QCD RESUMMATION (LARGE X)

Transcription

1 QCD RESUMMATION (LARGE X) Eric Laenen INT workshop on Gluon and Quark Sea at High Energies, Seattle, Sep. 3-29, 200

2 OUTLINE Basic issues, selected results Some new (eikonal) developments Next-to-eikonal corrections Path-integral methods, webs for multiple parton scattering Summary 2

3 RESUMMATION Catch-all phrase for summing something to all orders why would one do that? what can one sum? when should one do that? to what accuracy? Resummation: The art of constructing for some quantity out of a subset of perturbative terms, aided by extra insight, an all-order expression 3

4 WHY WOULD ONE RESUM? Rescue predictive power when perturbative series converges poorly Predicts terms in next order (approximate NNLO) Better physics description Less scale uncertainty CAVEATS New uncertainty from IR renormalons Sloppy definitions, casual use 4

5 PREDICTIVE POWER Ô = n c n α n + R n c n computed with Feynman diagrams Finite order: only take lowest few n. Check if α is small enough? (power correction is R n small enough?) c n does not grow too fast with n? 5

6 QCD HADRONIC OBSERVABLES PDF s O = φ Ô + P 0 Power correction Partonic observable Use equation in multiple ways Find weakest link, and update Predict O, or fit ϕ or infer P But c n often misbehave grow systematically with n Ô = n c n α n + R n 6

7 EXPONENTIAL SERIES IN CROSS SECTIONS Typical behavior of cross section π 2 =, 2, etc When good logs go bad - J. Huston Ô 2 = +α(l 2 + L + ) + α 2 (L 4 + L 3 + L 2 + L + ) +... Effective expansion parameter is αl 2 Fix: reorganize/resum such that Ô = +α s (L 2 + L + ) + αs(l L 3 + L 2 + L + ) +... = exp Lg (α s L) +g 2 (α s L) +α s g 3 (α s L)+... C(α s ) LL constants NLL + suppressed terms Just as systematically improvable as PT LO, NLO LL, NLL, etc 7

8 RESUMMATION OF WHAT? So many logs... ln( T ) ln(/x) ln(µ) ln(p T /m Z ) ln(k T ) ln( x) ln(b) ln(n) pt of Tevatron pt of Tevatron Thrust LEP Many, many results 8

9 RECOIL DOUBLE LOGS Eg. p T of Z-bosons produced at Tevatron Visible logs L 2 = ln 2 p 2 T /M 2 Z d /dp T *BR(Z ee) (pb/gev) DØ b-space (Ladinsky-Yuan) CTEQ4M g =0. GeV 2 g 2 =0.58 GeV 2 g 3 =-.5 GeV - q T -space (Ellis-Veseli) MRSR ã=0. GeV -2 q Tlim =4.0 GeV b-space (Davies-Webber-Stirling) MRSA g =0.5 GeV 2 g 2 =0.4 GeV 2 Fixed-order (O( 2 )) s MRSA p T (GeV) Z-boson get p T from recoil agains (soft) gluons 9

10 THRESHOLD DOUBLE LOGS Invisible logs can also plague perturbation theory S s Q 2 S > Q 2 Q2 s ln2 Q2 s 0

11 RESUMMATION IN A FEW EASY STEPS cross section for n extra gluons σ(n) = 2s Phase space measure Squared matrix element dφ n+ (P, k,...,k n ) M(P, k,...,k n ) 2 When emissions are soft, approximate phase space measure and matrix element dφ n+ (P, k,...,k n ) dφ(p ) n dφ (k) n! M(P, k,...,k n ) 2 M(P ) 2 M emission (k) 2 Sum over all orders σ(n) =σ(0) exp dφ (k) M emission (k) 2 n Incorporate Theta or Delta functions for specific cases but these must factorize similarly, or they cannot go into exponent

12 PHASE SPACE CONSTRAINTS k P Q i Kinematic condition expresses z in terms of gluon energies s = Q 2 2P K K 2 Transform factorizes the phase space k n δ Q2 s i 2ki 0 s 0 dw e wn δ w i w i = i exp( w i N) So can go into exponent Large logs: log(n) n σ(n) =σ(0) exp dφ (k) M emission (k) 2 (exp( wn) ) 2

13 ON THE ORIGIN OF LOGS Logarithms in cross sections are related to IR divergences (p + k) 2 = 2p k = 2E g E q ( cosθ qg ) Phase space integration α s d 4 2 k (2π) 4 p p p kp k α s K de g E g E g dθqg sin θ qg θ qg α s ( 2 + ln2 (K)). When one can find all IR divergences, one can find all the logs (and a good number of constants) 3

14 JOINT RECOIL AND THRESHOLD RES N EL, Sterman, Vogelsang Fix both k T and energy in exponent Can make k T integral visible (Higgs, Vector Boson) Kulesza, Sterman, Vogelsang Can make k T integral invisible (Prompt photon, heavy quark, sleptons) dσ dp T = C a,b dn 2πi φ a(n)φ b (N) δ(q T k T,a k T,b )e ib Q T dx 2 T (x 2 T ) N S 4(p T Q T /2) 2 d 2 b dk T,a N exp E(N,b,p T ) dk T,b EL, Sterman, Vogelsang, Banfi, Bozzi, Fuks, Klasen Key ingredient: PDF at fixed energy and k T R(z,k T )=c dλ 2π d 2 b 0 (2π) 2 e iλzp +ib k T p ψ(λ, b)γ + ψ(0) p Matching to TMD s? Or to NLO? 4

15 RESUMMATION AND FACTORIZATION Very generically, if a quantity factorizes, one can resum it Renormalization; factorizes UV modes Λ G B (g B, Λ,p)=Z µ,g R(µ) G R g R (µ), p µ Evolution equation Solving = resummation Type of factorization dictates resummation small x k T factorization µ d dµ ln G R g R (µ), p µ large x near-threshold factorization = µ d Λ dµ ln Z µ,g R(µ) Factorization is essentially separating degrees of freedom = γ(g R (µ)) p µ G R µ,g dλ R(µ) = G R (,g R (p)) exp p λ γ(g R(λ)) resummed Systematic approach in Soft Collinear Effective Theory Talk by Iain Stewart 5

16 RESUMMED CROSS SECTIONS Threshold-resummed Drell-Yan cross section dn dq 2 (z) = C 2π i z N ˆσ(N) σ(n) = exp dx xn 0 x dσ resum +D(α s (( x)q)) Sterman; Catani, Trentadue Q 2 ( x) 2 Q 2 dµ µ A(α s(µ)) ( + α s (Q 2 ) C F π +...) A. Vogt ˆσ DY (N,Q 2 ) = g 0 (Q 2 ) exp G N DY (Q 2 ) G N DY = ln Ng (λ)+g 2 (λ)+α s g 3 (λ)+..., λ = β 0 α s ln N 6 2 G N DY NNLL 0 (exp G DY f f ) / (f f) 4 s = 0.2, N f = 4 8 xf = x 0.5 ( x) 3 NLL 6 Good convergence in exponent 2 4 LL N x 6

17 NNNLL THRESHOLD DY AND HIGGS D(α s ) Single logs Moch, Vermaseren, Vogt EL, Magnea; Moch, Vermaseren Vogt; Ravindran; Idilbi, Ma, Yuan =4 B δ (α s ) 2 G(α s ) +β(α s ) d F MS (α s ) dα s virtual split.fn. form factor 7

18 SOME RULES OF THUMB Why increases in QCD resummation? σ partonic,resum (N) = σ hadronic(n) φ 2 (N) = exp( ln2 N) 2 = exp(+ ln 2 N) exp( ln 2 N Why scale cancellation? φ(n) exp[(a ln N + B)ln(µ F )] Sterman, Vogelsang σ partonic,resum (N) exp[e(ln 2 N) ln µ F (A ln N + B)] 8

19 GENERIC LARGE X BEHAVIOR For DY, DIS, Higgs, singular behavior when x δ( x) ln i ( x) x ln k ( x) singularity structure for plus distributions is organizable to all orders, perhaps also for divergent logarithms? After Mellin transform Constants ln i (N) ln k (N) N We know a lot about logs and constants, very little about /N 9

20 LN(N)/N TERMS Kraemer, EL, Spira; Catani, De Florian, Grazzini; Kilgore, Harlander Can be numerically important Kraemer, EL, Spira Moch,Vogt dl qq NLO (qq _ * ) [pb] d s = 4 TeV c 2,2 (N) all N 0 + all N exact 5 0 c 2,3 (N) all N 0 + all N exact Q [GeV] n f = 4 ( /60) N We know that the leading series ln i (N)/N exponentiates n f = 4 ( /2000) N by replacing in resummation formula +z 2 z 2 z 2 20

21 SUCCESFUL LN(N)/N ORGANIZATION γ qq (N) =A(α s )lnn + B(α s )+C(α s ) ln N N +... Moch, Vermaseren, Vogt noted a remarkable relation Dokshitzer, Marchesini, Salam Basso, Korchemsky C 2 = A 2 C 3 =2A 2 A DMS reproduced this by changing DGLAP equation µ 2 µ 2 ψ(x, µ2 )= x dz x µ 2 z ψ z, zµ2 P z,α s z P (z,α s )= A (α s) ( z) + + B δ (α s ) δ( z)+o (( z)) Can this be reproduced in threshold resummation? 2

22 EXTENDED THRESHOLD RESUMMATION Ansatz: modified resummed expression where +2 ln σ(n) ( z) 2 Q 2 /z dq 2 Q q 2 2 P s (n) (z) = z z A(n) + C γ (n) ln( z)+d (n) γ (We constructed a similar expression for DIS). Structure: σ(n) = = F DY αs (Q 2 ) + g 2 2n n n=0 P s m=0 0 z,α s (q 2 ) dz z N ( z) α 2 z D Q 2 s z + a nm ln m N + 2n m=0 b nm ln m N N EL, Magnea, Stavenga + O N 2 CF 2 C A C F n f C F b b b 2 8ζ ζ 2 ζ ζ b 20 2 ζ ζ ζ ζ ζ ζ Must understand systematics beyond eikonal approximation 22

23 EIKONAL BASICS, QED Soft emission by charged particle Propagator: expand numerator & denominator in soft momentum, keep lowest order Vertex: expand in soft momentum, keep lowest order k p+k p (p + k) µ + p µ 2p k + k 2 2pµ 2p k 23

24 BASICS QED, CONT D k,µ k 2,µ 2 k n,µ n p Exact: Approx: (p + K ) 2 (2p + K 2 + K ) µ... 2pK 2p µ... 2pK n 2p µ n (p + K n ) 2 (2p + K n) µ n, K i = n m=i k m. Eikonal identity: p (k + k 2 ) p k 2 + p (k + k 2 ) p k = p k p k 2 Sum over all permʼs: i p µ i p k i. Independent, uncorrelated emissions, Poisson process 24

25 NON-ABELIAN EIKONAL APPROXIMATION Same methods as for QED, but organization harder: SU(3) generator at every vertex no obvious decorrelation Order the Ta according to λ Key object : Wilson line Φ n (λ 2, λ )=P exp ig λ2 λ dλ n A a (λn) T a Order by order in g, it generates QCD eikonal Feynman rules 25

26 EXPONENTIATON One loop vertex correction, in eikonal approximation p k A 0 d n k k 2 p p (p k)( p k) p Two loop vertex correction, in eikonal approximation p k k 2 A 0 2 p d n k k 2 2 p p (p k)( p k) Exponential series 26

27 NON-ABELIAN EXPONENTIATION: WEBS Take quark - antiquark line, connect with soft gluons in all possible ways, use eikonal approximation Exponentiation still occurs, without path ordering! Gatheral; Frenkel, Taylor; Sterman A selection of diagrams in exponent, but with modified color weights: webs Webs are two-eikonal line irreducible Proof by induction; recursive definition of color weights How can we extend this to include next-to-eikonal terms? 27

28 PATH INTEGRAL METHOD EL, Stavenga, White Represent propagator as particle path integral, between coord. and momentum states F (p 2 f )= 2 0 dt p f U(T ) x i p f x i i = p 2 f + m2 iε where p f U(T ) x i = e ip f x i i 2 (p2 f +m2 )T p(t )=0 x(0)=0 DpDxe i R T 0 dt(pẋ 2 p2 ) Add an (abelian) gauge field p(t )=pf p f U(T ) x i = DpDx exp x(0)=x i m 2 )+p A + i 2 A 2 A2 ) ip(t )x(t )+i T 0 dt(pẋ 2 (p2 + n-point Green s function G(p,...,p n )= DA µ s H(x,...,x n ) S S p ((p A s ) 2 iε) x...p n (p A s ) 2 iε) x n H S 28

29 PATH INTEGRAL METHOD Truncate external lines for S-matrix element i(p 2 f + m 2 )p f i((p A) 2 iε) x i = e ip f x i f( ) S(p,...,p n )= DA µ s H(x,...,x n ) e ip x f ( )...e ip nx n f n ( ) e is[a s] f( ) = x(0)=0 Dxe i R 0 dt ( 2 ẋ2 +(p f +ẋ) A(x i +p f t+x(t))+ i 2 A(x i+p f t+x)) Eikonal vertices act as sources for gauge bosons along path Disconnected QED: exponentiation now textbook result: all diagrams = exp (connected diagrams) Connected 29

30 REPLICA TRICK Can relate exponentiation of soft gauge fields to that of connected diagrams in QFT. Proof: replica trick (from stat. mech.) Consider a N copies of a scalar theory Z[J] N = Dφ...Dφ N e is[φ ]+..+is[φ N ]+Jφ +..Jφ N If Z is exponential, find out what contributes to log Z Z N =+N log Z + O(N 2 ) Amounts to diagrams that allow only one replica connected! 30

31 REPLICA TRICK Can relate exponentiation of soft gauge fields to that of connected diagrams in QFT. Proof: replica trick (from stat. mech.) Consider a N copies of a scalar theory Z[J] N = Dφ...Dφ N e is[φ ]+..+is[φ N ]+Jφ +..Jφ N If Z is exponential, find out what contributes to log Z Z N =+N log Z + O(N 2 ) Amounts to diagrams that allow only one replica connected! 30

32 APPLICATION TO QCD Amplitude for two colored lines S(p,p 2 )=H(p,p 2 ) Replicate, and introduce ordering operator DA s f( )e is[a s] f( ) =P exp dx A(x) N P exp dx A i (x) N = RP exp dx A i (x) i= i= Look for diagrams of replica order N. These will go into exponent x 2 x 2 x 2 i, A i, A j, B i, A j, B x x x (a) (b) (c) (a) is order N (b) for equal replica number (i=j): CF 2. For i j also CF 2. Sum: (c) for equal replica number (i=j): CF 2 -CF CA /2. N For i j CF 2. Term linear in N: 3 NC 2 F + N(N )C 2 F = N 2 C 2 F CF 2 C F C A +( N)CF 2 = N C F C A 2 2

33 APPLICATION TO QCD Amplitude for two colored lines S(p,p 2 )=H(p,p 2 ) Replicate, and introduce ordering operator DA s f( )e is[a s] f( ) =P exp dx A(x) N P exp dx A i (x) N = RP exp dx A i (x) i= i= Look for diagrams of replica order N. These will go into exponent x 2 x x 2 x 2 i, A j, B i, A i, A j, B x x (a) (b) (c) Web Modified color factor (a) is order N (b) for equal replica number (i=j): CF 2. For i j also CF 2. Sum: (c) for equal replica number (i=j): CF 2 -CF CA /2. N For i j CF 2. Term linear in N: 3 NC 2 F + N(N )C 2 F = N 2 C 2 F CF 2 C F C A +( N)CF 2 = N C F C A 2 2

34 WEBS TO ALL ORDERS Can give an all-order, non-recursive formula for modified color factors Consider general diagram G, with a number of connected pieces Distribute replica numbers in all possible ways, and count for each such partition P the multiplicity to linear order in N C(G) = P ( ) n(p ) (n(p ) )! g C(g) Examples C(X) =C(X) C(I)C(I) =(C 2 F 2 C AC F ) C 2 F C(IX) =C(IX) C(I)C(X) =0 32

35 NEXT-TO-EIKONAL EXPONENTIATION Wilson lines are classical solutions of path integral Fluctuations around classical path lead to NE corrections This class of NE corrections exponentiates Keep track via scaling variable λ f( ) = x(0)=0 Dx exp + i 2λ A(x i + p f t + x) Use -D field theory propagators p µ = λn µ λ i dt 0 2 ẋ2 +(n +ẋ) A(x i + nt + x) x(t)x(t ) = G(t, t )= i λ min(t, t ) ẋ(t)ẋ(t ) = i λ δ(t t ) 33

36 FEYNMAN RULES f( ) =exp d d k n µ (2π) d n k õ(k)+ 2λ d d k k µ (2π) d n k õ(k)+ + Both 2-point correlators and tadpoles contribute NE Feynman rules k k l k l p p p k µ 2p k p µ k2 2(p k) 2 + ηµν p (k + l) lµ p ν p k + k ν p µ p l p (k + l)p kp l 34

37 LOW-BURNETT-KROLL One soft emission determined by elastic amplitude to eikonal and next-to-eikonal order p 2 p 2 p 2 = H + H + k k H k p p p Γ µ = (2p k) µ 2p k + (2p 2 + k) µ p µ Γ+ (k p 2 k p ) 2p 2 k p k + pµ 2 (k p k p 2 ) p 2 k Γ p p 2 Analyzed in context of jet-soft factorization by Del Duca S One emission from H still missing in our approach H S S 35

38 LOW-BURNETT-KROLL Path integral method provides elegant way to derive Low s theorem S(p,...,p n )= DA s H(x,...,x n ; A s )e ip x f(x,p ; A s )...e ip nx n f(x,p ; A s )e is[a s] H S S S Gauge transformation must cancel between f s and H f(x i,p f ; A) f(x i,p f ; A + Λ) =e iqλ(x i) f(x i,p f ; A) Opposite transformation in H, expand to first order in A and Λ Low s contribution is then: S(p,...,p n )= DA f(0,p ; A)...f(0,p n ; A) d d k (2π) d n j n µ j q j n j k k ν H(p,...,p n )A µ (k) p j ν p j µ First term is due to displacement of f(x,p,a) Analagous result in non-abelian case, for n=2 36

39 UPSHOT Exponentiation of soft emissions for matrix elements as connected diagrams For both eikonal and next-to-eikonal contributions from external lines Replica trick both for exponentiation, and for explicit expression for webs. New NE Webs. emission from hard part also included QCD: 2 lines ok. 3 lines also easy. 4+ (later) Can we arrive at the same results using diagrams, and inductive reasoning? Combinatorics challenging 37

40 DIAGRAMMATIC APPROACH EL, Magnea, Stavenga, White k,µ k 2,µ 2 k n,µ n Recall: Abelian case, multiple emission, and sum over permutations p Eikonal identity: For many emissions + p (k + k 2 ) p k 2 i p µ i p k i. p (k + k 2 ) p k = p k p k 2 Non-abelian case requires web: two-eikonal irreducible graph group : projection of web on external line analogue of eikonal identity for permutations that leaves ordering in group invariant (shuffle product)... 2p k π 2p (k π + k π2 )... p (k π k πn ) = π g 2p (k g k gm ) 2p k g 2p (k g + k g2 ) 38

41 EXPONENTIATION BY INDUCTION Can also use in reverse, as merging B 2 2 E(H ) E(H 2 )= π A π B E(H πb πa H 2 ) A = 3 2 Collect identical diagrams E(H ) E(H 2 )= G E(G) N G H H 2 exp c H E(H) i = H n n! [ c HE(H)] n = G c G E(G) Prove that, for normal color factors on rhs, those on left side are those of webs Proof uses induction combinatorics simplicity of color structure 39

42 NEXT-TO-EIKONAL CASE x 2 x 2 i, A x x (a) (b) x 2 x 2 Identify next-to-eikonal vertices x (c) x (d) show that they decorrelate, once summed over all perm s. Use induction again as eikonal webs, but now with a special vertex for fermions: they become spin-sensitive new correlations between eikonal webs NE webs 40

43 DRELL-YAN CHECK Check use of NE Feynman rules for Drell-Yan double real emission for amplitude, expand A = A (0) Ā()E exp + Ā()NE + Ā(2)NE to 2nd order, and integrate each term with exact 3-particle phase space. Cross term leads to αs C F 4π log 3 ( z) Also need special vertex 3 52 log2 ( z) 52 log( z) R µν (p; k,k 2 )= (p k 2)p µ k ν +(p k )k µ 2 pν (p k )(p k 2 )g µν (k k 2 )p µ p ν p (k + k 2 ) p k p k q q p _ k 2 p _ k 2 (a) (b) 4

44 DRELL-YAN CHECK Combine with exact phase space K (2)NE = αs C F 4π 2 024D3 + 52D 2 52 log 2 ( z) log( z) + 256D Agrees with equivalent exact result log3 ( z) log 2 ( z) + 52D 52 log( z) 2 D i = log i ( z) z + 42

45 MULTIPLE COLORED LINES j j 3 W W H W 2 i i 3 W 3 4 ZIJ N = DA...DA N e i P S[A i ] (W ()...W () N )... i i 2 4 j j 2 4 Gardi, EL, Stavenga, White Mitov, Sterman, Sung Replica trick for multiple colored lines; find again order N terms even when present, these may be kinematically zero Aybat, Dixon, Sterman 43

46 MULTIPLE COLORED LINES We find F(D)C(D) = exp[ example: closed sets of diagrams Matrix with unusual properties d,d F(d)R dd C(d )] d R dd =0 Eigenvalues 0 or (3a) (3b) (3c) (3d) C(3a) C(3b) C(3c) +C(3d) M(3a) 2M(3b) 2M(3c)+M(3d) 6 Closed form solution for modified color factor C(G) = P n(p ) ( ) n(p ) C(g π )...C(g πn (P )) π 44

47 SUMMARY QCD (large-x) resummation consistent framework, generally succesful Eikonal approximation important ingredient Next-to-eikonal contributions form new webs, and exponentiate using path integrals, or diagrams Feynman rules for exponent of scattering amplitude classified Low s theorem contributions Multi-parton webs defined To do: application to cross sections 45