part i/v stefan sint - saha institute of nuclear...

Non-perturbative Renormalization of Lattice QCDPart I/V

Stefan Sint

Trinity College Dublin & NIC@DESY-Zeuthen

Saha Institute of Nuclear Physics

Kolkata, 2 December 2013

Some references

1 P. Weisz, ”Renormalization and lattice artifacts”, Les HouchesSummer School 2009, arXiv:1004.3462v1 [hep-lat];

2 R. Sommer, “Non-perturbative renormalisation of QCD”,Schladming Winter School lectures 1997, hep-ph/9711243v1;“Non-perturbative QCD: Renormalization, O(a) improvementand matching to heavy quark effective theory” Lectures atNara, November 2005 hep-lat/0611020

3 M. Luscher: “Advanced lattice QCD”, Les Houches SummerSchool lectures 1997 hep-lat/9802029

4 S. Capitani, “Lattice perturbation theory” Phys. Rept. 382(2003) 113-302 hep-lat/0211036

5 S. Sint “Nonperturbative renormalization in lattice fieldtheory” Nucl. Phys. (Proc. Suppl.) 94 (2001) 79-94,hep-lat/0011081

Contents

1 QCD and the Standard Model

2 Non-perturbative definition of QCD

3 Hadronic renormalization schemes

4 From bare to renormalized parameters

5 Non-perturbatively defined renormalized parameters

6 Lattice QCD with Wilson quarks

7 Approach to the continuum limit & Symanzik’s effectivetheory

8 The 2d O(n) model as a test laboratory

9 Callan-Symanzik equation, Λ-parameter and RGI quark masses

10 The problem of large scale differences

QCD and the Standard Model of particle physics

The Standard Model (SM):

describes strong, weak and electromagnetic interactions;gauge theory SU(3)×SU(2)×U(1)

large scale differences, for instance:

mt ,mH ,mZ ,mW = O(100GeV) mb,mc = O(1GeV)

with light quark masses still much lighter.

⇒ SM for energies mW reduces to QCD + QED + tower ofeffective weak interaction vertices (4-quark-operators, 6-quarkoperators ...).

⇒ the structure of this effective“weak hamiltonian” is obtainedperturbatively e.g in MS scheme.

⇒ QCD + effective 4-quark operators a priori defined inperturbative framework at high energies.

Non-perturbative definition of QCD (1)

To define QCD as a QFT beyond perturbation theory it is notenough to write down its classical Lagrangian:

LQCD(x) =1

2g 2tr Fµν(x)Fµν(x)+

Nf∑i=1

ψi (x) (D/+ mi )ψi (x)

One needs to define the functional integral:

Introduce a Euclidean space-time lattice and discretise thecontinuum action such that the doubling problem is solved

Consider a finite space-time volume ⇒ the functional integralbecomes a finite dimensional ordinary or Grassmann integral,i.e. mathematically well defined!

Take the infinite volume limit L→∞Take the continuum limit a→ 0


The infinite volume limit is reached with exponentialcorrections ⇒ usually not a major problem.

Continuum limit: existence only established order by order inperturbation theory; only for selected lattice regularisations:

lattice QCD with Wilson quarks [Reisz ’89 ]lattice QCD with overlap/Neuberger quarks [Reisz, Rothe ’99 ]not (yet ?) for lattice QCD with staggered quarks [cf. Giedt’06 ]

From asymptotic freedom expect

g 20 = g 2

0 (a)a→0∼ −1

2b0 ln a, b0 = 11N

3 − 23 Nf


Working hypothesis: the perturbative picture is essentially correct:

The continuum limit of lattice QCD exists and is obtained bytaking g0 → 0

Hence, QCD is asymptotically free, naive dimensional analysisapplies: Non-perturbative renormalisation of QCD is based onthe very same counterterm structure as in perturbation theory!

Absence of analytical methods: try to take the continuumlimit numerically, i.e. by numerical simulations of lattice QCDat decreasing values of g0.

WARNING:

Perturbation Theory might be misleading (cp. triviality ofφ4

4-theory)

Renormalisation of QCD

The basic parameters of QCD are g0 and mi , i = u, d , . . ..

To renormalise QCD one must impose a correspondingnumber of renormalisation conditions

If we only consider gauge invariant observables

⇒ no need to renormalize quark, gluon, ghost field and gaugeparameter.

All physical information (particle masses and energies, particleinteractions) is contained in the (Euclidean) correlationfunctions of gauge invariant composite, local fields φi (x)

〈φ1(x1) · · ·φn(xn)〉

a priori each φi requires renormalisation, and thus furtherrenormalisation conditions.

What would we like to achieve?Natural question to ask:

What are the values of the fundamental parameters of QCD (andthus of the Standard Model!),

αs , mu ≈ md , ms , mc , mb

if we renormalise QCD in a hadronic renormalization scheme,i.e. by choosing the same number of experimentally well-measuredhadron properties: Fπ, mπ,mK , mD , mB ?

QCD is regarded as a low energy approximation to theStandard Model; e.m. effects/isospin breaking effects aresmall (αe.m. = 1/137) and must be subtracted fromexperimental results.

conceptually clean, natural question for lattice QCD

alternative: combination of perturbation theory +assumptions (”quark hadron duality”, sum rules,hadronisation Monte-Carlo, . . .).

Renormalization of QCD in hadronic scheme

Sketch of the procedure, using e.g. hadronic observablesFπ, mπ,mK , mD , mB :

1 Choose a value of the bare coupling g 20 = 6/β; this

determines the lattice spacing (i.e. massless scheme)

2 Compute the hadronic input parameters (spatial latticevolume (L/a)3 must be large enough ⇒ constraint for choiceof g0 or β in 1.);

3 tune the bare quark mass parameters such that mπ/Fπ,mK/Fπ, mD/Fπ take their desired values (e.g. experimentalones, but not necessarily!)

4 The lattice spacing is obtained from a(β) = (aFπ)(β)/Fπ|exp.

5 Reduce the value of g 20 (i.e. increase β)

6 Repeat steps 1 – 5 until you run out of resources...

Auxiliary scale parameters r0, t0, w0

For technical reasons one often introduces an auxiliary scaleparameter:

serves as a yardstick for precise tuning or scaling studies.

should be easily computable (in any case easier than say Fπ)

should have a mild dependence on the quark masses

Example: Sommer’s scale r0 obtained from the force F (r)between static quarks:

r 20 F (r0) = 1.65 ⇒ r0 ≈ 0.5 fm

Idea: at finite a use r0/a rather than aFπ but also determiner0Fπ(β); Conversion to physical units from Fπ is thenpostponed to the continuum limit.

Advantage: constant physics conditions can be satisfied moreprecisely.

Future: a very convenient scale t0 is based on the gradient flow[M. Luscher ’10 ]; (later also a variant w0 by the BMW coll.)

From bare to renormalised parameters

For g 20 (or β) in some interval one obtains:

Fπ,mπ,mK ,mD ⇒ g0, am0,l(g0), am0,s(g0), am0,c(g0)

These are bare parameters, their continuum limit vanishes!

N.B.: due to quark confinement there is no natural definitionof “physical” quark masses or the coupling constant fromparticle masses or interactions

At high energy scales, µ mp, one may use perturbativeschemes to define renormalised parameters (e.g. dimensionalregularisation and minimal subtraction)

How can we relate the bare lattice parameters to therenormalised ones in, say, the MS scheme?

basic idea: introduce an intermediate renormalisation schemewhich can be evaluated both perturbatively andnon-perturbively.

Why not use perturbation theory directly?

Shortcut: try to relate the bare parameters directly to MS scheme,e.g. coupling: Allowing for a constant d = O(1),

αMS(d/a) = α0(a) + c1α20(a) + c2α

30(a) + . . . , α0 =

g 20

4π

mMS(d/a) = m(a)(

1 + Z(1)m α0(a) + Z

(2)m α2

0(a) + . . .

Main difficulties:

Setting µ ∝ a−1 means that cutoff effects and renormalisationeffects cannot be disentangled; any change in the scale is atthe same time a change in the cutoff.

One needs to assume that the cutoff scale d/a is in theperturbative region, higher order effects negligible.

One furthermore assumes that cutoff effects are negligible

⇒ how reliable are the error estimates?

Non-perturbatively defined renormalized parametersExample for a renormalised coupling

Consider the force F (r) between static quarks at a distance r , anddefine

αqq(r) = r 2F (r)|mq,i=0

at short distances:

αqq(r) = αMS(µ) + c1(rµ)α2MS

(µ) + . . .

at large distances:

limr→∞

αqq(r) =

∞ for Nf = 0

0 for Nf > 0

NB: renormalization condition is imposed in the chiral limit ⇒αqq(r) and its β-function are quark mass independent.

Example for a renormalised quark mass

Use PCAC relation as starting point:

∂µ(AR)aµ = 2mR(PR)a

Aaµ, Pa: isotriplet axial current & density

The normalization of the axial current is fixed by currentalgebra (i.e. axial Ward identities) and scale independent!

⇒ Quark mass renormalization is inverse to the renormalizationof the axial density:

(PR)a = ZPPa, mR = Z−1P mq.

⇒ Impose renormalization condition for the axial density ratherthan for the quark mass

Renormalization condition for axial density

Define 〈PaR(x)Pb

R(y)〉 = δabGPP(x − y), and impose the condition

GPP(x)∣∣∣µ2x2=1,mq,i=0

= − 1

2π4(x2)3

GPP(x) is defined at all distances:

GPP(x)x2→0∼ − 1

2π4(x2)3+O(g 2), GPP(x)

x2→∞∼ − 1

4π2x2G 2π+. . .

⇒ ZP is defined at all scales µ:

at large µ (but µ 1/a):

ZP(g0, aµ) = 1 + g 20 d0 ln(aµ) + . . . ,

at low scales µ:ZP(g0, aµ) ∝ µ2

Lattice QCD with Wilson quarks

The action S = Sf + Sg is given by

Sf = a4∑x

ψ(x) (DW + m0)ψ(x), Sg = 1g2

0

∑µ,ν

tr 1− Pµν(x)

DW = 12

(∇µ +∇∗µ

)γµ − a∇∗µ∇µ

Symmetries: U(Nf)V (mass degenerate quarks), P,C ,T andO(4,ZZ)

⇒ Renormalized parameters:

g 2R = Zgg 2

0 , mR = Zm (m0 −mcr) , amcr = amcr(g0).

In general: Z = Z (g 20 , aµ, am0);

Quark mass independent renormalisation schemes:Z = Z (g 2

0 , aµ)

Simple non-singlet composite fields, e.g. Pa = ψγ5τaψ

renormalise multiplicatively, PaR = ZP(g 2

0 , aµ, am0)Pa

Approach to the continuum limit (1)

Suppose we have renormalised lattice QCD non-perturbatively, howis the the continuuum limit approached?Symanzik’s effective continuum theory [Symanzik ’79 ]:

purpose: render the a-dependence of lattice correlationfunctions explicit. ⇒ structural insight into the nature ofcutoff effects

at scales far below the cutoff a−1, the lattice theory iseffectively continuum like; the influence of cutoff effects isexpanded in powers of a:

Seff = S0 + aS1 + a2S2 + . . . , S0 = ScontQCD

Sk =

∫d4x Lk(x)

Lk(x): linear combination of fields

with canonical dimension 4 + kwhich share all the symmetries with the lattice action


A complete set of dimension 5 fields for L1 is given by:

ψσµνFµνψ, ψDµDµψ, mψD/ψ, m2ψψ, m tr FµνFµν

The same procedure applies to composite fields:

φeff(x) = φ0 + aφ1 + a2φ2 . . .

for instance: φ(x) = Pa(x), basis for φ1:

mψγ512τ

aψ, ψD/←γ5

12τ

aψ − ψγ512τ

aD/ψ

Consider renormalised, connected lattice n-point functions of amultiplicatively renormalisable field φ

Gn(x1, . . . , xn) = Znφ 〈φ(x1) · · ·φ(xn)〉con


Effective field theory description:

Gn(x1, . . . , xn) = 〈φ0(x1) . . . φ0(xn)〉con

+ a

∫d4y 〈φ0(x1) . . . φ0(xn)L1(y)〉con

+ an∑

k=1

〈φ0(x1) . . . φ1(xk) . . . φ0(xn)〉con + O(a2)

〈· · · 〉 is defined w.r.t. continuum theory with S0

the a-dependence is now explicit, up to logarithms, which arehidden in the coefficients.

In perturbation theory one expects at l-loop order:

P(a) ∼ P(0) +∞∑n=1

l∑k=1

cnkan(ln a)k

where e.g. P(a) = Gn at fixed arguments.


Conclusions from Symanzik’s analysis:

Asymptotically, cutoff effects are powers in a, modified bylogarithms;

In contrast to Wilson quarks, only even powers of a areexpected for

bosonic theories (e.g. pure gauge theories, scalar field theories)fermionic theories which retain a remnant axial symmetry(overlap, Domain Wall Quarks, staggered quarks, Wilsonquarks with a twisted mass term, etc.)

In QCD simulations a is typically varied by a factor 2

⇒ logarithms vary too slowly to be resolved; linear or quadraticfits (in a resp. a2) are used in practice.

Example 1: quenched hadron spectrum

Linear continuum extrapolation of the quenched hadron spectrum;standard Wilson quarks with Wilson’s plaquette action:[CP-PACScoll., Aoki et al. ’02 ] a = 0.05− 0.1 fm, experimental input:mK , mπ, mρ

0.8

0.9

1.0

1.1

K*

φ

0.80.91.01.11.21.3

N

ΛΣΞ

0.0 0.1 0.2 0.3 0.4 0.5 0.61.2

1.3

1.4

1.5

1.6

∆Σ*Ξ*Ω

Example 2: pion mass in Nf = 2 tmQCD

[ETM coll. Baron et al ’09 ]

= 0.045= 0.090

rχ0µR = 0.130(rχ0mPS)

2

(a/rχ0 )2

0.060.040.020

1.4

1.0

0.6

0.2

Example 3: O(a) improved charm quark mass (quenched)

[ALPHA coll. J. Rolf et al ’02 ]

Example 3: Step Scaling Function for SF coupling (Nf = 2)

[ALPHA coll., Della Morte et al. 2005 ]

The 2d O(n) sigma model: a test laboratory for QCD?

S = N2γ

∑x ,µ

(∂µs)2, s = (s1, . . . , sN) s2 = 1

like QCD the model has a mass gap and is asymptotically free

many analytical tools: large N expansion, Bethe ansatz, formfactor bootstrap, etc.

efficient numerical simulations due to cluster algorithms.

⇒ very precise data over a wide range of lattice spacing (a canbe varied by 1-2 orders of magnitude).

Symanzik: expect O(a2) effects, up to logarithms

Large N, at leading [Caracciolo, Pelissetto ’98 ] andnext-to-leading [Knechtli, Leder, Wolff ’05 ]:

P(a) ∼ P(0) +a2

L2(c1 + c2 ln(a/L))

A sobering result (1):

Numerical study of renormalised finite volume coupling to highprecision (n = 3) [Hasenfratz, Niedermayer ’00, Hasenbusch et al.’01, Balog et al. ’09 ]

Cutoff effects (blue points) seem to be almost linear in a!

Is this just an unfortunate case?

A sobering result (2):

[Balog, Niedermayer & Weisz ’09 ]

0 0.05 0.1a/L

1.25

1.27

1.29

1.31

Σ(2,

u 0,a/L

)

Fits with a and a ln a terms, lattice sizes L/a = 10, . . . , 64

A closer look (1)

[Knechtli, Leder, Wolff ’05 ], plot of cutoff effects vs. a2/L2,various n:

0 0.0005 0.001 0.0015 0.002

1

1.002

1.004

1.006 N = 3N = 4N = 8N = ∞

(a/L)2

Σ(u

0,N

,a/L

)/σ(u

0,N

)

Asymptotic behaviour for larger n according to expectation, whatabout n = 3?

A closer look (2)

Continuum limit for mass gap m(L) known analytically [Balog &Hegedus ’04 ]! Subtract & study pure cutoff effect [Balog,Niedermayer, Weisz ’09 ]

0 0.05 0.1a/L

0

0.01

0.02

0.03

0.04

δΣ(2

,u0,a

/L)

O(3), STO(3), D(1/3)O(3), D(-1/4)

A closer look (3)

Continuum limit for mass gap m(L) known analytically [Balog &Hegedus ’04 ]!Subtract & study pure cutoff effect: Σ(2, u0, a/L)− σ(2, u0):

0 0.05 0.1 0.15a/L

0

0.01

0.02

0.03

Σ(2,

u 0,a/L

) -

σ(2,

u 0)

0 0.020

0.002

0.004

c1a + c2a ln a + c3a2 (dashed) vs. c1a2 + c2a2 ln a + c3a4 (solid)

A closer look (4) & solution of puzzle

[Balog, Niedermayer & Weisz ’09 ]

performed two-loop calculation with both effective Symanziktheory and lattice theory (various actions)

Matching of both sides and subsequent RG considerations

⇒ Symanzik theory predicts for O(n) model, leading O(a2

behaviour:δ(a) ∝ a2

(ln a2

)n/(n−2)

(compatible with previous large n result)

For O(3) model:

δ(a) ∝ a2(ln3(a2) + c1 ln2(a2) + c2 ln(a2) + c4

)+ O(a4)

A closer look (5)

Coefficient of O(a2) term [Balog, Niedermayer & Weisz ’09 ]:

1.6 1.7 1.8 1.9β

0

1

2

3

4

5

6

7

[Σ(2

,u0,a

/L)-

σ(2,

u 0)] L

2

Not exactly constant! Multiplied with a2 obtain “fake” linearbehaviour in a!

Conclusion

Symanzik’s analysis is applicable beyond perturbation theory

In QCD numerical results seem to confirm expectations;

However, The Symanzik expansion is only asymptotic, andpowers of a are accompanied by (powers of) logarithms,

Lesson from σ model: logarithmic corrections to powers in anot always negligible!

It helps to combine results from different regularisations:renormalised quantities must agree in the continuum limit(assuming universality)

Renormalization group functions

The renormalized coupling and quark mass are definednon-perturbatively at all scales⇒ Renormalization group functions are defined non-perturbatively,too:

β-function

β(g) = µ∂g(µ)

∂µ, g 2(µ) = 4παqq(1/µ)

quark mass anomalous dimension:

τ(g) =∂ ln m(µ)

∂ lnµ= − lim

a→0

∂ ln ZP(g0, aµ)

∂ ln aµ

∣∣∣∣g(µ)

Asymptotic expansion for weak couplings:

β(g) ∼ −g 3b0 − g 5b1 . . . , b0 =

113 N − 2

3 Nf

(4π)−2, ...

τ(g) ∼ −g 2d0 − g 4d1 . . . , d0 = 3(N − N−1)(4π)−2, . . .

The Callan-Symanzik equation

Physical quantities P are independent of µ, and thus satisfy theCS-equation:

µ∂

∂µ+ β(g)

∂

∂g+ τ(g)m

∂

∂m

P = 0

Λ and Mi are special solutions:

Λ = µ (b0g 2)−b1/2b20 exp

− 1

2b0g 2

× exp

−∫ g

0dx

[1

β(x)+

1

b0x3− b1

b20x

]

Mi = mi (2b0g 2)−d0/2b0 exp

−∫ g

0dx

[τ(x)

β(x)− d0

b0x

]N.B. no approximations involved!

Λ and Mi as fundamental parameters of QCD

defined beyond perturbation theory

scale independent

scheme dependence? Consider finite renormalization:

g ′R = gRcg (gR), m′R,i = mR,icm(gR)

with asymptotic behaviour c(g) ∼ 1 + c(1)g 2 + . . .⇒ find the exact relations

M ′i = Mi , Λ′ = Λ exp(c(1)g /b0).

⇒ ΛMS can be defined indirectly beyond PT; to obtain Λ inany other scheme requires the one-loop matching of therespective coupling constants.

Strategy to compute Λ and Mi

At fixed g0 determine the bare parameters corresponding tothe experimental input.

Determine αqq(1/µ) and ZP(g0, aµ) at the same g0 in thechiral limit

use ZP to pass from bare to renormalised quark masses

do this for a range of µ-values

repeat the same for a range of g0-values and take thecontinuum limit

lima→0

Z−1P (g0, aµ)mi (g0), lim

a→0αqq(1/µ)

check wether perturbative scales µ have been reached

if this is the case, use the perturbative β- and τ -function toextrapolate to µ =∞; extract Λ and Mi (equivalently convertto MS scheme deep in perturbative region).

Example: running of the coupling (SF scheme, Nf = 2)

[ALPHA, M. Della Morte et al. 2005 ]

The problem of large scale differences

Λ and Mi refer to the high energy limit of QCD

The scale µ must reach the perturbative regime: µ ΛQCD

The lattice cutoff must still be larger: µ a−1

The volume must be large enough to contain pions:L 1/mπ

Taken together a naive estimate gives

L/a µL mπL 1 ⇒ L/a ' O(103)

⇒ widely different scales cannot be resolved simultaneously on afinite lattice!

In practice ...

This estimate may be a little too pessimistic:

Lmπ ≈ 3− 4 often sufficient

if cutoff effects are quadratic one only needs a2µ2 1.

when working in momentum space one may argue that thecutoff really is π/a;

in any case, one must satisfy the requirement µ ΛQCD

Heavy quark thresholds

Λ and Mi implicitly depend on Nf the number of active flavours! Ifcomputed for Nf = 2, 3 one needs to perform a matching acrossthe charm and bottom thresholds to match the real world at highenergies.

part i/v stefan sint - saha institute of nuclear...

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