Introduction to the Soft - Collinear Effective Theory

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1 Introduction to the Soft - Collinear Effective Theory An effective field theory for energetic hadrons & jets Lecture 3 E Λ QCD Iain Stewart MIT Heavy Quark Physics Dubna International Summer School August, 28 1

2 So far Lecture I Introduction to SCET1, SCET11 Collinear & Soft degrees of freedom Construction of HQET SCET1 propagators, field power counting Leading Lagrangian Lecture II Heavy-light current and Wilson lines Gauge symmetry and reparameterizations in SCET Wilson coefficients & hard-collinear factorization Field redefinition & ultrasoft-collinear factorization One-Loop ultrasoft and collinear graphs, IR divergences 2

3 Lecture 3 Outline Renormalization group evolution & Sudakov logs B X s γ Factorization Theorem More on large logs, Evolution with Convolutions SCET11, building blocks, exploiting SCET1 Factorization for B Dπ, B πl ν eg. of power corrections in SCET1 Jet Production e + e J n J n X 3

4 Renormalization in SCET & Summing Sudakov Logs 4

5 Renormalize Heavy to Light Current in SCET C(ω,µ) [ ( ξ n W ) ω Γh v ] graph sum = α [ ( s ln 2 p 2 ) 3π m ( p 2 ) b 2 ln m 2 b need 1 ɛ 2 UV Z c =1 α s(µ)c F 4π 5 2ɛ UV 2 ɛ UV ln ( 1 ɛ ɛ + 2 ɛ ln µ ω ) C bare = C +(Z c 1)C + 1 ɛ IR ( µ m b ) Compute the Anomalous Dimension 2 ln 2 ( µ m b ) µ d dµ Cbare = = µ d dµ C(ω,µ)=γ c(ω,µ)c(ω,µ) γ c = Zc 1 µ d dµ Z c = µ d = α s(µ)c F 4π ( 2 ɛ dµ 5 4 ln µ ω + 2 ɛ α s (µ)c F 4π ) 3 2 ln ( µ 2 m 2 b to remove UV divergences ( 1 ɛ ɛ + 2 ɛ ln µ ω ) = α s(µ)c F π LL ( ln µ ω + 5 ) 4 ) ω = m b ] + constants µ d dµ α s(µ) = 2ɛα s (µ)+β[α s ] part of NLL 5

LL solution cusp anomalous dimension Solve µ d dµ ln C(ω,µ)= α s(µ)c F π ln µ ω, µ d dµ α s(µ) = β 2π α2 s(µ) use d ln(µ) = 2π β dα s α 2 s and integrate to obtain the solution C(ω,µ)=C(ω,µ ) exp

6 LL solution cusp anomalous dimension Solve µ d dµ ln C(ω,µ)= α s(µ)c F π ln µ ω, µ d dµ α s(µ) = β 2π α2 s(µ) use d ln(µ) = 2π β dα s α 2 s and integrate to obtain the solution C(ω,µ)=C(ω,µ ) exp boundary condition, no large logs for µ ω [ 4πCF ( 1 ) ] ( β 2α s(µ ) z 1 + ln z µ ω exp(α s ln 2 +α 2 s ln ) ) 2CF ln z/β z α s(µ) α s (µ ) If β and α s = constant, then [ αs C F C(ω,µ)=C(ω,µ ) exp π ( 1 µ 2 ln2 + ln µ ln µ ) ] µ µ ω Sudakov double logs exponentiated 6

7 Exercise q c!,b µ,a q c!,b µ,a!,b FIG. 1: QCD graphs with collinear, q c, and soft, q s, momenta. q c µ,a Problem 4) SCET Loops for Two-Jet Production Consider the two-jet production process through a virtual photon in SCET, namely e + e J n J n X us where J n is a jet in the n = (1,,, 1) direction, J n is a jet in the n = (1,,, 1) direction, and any remaining particles in the final state are ultrasoft, contained in X us. a) Write down two collinear quark Lagrangians, one for ξ n fields and one for ξ n fields. Interactions between these two types of collinear fields are hard, and so do not effect your analysis. What are the Feynman rules for the ultrasoft gluon coupling to each of these collinear quarks? b) Start with J QCD = ψγ µ ψ and determine the appropriate LO SCET current J SCET = ξ n ξ n, ie. fill in the dots with appropriate collinear Wilson lines and Dirac structure. c) Draw the five one-loop Feynman diagrams that are non-zero for e + e q n q n (use Feynman gauge for all gluons when determining which graphs are zero). Here q n has n-collinear momentum p, and q n has n-collinear momentum p and you should work in the CM frame. All graphs but one can be directly read off using the loop computations done in lecture (or given in the handout notes), as long as you use the same IR regulator. That is, you should keep both collinear quarks offshell, p 2 and p 2. Compute the divergent terms in the one remaining ultrasoft graph using dimensional regularization in the UV. d) Add up the 1/ɛ terms from the graphs in c) and determine the lowest order anomalous dimension equation for C the Wilson coefficient of J SCET. Solve this equation keeping only the ln µ/q term and using a fixed coupling α s, and then with a running coupling α s (µ). (Voilá, Sudakov double logs resummed.) 7

8 SCET I Construction of operators (using power counting, ultrasoft & collinear gauge invariance, RPI) We built gauge invariant operators with nice power counting: eg. LO heavy-to-light current J () = [ dω C(ω, µ) ( ξ n W )δ(ω P ] )Γ(Y n h v ) = dω C(ω, µ) χ n,ω Γ H n v eg. a subleading current suppressed by J (1) = dω dω C (1) (ω, ω,µ) χ n,ω ig /B ω Γ H n v λ igb µ ω = 1 P W [ i n D n, idn µ ] W = ga µ n, ω +... δ(ω P ) 8

9 Endpoint B X s γ Events/1 MeV Belle Optical Thm: E*! [GeV] q q b p B s P X b standard OPE endpoint region resonance region For EndPoint:, collinear, usoft, We want to prove that the Decay rate is given by factorized form Korchemsky, Sterman ( 94) 9

10 Match: B.P.S. label conservation Factor usoft: ξ n W Γ µ h v ξ n W Γ µ Y n h v T µ µ = C(mb ) 2 d 4 xe i(m b n q) x 2 B T [ h v Y ](x)[y h v ]() B T [W ξ n ](x)[ ξ n W ]() [Γ µ Γ µ ] = C(m b ) 2 d 4 x d 4 k (2π) 4 ei(m b J P (k) [Γ µ Γ µ ] n 2 q k) x B T [ h v Y ](x)[y h v ]() B 1

11 Convolution J P (k) =J P (k + ) Im T µ µ = C(m b ) 2 d 4 x ImJ P (k + ) = C(m b ) 2 dk + [ dx ImJ P (k + ) = C(mb ) 2 d 4 k (2π) 4 ei(m b n 4π ei(m b 2E γ k 2 q k) x B dk + S(2E γ m b + k + )ImJ P (k + ) T [ h v Y ](x)[y h v ]() B + )x /2 B T [ h v Y ](x)[y ] h v ]() B as desired 1 Γ calculable dγ de γ = H(m b,µ) calculable nonpert. shape function dk + J(k +,µ) S(2E γ m b + k +,µ) p 2 m 2 b p 2 m b Λ QCD p 2 Λ 2 QCD µ 2 h µ 2 J µ 2 Λ To minimize large logs we want to evaluate these functions at different µ s 11

12 our result for the RGE for C, allows us to write p- H(m b,µ J )=H(m b,µ h ) U H (m b,µ h,µ J ) Q! cn hard µ h need to be able to run the shape function up to µ J Q! 2 usoft 2 µ Λ Q Q!! µ J p + or we could run the jet and hard functions down to µ Λ Lets consider the jet function & its RGE 12

13 J n (Qr + n,µ)= 1 8πN c Q Disc The Jet Function X n 1 4N c tr / nχ n (x) Xn Xn χn,q () = Q d 4 xe ir n x T χ n,q ()/ˆ nχ n (x) d 4 r n (2π) 3 e ir n x J n (Qr + n,µ) tree level: one loop: a) b) c) d) RGE: µ d J(s, µ) = dµ solution J(s, µ) = ds γ J (s s,µ) J(s,µ) ds U J (s s, µ, µ ) J(s,µ ) U J (s s, µ, µ )= ek ( e γ E ) ω µ 2 Γ( ω) [ (µ 2 ) 1+ω θ(s s ] ) (s s ) 1+ω + 13

14 { More examples which involve convolutions twist 2 operators J () = label on 2nd block of fields is fixed by mom.cons. in m.elt. dω C(ω, µ) χ n,ω / nχ n p- Q! Q! 2 cn hard O(ω) 2 Q Q!! p + Matrix Elements π light-cone distrib. πn (p π ) J () = dω C(ω, µ) φ π (ω/p π,µ) = p π 1 dx C(xp π,µ) φ π (x, µ) DIS p.d.f pn (p ) J () pn (p ) = = Q x dω C(ω, Q, µ) f i/p (ω/p,µ) 1 x dξ C( Qξ x, Q, µ) f i/p(ξ, µ) p = Q x 14

15 1 Γ dγ de γ = H(m b,µ) dk + J(k +,µ) S(2E γ m b + k +,µ) +... Factorization formulas of this type have also been derived for the power corrections using SCET 15

16 SCET II So far we have considered inclusive processes with jets, or processes with only one identified hadron like DIS SCET II eg. allows us to treat cases with two or more hadrons B Dπ, B πl ν, B ππ λ = Λ Q p- modes Q! c n hc n hard Q! s perturbative hc n p 2 = " 2 QCD Q! 2 c n 2 Q! Q!! Q p + 16

17 SCET II So far we have considered inclusive processes with jets, or processes with only one identified hadron like DIS SCET II eg. allows us to treat cases with two or more hadrons B Dπ, B πl ν, B ππ λ = Λ Q p- modes Q! c n hc n hard Q! s perturbative hc n p 2 = " 2 QCD Q! 2 c n 2 Q! Q!! Q p + 17

18 SCET II So far we have considered inclusive processes with jets, or processes with only one identified hadron like DIS SCET II eg. allows us to treat cases with two or more hadrons B Dπ, B πl ν, B ππ λ = Λ Q p- modes Q! c n hc n hard Q! s perturbative hc n p 2 = " 2 QCD Q! 2 c n 2 Q! Q!! Q p + 18

19 Constructing SCET II Operators Factorization of Soft Gluons For simplicity consider a collinear (c n ) and a soft (s) mode soft gluons can not couple to collinear particles in a local way We can construct operators directly from QCD by integrating out the offshell modes (i) soft gauge invariance requires a soft Wilson line q = q s + q n Q(λ, 1,λ) q 2 Qλ Λ 2 S C e.g. S C in h.c. (ii) built up by integrating out offshell flucutations S C S C +... builds up builds up ξn S n Γ W q s soft Wilson line S S switches order switch order ξ n W Γ S n q s C Can be shown to all orders C +... Soft & Collinear Gauge Invariant Soft-Collinear Factorization Iain Stewart p.29 19

20 A Simpler Method: 1) Match QCD onto SCET I 2) Factorize usoft with field redefinition 3) Match onto SCET II {hc n, us} {c n, s} use factorization in SCET1 Q p- Q! Q!! 2 c n hc n s=us p 2 = " 2 QCD eg. J =( ξ n W )Γh v 2 Q! Q!! Q p + =( ξ n W )Γ(Y n h v ) J =( ξ n W )Γ(S nh v ) In this matching, the power of λ can only increase and does so due to change in scaling to uncontracted fields 2

21 Exclusive Example B Dπ Proof for Dπ Proof for B Dπ Proof for B Dπ -! Steps Steps Steps Match at µ 2 Q 2 onto SCET I [Decouple ξ Y ξ () ] Match at µ 2 Q 2 onto SCET I [Decouple ξ Y ξ () ] Match ][ ]] at µ { d [ h(c) d u ] v h (b)][ ξ() = v n,p W () C ( P + )W () ξ n,p ] [ c 2 } Q 2 { onto SCET d [ h(c) I [Decouple [ h(c) v Y T A Y (b)][ ξ() v n,p W () C 8 ( P + )T A W () ξ n,p () ] [ c A ][ d A ] v h (b)][ ξ() ξ Y ξ () = v n,p W () C ( P ] + )W () ξ () ] ][ ] } { n,p [ c b d [ h(c) [ h(c) u v Y T A Y h (b)][ ξ() v n,p W () C 8 ( P + )T A W () ξ () ] n,p [ c T A b ][ d T A u ] v h (b)][ ξ() = v n,p W () C ( P + )W () ξ () ] [ h(c) n,p at µ 2 v Y T A Y h (b)][ ξ() v n,p W () C 8 ( P + )T A W () ξ () ] n,p Match at 2 QΛ QΛonto onto SCET SCET II II Match at µ 2 QΛ [ h(c) onto SCET [ h(c) II v h (b) v ][ ξn,p W C ( P + )W ] ξ n,p [ h(c) v (b) v ][ ξn,p W C ( P + )W ] [ h(c) ξ v ST A S h (b) v ][ ξn,p W C 8 ( P + )T A W ] n,p [ h(c) ξ n,p v ST A v h (b) (b) v ][ ξn,p W C ( P + )W ] ξ n,p v ][ ξn,p W 8 ( P + )T A ] [ h(c) ξ octet m.elt. n,p Take matrix elements v ST A S h (b) v ][ ξn,p W C 8 ( P + )T A W ] ξ n,p will vanish Take matrix elements πn ξ() n,p W () C ( P + )W () ξ n,p () = i Take matrix elements 2 f πe π dx C[2E π (2x 1)]φ π (x) πn ξ() n,p () P + )W () ξ n,p () i 2 f πe π dx C[2E π (2x 1)]φ π (x) πn ξ() n,p W () C ( P + )W () ξ n,p () = i 2 f πe π dx C[2E π (2x 1)]φ π (x) D v h v Γ h h v B v = F B D () D v h v Γ v B v F B D () D v h v Γ h h v B v = F B D () 1 Dπ cbūd B = N F B D dx T 1 (x, µ) φ π (x, µ) Dπ cbūd B Dπ cbūd B = N F B D B D 1 dx T (x, µ) φ π (x, µ) B dx T (x, µ) φ π (x, µ) Iain Stewart p.21 Factorized! + power corrections Iain Iain Stewart Stewart p.21 p.21 D 21

22 Power Corrections & Color Suppressed Decays B D π T {O, B D π A D( ) = N ( ) L (1) ξq (a) () L (1) ξq L(1) ξq } = ( qy )ig /B n,ω χ n dx dz dk 1 + dk+ 2 T (i) (z) J (i) (z, x, k 1 +, k+ 2 ) S(i) (k 1 +, k+ 2 ) φ M (x) b d c u d d ( s ) Comparison to Data δ(dπ) = 3.4 ± 4.8 δ(d π) = 31. ± 5. A(D*M) A(D M) color allowed color suppressed * # + # D! D " D K D ' " D $ D # *.5! - D +! -D D + Ḏ % - % D + $ - D $ - LO SCET prediction. 22

23 Another Exclusive Example B πl ν (B ππ) SCET1 needs time-ordered products of with Q () = χ n,ω ΓH n v Q (1) = χ n,ω ig /B n,ω ΓH n v L (1) ξq = ( qy )ig /B n,ω χ n,... f(e) = dz T (z, E) ζ BM J B p 2 2 ~ Λ p 2 ~ Q 2 p 2 ~ Requires a power suppressed interaction (z, E) + C(E) ζ BM (E) QΛ similar M p 2 ~ 2 Λ same functions in B ππ universality at EΛ SCET11 (further factorization) 1 J (z) = f M f B dx ζ BM ζ BM =? ) dk + J(z, x, k +, E)φ M (x)φ B (k + ) has endpoint singularities 1 (n L ")!q dx φ π(x) x 2, 23

24 eg. e + e 2 jets!! event shapes in two jet region n-collinear jet usoft particles n-collinear jet dσ de = SCET I p- 1 Q 2 X L µν J ν () X X J µ () δ(e e(x))δ 4 (q p X ) SCET I Q! cn hard X = X n X n X us Q! 2 u cn 2 Q Q!! p + 24

25 What observable? d 2 σ dm 2 dm 2 Hemisphere Invariant Masses M 2 = ( p µ i ) 2 M 2 = ( p µ i ) 2 i a soft particles i b n-collinear n-collinear thrust axis hemisphere-a hemisphere-b Let: Dijet region: M 2, M 2 Q 2 s M 2 s M 2 25

26 In QCD: σ = res. X The full cross-section is a restricted set of states: (2π) 4 δ 4 (q p X ) i=a,v s M 2 Q 2 L i µν J ν i () X X J µ i () lepton tensor, γ & Z exchange by using EFT s we will be able to move these restrictions into the operators In SCET: J µ i () = dω d ω C(ω, ω, µ)j ()µ i (ω, ω, µ) Momentum conservation: C(Q, Q, µ) Wilson coefficient SCET current ( ξ n W n ) ω Y n Γ µ Y n (W nξ n ) ω χ n,ω Y n Γ µ Y n χ n, ω 26

27 SCET cross-section: X = X n X n X s σ = K n res. X n X n X s (2π) 4 δ 4 (q P Xn P X n P Xs ) Y n Y n X s X s Y n C(Q, µ) 2 Y n /ˆ nχ n,ω X n X n χ n,ω χ n, ω X n X n /ˆnχ n, ω QCD a) b) p p ] all-orders one-loop a) b) c) d) SCET e) difference gives one-loop matching: C(Q, µ) = 1 + α sc F 4π [ 3 log Q2 i log 2 Q 2 i ] 8 + π2 µ 2 µ

28 Specify hemisphere invariant masses for the jets: total soft momentum is the sum of momentum in each hemisphere K Xs = k a s + k b s ˆP a X s = k a s X s, ˆPb X s = k b s X s Insert: 1= hemisphere projection operators ( ds δ (p n + ks a ) 2 s) ( ) d s δ (p n + ks) b 2 s expand: ( ) δ (p n + ks a ) 2 s = 1 ( Q δ k n + + k s +a ( ) δ (p n + ks) b 2 s = 1 ( Q δ kn + ks b ) s Q s ) Q... Some Algebra... 28

29 d 2 σ ds d s = σ Q 2 C(Q, µ) 2 X n 1 2π X n 1 2π ( dk n + dk n dl + dl δ k n + + l + s ) ( δ k n + l s ) Q Q d 4 xe ik+ n x /2 tr /ˆ nχ n (x) Xn Xn χn,q () d 4 ye ik n y+ /2 tr χ n (y) X n X n /ˆnχ n, Q () 1 δ(l + k s +a )δ(l ks b )tr Y n Y n () X s X s Y n Y n() N c X s Factorization Theorem: d 2 σ = σ H Q (Q, µ) ds d s Hard Function n H Q (Q, µ) = C(Q, µ) 2 + dl + dl J n (s Ql +,µ) J n ( s Ql,µ) S hemi (l +, l,µ) Quark Jet Function Anti-quark Jet Function Soft radiation Function universal 29

30 S hemi (l +, l,µ) S hemi (l +, l, µ) = 1 N c a) δ(l + k s +a X s )δ(l ks b ) Y n Y n () X s X s Y n Y n() b) c) d) Y n Y n n n n n n n Y n Y n n n n n n n Soft function is perturbative if and is nonperturbative if l +, l Λ QCD l +, l Λ QCD It is also universal, it appears in many different event shapes (thrust, heavy-jet mass,...) for both massless and massive jets 3

31 A very popular event shape is thrust Thrust T =1 dijet T = max ˆt T = 1 2 i ˆt p i Q Insert: 1= ( dt δ 1 T s + s ) Q 2 Factorization theorem dσ dt = σ H(Q, µ) ds J T (s, µ) S thrust ( Q(1 T ) s Q,µ ) with S thrust (l,µ)= dl + dl δ(l l + l ) S hemi (l +, l,µ) 31

32 SCET is a field theory which: explains how soft & collinear degrees of freedom communicate with each other, and with hard interactions organizes the interactions in a series expansion in λ which measures how collinear/soft the particles are λ = ΛQCD m b λ = Λ QCD m b 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 results are constrained by symmetries λ 2 = m2 X Q 2 scale separation & decoupling n µ! 32

33 How is SCET used? cleanly separate short and long distance effects in QCD derive new factorization theorems find universal hadronic functions, exploit symmetries predict decay rates and cross sections model independent, systematic expansion study power corrections keep track of µ dependence sum large logarithms The End 33

Effective Field Theory

Effective Field Theory Iain Stewart MIT The 19 th Taiwan Spring School on Particles and Fields April, 2006 Outline Lecture I Principles, Operators, Power Counting, Matching, Loops, Using Equations of Motion,