perturbatively improving renormalization constants...1university of cyprus, 2universität...
TRANSCRIPT
![Page 1: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/1.jpg)
Perturbatively improving renormalization constants
M. Constantinou1, M. Costa1, M. Göckeler2, R. Horsley3,H. Panagopoulos1, H. Perlt4, P. E. L. Rakow5, G. Schierholz6 and
A. Schiller4
1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig,5University Liverpool, 6DESY
Lattice 2013, Mainz, Germany, July 29 - August 3, 2013
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 1 / 18
![Page 2: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/2.jpg)
Outline
1 Introduction
2 Renormalization constants in RGI scheme
3 Subtraction of O(a2) lattice artifacts in one-loop
4 Summary
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 2 / 18
![Page 3: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/3.jpg)
Outline
1 Introduction
2 Renormalization constants in RGI scheme
3 Subtraction of O(a2) lattice artifacts in one-loop
4 Summary
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 2 / 18
![Page 4: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/4.jpg)
Outline
1 Introduction
2 Renormalization constants in RGI scheme
3 Subtraction of O(a2) lattice artifacts in one-loop
4 Summary
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 2 / 18
![Page 5: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/5.jpg)
Outline
1 Introduction
2 Renormalization constants in RGI scheme
3 Subtraction of O(a2) lattice artifacts in one-loop
4 Summary
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 2 / 18
![Page 6: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/6.jpg)
Introduction
Motivation I
Talk is based on our recent paperM. Constantinou et al. PR D87 (2013) 096019
Connection between "lattice world" and "real world":renormalization constants ZMust know them as accurate as possiblePerturbative approach: complicated, slow convergence, mixingproblems, ...Nonperturbative approach: widely used scheme is RI-MOMscheme
Simple implementationGauge fixing required
Simulations at finite lattice constant a→ problem of latticesartifacts
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 3 / 18
![Page 7: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/7.jpg)
Introduction
Motivation I
Talk is based on our recent paperM. Constantinou et al. PR D87 (2013) 096019
Connection between "lattice world" and "real world":renormalization constants ZMust know them as accurate as possiblePerturbative approach: complicated, slow convergence, mixingproblems, ...Nonperturbative approach: widely used scheme is RI-MOMscheme
Simple implementationGauge fixing required
Simulations at finite lattice constant a→ problem of latticesartifacts
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 3 / 18
![Page 8: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/8.jpg)
Introduction
Motivation I
Talk is based on our recent paperM. Constantinou et al. PR D87 (2013) 096019
Connection between "lattice world" and "real world":renormalization constants ZMust know them as accurate as possiblePerturbative approach: complicated, slow convergence, mixingproblems, ...Nonperturbative approach: widely used scheme is RI-MOMscheme
Simple implementationGauge fixing required
Simulations at finite lattice constant a→ problem of latticesartifacts
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 3 / 18
![Page 9: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/9.jpg)
Introduction
Motivation I
Talk is based on our recent paperM. Constantinou et al. PR D87 (2013) 096019
Connection between "lattice world" and "real world":renormalization constants ZMust know them as accurate as possiblePerturbative approach: complicated, slow convergence, mixingproblems, ...Nonperturbative approach: widely used scheme is RI-MOMscheme
Simple implementationGauge fixing required
Simulations at finite lattice constant a→ problem of latticesartifacts
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 3 / 18
![Page 10: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/10.jpg)
Introduction
Motivation I
Talk is based on our recent paperM. Constantinou et al. PR D87 (2013) 096019
Connection between "lattice world" and "real world":renormalization constants ZMust know them as accurate as possiblePerturbative approach: complicated, slow convergence, mixingproblems, ...Nonperturbative approach: widely used scheme is RI-MOMscheme
Simple implementationGauge fixing required
Simulations at finite lattice constant a→ problem of latticesartifacts
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 3 / 18
![Page 11: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/11.jpg)
Introduction
Motivation I
Talk is based on our recent paperM. Constantinou et al. PR D87 (2013) 096019
Connection between "lattice world" and "real world":renormalization constants ZMust know them as accurate as possiblePerturbative approach: complicated, slow convergence, mixingproblems, ...Nonperturbative approach: widely used scheme is RI-MOMscheme
Simple implementationGauge fixing required
Simulations at finite lattice constant a→ problem of latticesartifacts
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 3 / 18
![Page 12: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/12.jpg)
Introduction
Motivation I
Talk is based on our recent paperM. Constantinou et al. PR D87 (2013) 096019
Connection between "lattice world" and "real world":renormalization constants ZMust know them as accurate as possiblePerturbative approach: complicated, slow convergence, mixingproblems, ...Nonperturbative approach: widely used scheme is RI-MOMscheme
Simple implementationGauge fixing required
Simulations at finite lattice constant a→ problem of latticesartifacts
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 3 / 18
![Page 13: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/13.jpg)
Introduction
Motivation I
Talk is based on our recent paperM. Constantinou et al. PR D87 (2013) 096019
Connection between "lattice world" and "real world":renormalization constants ZMust know them as accurate as possiblePerturbative approach: complicated, slow convergence, mixingproblems, ...Nonperturbative approach: widely used scheme is RI-MOMscheme
Simple implementationGauge fixing required
Simulations at finite lattice constant a→ problem of latticesartifacts
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 3 / 18
![Page 14: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/14.jpg)
Renormalization constants in RGI scheme
RGI scheme
ORGI = ∆ ZS(M)OS(M) = ∆ ZS(M) ZSbare(M,a)Obare = Z RGI(a)Obare
S = RI′ −MOM is not covariant for most operators→ not suitablefor computing anomalous dimensions (needed in ∆ ZS(M))Intermediate scheme S: RI′ −MOM→ S → RGI
Z RGI(a) = ∆ Z M̃OMgg(M = µp) Z M̃OMggRI′−MOM(M = µp) Z RI′−MOM
bare (µp,a)
∆ Z M̃OMgg and Z M̃OMggRI′−MOM(M = µp) computed in continuum PT
Z RI′−MOMbare (µ,a) is the nonperturbatively measured Z-factor
In MC simulations (aµ) is not small→ lattice artifacts!Artifacts under control→ determination of Z RGI with betteraccuray!
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 4 / 18
![Page 15: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/15.jpg)
Renormalization constants in RGI scheme
RGI scheme
ORGI = ∆ ZS(M)OS(M) = ∆ ZS(M) ZSbare(M,a)Obare = Z RGI(a)Obare
S = RI′ −MOM is not covariant for most operators→ not suitablefor computing anomalous dimensions (needed in ∆ ZS(M))Intermediate scheme S: RI′ −MOM→ S → RGI
Z RGI(a) = ∆ Z M̃OMgg(M = µp) Z M̃OMggRI′−MOM(M = µp) Z RI′−MOM
bare (µp,a)
∆ Z M̃OMgg and Z M̃OMggRI′−MOM(M = µp) computed in continuum PT
Z RI′−MOMbare (µ,a) is the nonperturbatively measured Z-factor
In MC simulations (aµ) is not small→ lattice artifacts!Artifacts under control→ determination of Z RGI with betteraccuray!
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 4 / 18
![Page 16: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/16.jpg)
Renormalization constants in RGI scheme
RGI scheme
ORGI = ∆ ZS(M)OS(M) = ∆ ZS(M) ZSbare(M,a)Obare = Z RGI(a)Obare
S = RI′ −MOM is not covariant for most operators→ not suitablefor computing anomalous dimensions (needed in ∆ ZS(M))Intermediate scheme S: RI′ −MOM→ S → RGI
Z RGI(a) = ∆ Z M̃OMgg(M = µp) Z M̃OMggRI′−MOM(M = µp) Z RI′−MOM
bare (µp,a)
∆ Z M̃OMgg and Z M̃OMggRI′−MOM(M = µp) computed in continuum PT
Z RI′−MOMbare (µ,a) is the nonperturbatively measured Z-factor
In MC simulations (aµ) is not small→ lattice artifacts!Artifacts under control→ determination of Z RGI with betteraccuray!
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 4 / 18
![Page 17: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/17.jpg)
Renormalization constants in RGI scheme
RGI scheme
ORGI = ∆ ZS(M)OS(M) = ∆ ZS(M) ZSbare(M,a)Obare = Z RGI(a)Obare
S = RI′ −MOM is not covariant for most operators→ not suitablefor computing anomalous dimensions (needed in ∆ ZS(M))Intermediate scheme S: RI′ −MOM→ S → RGI
Z RGI(a) = ∆ Z M̃OMgg(M = µp) Z M̃OMggRI′−MOM(M = µp) Z RI′−MOM
bare (µp,a)
∆ Z M̃OMgg and Z M̃OMggRI′−MOM(M = µp) computed in continuum PT
Z RI′−MOMbare (µ,a) is the nonperturbatively measured Z-factor
In MC simulations (aµ) is not small→ lattice artifacts!Artifacts under control→ determination of Z RGI with betteraccuray!
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 4 / 18
![Page 18: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/18.jpg)
Renormalization constants in RGI scheme
RGI scheme
ORGI = ∆ ZS(M)OS(M) = ∆ ZS(M) ZSbare(M,a)Obare = Z RGI(a)Obare
S = RI′ −MOM is not covariant for most operators→ not suitablefor computing anomalous dimensions (needed in ∆ ZS(M))Intermediate scheme S: RI′ −MOM→ S → RGI
Z RGI(a) = ∆ Z M̃OMgg(M = µp) Z M̃OMggRI′−MOM(M = µp) Z RI′−MOM
bare (µp,a)
∆ Z M̃OMgg and Z M̃OMggRI′−MOM(M = µp) computed in continuum PT
Z RI′−MOMbare (µ,a) is the nonperturbatively measured Z-factor
In MC simulations (aµ) is not small→ lattice artifacts!Artifacts under control→ determination of Z RGI with betteraccuray!
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 4 / 18
![Page 19: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/19.jpg)
Renormalization constants in RGI scheme
RGI scheme
ORGI = ∆ ZS(M)OS(M) = ∆ ZS(M) ZSbare(M,a)Obare = Z RGI(a)Obare
S = RI′ −MOM is not covariant for most operators→ not suitablefor computing anomalous dimensions (needed in ∆ ZS(M))Intermediate scheme S: RI′ −MOM→ S → RGI
Z RGI(a) = ∆ Z M̃OMgg(M = µp) Z M̃OMggRI′−MOM(M = µp) Z RI′−MOM
bare (µp,a)
∆ Z M̃OMgg and Z M̃OMggRI′−MOM(M = µp) computed in continuum PT
Z RI′−MOMbare (µ,a) is the nonperturbatively measured Z-factor
In MC simulations (aµ) is not small→ lattice artifacts!Artifacts under control→ determination of Z RGI with betteraccuray!
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 4 / 18
![Page 20: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/20.jpg)
Renormalization constants in RGI scheme
RGI scheme
ORGI = ∆ ZS(M)OS(M) = ∆ ZS(M) ZSbare(M,a)Obare = Z RGI(a)Obare
S = RI′ −MOM is not covariant for most operators→ not suitablefor computing anomalous dimensions (needed in ∆ ZS(M))Intermediate scheme S: RI′ −MOM→ S → RGI
Z RGI(a) = ∆ Z M̃OMgg(M = µp) Z M̃OMggRI′−MOM(M = µp) Z RI′−MOM
bare (µp,a)
∆ Z M̃OMgg and Z M̃OMggRI′−MOM(M = µp) computed in continuum PT
Z RI′−MOMbare (µ,a) is the nonperturbatively measured Z-factor
In MC simulations (aµ) is not small→ lattice artifacts!Artifacts under control→ determination of Z RGI with betteraccuray!
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 4 / 18
![Page 21: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/21.jpg)
Renormalization constants in RGI scheme
RGI scheme
ORGI = ∆ ZS(M)OS(M) = ∆ ZS(M) ZSbare(M,a)Obare = Z RGI(a)Obare
S = RI′ −MOM is not covariant for most operators→ not suitablefor computing anomalous dimensions (needed in ∆ ZS(M))Intermediate scheme S: RI′ −MOM→ S → RGI
Z RGI(a) = ∆ Z M̃OMgg(M = µp) Z M̃OMggRI′−MOM(M = µp) Z RI′−MOM
bare (µp,a)
∆ Z M̃OMgg and Z M̃OMggRI′−MOM(M = µp) computed in continuum PT
Z RI′−MOMbare (µ,a) is the nonperturbatively measured Z-factor
In MC simulations (aµ) is not small→ lattice artifacts!Artifacts under control→ determination of Z RGI with betteraccuray!
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 4 / 18
![Page 22: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/22.jpg)
Renormalization constants in RGI scheme
Settings, parameters
Following settings are used for the discussed examples
Operators: OS, OVµ , OA
µ , OTµν , Ov2,a , Ov2,b
Clover improved Wilson fermions + plaquette gauge action, Nf = 2Landau gaugeβ = (5.20,5.25,5.29,5.40)
r0 ΛMS = (0.700,0.789)→ gS → ∆ ZS(M)
r0 = 0.501fm, r0/a = 6.050(β = 5.20),6.603(β =5.25),7.004(β = 5.29),8.285(β = 5.40)
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 5 / 18
![Page 23: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/23.jpg)
Renormalization constants in RGI scheme
Settings, parameters
Following settings are used for the discussed examples
Operators: OS, OVµ , OA
µ , OTµν , Ov2,a , Ov2,b
Clover improved Wilson fermions + plaquette gauge action, Nf = 2Landau gaugeβ = (5.20,5.25,5.29,5.40)
r0 ΛMS = (0.700,0.789)→ gS → ∆ ZS(M)
r0 = 0.501fm, r0/a = 6.050(β = 5.20),6.603(β =5.25),7.004(β = 5.29),8.285(β = 5.40)
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 5 / 18
![Page 24: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/24.jpg)
Renormalization constants in RGI scheme
Settings, parameters
Following settings are used for the discussed examples
Operators: OS, OVµ , OA
µ , OTµν , Ov2,a , Ov2,b
Clover improved Wilson fermions + plaquette gauge action, Nf = 2Landau gaugeβ = (5.20,5.25,5.29,5.40)
r0 ΛMS = (0.700,0.789)→ gS → ∆ ZS(M)
r0 = 0.501fm, r0/a = 6.050(β = 5.20),6.603(β =5.25),7.004(β = 5.29),8.285(β = 5.40)
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 5 / 18
![Page 25: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/25.jpg)
Renormalization constants in RGI scheme
Settings, parameters
Following settings are used for the discussed examples
Operators: OS, OVµ , OA
µ , OTµν , Ov2,a , Ov2,b
Clover improved Wilson fermions + plaquette gauge action, Nf = 2Landau gaugeβ = (5.20,5.25,5.29,5.40)
r0 ΛMS = (0.700,0.789)→ gS → ∆ ZS(M)
r0 = 0.501fm, r0/a = 6.050(β = 5.20),6.603(β =5.25),7.004(β = 5.29),8.285(β = 5.40)
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 5 / 18
![Page 26: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/26.jpg)
Renormalization constants in RGI scheme
Settings, parameters
Following settings are used for the discussed examples
Operators: OS, OVµ , OA
µ , OTµν , Ov2,a , Ov2,b
Clover improved Wilson fermions + plaquette gauge action, Nf = 2Landau gaugeβ = (5.20,5.25,5.29,5.40)
r0 ΛMS = (0.700,0.789)→ gS → ∆ ZS(M)
r0 = 0.501fm, r0/a = 6.050(β = 5.20),6.603(β =5.25),7.004(β = 5.29),8.285(β = 5.40)
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 5 / 18
![Page 27: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/27.jpg)
Renormalization constants in RGI scheme
Settings, parameters
Following settings are used for the discussed examples
Operators: OS, OVµ , OA
µ , OTµν , Ov2,a , Ov2,b
Clover improved Wilson fermions + plaquette gauge action, Nf = 2Landau gaugeβ = (5.20,5.25,5.29,5.40)
r0 ΛMS = (0.700,0.789)→ gS → ∆ ZS(M)
r0 = 0.501fm, r0/a = 6.050(β = 5.20),6.603(β =5.25),7.004(β = 5.29),8.285(β = 5.40)
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 5 / 18
![Page 28: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/28.jpg)
Subtraction of O(a2) lattice artifacts in one-loop
One-loop subtraction O(a2)
In M. Göckeler et al. PR D82 (2010) 114511 we proposed tosubtract the complete one-loop lattice artifactsVery efficient, but being a purely numerical procedure ratherinvolvedLook for procedure which can be applied to more general cases -eventually with "less" correction effectCyprus group pioneered diagrammatic O(g2a2) approach (seee.g., M. Constantinou, V. Lubicz, H. Panagopoulos andF. Stylianou, JHEP 0910 (2009) 064)Results for local and one-link bilinears and different actions andgeneral mass terms; higher derivative operators are possible (butalso not very easy)
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 6 / 18
![Page 29: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/29.jpg)
Subtraction of O(a2) lattice artifacts in one-loop
One-loop subtraction O(a2)
In M. Göckeler et al. PR D82 (2010) 114511 we proposed tosubtract the complete one-loop lattice artifactsVery efficient, but being a purely numerical procedure ratherinvolvedLook for procedure which can be applied to more general cases -eventually with "less" correction effectCyprus group pioneered diagrammatic O(g2a2) approach (seee.g., M. Constantinou, V. Lubicz, H. Panagopoulos andF. Stylianou, JHEP 0910 (2009) 064)Results for local and one-link bilinears and different actions andgeneral mass terms; higher derivative operators are possible (butalso not very easy)
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 6 / 18
![Page 30: Perturbatively improving renormalization constants...1University of Cyprus, 2Universität Regensburg, 3University Edinburgh, 4Universität Leipzig, 5University Liverpool, 6DESY Lattice](https://reader036.vdocuments.net/reader036/viewer/2022070713/5ed27d73995ce412b22babca/html5/thumbnails/30.jpg)
Subtraction of O(a2) lattice artifacts in one-loop
One-loop subtraction O(a2)
In M. Göckeler et al. PR D82 (2010) 114511 we proposed tosubtract the complete one-loop lattice artifactsVery efficient, but being a purely numerical procedure ratherinvolvedLook for procedure which can be applied to more general cases -eventually with "less" correction effectCyprus group pioneered diagrammatic O(g2a2) approach (seee.g., M. Constantinou, V. Lubicz, H. Panagopoulos andF. Stylianou, JHEP 0910 (2009) 064)Results for local and one-link bilinears and different actions andgeneral mass terms; higher derivative operators are possible (butalso not very easy)
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 6 / 18
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Subtraction of O(a2) lattice artifacts in one-loop
One-loop subtraction O(a2)
In M. Göckeler et al. PR D82 (2010) 114511 we proposed tosubtract the complete one-loop lattice artifactsVery efficient, but being a purely numerical procedure ratherinvolvedLook for procedure which can be applied to more general cases -eventually with "less" correction effectCyprus group pioneered diagrammatic O(g2a2) approach (seee.g., M. Constantinou, V. Lubicz, H. Panagopoulos andF. Stylianou, JHEP 0910 (2009) 064)Results for local and one-link bilinears and different actions andgeneral mass terms; higher derivative operators are possible (butalso not very easy)
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 6 / 18
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Subtraction of O(a2) lattice artifacts in one-loop
One-loop subtraction O(a2)
In M. Göckeler et al. PR D82 (2010) 114511 we proposed tosubtract the complete one-loop lattice artifactsVery efficient, but being a purely numerical procedure ratherinvolvedLook for procedure which can be applied to more general cases -eventually with "less" correction effectCyprus group pioneered diagrammatic O(g2a2) approach (seee.g., M. Constantinou, V. Lubicz, H. Panagopoulos andF. Stylianou, JHEP 0910 (2009) 064)Results for local and one-link bilinears and different actions andgeneral mass terms; higher derivative operators are possible (butalso not very easy)
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 6 / 18
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Subtraction of O(a2) lattice artifacts in one-loop
OS
Example: scalar operator OS
Z S(a,p) = 1 +g2CF
16π2
(− 23.3099453215 + 3 log(a2S2)
+(a2S2)(
1.6408851782248− 239240
log(a2S2)
+S4
(S2)2
(1.9510436778− 101
120log(a2S2)
)))
≡ 1 +g2CF
16π2 Z (1)S + g2 a2Z (a2)
S,1−loop
with Sn =∑4
λ=1 pnλ
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 7 / 18
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Subtraction of O(a2) lattice artifacts in one-loop
OS −O(a2)artifacts
g2 a2Z (a2)S,1−loop for a general momentum set (243 × 48)
0 1 2 3 4 5 6 7 8 9 10
(a p)2
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
arbitrary momenta
diagonal
actual momenta
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Subtraction of O(a2) lattice artifacts in one-loop
Subtraction procedures
Following subtraction procedures have been investigated
Z RI′−MOMbare (p,a)MC,sub,s
(s)= Z RI′−MOM
bare (p,a)MC − a2 g2? Z (a2)
1−loop(p,a)
Z RI′−MOMbare (p,a)MC,sub,m
(m)= Z RI′−MOM
bare (p,a)MC ×(
1− a2 g2? Z (a2)
1−loop(p,a)),
g? can be chosen to be either the bare lattice coupling g or theboosted coupling gB.
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 9 / 18
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Subtraction of O(a2) lattice artifacts in one-loop
Subtraction procedures
Following subtraction procedures have been investigated
Z RI′−MOMbare (p,a)MC,sub,s
(s)= Z RI′−MOM
bare (p,a)MC − a2 g2? Z (a2)
1−loop(p,a)
Z RI′−MOMbare (p,a)MC,sub,m
(m)= Z RI′−MOM
bare (p,a)MC ×(
1− a2 g2? Z (a2)
1−loop(p,a)),
g? can be chosen to be either the bare lattice coupling g or theboosted coupling gB.
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 9 / 18
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Subtraction of O(a2) lattice artifacts in one-loop
Effect of subtraction
For the scalar operator we get
0.45
0.5
0.55
0 20 40 60 80 100
ZRGI
S
p2 [GeV2]
no subtraction
O(a2) subtraction (s)
O(a2) subtraction (m)
complete subtraction
Figure: ZS and ZS,sub,i as function of a2p2.
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 10 / 18
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Subtraction of O(a2) lattice artifacts in one-loop
Fit of lattice artifacts
Parametrizing the remaining lattice artifacts(Z RGI(a) = ∆ Z M̃OMgg(p) Z M̃OMgg
RI′−MOM(p) Z RI′−MOMbare (p,a)MC,sub
)
ZSRI′−MOM(p)Z RI′−MOMbare (p,a)MC,sub =
Z RGI(a)
∆ZS(p)[1 + b1 (gS)8
] +
a2(
c1 S2 + c2S4
S2+ c3
S6
(S2)2
)+a4
(c4 (S2)2 + c5 S4
)+a6
(c6 (S2)3 + c7 S4 S2 + c8 S6
).
ci fit the lattice artifacts (Sn =∑
i pni )
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 11 / 18
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Subtraction of O(a2) lattice artifacts in one-loop
Fit procedure
1 In order to avoid the region of breakdown of perturbation theorywe fit for p2 ≥ p2
min = 10 GeV2.2 We vary the type of subtraction ((s),(m)) and type of coupling (g,
gB)3 We vary the number/kind of parameters (ci )→ strength of
hypercubic lattice artifacts4 Assuming that one-loop complete subtraction is “benchmark”→
compare the O(a2) subtraction results with these numbers5 Fitting routines: Nelder-Mead/differential evolution algorithms
cross checked with MINUIT
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 12 / 18
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Subtraction of O(a2) lattice artifacts in one-loop
Fit procedure
1 In order to avoid the region of breakdown of perturbation theorywe fit for p2 ≥ p2
min = 10 GeV2.2 We vary the type of subtraction ((s),(m)) and type of coupling (g,
gB)3 We vary the number/kind of parameters (ci )→ strength of
hypercubic lattice artifacts4 Assuming that one-loop complete subtraction is “benchmark”→
compare the O(a2) subtraction results with these numbers5 Fitting routines: Nelder-Mead/differential evolution algorithms
cross checked with MINUIT
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 12 / 18
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Subtraction of O(a2) lattice artifacts in one-loop
Fit procedure
1 In order to avoid the region of breakdown of perturbation theorywe fit for p2 ≥ p2
min = 10 GeV2.2 We vary the type of subtraction ((s),(m)) and type of coupling (g,
gB)3 We vary the number/kind of parameters (ci )→ strength of
hypercubic lattice artifacts4 Assuming that one-loop complete subtraction is “benchmark”→
compare the O(a2) subtraction results with these numbers5 Fitting routines: Nelder-Mead/differential evolution algorithms
cross checked with MINUIT
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 12 / 18
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Subtraction of O(a2) lattice artifacts in one-loop
Fit procedure
1 In order to avoid the region of breakdown of perturbation theorywe fit for p2 ≥ p2
min = 10 GeV2.2 We vary the type of subtraction ((s),(m)) and type of coupling (g,
gB)3 We vary the number/kind of parameters (ci )→ strength of
hypercubic lattice artifacts4 Assuming that one-loop complete subtraction is “benchmark”→
compare the O(a2) subtraction results with these numbers5 Fitting routines: Nelder-Mead/differential evolution algorithms
cross checked with MINUIT
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 12 / 18
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Subtraction of O(a2) lattice artifacts in one-loop
Fit procedure
1 In order to avoid the region of breakdown of perturbation theorywe fit for p2 ≥ p2
min = 10 GeV2.2 We vary the type of subtraction ((s),(m)) and type of coupling (g,
gB)3 We vary the number/kind of parameters (ci )→ strength of
hypercubic lattice artifacts4 Assuming that one-loop complete subtraction is “benchmark”→
compare the O(a2) subtraction results with these numbers5 Fitting routines: Nelder-Mead/differential evolution algorithms
cross checked with MINUIT
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 12 / 18
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Subtraction of O(a2) lattice artifacts in one-loop
Results, OS as example: subtraction type
0.435
0.44
0.445
0.45
0.455
1 2 3 4 5
ZRGI
S
subtype
1: complete subtraction with gB, 2: (s) with gB, 3: (m) with gB, 4: (s) with g,5: (m) with gFit using all ci , Shaded area: 1% deviation from 1.
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 13 / 18
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Subtraction of O(a2) lattice artifacts in one-loop
Results, OS as example: parameter type
0.435
0.44
0.445
0.45
0.455
1 2 3 4 5
ZRGI
S
partype
1: complete subtraction, 2: all ci , 3: O(4)-invariants, 4: a2,a4 - invariants,5: a4,a6 - invariantsFit for (s) and gB, Shaded area: 1% deviation from 1.
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 14 / 18
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Subtraction of O(a2) lattice artifacts in one-loop
Results, OS as example: final comparison
0.438
0.44
0.442
0.444
0.446
0.448
0.45
0.452
0.454
0.456
0.458
5.2 5.25 5.29 5.4
ZRGI
S
β
a2 subtraction
complete subtraction
Z RGIS at r0 ΛMS = 0.700
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 15 / 18
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Subtraction of O(a2) lattice artifacts in one-loop
Results for a2 subtraction
Op. r0 ΛMS Z RGI|β=5.20 Z RGI|β=5.25 Z RGI|β=5.29 Z RGI|β=5.40
OS 0.700 0.4530(34) 0.4475(33) 0.4451(32) 0.4414(30)
0.789 0.4717(44) 0.4661(65) 0.4632(54) 0.4585(27)
OV 0.700 0.7163(26) 0.7253(26) 0.7308(25) 0.7451(24)
0.789 0.7238(72) 0.7319(94) 0.7365(99) 0.7519(50)
OA 0.700 0.7460(41) 0.7543(40) 0.7590(39) 0.7731(37)
0.789 0.7585(46) 0.7634(77) 0.7666(81) 0.7805(30)
OT 0.700 0.8906(43) 0.9036(42) 0.9108(41) 0.9319(39)
0.789 0.8946(85) 0.9041(111) 0.9075(120) 0.9316(49)
Ov2,a 0.700 1.4914(55) 1.5131(55) 1.5266(54) 1.5660(53)
0.789 1.4635(108) 1.4776(112) 1.4926(90) 1.5397(58)
Ov2,b 0.700 1.5061(37) 1.5218(37) 1.5329(36) 1.5534(35)
0.789 1.4601(151) 1.4727(206) 1.4863(165) 1.5115(140)
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Summary
Summary
Described subtraction procedures for getting rid of lattice artifactsNumerical procedure for subtraction of all lattice artifacts inone-loop very efficientNot suited for more complicated operators and/or actionsFit procedure must be performed and adjusted for each operatorindividuallyUsed data do not favour strongly an unique choice of g,subtraction type or parameter set ci - compared to the completeone-loop subtraction results“External” arguments from boosted perturbation theory (→ gB),simplicity considerations (→ (s)) and data momenta (→ all ci )served as guide linesZ RGI from a2 subtraction differ at most by 2% from thecorresponding results obtained from the complete one-loopsubtraction
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 17 / 18
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Summary
Summary
Described subtraction procedures for getting rid of lattice artifactsNumerical procedure for subtraction of all lattice artifacts inone-loop very efficientNot suited for more complicated operators and/or actionsFit procedure must be performed and adjusted for each operatorindividuallyUsed data do not favour strongly an unique choice of g,subtraction type or parameter set ci - compared to the completeone-loop subtraction results“External” arguments from boosted perturbation theory (→ gB),simplicity considerations (→ (s)) and data momenta (→ all ci )served as guide linesZ RGI from a2 subtraction differ at most by 2% from thecorresponding results obtained from the complete one-loopsubtraction
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 17 / 18
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Summary
Summary
Described subtraction procedures for getting rid of lattice artifactsNumerical procedure for subtraction of all lattice artifacts inone-loop very efficientNot suited for more complicated operators and/or actionsFit procedure must be performed and adjusted for each operatorindividuallyUsed data do not favour strongly an unique choice of g,subtraction type or parameter set ci - compared to the completeone-loop subtraction results“External” arguments from boosted perturbation theory (→ gB),simplicity considerations (→ (s)) and data momenta (→ all ci )served as guide linesZ RGI from a2 subtraction differ at most by 2% from thecorresponding results obtained from the complete one-loopsubtraction
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 17 / 18
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Summary
Summary
Described subtraction procedures for getting rid of lattice artifactsNumerical procedure for subtraction of all lattice artifacts inone-loop very efficientNot suited for more complicated operators and/or actionsFit procedure must be performed and adjusted for each operatorindividuallyUsed data do not favour strongly an unique choice of g,subtraction type or parameter set ci - compared to the completeone-loop subtraction results“External” arguments from boosted perturbation theory (→ gB),simplicity considerations (→ (s)) and data momenta (→ all ci )served as guide linesZ RGI from a2 subtraction differ at most by 2% from thecorresponding results obtained from the complete one-loopsubtraction
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 17 / 18
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Summary
Summary
Described subtraction procedures for getting rid of lattice artifactsNumerical procedure for subtraction of all lattice artifacts inone-loop very efficientNot suited for more complicated operators and/or actionsFit procedure must be performed and adjusted for each operatorindividuallyUsed data do not favour strongly an unique choice of g,subtraction type or parameter set ci - compared to the completeone-loop subtraction results“External” arguments from boosted perturbation theory (→ gB),simplicity considerations (→ (s)) and data momenta (→ all ci )served as guide linesZ RGI from a2 subtraction differ at most by 2% from thecorresponding results obtained from the complete one-loopsubtraction
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 17 / 18
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Summary
Summary
Described subtraction procedures for getting rid of lattice artifactsNumerical procedure for subtraction of all lattice artifacts inone-loop very efficientNot suited for more complicated operators and/or actionsFit procedure must be performed and adjusted for each operatorindividuallyUsed data do not favour strongly an unique choice of g,subtraction type or parameter set ci - compared to the completeone-loop subtraction results“External” arguments from boosted perturbation theory (→ gB),simplicity considerations (→ (s)) and data momenta (→ all ci )served as guide linesZ RGI from a2 subtraction differ at most by 2% from thecorresponding results obtained from the complete one-loopsubtraction
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 17 / 18
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Summary
Summary
Described subtraction procedures for getting rid of lattice artifactsNumerical procedure for subtraction of all lattice artifacts inone-loop very efficientNot suited for more complicated operators and/or actionsFit procedure must be performed and adjusted for each operatorindividuallyUsed data do not favour strongly an unique choice of g,subtraction type or parameter set ci - compared to the completeone-loop subtraction results“External” arguments from boosted perturbation theory (→ gB),simplicity considerations (→ (s)) and data momenta (→ all ci )served as guide linesZ RGI from a2 subtraction differ at most by 2% from thecorresponding results obtained from the complete one-loopsubtraction
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 17 / 18
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Summary
Acknowledgements
We thank:
SFB/TR55 (Hadron Physics from Lattice QCD)
EU grant 283286 (HadronPhysics3)
Cyprus Research Promotion Foundation under Contract No.TECHNOLOGY/ΘEΠIΣ/ 0311(BE)/16
Talk H. Perlt (Leipzig) LPT improved Z factors Lattice 2013 18 / 18