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Introduction to Lattice QCD II

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Introduction to Lattice QCD II Introduction to Lattice QCD II Karl Jansen Task: compute the proton mass • { need an action { need an algorithm { need an observable { need a supercomputer Anomalous magnetic moment of Muon • Quantum Chromodynamics Quantization via Feynman path integral R − − = A Ψ Ψ¯ e Sgauge Sferm Z D µD D Fermion action R 4 Sferm = d xΨ(¯ x)[γµDµ + m] Ψ(x) gauge covriant derivative D Ψ(x) (@ ig A (x))Ψ(x) µ ≡ µ − 0 µ with Aµ gauge potential, g0 bare coupling, m bare quark mass S = R d4xF F ;F (x) = @ A (x) @ A (x) + i [A (x);A (x)] gauge µν µν µν µ ν − ν µ µ ν 1 Lattice Gauge Theory had to be invented QuantumChromoDynamics ! asymptotic confinement freedom distances 1fm distances 1fm & world of quarks world of hadrons and gluons and glue balls perturbative non-perturbative description methods Unfortunately, it is not known yet whether the quarks in quantum chromodynamics actually form the required bound states. To establish whether these bound states exist one must solve a strong coupling problem and present methods for solving field theories don't work for strong coupling. Wilson, Cargese Lecture notes 1976 2 Reminder: Wilson fermions introduce a 4-dimensional lattice with lattice spacing a fields Ψ(x), Ψ(¯ x) on the lattice sites x = (t; x) integers discretized fermion action S a4 P Ψ[¯ γ @ r @2 +m] Ψ(x) ;@ = 1 ∗ + ! x µ µ − µ µ 2 rµ rµ r|{z}∗ r discrete derivatives µ µ Ψ(x) = 1 [Ψ(x + aµ^) Ψ(x)] ; ∗ Ψ(x) = 1 [Ψ(x) Ψ(x aµ^)] rµ a − rµ a − − second order derivative remove doubler break chiral symmetry ! Nielsen-Ninimiya theorem: The theorem simply states the fact that the Chern number is a cobordism invariant (Friedan) 3 Reminder: Gauge fields Introduce group-valued gauge field U(x; µ) SU(3) 1¡1¡1 3¡3¡34¡4 5¡5¡56¡6 7¡7¡78¡8 9¡9¡9:¡: ;¡;¡;<¡< 2 2¡2 Relation to gauge potential −−− %¡%¡% '¡'¡'(¡( )¡)¡)*¡* +¡+¡+,¡, -¡-¡-.¡. /¡/¡/0¡0 &¡& # U(x; µ) = exp(iaAb (x)T b) = 1 + iaAb (x)T b + ::: ¡¡ ¡¡¡ ¡¡¡ ¡¡ ¡ !¡!¡!"¡" #¡#¡#$¡$ µ µ ¡ L=aN ¡ ¡ ¡ ¡¡¡ ¡¡¡ ¡¡¡ ¡¡¡ ¡¡¡ a ¡ ¡ ¢¡¢ £¡£¡£¤¡¤ ¥¡¥¡¥¦¡¦ §¡§¡§¨¡¨ ©¡©¡© ¡ ¡ ¡ ¡ x −−−" Discretization of the field strength tensor x+ µa principle of local gauge invariance ( U(x, µ ) U(x; µ)U(x + aµ,^ ν) U(x; ν)U(x + aν;^ µ) = ia2F (x) + O(a3) − µν F (x) = @ A (x) @ A (x) + i[A (x);A (x)] µν µ ν − ν µ µ ν 2 Action for the gauge field,( β = 6=g ), plaquettes (K.G. Wilson, 1974) ! S (U) = P β 1 1Re tr (U ) a 0 1 a4 P tr (F (x)2) + O(a6) w − 3 −! 2g2 x µν 4 Implementing gauge invariance Wilson's fundamental observation: introduce Paralleltransporter connecting the points x and y = x + aµ^ : U(x; µ) = eiaAµ(x) 1 lattice derivatives µΨ(x) = [U(x; µ)Ψ(x + µ) Ψ(x)] ) r a − ∗ Ψ(x) = 1 Ψ(x) U −1(x µ, µ)Ψ(x µ) rµ a − − − action gauge invariant under Ψ(x) g(x)Ψ(x); Ψ(¯ x) Ψ(¯ x)g∗(x); ! ! U(x; µ) g(x)U(x; µ)g∗(x + µ) ! 5 Lattice Quantum Chromodynamics Wilson Dirac operator (K.G. Wilson, 1974) 1 ? ? D = m + γµDµ Dw = mq + 2 γµ( µ + µ) a µ µ ! f| r{z r} − |r{zr}g naive Wilson bare quark mass mq gauge covariant lattice derivatives Ψ(x) = 1 [U(x; µ)Ψ(x + µ) Ψ(x)] rµ a − ∗ Ψ(x) = 1 Ψ(x) U −1(x µ, µ)Ψ(x µ) rµ a − − − 4 P ¯ Sfermion = a x Ψ(x)DwΨ(x) 6 Physical Observables expectation value of physical observables O Z 1 −S = Z e hOi fields O | {z } lattice discretization # 01011100011100011110011 # 7 Monte Carlo Method 2 f(x) = R dxf(x)e−x h i importance sampling: ! select points x ; i = 1; N with x Gaussian random number i ··· i f(x) 1 P f(x ) )h i ≈ N i i Quantum Field Theory/Statistical Physics: sophisticated methods to generate the Boltzmann distribution e−S • x become field configurations • i : become physical observables •h i 8 There are dangerous lattice animals 9 Wilson's Lattice Quantum Chromodynamics S = SG + Snaive + Swilson |{z} | {z } | {z } O(a2) O(a2) O(a) lattice artefacts appear linear in a possibly large lattice artefacts ! need of fine lattice spacings ) large lattices (want) L = N a = 1fm fixed) · simulation costs 1=a6−7 ! / present (Wilson-type) solutions: clover-improved Wilson fermions • maximally twisted mass Wilson fermions • overlap/domainwall fermions exact (lattice) chiral symmetry • 10 Realizing O(a)-improvement Continuum lattice QCD action S = Ψ[¯ m + γµDµ]Ψ an axial transformation: Ψ ei!γ5τ3=2Ψ ; Ψ¯ Ψ¯ ei!γ5τ3=2 ! ! changes only the mass term: p m mei!γ5τ3 m0 + iµγ τ ; m = m02 + µ2 ; tan ! = µ/m ! ≡ 5 3 generalized form of continuum action ! ! = 0 : standard QCD action • ! = π=2 : S = Ψ[¯ iµγ τ + γ D ]Ψ • 5 3 µ µ general ! : smooth change between both actions • 11 Wilson (Frezzotti, Rossi) twisted mass QCD (Frezzotti, Grassi, Sint, Weisz) D = m + iµτ γ + 1γ + ∗ ar1 ∗ tm q 3 5 2 µ rµ rµ − 2rµrµ quark mass parameter mq , twisted mass parameter µ difference to continuum situation: Wilson term not invariant under axial transformations Ψ ei!γ5τ3=2Ψ ; Ψ¯ Ψ¯ ei!γ5τ3=2 ! ! h i−1 2-point function: m + iγ sin p a + r P (1 cos p a) + iµτ γ q µ µ a µ − µ 3 5 h i2 (sin p a)2 + m + r P (1 cos p a) + µ2 / µ q a µ − µ 2 2 2 X lima!0 : pµ + mq + µ + amq pµ µ | {z } O(a) setting m = 0 (! = π=2) : no O(a) lattice artefacts • q quark mass is realized by twisted mass term alone • 12 O(a) improvement Symanzik expansion = [ξ(r) + am η(r)] cont + aχ(r) cont hOij(mq;r) q hOijmq hO1ijmq = [ξ( r) am η( r)] cont + aχ( r) cont hOij(−mq;−r) − − q − hOij−mq − hO1ij−mq 3 Using symmetry: R (r r) (m m ) ;R = ei!γ5τ 5 × ! − × q ! − q 5 h i mass average: 1 + = cont + O(a2) • 2 hOijmq;r hOij−mq;r hOijmq h i Wilson average: 1 + = cont + O(a2) • 2 hOijmq;r hOijmq;−r hOijmq automatic O(a) improvement • special case of mass average: m = 0 ! q = cont + O(a2 ) ) hOijmq=0;r hOijmq 13 A first test: experiments in the free theory (K. Cichy, J. Gonzales Lopez, A. Kujawa, A. Shindler, K.J.) free fields: imagine study system for L[fm] < 1 L = N a a 0 N ) · ! ! $ ! 1 1 N·mq = 0.5, r = 1.0 data Scaling for pure Wilson fermions 0.99 fit 0.98 at Nmq = 0:5 N·m 0.97 ps N m versus 1=N = a · π 0.96 0.95 0.94 0 0.05 0.1 0.15 0.2 0.25 1/N 14 Averaging over the mass 1.08 |N·mq| = 0.5, r = 1.0 1.06 data + fit + data - fit - 1.04 Wilson fermions at Nmq = 0:5 1.02 ± N·mps 1 N m versus 1=N = a · π 0.98 0.96 0.94 0 0.05 0.1 0.15 0.2 0.25 1/N 1.006 1.005 1.004 Averaging Wilson fermions at Nm = 0:5 q ± N·m 1.003 ps 2 2 N mπ versus 1=N = a 1.002 · 1.001 Mass Average: |N·mps| = 0.5, r = 1.0 data fit 1 0 0.01 0.02 0.03 0.04 0.05 0.06 2 1/N 15 Twisted mass at maximal twist choosing m = 0 ! = π=2 q ) 1 MAXIMAL TM: N·mq = 0.5, r = 1.0 0.9999 data maximally twisted mass fermions fit 0.9998 at Nµ = 0:5 0.9997 2 N·m 2 ps N mπ versus 1=N = a 0.9996 · 0.9995 0.9994 0.9993 0 0.01 0.02 0.03 0.04 0.05 0.06 2 1/N 16 Overlap fermions: exact lattice chiral symmetry starting point: Ginsparg-Wilson relation γ D + Dγ = 2aDγ D D−1γ + γ D−1 = 2aγ 5 5 5 ) 5 5 5 Ginsparg-Wilson relation implies an exact lattice chiral symmetry (L¨uscher): for any operator D which satisfies the Ginsparg-Wilson relation, the action S = ¯D is invariant under the transformations δ = γ (1 1aD) ;δ ¯ = ¯(1 1aD)γ 5 − 2 − 2 5 almost continuum like behaviour of fermions ) one local (Hern´andez,L¨uscher,K.J.) solution: overlap operator Dov (Neuberger) D = 1 A(AyA)−1=2 ov − with A = 1 + s D (m = 0); s a tunable parameter, 0 < s < 1 − w q 17 The \No free lunch theorem" A cost comparison T. Chiarappa, K.J., K. Nagai, M. Papinutto, L. Scorzato, A. Shindler, C. Urbach, U. Wenger, I. Wetzorke V; mπ Overlap Wilson TM rel. factor 124; 720Mev 48.8(6) 2.6(1) 18.8 124; 390Mev 142(2) 4.0(1) 35.4 164; 720Mev 225(2) 9.0(2) 25.0 164; 390Mev 653(6) 17.5(6) 37.3 164; 230Mev 1949(22) 22.1(8) 88.6 timings in seconds on Jump nevertheless chiral symmetric lattice fermions can be advantageous • { e.g., Kaon Physics, BK, K ππ { -regime of chiral perturbation! theory { topology { use in valence sector 18 Decide for an action ACTION ADVANTAGES DISADVANTAGES clover improved Wilson computationally fast breaks chiral symmetry needs operator improvement twisted mass fermions computationally fast breaks chiral symmetry automatic improvement violation of isospin staggered computationally fast fourth root problem complicated contraction domain wall improved chiral symmetry computationally demanding needs tuning overlap fermions exact chiral symmetry computationally expensive For all actions: O(a)-improvement Olatt = Olatt + O(a2) )h physi h conti Here, example of twisted mass fermions 19 Wilson (Frezzotti, Rossi) twisted mass QCD (Frezzotti, Grassi, Sint, Weisz) D = m + iµτ γ + 1γ + ∗ a1 ∗ tm q 3 5 2 µ rµ rµ − 2rµrµ quark mass parameter mq , twisted mass parameter µ mq = mcrit O(a) improvement for • hadron masses,! matrix elements, form factors, decay constants 2 2 det[Dtm] = det[DWilson + µ ] • protection against small eigenvalues ) computational cost comparable to Wilson • simplifies mixing problems for renormalization • no operator specific improvement coefficients • natural to twist • ? based on symmetry arguments Drawback: explicit breaking of isospin symmetry for any a > 0 possibly large cut-off effects in isospin zero sector ) 20 Need an algorithm 21 Molecular Dynamics follow discussion of S.
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