Effective Field Theory

Transcription

1 Effective Field Theory Iain Stewart MIT The 19 th Taiwan Spring School on Particles and Fields April, 2006

2 Outline Lecture I Principles, Operators, Power Counting, Matching, Loops, Using Equations of Motion, Renormalization and Decoupling Lecture II Summing Logarithms, matching, HQET Lecture III Soft - Collinear Effective Theory α s Weak Interactions at low energy, [power counting velocity NRQCD] An effective field theory for energetic hadrons & jets E Λ QCD

3 Non-relativistic QCD

4 VII. NRQCD Systems with Two Heavy Particles eg. e + e positronium (NRQED) pe Hydrogen (NRQED) b b, c c Υ, J/Ψ (NRQCD) t t e + e t t (NRQCD) N N deuteron (few nucleon EFT) (66) E = p 2 /(2m) v 2, count powers of v (and α s ) Momentum Regions i µ ψ p (x) (mv 2 )ψ p (x) k 0 ψ(x) = p k hard: m m integrate these out e ip x ψ p (x) (67) treat mv 2 Λ QCD L NRQCD = L ultrasoft + L potential + L soft (68) potential: mv 2 mv ptnl gluons are not propagating soft: mv mv radiative corrections, binding ultrasoft: mv 2 mv 2 need multipole expansion ψ, χ A µ s, q s A µ us

5 Simplify p.c. and Implement multipole expansion a) m v p index ψ(x) = p continuous e ip x ψ p (x) m v p k i µ ψ p (x) (mv 2 )ψ p (x) p m v 2

6 Power Counting Power Counting O(v 0 ) Kinetic Terms give potential quarks ψ, χ v 3/2 soft gluons A µ s v (scale µ S ) order in v ultrasoft gluons A µ us v 2 (scale µ U ) Manifest power counting means each graph contributes at one Basic Threshold Dynamics Manifest power counting means each graph contributes at one order in v power counting requires separating soft and ultrasoft fields (seen in threshold expansion for any µ < m) and Can power move associate counting all powers of v with vertices = 5 + with non-relativistic requires separating velocities soft and ultrasoft fields (seen in threshold ( 5) P + ( 8) U + ( 4 δ = 5 + expansion for any µ < m) Can associate all powers P U k (k 5)V of v with k + (k vertices 8)V(Luke et al.) k δ = 5 + P U S k (k 5)Vk + (k 8)Vk + (k 4)Vk N s Can associate all powers of v with vertices (Luke et al.) S + (k 4)V N s k (board) Power counting of operators implies power counting of states (Bodwin, Braaten, Lepage) Power counting of operators implies power counting of states (Bodwin, Braaten, Lepage) Coulombic Singularities sum Power counting implies mv 2 insertions of mv correlation Power counting implies mv 2 mv correlation! t in dimensional regularization (in vnrqcd) in dimensional regularization (in vnrqcd)... - t ψ (mv) 3/2 ɛ, A µ s (mv) 1 ɛ, A µ us (mv 2 ) 1 ɛ Manohar, I.S., Hoang ψ (mv) 3/2 ɛ, A µ s (mv) 1 ɛ, A µ us (mv 2 ) 1 ɛ implies we need µ U = µ 2 S /m mv2, so take µ U = mν 2, µ S = mν Manohar, I.S., Hoang resonances uniquely implies overlap fixeswe powers need of µ remnant = µ 2 S /m mv2, so take µ U = mν 2, µ S = mν U, µ S in operators only uniquely fixes powers of µ U, µ S in operators LO: An Effective Fie Non-Relativistic systematically include re corrections from QCD Coulombic: Want an expansion: The rest of the NRQCD discussion can be found at the Iain Stewart p.5 end of Lecture II on the website LO NLO Iain Stewart p

7 Lecture III Outline d.o.f., SCET1 & SCET11, Lagrangians Label Operators, Gauge Invariance, Wilson Lines, RPI, multipole expansion Hard-Collinear and Ultrasoft-Collinear Factorization IR divergences, Running B X s γ

8 m W? m b E m c SCET can be used for: B - decays by weak interactions: B X u l ν B πl ν B D η B Dπ B X s γ B ρρ B ππ B Kπ The B is heavy, so many of its decay products are energetic, E b B K γ B ργ B γl ν B Λ QCD m s m u,d Any other QCD process with large energy transfer: e p e X e γ e π 0 e + e jets p p Xl + l γ M M e + e J/ΨX Υ Xγ

9 Degrees of Freedom eg.! B D Pion has: p µ π = (2.310 GeV, 0, 0, GeV) = Qnµ Q Λ QCD n µ = (1, 0, 0, 1) Light - Cone coordinates: Basis vectors n µ, n µ with n 2 = 0, n 2 = 0, n n = 2 p µ = nµ 2 = nµ n ν 2 g µν n p + nµ 2 n p + pµ + nµ n ν 2 + g µν p + n p, p n p eg. n µ = (1, 0, 0, 1)

10 eg.! B D Collinear constituents: p µ c = (p +, p, p ) Just a boost of (p +, p, p ) (Λ, Λ, Λ) ( Λ 2 Q, Q, Λ ) Q(λ 2, 1, λ) n µ! λ = Λ Q Soft Constituents p µ s = (p +, p, p ) (Λ, Λ, Λ) D and B SCET II Energetic hadrons λ = Λ Q modes p µ = (+,, ) p 2 fields collinear Q(λ 2, 1, λ) Q 2 λ 2 ξ n, A µ n soft Q(λ, λ, λ) Q 2 λ 2 q s, A µ s

11 eg. B X s γ m 2 X m2 B OPE in 1/m b (not SCET) X m 2 X ΛQ Λ 2 QΛ Q 2 n µ m 2 X Λ2 not inclusive X B γ Jet constituents: p µ (Λ, Q, QΛ) Q(λ 2, 1, λ) SCET I Energetic jets usoft p µ Λ collinear p 2 c QΛ, λ = Λ/Q modes p µ = (+,, ) p 2 fields collinear Q(λ 2, 1, λ) Q 2 λ 2 ξ n, A µ n soft Q(λ, λ, λ) Q 2 λ 2 q s, A µ s usoft Q(λ 2, λ 2, λ 2 ) Q 2 λ 4 q us, A µ us

12 What makes this EFT different? p 2 = p + p + p 2 Usually m 1 Λ a d n i=1 C i(µ, m 1 ) O i (µ, Λ) p+ p 2 m 2 1 p 2 soft Λ2 p - In SCET constituent p m b E π p+ dω C(ω) O(ω) p 2 Q 2 p 2 c Λ2 p -

13 Collinear Propagator (board) Power Counting for Collinear Fields (board)

14 Currents eg. ū Γ b (board) involves both collinear and ultrasoft objects u ū Γ b ξ n W Γ h v W = P exp ( ig y ds n A n(s n µ ) ) b Interaction of modes: Offshell versus Onshell (board)

15 i µ e ip x φ p (x) = e ip x (P µ + i µ )φ p (x) Separate Momenta Separate Momenta (multipole expansion) label residual HQET P µ = m b v µ + k µ h v v (x) (x) (Georgi) SCET P µ = p µ + k µ ξξ n,p (x) Collinear Quarks ψ(x) p e ip x ξ n,p (x) n/ ξ n,p = 0 troduce µ Label ξ n,p Operator (Qλ 2 ) ξ n,p (1, λ) (1, λ) Q! Separate Momenta Introduce Label Operator q But labels are changed Can pull by phases SCET interactions to front of operatorsp p! p Separate usual Momenta derivative P µ( φ q 1 φ p1 ) = (p µ qµ 1...)( φ q 1 φ p1 ) P µ( φ q 1 φ p1 ) = (p µ qµ 1...)( φ q 1 φ p1 ) Cani pull µ e ip x phases φ p (x) to= front e ip x of operators (P µ + i µ )φ p (x) kk Q Q Q Q derivative for labels! 2! 2

16 Labels are changed by collinear interactions q collinear p p! Labels are preserved by ultrasoft interactions p p ultrasoft

17 Power Counting Summary Type (p +, p, p ) Fields Field Scaling collinear (λ 2, 1, λ) ξ n,p λ In an arbitrary graph, (A + n,p, Euler A n,p, identities A n,p) allow(λ all 2, 1, powers λ) of associated just with vertices soft (λ, λ, λ) q s,p λ 3/2 A µ s,p! λ usoft (λ 2, λ 2, λ 2 ) q us λ 3 A µ us λ 2 Make kinetic terms order λ 0 d 4 X ξ n,p Power Counting n/ ( ) B 2 ( ) in +... ξ n,p λ 0 = λ 4 λ λ 2 λ At leading power only At leading λ 0 interactions power only are required ie. interactions are required external currents and Lagrangians Power counting can be assigned to vertices D to be

18 LO SCET Lagrangian ū Γ b L QCD = ψ id/ ψ Write ψ = ξ n + χ n L = ( χ n + ξ ) [ ( ) ( n i n/ ] (ξn 2 n D + in/ 2 n D + id/ ) + χ n ξ n = n/ n/ 4 ψ χ n = n/n/ 4 ψ = n/ ) ( ξn 2 in D ξ n + ( n/ ) χ n i n D χ n 2 + ( ξn id/ χ n ) + ( χ n id/ ξ n ) e.o.m: δ δ χ n : i n Dχ n + n/ 2 id/ ξ n = 0 χ n = 1 i n D id/ n/ 2 ξ n L = ξ ( 1 ) n/ n i n D + id/ i n D id/ 2 ξ n (board) L (0) c = ξ n {n id us + gn A n + i /D c 1 } / n i /D c i n D c 2 ξ n

19 That was tree level. Use Symmetries Power counting, Gauge symmetry, Discrete, Lorentz invariance (?) Gauge symmetry U(x) = exp [ iα A (x)t A] need to consider U s which leave us in the EFT collinear usoft i µ U c (x) p µ c U c(x) A µ n,q i µ U us (x) p µ c U us(x) A µ us Object Collinear U c Usoft U us ξ n U c ξ n U us ξ n U c ga µ n U c [ + U c id µ, U c ] U us ga µ n U us W U c W U us W U us ga µ n q us q us U us q us ga µ us ga µ us U us ga µ usu us + U us [i µ, U us] Y Y U us Y reconsider current (board)

20 Reparameterization Invariance (RPI) n, n break Lorentz invariance, restored within collinear cone n µ n µ + µ (I) n µ n µ by RPI, three types n µ n µ (II) n µ n µ + ε µ n µ (1 + α) n µ (III) n µ (1 α) n µ eg. rules out ξ n id µ 1 i n D id µ n/ 2 ξ n L (0) c = ξ n {n id us + gn A n + i /D c 1 } / n i /D c i n D c 2 ξ n

21 Wilson Coefficients and Hard - Collinear Factorization ( ) [ ( ) C( P, µ): they depend on large momenta picked out by P = n P λ 0 eg. C( P, µ) ( ξn W ) Γh v = ( ξn W ) Γh v C( P, µ) Write { ( ) ( ) ( ξn W ) Γh v C( P [, µ) = dω C(ω, µ) ( ξn W ) δ(ω P ] [ )Γh ( ) ] v only the product is gauge invariant = dω C(ω, µ)o(ω, µ) In general: hard-collinear factorization follows f(i n D c ) = W f( P)W from properties of = dω f(ω) [ W δ(ω P)W ] SCET operators in collinear hard coefficient p 2 Q 2 operator p 2 Q 2 λ 2

22 Multipole Expansion and Ultrasoft - Collinear Factorization Multipole Expansion: (board) Field Redefinition: L (0) c = ξ n {n id us + gn A n + i /D c ξ n Y ξ n A n Y A n Y, gives: L (0) 1 } / n i /D c i n D c 2 ξ n usoft gluons have eikonal Feynman rules and induce eikonal propagators (board) L L ( ) ( Y (x) = P exp ig 0 [ ] [ ] n D us [ Y =0, Y Y ] =1 c = ξ } n/ n {n id us ξ n ξ { n n id c + id/ c ) ds n A us (x+ns) 1 } n/ id/ c i n D c 2 ξ n Moves all usoft gluons to operators, simplifies cancellations

23 Field Theory gives the same results pre- and post- field redefinition, { but the } organization { is different } Ultrasoft - Collinear Factorization: eg1. J = ( ξ n W ) ω Γh v ( ξ n Y Y W Y ) ω Γh v = ( ξ n W ) ω Γ(Y h v ) note: not upset by hard-collinear mometum fraction since ultrasoft gluons carry no hard momenta (board) so usoft-collinear factorization is also simply a property of SCET eg2. No ultrasoft fields J = ( ξ n W ) ω1 Γ(W ξ n ) ω2 ( ξ n W ) ω1 Y Y Γ(W ξ n ) ω2 = ( ξ n W ) ω1 Γ(W ξ n ) ω2 color transparency

24 b IR divergences, Matching, & Running X. IR DIVERGENCES J SCET AND = ( ξ LOOPS n W ) ω Γh v Γ = σ µν J SCET = ( ξ n W ) ω Γh J QCD = s Γ b a) J SCET a) c) c) e) = ( ξ n Γ W = ) ω σ Γh µν v Γ = σ µν nw ) ω Γh v QCD sum= SCET sum= b a) α [ s 3π a) b) b) c) d) d) f) e) s s FIG. 6. Order λ 0 effective FIG. theory6. diagrams Order λ 0 for effective the heavy theory to light diagrams currentfor atthe oneheavy loop. to li ( ln 2 p 2 ) + 3 ( p 2 ) 2 ln + 1 From Eq. (35) we see that the logarithms in diagrams with collinear gluons are small at a From Eq. (35) we see that the logarithms in diagrams with c b) scale µ p 2 Qλ. For the graphs d) with soft gluons the logarithms aref) small at a differen m 2 scale µ p 2 scale µ p 2 /( n p) Qλ 2 Qλ. For the graphs with soft gluons the logari b m ln b ɛ IR m 2 + constants. Running the collinear-soft b scale µ p 2 theory from µ = Q to µ = Qλ /( n p) Qλ 2. Running the collinear-soft theo therefore sums all logarithms therefore originating sums all from logarithms collinear originating effects and part from of collinear the logarithm effects from soft exchange. At µ = Qλ collinear gluons may be integrated out and one matches onto from soft exchange. At µ = Qλ collinear gluons may be integrat FIG. 6. a theory Order containing λ 0 effective only theory soft c) degrees diagrams of freedom. for the Theheavy runningto inlight this soft current theoryat include one l a theory containing only soft degrees of freedom. The running a) the b) remaining logarithms from soft exchange, which would d) need to be taken into account to the remaining logarithms from soft exchange, which would need α [ ( s ln 2 p 2 ) sum all Sudakov logarithms. 3π Fromm Eq. 2 (35) + 3 ( p 2 ) we see that the logarithms in diagrams with collinear gluons are sum all Sudakov logarithms. b 2 ln m ( µ 2 ) ] + 2 ln To run between Q and Qλ we add up the ultraviolet divergences in the soft and collinea scale µ p 2 b ɛ IR m 2 + constants b Qλ. For the graphs To runwith between softq gluons and Qλthe we add logarithms up the ultraviolet are small diverge at diagrams in Eqs. (35) and (36). This gives the counterterm in the effective theory scale µ p 2 /( n p) Qλ 2. diagrams Running Eqs. the(35) collinear-soft and (36). This theory gives the from counterterm µ = Qinto Z i = 1 + α [ s(µ)c F 1 4π ɛ ɛ ln ( µ ) ] α [ ( s ln 5 +. (37 Z i = n 1 + P α [ therefore 2 p 2 ) s(µ)c 2ɛ F 1 4π ɛ ɛ ln ( µ ) 3π m ( p 2 ) sums all logarithms originating from collinear effects and part of the5l b 2 ln m same IR b ɛ IR + from soft exchange. n P 2ɛ For b sγ, n At P µ = = m b Qλ andcollinear Eq. (37) agrees gluons withmay Ref. [6]. besince integrated µ > Qλ the outcounterterm and one ma can 1 depend the label n P For Q, b but sγ, does n P not = depend m b and on Eq. P (37) Qλ. agrees Z with i could Ref. also [6]. have Since a theory ɛ ( µ ) ( ln 2 ln 2 µ ) 3 ( µ 2 ) ] containing only soft degrees of freedom. The running in this soft theor been UV 2ɛ UV ɛ UV m b m b 2 ln m 2 + constants b calculated directly from the depend matching on the result label in Eq. n P(31). Q, Since butthe does effective not depend theory reproduce on P Q -quark 1/ɛ IR for b-quark the remaining logarithms from soft exchange, which would need to be taken into b α [ IR regulator ( s ln 2 p 2 3π 0 for s-quark 1 p 2 0 for s-quark 1/ɛ IR for b-quark ( µ 2 ) ]

25 Running dω C(ω) O(ω) ( ) ( ) counterterm ω = m b in B X s γ Z = 1 + α s 3π [ ( ) ] [ 1 ɛ ɛ ln ( µ ω ) + 5 2ɛ ] RGE µ µ C(µ) = 4α s(µ) 3π ln ( µ ω ) C(µ) +... (board)

26 Endpoint B X s γ (board)

27 Endpoint B X s γ Optical Thm: q q b p B s P X b standard OPE endpoint region resonance region For EndPoint:, collinear, usoft, Decay rate is given by factorized form Korchemsky, Sterman ( 94)

28 Match: B.P.S. label conservation Factor usoft:

29 ground subtracted E*! [GeV] ra in the Υ(4S) frame. Convolution: nd MC using appropriate the MC background samese efficiencies. The effifor non-π 0, non-η photons e photon from a well reto without using the other eto efficiency is measured ming from measured π 0 nstructed D + D 0 π +, the π 0 is replaced by the struction. The η veto efand event-shape criteria a π 0 anti-veto sample. It l selection criteria except d with another photon in d larger than Other the signal sample. efficiencies versus E γ are er polynomials, which are C. Most are found to be nity. An exception is the at 95% of the energy has ine cells of the 5 5 clusrly modelled by our MC e estimate the efficiency ndidate photons in OFFacting the known contris effectively increases the %. kground categories, after the above described prohe OFF-subtracted specig ± 2140 ± 1260 events in the B B background scaling. We correct this spectrum for the signal selection efficiency function obtained from signal MC, applying the same data/mc correction factors as for the generic photon background category (iii). The average signal selection efficiency is 23%. The efficiency-corrected spectrum is shown in Figure 2. The two error bars for each point show the statistical and the total error, including the systematic error which Define is correlated Fourier among the points. Transforms As expected, the spectrum above the endpoint for decays of B mesons from the Υ(4S) at about 3 GeV, is consistent with zero. Integrating this spectrum from 1.8 to 2.8 GeV, we obtain a partial branching fraction of ( 3.55 ± ) The systematic error contains the contribution from the fits to data/mc efficiency ratios (±5.9%) to which we add the following contributions in quadrature. The uncertainty on the number of B B events, which also affects the weight applied to OFF events, contributes ( )%. We estimate the error on the OFF data subtraction using the result of the fit to the spectrum above the endpoint. We integrate the resulting function in the GeV range and obtain a yield of +40 ± 160. We add ±200 to the systematic error (±0.8%). For the choice of the where polynomial functions in the data/mc efficiency ratio fits, we perform the same fit increasing the polynomial order by one. The contribution is ±1.3%. As we do not measure the yields of photons from sources other than π 0 s and η s in B B events, we vary the expected yields by ±20% to estimate the systematic error and obtain a Events/100 MeV Belle E*! [GeV] FIG. 2: Efficiency-corrected photon energy spectrum. two error bars show the statistical and total errors. The CLEO 01 Weights / 100 MeV 40 0 Data Spectator Model E (GeV)

30 SCET is a field theory which: explains how these degrees of freedom communicate with each other, and with hard interactions organizes the interactions in a series expansion in Λ QCD E provides a simple operator language to derive factorization theorems in fairly general circumstances eg. unifies the treatment of factorization for exclusive and inclusive QCD processes new symmetry constraints scale separation & decoupling n µ!

31 How is SCET used? cleanly separate short and long distance effects in QCD derive new factorization theorems find universal hadronic functions, exploit symmetries & relate different processes model independent, systematic expansion study power corrections keep track of µ dependence sum logarithms, reduce uncertainties The End