We employ the chirally rotated Schr"odinger functional ($chi$SF) to study

two-point fermion bilinear correlation functions used in the determination of

$Z_A,V,S,P,T$ on a series of well-tuned ensembles. The gauge configurations,

which span renormalisation scales from 4 to 70~GeV, are generated with $N_
m

f=3$ massless flavors and Schr"odinger Functional (SF) boundary conditions.

Valence quarks are computed with $chi$SF boundary conditions. We show

preliminary results on the tuning of the $chi$SF Symanzik coefficient $z_f$

and the scaling of the axial current normalization $Z_
m A$. Moreover we

carry out a detailed comparison with the expectations from one-loop

perturbation theory. Finally we outline how automatically

$mathrmO(a)$-improved $B_
m K$ matrix elements, including BSM

contributions, can be computed in a $chi$SF renormalization scheme

Campos, I

Dalla Brida, M

DE DIVITIIS, GM

Lytle, A

Papinutto, M

Vladikas, A

We employ the chirally rotated Schr"odinger functional ($chi$SF) to study two-point fermion bilinear correlation functions used in the determination of $Z_{A,V,S,P,T}$ on a series of well-tuned ensembles. The gauge configurations, which span renormalisation scales from 4 to 70~GeV, are generated with $N_{mf}=3$ massless flavors and Schr"odinger Functional (SF) boundary conditions. Valence quarks are computed with $chi$SF boundary conditions. We show preliminary results on the tuning of the $chi$SF Symanzik coefficient $z_f$ and the scaling of the axial current normalization $Z_{m A}$. Moreover we carry out a detailed comparison with the expectations from one-loop perturbation theory. Finally we outline how automatically $mathrm{O}(a)$-improved $B_{m K}$ matrix elements, including BSM contributions, can be computed in a $chi$SF renormalization scheme

Isabel Campos

Mattia Dalla Brida

Giulia Maria de Divitiis

Andrew Lytle

Mauro Papinutto

Anastassios Vladikas

Archivio della ricerca- Università di Roma La Sapienza

PoS(LATTICE2019)202χSF near the electroweak scaleIsabel CamposInstituto de Física de Cantabria - IFCA-CSIC, Avda. de Los Castros s/n, 39005 Santander, SpainMattia Dalla BridaUniversità di Milano-Bicocca, Dipartimento di Fisica, and INFN, sezione di Milano-Bicocca,Piazza della Scienza 3, I-20126 Milano, ItalyGiulia Maria de DivitiisUniversità di Roma “Tor Vergata”, Dipartimento di Fisica and INFN sezione di RomaTor Vergata, Via della Ricerca Scientifica 1, I-00133 Rome, ItalyAndrew Lytle∗INFN, Sezione di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma RM, ItalyE-mail: andrew.lytle@roma2.infn.itMauro PapinuttoUniversità di Roma “La Sapienza”, Dipartimento di Fisica and INFN sezione di Roma1,Piazzale Aldo Moro 2, I-00185 Roma, ItalyAnastassios VladikasINFN, sezione di Roma Tor Vergata, c/o Dipartimento di Fisica, Via della Ricerca Scientifica 1,I-00133 Rome, ItalyWe employ the chirally rotated Schrödinger functional (χSF) to study two-point fermion bilinearcorrelation functions used in the determination of ZA,V,S,P,T on a series of well-tuned ensembles.The gauge configurations, which span renormalisation scales from 4 to 70 GeV, are generated withNf = 3 massless flavors and Schrödinger Functional (SF) boundary conditions. Valence quarksare computed with χSF boundary conditions. We show preliminary results on the tuning of theχSF Symanzik coefficient z f and the scaling of the axial current normalization ZA. Moreover wecarry out a detailed comparison with the expectations from one-loop perturbation theory. Finallywe outline how automatically O(a)-improved BK matrix elements, including BSM contributions,can be computed in a χSF renormalization scheme.The 37th Annual International Symposium on Lattice Field Theory - LATTICE201916-22 June, 2019Wuhan, China.∗Speaker.c© 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/PoS(LATTICE2019)202χSF near the electroweak scale Andrew LytleThe chirally rotated Schrödinger functional (χSF) with massless Wilson fermions is a lat-tice regularization which endows the Schrödinger functional (SF) with the property of automaticO(a)-improvement. The χSF framework is effective in reducing lattice artefacts in correlationand step scaling functions, but especially it offers new strategies to study and simplify the patternof renormalization. The price to pay for the automatic O(a)-improvement is the nonperturbativetuning of coefficients of new boundary counterterms. This tuning is the first phase of a long-termproject, aiming at the computation of BK low-energy contributions beyond the Standard Model(BSM), with Wilson fermion Nf = 2+1 lattice QCD in a non-unitary (mixed-action) framework.1. The χSF setupFollowing ref. [1], the fermion flavour doublet ψ =(ψuψd)satisfies χSF boundary conditionsQ̃± ≡12(1± iγ0γ5τ3){Q̃+ψ(x)|x0=0 = 0 Q̃−ψ(x)|x0=T = 0ψ̄(x)Q̃+|x0=0 = 0 ψ̄(x)Q̃−|x0=T = 0in time and periodic ones in space. The massless fermion action isS f = a4T∑x0=0∑xψ̄(x)(DW +δDW )ψ(x) ,with DW the standard Wilson fermion matrix and the boundary termδDW ψ(x) = (δx0,0 +δx0,T )[(z f −1)+(ds−1)aDs]ψ(x) .The χSF boundary conditions can be derived from the standard SF boundary conditions by apply-ing a non-anomalous chiral flavour rotation on the fermion doublet as follows:R = exp(iα2γ5τ3)∣∣∣α=π/2{ψ → ψ ′ = Rψψ̄ → ψ̄ ′ = ψ̄R .Consequently composite operators, which depend on fermion fields, are also rotated:O[ψ, ψ̄]→ Q[ψ, ψ̄] = O[Rψ, ψ̄R ] .SF and χSF correlation functions of composite operators O,Q defined in the bulk and O,Q definedon a time boundary obey the following universality relation:〈OO〉cont(SF) = lima→0[ZOZO 〈OO〉(SF)+O(a)]= lima→0[ZQZQ 〈QQ〉(χSF)+O(a2)] .Note that χSF incorporates automatic O(a) improvement. The price to pay is the introduction ofboundary counterterms. We tune non-perturbatively the boundary counterterm coefficient z f , whilewe fix the others at their tree-level value.1PoS(LATTICE2019)202χSF near the electroweak scale Andrew Lytle1.1 Correlation functionsAs in ref. [2], we consider the set of fermion bilinear operators; e.g.V f1 f2µ (x) = ψ f1(x)γµψ f2(x), Af1 f2µ (x) = ψ f1(x)γµγ5ψ f2(x),with flavours f1, f2 ∈ {u,d,u′,d′}, and determine the χSF bulk-to-boundary correlation functionsg f1 f2X (x0) =−12〈X f1 f2(x)Q f2 f15〉, X =V0,A0,S,P ,l f1 f2Y (x0) =−163∑k=1〈Y f1 f2k (x)Qf2 f1k〉, Yk =Vk,Ak,Tk0, T̃k0 .The complete list of boundary operators Q f2 f15 ,Qf2 f1k can be found in [5].Up to discretization effects and boundary fields renormalization they are related to the standard SFcorrelation functions fX and kY by universalityf contA =ZAguu′A =ZAgdd′A =−iZVgudV = iZVgduV , (1.1)f contV =ZVguu′V =ZVgdd′V =−iZAgudA = iZAgduA , (1.2)kcontV = ZVluu′V = ZVldd′V = −iZAludA = iZAlduA , (1.3)kcontA = ZAluu′A = ZAldd′A = −iZVludV = iZVlduV . (1.4)The χSF correlation functions in eqs. (1.2), (1.4) are O(a), since they become f contV ,kcontA in thecontinuum, which are parity odd. The local vector current can be replaced by the exactly conservedone Ṽµ(x) with normalization ZṼ = 1. Therefore ZA may be obtained from the ratiosZgA =−igudṼ(x0)guu′A (x0)∣∣∣∣x0=L/2or ZlA =iluu′Ṽ(x0)ludA (x0)∣∣∣∣x0=L/2. (1.5)2. Computational setup and resultsWe obtain results for Nf = 3 QCD in a non-unitary setup. Valence quark propagators areinverted with χSF boundaries on the configuration ensembles of [3], generated on lattices withstandard SF boundary conditions. These configurations have been used for the RG-running of thequark mass in a range of scales 2GeV . µ . 128GeV , in the standard framework of finite-sizescaling L→ 2L.2.1 TuningWe must ensure that massless QCD with χSF boundary conditions is correctly regularized.This is achieved by tuning the bare mass parameter m0 to its critical value, mcr, where the axialcurrent is conserved, and by tuning the boundary counterterm coefficient z f so that physical parity2PoS(LATTICE2019)202χSF near the electroweak scale Andrew Lytleis restored. At present we choose to set the PCAC mass to zero in terms of the SF correlationfunctions, so taking the mcr value from [3], and to set the χSF correlation function gudA to zero:m =∂̃0 f udA (x0)2 f udP (x0)∣∣∣∣x0=L/2= 0, mcr tuning ,gudA (x0)∣∣∣∣x0=L/2= 0 , z f tuning . (2.1)By requiring that eq. (2.1) be satisfied, z f is tuned for each ensemble as shown in Fig. 1.6.5 7.0 7.5 8.0 8.5 9.0β1.001.051.101.151.201.251.30zfL/a = 6L/a = 8L/a = 12 PRELIMINARYFigure 1: Results of nonperturbative tuning of z f , according to eq. (2.1).2.2 O(a)-improvementUsing eqs. (1.5) we obtain two estimates for ZAZgA(β ) = ZlA(β )+O(a2)which differ by discretization errors. In Fig. 2 we show the ratio of these two definitions and weconfirm that, after tuning z f , the ratio scales as a2 and goes to 1 in the continuum.0.000 0.005 0.010 0.015 0.020 0.025 0.030(a/L)20.991.001.011.021.03Zg A/Zl APRELIMINARYFigure 2: Ratio of two different definitions of ZA (see eq. (1.5)) calculated on our ensembles with 1/L = 4GeV.3PoS(LATTICE2019)202χSF near the electroweak scale Andrew LytleFinally we study ZlA(g20) over the full range of ensembles. As seen in Fig. 3 a fit to the datamatches onto the asympotic perturbative result in the limit g20→ 0.0.0 0.2 0.4 0.6 0.8 1.0g200.800.850.900.951.00ZApt O(g20)fit O(g60)PRELIMINARYFigure 3: ZlA(g20) calculated across the full range of ensembles available.3. Outlook for 4 fermions3.1 RenormalizationχSF framework is especially valuable in simplifying the renormalization of four fermion opera-tors. They enter the most general expression of the effective Hamiltonian which describes flavourphysics processes at low energy in the Standard Model (SM) and its extensions (BSM). Here wefocus on ∆F = 2 transitions. The 4-quark operators with four distinct flavoursO±XY ≡12[(ψ1ΓX ψ2)(ψ3ΓY ψ4)± (2↔ 4)]can be classified as parity even and parity odd:Oe,±k ∈{O±VV+AA,O±VV−AA,O±SS−PP,O±SS+PP,O±T T},Oo,±k ∈{O±VA+AV ,O±VA−AV ,O±SP−PS,O±SP+PS,O±T T̃}.Due to the explicit breaking of chiral symmetry of the Wilson regularisation, the operators in ge-neral mix as follows:Oe,±i = ∑jmZe,±i j (δ jm +∆e,±jm )Oe,±m ,Oo,±i = ∑jmZo,±i j (δ jm +∆o,±jm )Oo,±m .The parity-odd sector has a simpler, continuum-like mixing pattern (∆o,±jm = 0) [6]. In previousWilson fermion computations [7, 8], standard SF renormalization conditions were imposed onparity-odd operators by setting suitable renormalized correlation functions equal to their tree levelvalues at the scale µ = L−1:Fi(x0) = 〈O ′455 Oo,1234i (x0)O215 O535 〉 . (3.1)4PoS(LATTICE2019)202χSF near the electroweak scale Andrew LytleNote that five distinct valence flavours are required; see the left diagram in Fig. 4. These four-pointcorrelation functions are parity even, but suffer from large statistical fluctuations. Moreover, theyhave bulk O(a) discretization errors.Alternatively we plan to employ a new renormalization scheme on χSF three-point correlationfunctionsGi(x0) = 〈Q′215 Qo,1234i (x0)Q435 〉 , (3.2)which are statistically less noisy and automatically O(a) improved in the bulk. Performing suitablechiral rotations, the χSF flavours are rotated into the physical flavours [4]. Thus we can map therenormalized [Gi(x0)]R into the continuum correlation function[Gi(x0)]R→ 〈O′215 Oe,1234i (x0)O435 〉cont.SF χSFFigure 4: Correlation functions used in two renormalization schemes. Left: parity odd operators withstandard SF boundary conditions; right: parity odd operators with χSF boundary conditions. Differentcolors stand for different valence flavors.3.2 Physical determinationsThe renormalization program will be employed in the lattice computation of the physical BK-parameter which controls the K̄0−K0 meson oscillations, towards a better understanding of thephysics of CP violation in the SM and BSM. Whereas there is general agreement among variouscollaborations on BK in the SM, the situation is somewhat unclear for the BSM contributions [11].In the SF approach, BK has been computed by combining the renormalization parametersbased on eq. (3.1) with bare four-point correlation functions of the parity odd operators in a twistedmass QCD setup [9, 10].We plan to use the χSF renormalization conditions based on eq. (3.2). For the bare ma-trix elements we will employ the CLS Nf = 2+1 ensembles, characterised by large physical vol-umes with open boundary conditions and by non-zero quark masses [12, 13]. The sea quarks areWilson/Clover. Valence fermions are fully twisted [14], with three flavours tuned at twisted an-gle α = π/2 and the fourth one at α = −π/2. Unitarity is lost at finite lattice spacing, but itis recovered in the continuum limit. Performing distinct Osterwalder-Seiler chiral rotations foreach flavour, correlation functions with parity-odd operators (renormalized in the χSF scheme asoutlined above) are mapped onto the 3-point correlation functions of parity even operators withpseudoscalar sources, from which the B-parameters are readily extracted:BKi(µ) ∝ 〈K̄0| [Oei (µ)]R |K0〉 .5PoS(LATTICE2019)202χSF near the electroweak scale Andrew Lytle4. AcknowledgementsWe gratefully acknowledge help from Patrick Fritzsch, Carlos Pena, David Preti, and AlbertoRamos. AL would like to thank the conference organisers, and F. Joswig, S. Kuberski, and R.Sommer for useful discussions. We acknowledge the Santander Supercomputacion support groupat the University of Cantabria which provided access to the Altamira Supercomputer at the Insti-tute of Physics of Cantabria (IFCA-CSIC), member of the Spanish Supercomputing Network, forperforming simulations/analyses. Special thanks go to Aida Palacio Hoz for her constant help.References[1] S. Sint, Nucl. Phys. B 847 (2011) 491 doi:10.1016/j.nuclphysb.2011.02.002 [arXiv:1008.4857[hep-lat]].[2] M. Dalla Brida, T. Korzec, S. Sint and P. Vilaseca, Eur. Phys. J. C 79 (2019) no.1, 23doi:10.1140/epjc/s10052-018-6514-5 [arXiv:1808.09236 [hep-lat]].[3] I. Campos et al. [ALPHA Collaboration], Eur. Phys. J. C 78 (2018) no.5, 387doi:10.1140/epjc/s10052-018-5870-5 [arXiv:1802.05243 [hep-lat]].[4] P. V. Mainar, M. Dalla Brida and M. Papinutto, PoS LATTICE 2015 (2016) 252.doi:10.22323/1.251.0252[5] M. Dalla Brida, S. Sint and P. Vilaseca, JHEP 1608 (2016) 102 doi:10.1007/JHEP08(2016)102[arXiv:1603.00046 [hep-lat]].[6] A. Donini, V. Gimenez, G. Martinelli, M. Talevi and A. Vladikas, Eur. Phys. J. C 10 (1999) 121doi:10.1007/s100529900097 [hep-lat/9902030].[7] M. Guagnelli et al. [ALPHA Collaboration], JHEP 0603 (2006) 088doi:10.1088/1126-6708/2006/03/088 [hep-lat/0505002].[8] M. Papinutto, C. Pena and D. Preti, Eur. Phys. J. C 77 (2017) no.6, 376 Erratum: [Eur. Phys. J. C 78(2018) no.1, 21] doi:10.1140/epjc/s10052-018-5522-9, 10.1140/epjc/s10052-017-4930-6[arXiv:1612.06461 [hep-lat]].[9] P. Dimopoulos, H. Simma and A. Vladikas, JHEP 0907 (2009) 007doi:10.1088/1126-6708/2009/07/007 [arXiv:0902.1074 [hep-lat]].[10] R. Frezzotti and G. C. Rossi, JHEP 0410 (2004) 070 doi:10.1088/1126-6708/2004/10/070[hep-lat/0407002].[11] S. Aoki et al. [Flavour Lattice Averaging Group], arXiv:1902.08191 [hep-lat].[12] M. Bruno et al., JHEP 1502 (2015) 043 doi:10.1007/JHEP02(2015)043 [arXiv:1411.3982 [hep-lat]].[13] M. Bruno, T. Korzec and S. Schaefer, Phys. Rev. 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English

$χ$SF near the electroweak scale

We employ the chirally rotated Schr"odinger functional ($chi$SF) to study
two-point fermion bilinear correlation functions used in the determination of
$Z_A,V,S,P,T$ on a series of well-tuned ensembles. The gauge configurations,
which span renormalisation scales from 4 to 70~GeV, are generated with $N_m
f=3$ massless flavors and Schr"odinger Functional (SF) boundary conditions.
Valence quarks are computed with $chi$SF boundary conditions. We show
preliminary results on the tuning of the $chi$SF Symanzik coefficient $z_f$
and the scaling of the axial current normalization $Z_m A$. Moreover we
carry out a detailed comparison with the expectations from one-loop
perturbation theory. Finally we outline how automatically
$mathrmO(a)$-improved $B_m K$ matrix elements, including BSM
contributions, can be computed in a $chi$SF renormalization scheme

https://iris.uniroma1.it/bitstream/11573/1392042/1/Papinutto_Electroweak-scale.pdf

$χ$ SF near the electroweak scale

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χχχSF near the electroweak scale

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$χ$ SF near the electroweak scale