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Logarithmic corrections Yang Mills theory Nikolai Husung with Peter Marquard, Rainer Sommer Wuhan, China, June 17th, 2019 to a 2 scaling in lattice Motivation Figure: Deviation of the step scaling function from its continuum counterpart, O g g

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Logarithmic corrections to a2 scaling in lattice Yang Mills theory

Nikolai Husung with Peter Marquard, Rainer Sommer Wuhan, China, June 17th, 2019

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Motivation

0.05 0.1 0.15 0.2 0.01 0.02 0.03 a/L δΣ(a) = Σ(a) − Σ(0)

Figure: Deviation of the step scaling function from its continuum counterpart, as an example, in the 2-dimensional O(3) model [Balog et al., 2009, 2010].

O(3) model: “Worst case” example [Balog et al., 2009, 2010]

const.a g − g O g

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 2

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Motivation

1.6 1.8 2 2 4 6 8 1/g2 L2 Σ(a) − Σ(0) a2 2 parameter fjt

Figure: Deviation of the step scaling function from its continuum counterpart, as an example, in the 2-dimensional O(3) model [Balog et al., 2009, 2010].

O(3) model: “Worst case” example [Balog et al., 2009, 2010]

δΣ = const.a2 (g2

0)−3−1.1386(g2 0)−2 + O((g2 0)−1)

  • DESYª

| Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 2

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Symanzik efgective theory I

Idea: Parametrise lattice artifacts originating from the lattice action [and for a fjeld Φ] by a minimal basis of operators living in a continuous Symanzik efgective theory [Symanzik, 1980, 1981, 1983a,b] Leff = L + a2δL + O(a4) , (1) Φeff = Φ + a2δΦ + O(a4) , (2) where a is the lattice spacing and δL =

  • i

biOi , (3) δΦ =

  • i

ciΦi , (4) with free coeffjcients bi and ci.

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 3

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Symanzik efgective theory II

Occurring operators Oi and Φi must comply with symmetries, i.e. for the lattice pure gauge action > Local SU(N) gauge symmetry, > C-, P- and T -symmetry, > discrete rotation and translation invariance (at least in infjnite volume) ⇒ broken O(4) symmetry. Require minimal basis for physical energies and matrix elements (“on-shell”) ⇒ use EOMs to reduce set of operators [Lüscher and Weisz, 1985; Georgi, 1991].

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 4

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Symanzik efgective theory III

The only operators Oi left are [Lüscher and Weisz, 1985] O0 = 1 g2 tr(DµFνρDµFνρ), O1 = 1 g2

  • µ

tr(DµFµρDµFµρ), O2 = 1 g2 tr(DµFµνDρFρν)

EOM

= 0 . (5) We then fjnd for a fjeld Φ in analogy to [Balog et al., 2009, 2010] Φlatt(a) = Φcont

  • 1 + a2
  • i

ciδΦ

i (a) − 1

  • j=0

bjδO

j (a)

  • + O(a4)
  • . (6)

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 5

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Symanzik efgective theory IV

Leading lattice artifacts

Φlatt(a) Φcont = 1 + a2

  • i

¯ ciδΦ

i (a) − 1

  • j=0

¯ bjδO

j (a)

  • × [1 + O(α(1/a))] + O(a4)

with tree-level coeffjcients ¯ ci and ¯ bj. Need to understand (leading) lattice spacing dependence of δO

j (a) =

  • d4x
  • ΦOj(x)cont;R

Φcont;R − Oj(x)cont;R

  • µ=1/a

, δΦ

i (a) =

Φicont;R Φcont;R

  • µ=1/a

. ⇒ Renormalisation Group

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 6

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Symanzik efgective theory V

Leading lattice artifacts

Φlatt(a) Φcont = 1 + a2

  • i

¯ ciδΦ

i (a) − 1

  • j=0

¯ bjδO

j (a)

  • × [1 + O(α(1/a))] + O(a4)

Remarks: > Tree-level coeffjcients ¯ ci and ¯ bj can be obtained from classical expansion in the lattice spacing. > We limit ourselves to the leading behaviour as a ց 0, i.e. we do not require 1-loop coeffjcients. However, if tree-level coeffjcient is zero 1-loop might be needed to obtain leading logarithms. > We consider only the case ci = 0 to all orders, i.e. no additional artifacts from observables.

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 7

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Renormalisation Group I

Use Renormalisation Group Equations (RGEs) to determine renormalisation scale dependence µ2 dδO

i (µ)

dµ2 = γijδO

j (µ),

β(α) = µ2 dα(µ) dµ2 = −α2

n≥0

βnαn, (7) where γ is the anomalous dimension matrix γij = µ2 d ln (Z)ij dµ2 = (γ0)ijα + O(α2), Oi;R = ZijOj;0 . (8)

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 8

renormalisation scheme independent

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Renormalisation Group II

We choose a basis such that γ0 = diag{(γ0)1, . . . , (γ0)n} and introduce the Renormalisation Group Invariant (RGI) DO

i (Λ) = lim µ→∞ [2β0α(µ)]ˆ γi δO i (µ) ,

ˆ γi = (γ0)i β0 , (9) with RGI scale Λ. This allows us to rewrite δO

i (µ) =(2β0α(µ)) −ˆ γi Pexp

  

α(µ)

  • dx

γ(x) β(x) + γ0 β0x

 

ij

DO

j (Λ)

(10)

i Di

O

i

Note: The renormalisation scale dependence is only in the prefactor of the RGI with leading power determined by

i.

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 9

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Renormalisation Group II

We choose a basis such that γ0 = diag{(γ0)1, . . . , (γ0)n} and introduce the Renormalisation Group Invariant (RGI) DO

i (Λ) = lim µ→∞ [2β0α(µ)]ˆ γi δO i (µ) ,

ˆ γi = (γ0)i β0 , (9) with RGI scale Λ. This allows us to rewrite δO

i (µ) =(2β0α(µ)) −ˆ γi Pexp

  

α(µ)

  • dx

γ(x) β(x) + γ0 β0x

 

ij

DO

j (Λ)

(10) =(2β0α(µ))−ˆ

γi DO i (Λ) + O

  • [α(µ)]1−ˆ

γi

. Note: The renormalisation scale dependence is only in the prefactor of the RGI with leading power determined by ˆ γi.

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 9

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Computing leading anomalous dimensions I

Renormalise operator basis at 1-loop by computing connected Green’s functions with operator insertion in continuum theory

  • ˜

A1;R(p1) . . . ˜ An;R(pn) ˜ Oi;R(q)

  • con =

= Z n

AZij

  • ˜

A1;0(p1) . . . ˜ An;0(pn) ˜ Oj;0(q)

  • con

with fundamental gauge fjelds ˜ A (gauge fjxed), and momenta pk, q. Tools: QGRAF, FORM p1 p2 q p1 p2 p3 q To obtain the anomalous dimension we extract only the UV-pole contributions following e.g. the procedure from [Misiak and Münz, 1995; Chetyrkin et al., 1998].

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 10

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Computing leading anomalous dimensions I

Renormalise operator basis at 1-loop by computing connected Green’s functions with operator insertion in continuum theory

  • ˜

A1;R(p1) . . . ˜ An;R(pn) ˜ Oi;R(q)

  • con =

= Z n

AZij

  • ˜

A1;0(p1) . . . ˜ An;0(pn) ˜ Oj;0(q)

  • con

with fundamental gauge fjelds ˜ A (gauge fjxed), and momenta pk, q. Tools: QGRAF, FORM p1 p2 q p1 p2 p3 q To obtain the anomalous dimension we extract only the UV-pole contributions following e.g. the procedure from [Misiak and Münz, 1995; Chetyrkin et al., 1998].

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 10

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Computing leading anomalous dimensions II

Common approach > “on-shell” momenta avoid gauge-variant contributions from gauge-fjxing. > If q = 0 total divergence

  • perators contribute, but 2- and

3-point functions are accessible for renormalisation. Background fjeld method [’t Hooft, 1975; Abbott, 1981, 1982; Lüscher and Weisz, 1995] > Only gauge-invariant operators relevant for renormalisation. > Can keep pk arbitrary and q , but EOM-vanishing

  • perators contribute (unless
  • n-shell).

Obtain relevant part of mixing matrix via Q

R

Z Z

Q

ZQQ Q (11) with class of “redundant” operators Q.

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 11

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Computing leading anomalous dimensions II

Common approach > “on-shell” momenta avoid gauge-variant contributions from gauge-fjxing. > If q = 0 total divergence

  • perators contribute, but 2- and

3-point functions are accessible for renormalisation. Background fjeld method [’t Hooft, 1975; Abbott, 1981, 1982; Lüscher and Weisz, 1995] > Only gauge-invariant operators relevant for renormalisation. > Can keep pk arbitrary and q = 0, but EOM-vanishing

  • perators contribute (unless
  • n-shell).

Obtain relevant part of mixing matrix via Q

R

Z Z

Q

ZQQ Q (11) with class of “redundant” operators Q.

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 11

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Computing leading anomalous dimensions II

Common approach > “on-shell” momenta avoid gauge-variant contributions from gauge-fjxing. > If q = 0 total divergence

  • perators contribute, but 2- and

3-point functions are accessible for renormalisation. Background fjeld method [’t Hooft, 1975; Abbott, 1981, 1982; Lüscher and Weisz, 1995] > Only gauge-invariant operators relevant for renormalisation. > Can keep pk arbitrary and q = 0, but EOM-vanishing

  • perators contribute (unless
  • n-shell).

Obtain relevant part of mixing matrix via O Q

  • R

= ZOO ZOQ ZQQ O Q

  • (11)

with class of “redundant” operators Q.

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 11

needed additionally

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Lattice artifacts originating from the action

We fjnd for the minimal basis

  • O0

O1

  • R

=

  • 1 + 7CA

3ǫ α 4π

− 7CA

15ǫ α 4π

1 + 21CA

5ǫ α 4π

  • O0

O1

  • .

(12) Thus the diagonal basis and leading anomalous dimensions are CA (13) CA [Narison and Tarrach, 1983; Alonso et al., 2014]

Leading lattice artifacts from the action

latt a cont

a

j

bj a

jDj

O a O a Leading anomalous dimensions improve the convergence as a .

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 12

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Lattice artifacts originating from the action

We fjnd for the minimal basis

  • O0

O1

  • R

=

  • 1 + 7CA

3ǫ α 4π

− 7CA

15ǫ α 4π

1 + 21CA

5ǫ α 4π

  • O0

O1

  • .

(12) Thus the diagonal basis and leading anomalous dimensions are B0 =O0 −ˆ γ0 = 7CA 12πβ0 = 7 11 ≈ 0.636 , B1 =O1 − 1 4O0 , (13) −ˆ γ1 = 21CA 20πβ0 = 63 55 ≈ 1.145 . [Narison and Tarrach, 1983; Alonso et al., 2014]

Leading lattice artifacts from the action

latt a cont

a

j

bj a

jDj

O a O a Leading anomalous dimensions improve the convergence as a .

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 12

due to O(4) symmetry

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Lattice artifacts originating from the action

We fjnd for the minimal basis

  • O0

O1

  • R

=

  • 1 + 7CA

3ǫ α 4π

− 7CA

15ǫ α 4π

1 + 21CA

5ǫ α 4π

  • O0

O1

  • .

(12) Thus the diagonal basis and leading anomalous dimensions are B0 =O0 −ˆ γ0 = 7CA 12πβ0 = 7 11 ≈ 0.636 , B1 =O1 − 1 4O0 , (13) −ˆ γ1 = 21CA 20πβ0 = 63 55 ≈ 1.145 . [Narison and Tarrach, 1983; Alonso et al., 2014]

Leading lattice artifacts from the action

Φlatt(a) Φcont = 1 − a2

  • 1
  • j=0

˜ bj[α(1/a)]−ˆ

γjDB j (Λ)

  • × [1 + O(α(1/a))] + O(a4)

⇒ Leading anomalous dimensions improve the convergence as a ց 0.

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 12

due to O(4) symmetry

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Conclusion

> No a2[α(1/a)]−3 behaviour like for the O(3) model nor naive a2, but a2 [α(1/a)]0.636 + d1[α(1/a)]1.145 + d2[α(1/a)]1.636 + . . .

  • is the leading behaviour.

> Leading anomalous dimensions of contributions from the lattice Yang Mills action improve convergence as a ց 0. > Short-cuts in perturbation theory can ease computational efgort. > Full Nf ×

j=1 U(1)V fmavour symmetric

  • lattice QCD is currently in

progress (well advanced). > Gradient fmow observables require additional operators on the 4D-boundary, also in pure gauge theory.

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 13

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References I

Janos Balog, Ferenc Niedermayer, and Peter Weisz. Logarithmic corrections to O(a2) lattice artifacts. Phys. Lett., B676:188–192, 2009. Janos Balog, Ferenc Niedermayer, and Peter Weisz. The Puzzle of apparent linear lattice artifacts in the 2d non-linear sigma-model and ’s solution. Nucl. Phys., B824:563–615, 2010.

  • K. Symanzik. CUTOFF DEPENDENCE IN LATTICE φ**4 in

four-dimensions THEORY. NATO Sci. Ser. B, 59:313–330, 1980.

  • K. Symanzik. Some Topics in Quantum Field Theory. In Mathematical

Problems in Theoretical Physics. Proceedings, 6th International Conference on Mathematical Physics, West Berlin, Germany, August 11-20, 1981, pages 47–58, 1981.

  • K. Symanzik. Continuum Limit and Improved Action in Lattice Theories.
  • 1. Principles and phi**4 Theory. Nucl. Phys., B226:187–204, 1983a.

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 14

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References II

  • K. Symanzik. Continuum Limit and Improved Action in Lattice Theories.
  • 2. O(N) Nonlinear Sigma Model in Perturbation Theory. Nucl. Phys.,

B226:205–227, 1983b. Martin Lüscher and Peter Weisz. On-shell improved lattice gauge

  • theories. Comm. Math. Phys., 97(1-2):59–77, 1985.

Howard Georgi. On-shell efgective fjeld theory. Nucl. Phys., B361: 339–350, 1991. Mikolaj Misiak and Manfred Münz. Two loop mixing of dimension fjve fmavor changing operators. Phys. Lett., B344:308–318, 1995. Konstantin G. Chetyrkin, Mikolaj Misiak, and Manfred Münz. Beta functions and anomalous dimensions up to three loops. Nucl. Phys., B518:473–494, 1998.

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 15

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References III

Gerard ’t Hooft. The Background Field Method in Gauge Field Theories. In Functional and Probabilistic Methods in Quantum Field Theory. 1. Proceedings, 12th Winter School of Theoretical Physics, Karpacz, Feb 17-March 2, 1975, pages 345–369, 1975.

  • L. F. Abbott. The Background Field Method Beyond One Loop. Nucl.

Phys., B185:189–203, 1981.

  • L. F. Abbott. Introduction to the Background Field Method. Acta Phys.

Polon., B13:33, 1982. Martin Lüscher and Peter Weisz. Background fjeld technique and renormalization in lattice gauge theory. Nucl. Phys., B452:213–233, 1995.

  • S. Narison and R. Tarrach. Higher dimensional renormalization group

invariant vacuum condensates in quantum chromodynamics. Physics Letters B, 125(2):217 – 222, 1983.

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 16

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References IV

Rodrigo Alonso, Elizabeth E. Jenkins, Aneesh V. Manohar, and Michael

  • Trott. Renormalization Group Evolution of the Standard Model

Dimension Six Operators III: Gauge Coupling Dependence and

  • Phenomenology. JHEP, 04:159, 2014.

DESYª | Logarithmic corrections to a2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 Page 17