Kaon mixing beyond the SM from N f = 2 tmQCD R. Frezzotti (on behalf - PowerPoint PPT Presentation

Kaon mixing beyond the SM from N f = 2 tmQCD R. Frezzotti (on behalf of ETM Collaboration) Univ. and INFN of Rome – Tor Vergata GGI workshop ”New Frontiers in Lattice Gauge Theory” Florence, September 19, 2012 R. Frezzotti (on behalf of ETM Collaboration)

K 0 oscillations and constraints on new physics (NP) K 0 –¯ Flavour physics processes vanishing at tree level in the SM (possibly also CKM- or chirality-suppressed) are a key tool to search for NP virtual particle effects. FCNC ∆ F = 2 transitions provided most stringent constraints on NP (e.g. technicolor) models. K 0 mixing in the framework of Here: parameters describing K 0 –¯ = � 5 i =1 c i (Λ /µ ) O i ( x µ ) + � 3 c i (Λ /µ ) ˜ H ∆ S =2 i =1 ˜ O i ( x µ ) , eff s α γ µ (1 − γ 5 ) d α ][¯ s β γ µ (1 − γ 5 ) d β ] O 1 = [¯ s α (1 − γ 5 ) d α ][¯ s β (1 − γ 5 ) d β ] O 2 = [¯ s α (1 − γ 5 ) d β ][¯ s β (1 − γ 5 ) d α ] O 3 = [¯ s α (1 − γ 5 ) d α ][¯ s β (1 + γ 5 ) d β ] O 4 = [¯ s α (1 − γ 5 ) d β ][¯ s β (1 + γ 5 ) d α ] O 5 = [¯ ˜ s α γ µ (1 + γ 5 ) d α ][¯ s β γ µ (1 + γ 5 ) d β ] O 1 = [¯ ˜ s α (1 + γ 5 ) d α ][¯ s β (1 + γ 5 ) d β ] O 2 = [¯ ˜ s α (1 + γ 5 ) d β ][¯ s β (1 + γ 5 ) d α ] O 3 = [¯ R. Frezzotti (on behalf of ETM Collaboration)

K 0 oscillations Bag parameters and ratios thereof relevant for K 0 –¯ Only the parity-even part of O i ( ˜ O i ), i = 1 , 2 , 3 , 4 , 5, matters i.e. s α γ µ d α ][¯ s β γ µ d β ] + [¯ s α γ µ γ 5 d α ][¯ s β γ µ γ 5 d β ] O 1 = [¯ s α d α ][¯ s β d β ] + [¯ s α γ 5 d α ][¯ s β γ 5 d β ] O 2 = [¯ s α d β ][¯ s β d α ] + [¯ s α γ 5 d β ][¯ s β γ 5 d α ] O 3 = [¯ s α d α ][¯ s β d β ] − [¯ s α γ 5 d α ][¯ s β γ 5 d β ] O 4 = [¯ s α d β ][¯ s β d α ] − [¯ s α γ 5 d β ][¯ s β γ 5 d α ] O 5 = [¯ K - ¯ K matrix elements in units of vacuum saturation approximation � ¯ K 0 | O 1 ( µ ) | K 0 � ξ 1 B 1 ( µ ) m 2 K f 2 = K m 2 K f K � 2 � � ¯ K 0 | O i ( µ ) | K 0 � = ξ i B i ( µ ) i = 2 , 3 , 4 , 5 , m s ( µ ) + m d ( µ ) ξ i = (8 / 3 , − 5 / 3 , 1 / 3 , 2 , 2 / 3). For accurate determinations define R i = � ¯ K 0 | O i | K 0 � / � ¯ K 0 | O 1 | K 0 � i = 2 , 3 , 4 , 5 Pioneering quenched lattice QCD studies (with two a ’s each): ⋆ Donini et al., Phys.Lett. B470 (1999) 233 (clover Wilson fermions) ⋆ Babich at al., Phys.Rev. D74 (2006) 073009 (overlap fermions) R. Frezzotti (on behalf of ETM Collaboration)

ETMC (arXiv:1207.1287) continuum N f = 2 results for B i , R i i 1 2 3 4 5 MS (3 GeV) 0.51(02) 0.51(02) 0.85(07) 0.82(04) 0.66(07) B i R i 1 -16.3(06) 5.5(04) 30.6(13) 8.2(05) RI-MOM (3 GeV) 0.50(02) 0.63(03) 1.07(09) 0.95(06) 0.75(09) B i R i 1 -15.4(06) 5.3(03) 26.9(12) 7.1(05) [ MS -scheme as in Buras, Misiak, Urban, Nucl.Phys. B586 (2000) 397] Quenching of s -quark: from comparison with N f = 2 + 1 results for B 1 ( a → 0) ⇒ systematic quenching error � 1 − 2%. Lattice artifacts are typically 5-10 times larger - depending on O i and action details ⇒ continuum limit crucial At one a ( ∼ 0 . 086 fm): N f = 2 + 1 results from RBC+UKQCD, arXiv:1206.5737 R. Frezzotti (on behalf of ETM Collaboration)

Update of the SM+NP UTfit-’08 analysis [JHEP 0803 (2008) 049] ... ... in arXiv:1207.1287 – triggered by our unquenched B i -estimates • Input: experimental and/or phenomenological determinations of heavy meson masses, decay widths and leptonic decay constants, CKM parameters, heavy quark masses, B K , D , B -parameters, ... • NP in ∆ F = 2 processes via N f = 3 effective weak Hamiltonian � � 5 � + � 3 c i ˜ H ∆ F =2 i =1 c i O fd O fd eff ; LO = � i =1 ˜ f = s , c , b i i neglecting non-local contributions and subleading local ones. • SM+NP UTfit results provide bounds on C i (of P -even O i ) C i ∼ F i L i / Λ 2 , with F i the (complex) NP coupling and L i a loop factor specific to the interaction that generates O i . • | ǫ K | ∝ Im [ � K 0 |H ∆ F =2; P − even | ¯ K 0 � ] ⇒ bounds on Im [ C i ] eff ; LO R. Frezzotti (on behalf of ETM Collaboration)

Switching on one C i at the time (with L i = F i = 1) yields ... 95% allowed same from lower bound same from range (GeV − 2 ) UTfit-2008 on Λ (TeV) UTfit-2008 Im C K [ − 2 . 8 , 2 . 6] · 10 − 15 [ − 4 . 4 , 2 . 8] · 10 − 15 1 . 9 · 10 4 1 . 5 · 10 4 1 Im C K [ − 1 . 6 , 1 . 8] · 10 − 17 [ − 5 . 1 , 9 . 3] · 10 − 17 24 · 10 4 10 · 10 4 2 [ − 6 . 7 , 5 . 9] · 10 − 17 [ − 3 . 1 , 1 . 7] · 10 − 16 12 · 10 4 5 . 7 · 10 4 Im C K 3 [ − 4 . 1 , 3 . 6] · 10 − 18 [ − 1 . 8 , 0 . 9] · 10 − 17 49 · 10 4 24 · 10 4 Im C K 4 [ − 1 . 2 , 1 . 1] · 10 − 17 [ − 5 . 2 , 2 . 8] · 10 − 17 29 · 10 4 14 · 10 4 Im C K 5 ⋆ models with tree-level FCNC from NP excluded up to 10 5 TeV ⋆ gluinos exchange in MSSM ⇒ L i > 1 ∼ α 2 s (Λ) ∼ 0 . 01 (Λ min = Λ tab min / 10) ⋆ loop-mediated weak FCNC ⇒ L i > 1 ∼ α 2 w (Λ) ∼ 10 − 3 (Λ min = Λ tab min / 30) ⋆ warped 5dim model with flavour hierarchy (RS scenario) F 4 = 2 m d m s ∗ v 2 , L 4 = ( g ∗ KKs ) 2 , Λ = M KKG ⇒ F 4 L 4 ∼ 10 − 8 (Λ min = Λ tab min / 10 4 ) Y 2 R. Frezzotti (on behalf of ETM Collaboration)

ETMC lattice computation of (renormalized) � K | O i | ¯ K � ... ... based on a lattice regularization of the correlator � z �P K ( y ) O i ( x ) P ¯ K ( z ) � that guarantees � y ,� • continuum-like renormalization pattern of O i ’s • O( a ) improvement of physical quantities (no artefacts ∼ a 2 k +1 ) • numerical efficiency ( ⇒ data at several a ’s, a 2 → 0 feasible) Mixed Action setup of maximally twisted mass (Mtm) lattice QCD S = S Mtm + S OS val + S OS ghost , ψ = ( u sea , d sea ) & valence q f ’s sea � � x ¯ S Mtm = a 4 � 1 2 γ µ ( ∇ µ + ∇ ∗ µ ) − i γ 5 τ 3 r sea W cr + µ sea ψ ( x ) ψ ( x ) sea � � S OS val = a 4 � 1 2 γ µ ( ∇ µ + ∇ ∗ x , f ¯ q f ( x ) µ ) − i γ 5 r f W cr + µ f q f ( x ) W cr ≡ M cr − a 2 ∇ ∗ µ ∇ µ M cr ≡ optimal critical m 0 R. Frezzotti (on behalf of ETM Collaboration)

MA setup of MtmLQCD (R.F., G.C. Rossi, JHEP10 (2004) 070) Two degenerate sea quarks with µ sea = µ ℓ & four valence quarks: q 1 , q 3 with µ 1 = µ 3 ≡ µ “ s ” , q 2 , q 4 with µ 2 = µ 4 ≡ µ ℓ and valence Wilson parameters r 1 = r 2 = r 3 = − r 4 , | r f | = 1 Evaluate two- and three-point correlators involving the fields P 12 = ¯ P 34 = ... , A 12 q 1 γ µ γ 5 q 2 , A 34 q 1 γ 5 q 2 , µ = ¯ µ = ... q β 3 γ µ q β q β 3 γ µ γ 5 q β O MA q α 1 γ µ q α q α 1 γ µ γ 5 q α �� � � �� 1[ ± ] = 2 [¯ 2 ][¯ 4 ] + [¯ 2 ][¯ 4 ] ± 2 ↔ 4 q β 3 q β q β 3 γ 5 q β O MA q α 1 q α q α 1 γ 5 q α �� � � �� 2[ ± ] = 2 [¯ 2 ][¯ 4 ] + [¯ 2 ][¯ 4 ] ± 2 ↔ 4 1 q β q β 1 γ 5 q β q β O MA q α 3 q α q α 3 γ 5 q α �� � � �� 3[ ± ] = 2 [¯ 2 ][¯ 4 ] + [¯ 2 ][¯ 4 ] ± 2 ↔ 4 O MA q α 1 q α q β 3 q β q α 1 γ 5 q α q β 3 γ 5 q β �� � � �� 4[ ± ] = 2 [¯ 2 ][¯ 4 ] − [¯ 2 ][¯ 4 ] ± 2 ↔ 4 q α 1 q β q β 3 q α q α 1 γ 5 q β q β 3 γ 5 q α O MA �� � � �� 5[ ± ] = 2 [¯ 2 ][¯ 4 ] − [¯ 2 ][¯ 4 ] ± 2 ↔ 4 in particular � a � 3 � x �P 43 2 O MA x , x 0 ) P 21 C i ( x 0 ) = i [+] ( � y 0 � , i = 1 , . . . , 5 � y 0 + T L R. Frezzotti (on behalf of ETM Collaboration)

In such a MA setup one finds (JHEP10 (2004) 070, arXiv:1207.1287) • the op.s O MA i [+] renormalize as in the formal QCD: ( b ) O MA O MA       Z 11 0 0 0 0 1[+] 1[+] O MA O MA 0 Z 22 Z 23 0 0     2[+] 2[+]       O MA   O MA 0 0 0 = Z 32 Z 33     3[+]   3[+]       O MA 0 0 0 O MA Z 44 Z 45       4[+] 4[+]     0 0 0 Z 54 Z 55 O MA O MA 5[+] 5[+] [mass-independent Z ij related to plain Wilson 4-fermion op. RC’s] • the relevant quark bilinear operators renormalize according to [ P 12 / 34 ] = Z S / P [ P 12 / 34 ] ( b ) , [ A 12 / 34 ] = Z A / V [ A 12 / 34 ] ( b ) µ µ • if µ 1 , 3 = µ s and µ 2 , 4 = µ u / d the m.e. � P 43 | O MA i [+] | P 12 � extracted from the correlators with insertion of O MA i [+] as a → 0 approaches (with rate a 2 ) the continuum QCD m.e. � ¯ K 0 | O i | K 0 � R. Frezzotti (on behalf of ETM Collaboration)

Lattice parameters for correlators at β = 3 . 80 , 3 . 90 and 4.05. β = 3 . 80, a ∼ 0 . 10 fm a − 4 ( L 3 × T ) a µ ℓ = a µ sea a µ “ s ” N stat 24 3 × 48 0.0080 0.0165, 0.0200, 0.0250 170 0.0110 “ “ 180 β = 3 . 90, a ∼ 0 . 09 fm 24 3 × 48 0.0040 0.0150, 0.0220, 0.0270 400 0.0064 “ “ 200 0.0085 “ “ 200 0.0100 “ “ 160 32 3 × 64 0.0030 “ 300 0.0040 “ “ 160 β = 4 . 05, a ∼ 0 . 07 fm 32 3 × 64 0.0030 0.0120, 0.0150, 0.0180 190 0.0060 “ “ 150 0.0080 “ “ 220 To improve signal-to-noise ratio: stochastic spatial-wall sources used for P 21 y 0 , P 43 y 0 + T / 2 and sum over spatial location of O i . R. Frezzotti (on behalf of ETM Collaboration)