Fundamental constants and electroweak phenomenology from the lattice

Transcription

1 Fundaental contant and electroweak phenoenology fro the lattice Lecture II: quark ae Shoji Hahioto INT uer chool 007, Seattle, Augut 007.

2 II. Quark ae. How to define Pole a; running a. Heavy quark ae: continuu extraction Quarkoniu u rule B eon eileptonic decay 3. Lattice calculation: baic trategy Input choice for heavy and light quark 4. Lattice calculation: cae tudy for heavy quark ae Perturbative and non-perturbative atching Botto and char quark S Hahioto KEK Aug 6, 007

3 II. Quark ae. How to define 3 S Hahioto KEK Aug 6, 007

4 Meauring the particle a J. J. Thoon 897 In QED electron a Meaure the drift of the particle in electric/agnetic field e/ v = e[ E + v B] R. Millikan 93 Together with the eaureent of e the coupling contant, the a i obtained. 4 S Hahioto KEK Aug 6, 007

5 In QCD, quark are confined No direct way to eaure it a alone. e + e - You ay conider a perturbative proce like e + e hadron. But the light quark a i negligible, thu no enitivity. Even for heavy quark, enitivity i lot in the region where PT can afely be ued. No way to eaure? See Sec. Let u conider it definition firt. 5 S Hahioto KEK Aug 6, 007

6 Quark a in perturbation theory Appear in the QCD lagrangian L QCD = q a q id/ q q Gν 4 Can be renoralized perturbatively i S p = p/ + Σ p The pole of the dreed propagator give a definition of quark a pole a p Σ p, 0 = / pole pole p = pole Infrared finite Gauge independent Renoralization cale independent Kronfeld, PRD58, S Hahioto KEK Aug 6, 007

7 Pole a p Σ p, / pole pole p = pole = 0 πiδ p p + iε Well-defined perturbatively, but hould not exit non-perturbatively. If it exit, quark ut appear a ayptotic tate. The pole of the propagator iplie a branch cut in cattering aplitude. Optical theore relate it to the ayptotic tate. 7 S Hahioto KEK Aug 6, 007

8 Another a definition Aug 6, 007 S Hahioto KEK 8 Conider the quark elf-energy at one-loop Renoralize by iply chop off the divergent ter which give the definition of the MSbar a. Thi a ut depend on to ake the phyical aplitude independent on ; thu the running quark a. Relation to the pole a i eaily obtained: + = Σ ln 4 ln 3 5 3, p p p p C p F ε π α ], [ ], [ p p p i p p i p S + Σ / + Σ = + Σ / = ln 4 γ π ε ε + + = =... ln 3 4 pole π α

9 Running quark a Aug 6, 007 S Hahioto KEK 9 Like the coupling contant, the quark a run. An integration contant i introduced: renoralization group invariant a + Λ Λ Λ =... / ln ] / ln[ln / ln β β β π α = d ' ' ' ' exp ] [ ˆ 0 0 / 0 α β γ α β α γ α α α β γ ˆ ; α γ α β α = = Plot fro ALPHA, NPB544,669999

10 Proble of the pole a Perturbative erie i divergent: pole α 4 α α = N f N f + 0.7N f π 3 π π In fact, the coefficient diverge a β 0 n!, if one u up a leading log diagra large β 0 liit dk n ~ f k [ β0α ln k / ] ~ 0 β α k Suation of the divergent erie ha an abiguity of order exp ~ β0α called a renoralon abiguity, which lead to an OΛ QCD abiguity of the pole a. Λ QCD n n! 0 S Hahioto KEK Aug 6, 007

11 Pole a cannot be deterined beyond the OΛ QCD level; conitent with the quark confineent. One ut ue oe perturbative definition to quote the quark a. MSbar i a preferred choice. Other definition are alo ued, epecially in the analyi of quarkoniu. Converion i poible no dangerou abiguity. S Hahioto KEK Aug 6, 007

12 II. Quark ae. Heavy quark ae continuu S Hahioto KEK Aug 6, 007

13 Heavy quark a ainly botto Ue the pair production e + e bb Only the reonance region i enitive to the quark a; but at the ae tie non-perturbative. α v ~ n α ~v: ut be reued. 3 S Hahioto KEK Aug 6, 007

14 Searing Aug 6, 007 S Hahioto KEK 4 Conider a eared R ratio: intead of IΠ. Iaginary oentu flow into the loop; can avoid the threhold ingularity, which lead to the α /v n reuation. Δ ut be larger than Λ QCD in order to avoid nonperturbative phyic. I = Π σ σ e e qq e e R [ ] ' ' ' ' ' ' ', 0 0 Δ Π Δ + Π = Δ Δ + = + Δ Δ Δ i i i i i R d i R d R π π Poggio, Quinn, Weinberg, PRD3, Quark-hadron duality

15 Su rule Ue the analytic propertie Vacuu polarization Optical theore Moent LHS: perturbatively calculable uing OPE for ufficiently all n. q g ν q q ν Π q n! = i d Π z = dπ z n dz 4 π xe z= 0 0 iqx 0 TV x V 0 0 RHS: ue experiental input R~IΠ; integral naturally eared over large range of, if n not too large. Continuu ore uppreed for larger n; ore enitivity to b. IΠ d z = π 0 ν IΠ d n+ 5 S Hahioto KEK Aug 6, 007

16 B eon eileptonic decay Fro incluive eileptonic B eon decay B X c lν, b and c can be deterined together with V cb. Ue the heavy quark expanion; earing correpond to the integral over final tate. Meaureent of hadronic a Xc and lepton energy l oent. Detailed dicuion will be given in Part IV. BaBar, PRL93, S Hahioto KEK Aug 6, 007

17 b quark a Copilation of any reult are found in the PDG review; an average b b = 4.0 ± 0.07 GeV excluding lattice. hep-ph/0458 Can lattice calculation copete with thi? How? Red circle: u rule; Green triangle: Υ S; Purple triangle: eileptonic; Blue quare: lattice 7 S Hahioto KEK Aug 6, 007

18 II. Quark ae 3. Lattice calculation: baic trategy 8 S Hahioto KEK Aug 6, 007

19 Input In general, lattice QCD iulation require input for Lattice cale /a deterine α /a Input dicued in Part I. Quark ae for each flavor up and down quark ud often aued to be degenerate fro peudo-calar eon a π, good enitivity becaue π ~ ud. Strange quark fro K, for the ae reaon. Char quark c either fro D heavy-light or J/ψ heavy-heavy a Botto quark b either fro B heavy-light or Υ heavy-heavy a 9 S Hahioto KEK Aug 6, 007

20 Up, down, trange Gell-Mann-Oake-Renner GMOR relation 968 At the leading order in q The lope yield u + d qq fπ π u + B 0 = qq qq f π Lattice calculation give B 0, then q i obtained with an input of the experiental value of π or K. q i in the lattice regularization; need a atching to obtain the MSbar value. f K K JLQCD 006 Dynaical overlap Beyond LO, uch ore coplicated; ee Lecture III. 0 S Hahioto KEK Aug 6, 007

21 Char and botto Heavy-light D, B + E H E Qq denote a binding energy. Siply calculate the eon a; tune Q until H reproduce the experiental value. Calculate E Qq, whoe Q dependence i ubleading. Then, H -E Qq give Q. Heavy Quark Syetry Heavy-heavy J/ψ, Υ = + + E = Q Qq H Q Q QQ E QQ denote a binding energy. Binding energy crucially depend on Q. S Hahioto KEK Aug 6, 007

22 Converion convert: MS =Z a lat a - Once the bare quark a i fixed on the lattice, it ut be converted to the continuu definition, becaue the pole a i not adequate. Jut like the converion of the coupling contant. May ue the perturbation theory Ue the renoralized coupling!. But, in ot cae, known only at the one-loop level. Exception are HQET, Aqtad, tochatic PT?. Ex. Oa-iproved Wilon ferion MS lat = a = [ +.05α +...] a Non-perturbative renoralization i deirable. S Hahioto KEK Aug 6, 007

23 II. Quark ae 4. Cae tudy: heavy quark a 3 S Hahioto KEK Aug 6, 007

24 Heavy quark on the lattice Additional coplication for heavy quark: Copton wave length ~/ Q i coparable to or horter than the lattice pacing. But, uch a hort ditance cale i irrelevant to the bound tate dynaic. Contruction of effective theorie HQET: the leading order in / Q expanion. Ued for heavy-light. Higher order ay be included a operator inertion, or in the Lagrangian a in NRQCD. NRQCD: expanion in v~α. Ued for heavy-heavy. heavy-light heavy-heavy Dedicated lecture by Kronfeld 4 S Hahioto KEK Aug 6, 007

25 Heavy Quark Effective Theory HQET Write the oentu of heavy quark a p= Q v+k v : four-velocity of the heavy quark. k: reidual oentu Heavy quark a liit: propagator p/ + v/ + + k/ + v/ i = i i p + iε v k+ k + iε v k+ iε Q Q Q Q Q Lagrangian L Q = Q v iv D Q ; Q x = e Q x v Heavy quark a drop out fro the dynaic = Heavy Quark Syetry i v x Q v Georgi 990, Eichten-Hill 990 Igur-Wie S Hahioto KEK Aug 6, 007

26 HQET on the lattice Dicretize the HQET lagrangian Auing v =,0: ret frae of the heavy quark S Q = x Q + + x[ U 4 x 4ˆ] Q x 4ˆ Heavy quark propagator becoe a tatic color ource. Heavy-light eon a: H = Q + EQq Calculate E Qq, then, H -E Qq give Q up to Λ QCD / Q correction. 6 S Hahioto KEK Aug 6, 007

27 Converion at two-loop Martinelli-Sachrajda, NPB559, Converion to the MSbar chee done to two-loop tatic theory i iple! b b = 4 3 α π α.66 π +... [ M E +.7α ln a.3 α +...] B b q Two-loop eential for table and thu precie deterination of b. Now, known to three-loop Di Renzo- Scorzato, JHEP b b b b b Continuu Pole Pole Lattice 7 S Hahioto KEK Aug 6, 007

28 Liitation of effective theory Obviouly, HQET at LO ignore the / Q effect. Higher order ter can be included. The leading correction: H = D σ B Q Q The coefficient of ter are contrained by the Lorentz invariance, thu giving / Q. But, in the quantu theory they are renoralized differently, ince the Lorentz invariance i violated by the choice of the reference frae v. The coefficient Z and c B ut be calculated non-perturbatively. The ae coplication arie at every order of the expanion. D H = Z Q c B σ B Z Q 8 S Hahioto KEK Aug 6, 007

29 State-of-the-art Della Morte, Garron, Papinutto, Soer, JHEP 070, / Q ter renoralized non-perturbatively. Thi i non-trivial, becaue it require a non-perturbative input. The atching wa done on a fine lattice againt the uual relativitic ferion in the region Q a<<. The lattice i necearily all there. Then atch onto coarer lattice uing the tep caling technique. Lecture by Sint 9 S Hahioto KEK Aug 6, 007

30 State-of-the-art atching to MSbar done non-perturbatively. The reult in the quenched theory. = [ ] GeV b b HQET / b Copared to the earlier reult, GeV, by Martinelli- Sachrajda in the heavy quark liit pert atching. PDG value 4.07 GeV 30 S Hahioto KEK Aug 6, 007

31 Della Morte et al S Hahioto KEK Aug 6, 007

32 Char quark HQET i of little ue. Ue the conventional lattice ferion with a a all a poible to avoid the a c error. Extrapolation in a will be eential. Non-perturbative atching of quark a i available for the Oa- iproved Wilon ferion. Again obtained with the tep caling technique. Quenched calculation Rolf-Sint, 00 gave Rolf-Sint, JHEP 0, = c c.303 GeV PDG value.59 GeV 3 S Hahioto KEK Aug 6, 007

33 Will becoe copetitive if done with dynaical quark! 33 S Hahioto KEK Aug 6, 007