear behavior in a wider range of t , which enables us to carry out a much more stable andreliable t → µ -scale to calculate the PCAC masses.The results of PCAC masses with µ -scale are summarized in Fig. 3 as function of flow-time t / a for the u quark (left panel) and for the s quark (right panel). We identify linear windowsas follows: First of all, we require the flow-time to satisfy a ≤ √ t ≤ min ( N t a / , N s a / ) , i.e. ,the smearing range √ t by the gradient flow should be larger than the minimal lattice sepatration4 alculation of PCAC mass with Wilson fermion using gradient flow Atsushi Baba to make the smearing effective, and smaller than the half of the smallest lattice extent to avoidfinite-size effects due to overlapped smearing. We then look for a range of t in which terms linearin t look dominating, and try linear extrapolation with various choices of the fitting range. Wethen select a (temporally) best linear fit whose fitting range is the widest under the condition that χ / N dof is smaller than a cutoff value. In this study, due to limitation of the statistics, we disregardcorrelations among data at different t . Thus the absolute value of χ / N dof does not have a strongsense — we vary the cutoff value widely. In this study, consulting the stability of the fit results, wechoose 1.0 as the cutoff value for PCAC mass. The linear window we adopt is shown by the twovertical lines in Fig. 3.To confirm the validity of the linear window and to estimate a systematic error due to the fitansatz, we also make additional non-linear fit of the form m f ( t , a ) = m f + tA + t B + a t C , using thedata within the linear window. Results of linear and non-linear fits are shown by red and blue linesin Fig. 3. We adopt the results of the linear fits as central values and take the difference betweenthe two fits as an estimate of the systematic error due to the fit ansatz. Our results of the PCACmasses are m SF t X u = . ± . , m SF t X s = . ± . , (4.1)in MeV unit, where statistical error and systematic error due to fit ansatz are included.
5. Summary and outlook
We studied PCAC mass in lattice QCD with ( + ) -flavors of dynamical Wilson quarks. Non-perturbative renormalization is carried out by the SF t X method based on the gradient flow. Our cal-culation was performed at heavy u , d quarks mass ( m π / m ρ (cid:39) .
63) and approximately physical s quark mass on a fine lattice with a (cid:39) .
07 fm. As the renormalization scale in the SF t X method, weadopt the recently proposed µ -scale. We found that the µ -scale is helpful to reduce uncertaintyin the t → u (or d ) quark and s quark are given in Eq. (4.1). Theseare consistent with the results of conventional Schrödinger functional method, Eq. (3.1), obtainedon the same configurations. By virtue of the gradient flow, statistical error is well suppressedcompered with the results of the Schrödinger functional method.We are extending the study to ( + ) -flavor QCD with physical u , d and s quarks [11]. Toobtain final results, we also have to repeat the calculation at different lattice spacings to carry outthe continuum extrapolation. Acknowledgments
This work was in part supported by JSPS KAKENHI Grant Numbers JP19K03819, JP19H05146,JP19H05598, JP18K03607, JP17K05442 and JP16H03982. This research used computational re-sources of COMA, Oakforest-PACS, and Cygnus provided by the Interdisciplinary ComputationalScience Program of Center for Computational Sciences, University of Tsukuba, K and other com-puters of JHPCN through the HPCI System Research Projects (Project ID:hp17208, hp190028,hp190036) and JHPCN projects (jh190003, jh190063), OCTOPUS at Cybermedia Center, Osaka5 alculation of PCAC mass with Wilson fermion using gradient flow
Atsushi Baba
University, and ITO at R.I.I.T., Kyushu University. The simulations were in part based on the latticeQCD code set Bridge++ [16].
References [1] R. Narayanan and H. Neuberger,
JHEP , 064 (2006)[2] M. Lüscher,
JHEP , 071 (2010), Erratum: [
JHEP , 092 (2014)][3] M. Lüscher and P. Weisz,
JHEP , 051 (2011)[4] M. Lüscher,
JHEP , 123 (2013)[5] H. Suzuki,
PTEP , 083B03 (2013), Erratum: [
PTEP , 079201 (2015)][6] T. Endo, K. Hieda, D. Miura and H. Suzuki,
PTEP , 053B03 (2015)[7] M. Kitazawa, T. Iritani, M. Asakawa, T. Hatsuda and H. Suzuki,
Phys. Rev. D , 114152 (2016)[8] Y. Taniguchi, S. Ejiri, R. Iwami, K. Kanaya, M. Kitazawa, H. Suzuki, T. Umeda and N. Wakabayashi, Phys. Rev. D , 014509 (2017)[9] Y. Taniguchi, K. Kanaya, H. Suzuki and T. Umeda, Phys. Rev. D , 054502 (2017)[10] R.V. Harlander, Y. Kluth and F. Lange, Eur. Phys. J. C , 944 (2018)[11] K. Kanaya, A. Baba, A. Suzuki, S. Ejiri, M. Kitazawa, H. Suzuki, Y. Taniguchi and T. Umeda, PoS(Lattice2019) 088. [arXiv:1910.13036 [hep-lat]][12] T. Ishikawa, S. Aoki, M. Fukugita, S. Hashimoto, K-I. Ishikawa, N. Ishizuka, Y. Iwasaki, K. Kanaya,T. Kaneko, Y. Kuramashi, M. Okawa, Y. Taniguchi, N. Tsutsui, A. Ukawa, N. Yamada and T. Yoshie(CP-PACS and JLQCD Collaborations), Phys. Rev. D , 011502(R) (2008)[13] T. Kaneko, S. Aoki, M. Della Morte, S. Hashimoto, R. Hoffmann and R. Sommer (CP-PACS/JLQCDand ALPHA Collaborations), JHEP (2007) 092[14] P.A. Baikov, K.G. Chetyrkin and J.H. Kühn,
Phys. Rev. Lett. , 082002 (2017)[15] P.A. Baikov, K.G. Chetyrkin and J.H. Kühn,
JHEP (2014) 076[16] http://bridge.kek.jp/Lattice-code/index_e.htmlhttp://bridge.kek.jp/Lattice-code/index_e.html