Nf=2+1+1 renormalisation of four-quark operators Julien Frison - - PowerPoint PPT Presentation

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Nf=2+1+1 renormalisation of four-quark operators Julien Frison - - PowerPoint PPT Presentation

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Order a 4 Intro. Strat. Results Concl. Backup slides Nf=2+1+1 renormalisation of four-quark operators Julien Frison University of Edinburgh For the RBC-UKQCD collaboration 33rd International Symposium on Lattice Field Theory Lattice15

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Nf=2+1+1 renormalisation of four-quark

  • perators

Julien Frison University of Edinburgh For the RBC-UKQCD collaboration 33rd International Symposium on Lattice Field Theory Lattice’15 - Kobe - July 15th, 2015

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RBC-UKQCD collaboration

BNL and RBRC Tomomi Ishikawa Taku Izubuchi Chulwoo Jung Christoph Lehner Meifeng Lin Shigemi Ohta (KEK) Taichi Kawanai Christopher Kelly Amarjit Soni Sergey Syritsyn CERN Marina Marinkovic Columbia University Ziyuan Bai Norman Christ Xu Feng Luchang Jin Bob Mawhinney Greg McGlynn David Murphy Daiqian Zhang University of Connecticut Tom Blum Edinburgh University Peter Boyle Luigi Del Debbio Julien Frison Richard Kenway Ava Khamseh Brian Pendleton Oliver Witzel Azusa Yamaguchi Plymouth University Nicolas Garron University of Southampton Jonathan Flynn Tadeusz Janowski Andreas Juettner Andrew Lawson Edwin Lizarazo Antonin Portelli Chris Sachrajda Francesco Sanfilippo Matthew Spraggs Tobias Tsang York University (Toronto) Renwick Hudspith

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1

Introduction Motivations Operators and chiral sectors

2

Strategy Step-scaling Charm threshold Ensembles RI-SMOM scheme

3

Numerical Results Wilson flow and topological charge Continuum limit

4

Including O(a4) discretisation terms Method BK Perspectives for other sectors

5

Conclusion

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Motivations

The story so far The Rome-Southampton method with momentum sources has proven to be very efficient We have presented last year a Nf = 2 + 1 + 1 strategy to treat the charm threshold ... and preliminary results for BK What more can we do? All systematics were not fully under control We need generalisation to SUSY BK and K → ππ

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Operators and chiral sectors

sector dim BK SUSY BK K → ππ I = 2 K → ππ I = 0 (27, 1) 1

  • (8, 8)

2

  • (8, 1)

4

  • (6, 6)

2

  • With our chiral action, we can consider each chiral sector

separately. Q(27,1) = [¯ sγµ(1 − γ5)d] [¯ sγµ(1 − γ5)d] (1) Q(8,8)

1

= [¯ sγµ(1 − γ5)d] [¯ sγµ(1 + γ5)d] (2) Q(8,8)

2

= [¯ s(1 − γ5)d] [¯ s(1 + γ5)d] (3) Q(6,6)

1

= [¯ sσµν(1 − γ5)d] [¯ sσµν(1 − γ5)d] (4) Q(6,6)

2

= [¯ s(1 − γ5)d] [¯ s(1 − γ5)d] (5)

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Step-scaling

If one chooses a scaling trajectory and compute at constant physical masses the continuum limit σ(p, p0 | m) = lim

a→0

Z(p | m) Z(p0 | m), (6) this is universal, i.e. only dependent on the continuum RGEs from

  • ne scale to another.

Does not depend on your scaling trajectory, your lattice action, the O(4) orbit of momenta, and so on. Therefore those ratios are convenient building-blocks that we can combine with as many ratios as we want into a telescopic product.

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Charm threshold

p m

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Ensembles

Nf = 2 + 1 Ensembles Matrix elements have been computed on a wide set of (M)DWF ensembles, including two ensembles at the physical quark masses, and lattice spacing going up to 3 GeV. Nf = 2 + 1 + 1 Ensembles β L3 × T × L5 ml mc a−1 5.70 323 × 64 × 12 0.0047 0.243, 0.1, 0.0186 3.0 GeV 5.70 323 × 64 × 12 0.002 0.243 3.0 GeV 5.77 323 × 64 × 12 0.0044 0.213 3.6 GeV 5.84 323 × 64 × 12 0.0041 0.183, 0.0146 4.3 GeV 5.84 323 × 64 × 12 0.002 0.183 4.3 GeV

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RI-SMOM scheme

p1 p2 p1 p2 2q

Kinematics Non-exceptional schemes avoid π pole p2

1 = p2 2 = (p1 − p2)2

no pi combination cancels out many orientations satisfy this condition but cont. limit is universal Renormalisation condition Z Tr [PijklGijkl] = Tr [PijklGijkl] |tree Pijkl = γiδijγkδkl or Pijkl = / qij/ qkl different schemes allow us to evaluate the truncation error Very versatile method, with many knobs to turn Cheap once we have configs, excellent signal/noise

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Wilson flow and topological charge

1700 1800 1900 2000 2100 2200 2300 2.6 2.8 3 3.2 3.4 sqrt(t0) w0

b5.70_2p2_mob2 flow history

(first configuration is b5.70_2p2, with b+c=5) 2200 2300 2400 2500 2600 2700 2800 3.5 4 4.5 5 5.5 sqrt(t0) w0

b5.84_2p2 flow history

excluding first configurations 2100 2150 2200 2250 2300

  • 3
  • 2
  • 1

1 2 3 1000 1500 2000 2500 3000

  • 1
  • 0.5

0.5 1 1.5 2

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Continuum limit (intro)

The following results are produced through this process:

1 Compute step-scaling from 3 GeV for each ensemble (and

  • rientation)

2 Interpolate them to a common set of p2 values 3 For each interpolated p2 and each orientation, extrapolate

independently in a2

4 The difference between two orientations is an estimate of the

systematics from higher order discretisation terms

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Continuum limit I

20 40 60 80 100 120 p

2

0,9 0,95 1 1,05 1,1 σ σcont[1,1,0,0] σfine[1,1,0,0] σcoarse[1,1,0,0] σcont[1,1,1,1] σfine [1,1,1,1] σcoarse[1,1,1,1] NLO LO

(27,1) γγ

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Continuum limit II

20 40 60 80 100 120 psq 0,9 0,95 1 1,05 1,1 1,15 σ σcont[1,1,0,0] σfine[1,1,0,0] σcoarse[1,1,0,0] σcont[1,1,1,1] σfine[1,1,1,1] σcoarse[1,1,1,1] NLO

(27,1) qq

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Continuum limit III

20 40 60 80 100 120 p

2

  • 0,4
  • 0,3
  • 0,2
  • 0,1

0,1 0,2 σ(p,3GeV) σcont[1,1,0,0] σfine[1,1,0,0] σcoarse[1,1,0,0] σcont[1,1,1,1] σfine[1,1,1,1] σcoarse[1,1,1,1] NLO LO

(8,8)12 gg

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Continuum limit IV

20 40 60 80 100 120 p

2

  • 0,005

0,005 0,01 0,015 σ(p,3GeV) σcont[1,1,0,0] σfine[1,1,0,0] σcoarse[1,1,0,0] σcont[1,1,1,1] σfine[1,1,1,1] σcoarse[1,1,1,1]

(6,6)12 gg

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Method

Because most of the running comes from low-order PT, discretisation effects might be well-described by PT However we do not need any explicit perturbative formula At high-p2, ΛQCD becomes irrelevant, dimensional analysis suggest discr is mostly (ap)n-dependent. More precisely: the boosted coupling constant only grows logarithmically (see hep-lat/1412.0834) When combined with this all-order theoretical argument, the p2-dependence gives additional information Of course it doesn’t replace a third ensemble, but the latter would be much more expensive than this project

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BK

20 40 60 80 100 p

2 (GeV 2)

0.72 0.74 0.76 0.78 BK b=5.70 sub gg b=5.77 sub gg b=5.84 sub gg b=5.70 sub qq b=5.77 sub qq b=5.84 sub qq

BK in RGI scheme

using only central value of BK(3GeV)

PRELIMINARY

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Perspectives for other sectors I

10 20 30 40 50 60 70 p

2

0,99 0,995 1 1,005 1,01 1,015 ratio of interpolations gg [1,1,0,0] gg [1,1,1,1] qq [1,1,0,0] qq [1,1,1,1]

Λ

coarse 11/Λ fine 11

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Perspectives for other sectors II

10 20 30 40 50 60 70 p

2

0,7 0,8 0,9 1 1,1 1,2 ratio of interpolations gg [1,1,0,0] gg [1,1,1,1] qq [1,1,0,0] qq [1,1,1,1]

Λ

coarse 12/Λ fine 12

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Perspectives for other sectors III

10 20 30 40 50 60 70 p

2

1,2 1,4 1,6 1,8 2 2,2 ratio of interpolations direction [1,1,0,0] direction [1,1,1,1]

Λ

coarse 34/Λ fine 34 gamma-gamma scheme

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We have presented preliminary results for the 5 × 5 SUSY BK step-scaling A tiny “Nf = 3/Nf = 4” has not been included here With O(a2) extrapolations the discretisation errors are huge A third ensemble would be most useful Generating new ensembles, with stats for the low-energy scale-setting, is painful It is important to extract as much info as possible from ensembles we already have

  • discr. effects in SUSY BK have a very smooth p2-dependence,

this looks like a relatively operator-independent observation.

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Thanks for your attention!

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20 40 60 80 100 120 p

2

0,9 0,95 1 1,05 1,1 σ σcont[1,1,0,0] σfine[1,1,0,0] σcoarse[1,1,0,0] σcont[1,1,1,1] σfine [1,1,1,1] σcoarse[1,1,1,1] NLO LO

(27,1) γγ

20 40 60 80 100 120 psq 0,9 0,95 1 1,05 1,1 1,15 σ σcont[1,1,0,0] σfine[1,1,0,0] σcoarse[1,1,0,0] σcont[1,1,1,1] σfine[1,1,1,1] σcoarse[1,1,1,1] NLO

(27,1) qq

20 40 60 80 100 120 psq 0,95 1 1,05 sigma sigma data 0 interp 0 data 1 interp 1 sigma data 0 interp 0 data 1 interp 1 NLO LO

results/StepCont_GGdir1_1_1.agr

20 40 60 80 100 120 psq 0,95 1 1,05 1,1 sigma sigma data 0 interp 0 data 1 interp 1 sigma data 0 interp 0 data 1 interp 1

results/StepCont_QQdir1_1_1.agr

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50 100 150 200 psq 0,5 1 1,5 2 sigma sigma data 0 data 1 sigma data 0 data 1 NLO LO

results/StepCont_GGdir1_2_2.agr

50 100 150 200 psq 0,5 1 1,5 2 sigma sigma data 0 interp 0 data 1 interp 1 sigma data 0 interp 0 data 1 interp 1

results/StepCont_QQdir1_2_2.agr

50 100 150 200 psq 0,5 1 1,5 2 sigma sigma data 0 interp 0 data 1 interp 1 sigma data 0 interp 0 data 1 interp 1

results/StepCont_GGdir1_3_3.agr

50 100 150 200 psq 0,5 1 1,5 2 sigma sigma data 0 interp 0 data 1 interp 1 sigma data 0 interp 0 data 1 interp 1

results/StepCont_GGdir1_4_4.agr

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20 40 60 80 100 120 p

2
  • 0,4
  • 0,3
  • 0,2
  • 0,1

0,1 0,2 σ(p,3GeV) σcont[1,1,0,0] σfine[1,1,0,0] σcoarse[1,1,0,0] σcont[1,1,1,1] σfine[1,1,1,1] σcoarse[1,1,1,1] NLO LO

(8,8)12 gg

20 40 60 80 100 120 p

2 (GeV 2)
  • 0,2
  • 0,1

0,1 0,2 σ σcont[1,1,0,0] σfine[1,1,0,0] σcoarse[1,1,0,0] σcont[1,1,1,1] σfine[1,1,1,1] σcoarse[1,1,1,1]

(8,8)1,2 qq

50 100 150 200 psq

  • 0,02
  • 0,01

0,01 0,02 0,03 0,04 sigma sigma data 0 interp 0 data 1 interp 1 sigma data 0 interp 0 data 1 interp 1 NLO LO

results/StepCont_GGdir1_2_1.agr

50 100 150 200 psq

  • 0,04
  • 0,02

0,02 0,04 sigma sigma data 0 interp 0 data 1 interp 1 sigma data 0 interp 0 data 1 interp 1

results/StepCont_QQdir1_2_1.agr

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20 40 60 80 100 120 p

2
  • 0,005

0,005 0,01 0,015 σ(p,3GeV) σcont[1,1,0,0] σfine[1,1,0,0] σcoarse[1,1,0,0] σcont[1,1,1,1] σfine[1,1,1,1] σcoarse[1,1,1,1]

(6,6)12 gg

50 100 150 200 psq

  • 0,3
  • 0,2
  • 0,1

0,1 0,2 0,3 sigma sigma data 0 interp 0 data 1 interp 1 sigma data 0 interp 0 data 1 interp 1

results/StepCont_GGdir1_4_3.agr