THE EVOLUTION OF SOFT-COLLINEAR EFFECTIVE THEORY

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1 THE EVOLUTION OF SOFT-COLLINEAR EFFECTIVE THEORY Christopher Lee, in lieu of Iain Stewart LANL QCD Evolution 2014, Santa Fe, New Mexico May 13, 2014

2 FLAVORS OF SCET SCETI: Hadronic Jets, measured with Thrust, jet masses, etc. SCETII: (Recoil-sensitive) Jet Broadening, kt-dependent distributions SCETG: SCET with Glaubers (relevant for heavy-ion collisions)

3 EFFECTIVE FIELD THEORIES Simplify full theory by integrating out heavy or UV degrees of freedom. Replace with effective theory of the accessible, low-energy degrees of freedom. e.g. effective EW theory (integrate out W,Z bosons), heavy quark effective theory (integrate out momentum modes ~ m Q ) EFT must reproduce same IR, low-energy behavior of full theory. Mismatch in UV encoded in Wilson or matching coefficients. EFT may possess additional symmetries or computational simplifications beyond full theory (e.g. heavy quark spin-flavor symmetry in HQET, soft-collinear decoupling in SCET.) EFT is an approximation to full theory to order-by-order in a power counting expansion in a small parameter

4 FLASHBACK: HEAVY QUARK EFFECTIVE THEORY Expansion of QCD in 1/m Q, HQET fields describes small momentum fluctuations around heavy quark at rest: see Isgur, Wise (90s); Manohar, Wise (2000) p Q = m Q v + k Q q Factor out large momentum phases, integrate out large momentum fluctuations: (x) =e im Qv x v (x) L Q = h id id hv v H v id id 2m Q H v h v (x) 1+v 2 v(x) k QCD O( QCD ) v =(1,~0) H v (x) 1 2 v v(x) integrate out H v Simpler Lagrangian at leading power for HQET field, enhanced spin-flavor symmetry, systematic power expansion L HQET = h v (x)(iv D)h v (x)+o QCD m Q

5 SOFT-COLLINEAR EFFECTIVE THEORY: MODES Similarly to HQET, SCET modes describe small fluctuations around lightlike momentum trajectories: p soft 2 Q -z +z Q n =(1, ~z), In 2-jet events, three EFT modes: n =(1, ~z) E,p z Q n 2 = n 2 =0, n n =2 p n = p n + k E + p z collinear a =0 Remove hard modes from theory hard p 2 Q 2 k Q 2 p n n p n 2 + p? Q Q label momentum soft Q 2 anti-collinear Q E p 2 Q 2 2 p 2 Q 2 4 p z Bauer, Fleming, Luke, Pirjol, Rothstein, Stewart (2000, 2001)

6 SCET LAGRANGIAN Bauer, Fleming, Pirjol, Stewart (2000, 2001) Factor out large momentum phase and project out heavy and light components of fermion fields: (x) = X p e i p x n,p(x) n,p = n n 4 n,p = n n Similarly for gluon fields. Resulting leading-power Lagrangian: 4 n,p n,p massless effective mass Q, integrate out L SCET = X n (L qn + L gn )+L s L qn = h n (x) in D + id c?w n (x) L gn = 1 2g 2 Tr id µ + ga n,id + ga n L s = L QCD [q s,a s ] 1 i n i n P W n(x)id c? 2 n(x) 2 D µ = n P nµ 2 + Pµ? P µ D c µ = P µ iga c µ + in D nµ 2 label momentum operator n (x) = X p e i p x n, p (x) similarly for collinear gluons

7 SCET FEYNMAN RULES Collinear quarks: Similarly for collinear gluons.

8 SCET OPERATOR MATCHING Besides Lagrangian, also need to match QCD currents onto SCET operators. e.g. for e + e - to 2 jets, DIS, or Drell-Yan: µ C 2 (µ)[ n1 W n1 ] p1 µ [W n 2 n2 ] p2 One-loop matching, difference of QCD and SCET loop graphs. IR divergences match, leaving over UV-dependent Wilson coefficient: QCD: SCET: C 2 (µ) =1+ s(µ)c F 4 h ln 2 µ 2 p 1 p 2 3ln µ 2 p 1 p i Manohar (2003); Bauer, CL, Manohar, Wise (2003)

9 SOFT-COLLINEAR DECOUPLING Bauer, Pirjol, Stewart (2001) At leading power, soft-collinear interactions are eikonal: n (in D s ) n = n + gn A s ) n Soft-collinear interactions resummed into Wilson lines: h Z 0 i Y n (x) =P exp ig ds n A s (ns + x) Perform field redefinition: n (x) =Y n (x) (0) n (x) n (in D s ) n 1 (0) n n (0) p collinear k soft p + k soft gluons see only direction and color of jet p ig µ t a (p + k) 2 p keep leading order part of diagram n µ t a n k n 2

10 FACTORIZATION FOR EVENT SHAPES Collins, Soper, Sterman Two-jet event shapes in e + e -, e.g. thrust: 1 0 Z d d = H 2(Q 2,µ) dt n dt n dk S t n + t n Q 2 k S Q J n (t n,µ)j n (t n,µ)s 2 (k S,µ) Fleming, Hoang, Mantry, Stewart (2007) Bauer, Fleming, CL, Sterman (2008)

11 RESUMMATION OF LARGE LOGARITHMS Computing in fixed-order perturbation theory, we encounter large logs in the two-jet kinematic region: ( ) = Z 0 d F 12 ln 2 F 11 ln + F 10 d d =1+ s 4 s 2 + F 24 ln 4 + F 23 ln 3 + F 22 ln 2 + F 21 ln + F Need to reorganize perturbative series to regain convergence: ln ( ) s (ln 2 +ln ) + 2 s(ln 3 +ln 2 +ln ) + 3 s(ln 4 +ln 3 +ln 2 +ln ) Leading Log (LL) Next-to- Leading Log (NLL) NNLL

12 RESUMMATION FROM RG EVOLUTION SCET RG gives equations for evolution of hard, jet, and soft functions in factorization theorem with energy scale µ. Q Full QCD H 1+ n s ln m µ Q evolve each function in factorization theorem from scale where logs are minimized Q Q evolution with calculable µ anomalous dimension soft + collinear EFT µ d dµ F (Q F,µ)= F (µ)f (Q F,µ) F (µ) = F [ s (µ)] ln Q2 F µ 2 + F [ s (µ)] J 1+ n s ln m µ Q p soft EFT S 1+ n s ln m µ Q Solutions of evolution equations contain logs resummed to all orders in s

13 SCET VS. DIRECT QCD RESUMMATION Resummed thrust distribution derived in traditional or direct h QCD: X 1 1/ ) eē(ln E(ln ) =2 Form derived from solutions of RGEs in SCET: K F (µ, µ F )= F (µ, µ F )= E (n) = Z µ dµ 0 n µ F µ 0 Z µ ( ) =N (Q)exp d n E d[ln(1/ )] n µ F dµ 0 µ 0 F [ s (µ 0 )] n=2 nē(n) 1 n Ē 0 Z Q o F [ s (µ 0 )] ln µ0 + F [ s (µ 0 )] µ F (1 Ē 0 (ln 1/ )) dµ 0 n µ 0 2 Q(e E ) µ 0 A[ s ]ln + BJ [ s ]o Q 1/2 Z Q(e E ) 1/2 dµ 0 n µ Q(e E ) µ 0 A[ s ]ln + BS [ s ]o Q(e 1 E ) ( ) =e K H+2K J +K µh H µj J S µ S S H2 (Q 2,µ Q Q 1/2 H ) Q J +ln µ2 2 Q 2,µ J +ln µ S e E Q,µ S (1 ) =2 J + S CTTW (1993) Berger, Kucs, Sterman (2003) cf. Becher, Neubert (2006)

14 SCET VS. DIRECT QCD RESUMMATION Showed that two forms are exactly equal with the dictionary: A[ s ]= cusp [ s ] B F [ s ]= F [ s ] d ln F (0,µ F ) d ln µ F Almeida, Ellis, CL, Sterman, Sung, Walsh, N (Q) =H(Q) J(0,Q) 2 S(0,Q) E =2E J + E S E F (µ, µ F )=K F (µ, µ F ) Z µ dµ 0 d ln F (0,µ 0 ) µ F µ 0 d ln µ 0 Leads to unified formula, generalizing dqcd to variable jet/soft scales, and SCET to more explicitly exponentiated form: µh R( ) = 0 H(µ H ) Q H e K H J(0,µ) 2 S(0,µ)e 2E J (µ,µ J )+E S (µ,µ S ) hx 1 exp n=2 1 n Q(e E ) 1/2 µ 2E (n) J (µ J)@ n 2E 0 J i + E (n) S (µ S)@ E n S 0 2E 0 J (µ,µ J ) Q E 0 S (µ,µ S) 1 µe E (1 + 2EJ 0 + E0 S )

15 NONPERTUBATIVE HADRONIZATION CORRECTIONS Soft power corrections shift mean values of event shapes hei = hei PT + c e 1 Q c e observable dependent, calculable coefficient conjecture from single soft gluon emission: Dokshitzer, Webber (1995, 1997) 1 universal nonperturbative parameter proof to all orders in soft gluon emission CL, Sterman (2006, 2007) SCET: First rigorous proof (and field theory definition of ) from factorization theorem and boost invariance of soft radiation: 1 = 1 Trh0 Y ny n E T ( )Y n Y n 0i N C 1 generalization to massive hadrons Mateu, Thaler, Stewart (2012) soft radiation sees only direction, not energy, of original collinear partons, invariant to boosts along z

16 HIGH PRECISION STRONG COUPLING EXTRACTION NNNLL perturbative prediction + nonperturbative soft power correction led to most precise extraction of strong coupling from event shapes Abbate, Fickinger, Hoang, Mateu, Stewart (2010) 1 σ dσ dτ Q=m Z Sum Logs, with S mod + gap N3 LL N3 LL NNLL NNLL NLL NNNLL resummed perturbative distribution Becher, Schwartz (2008) τ -7.5% shift from NP power corrections GenEvA Collaboration (2012) fit to LEP data gave consistent value (improved Monte Carlo with NLL and higher-order theory input) Abbate, Fickinger, Hoang, Mateu, Stewart (2010) Generically, better perturbative calculations + rigorous treatment of nonperturbative corrections gives smaller s

17 PROTON COLLISIONS: BEAM FUNCTIONS AND PDFS Nonperturbative factorization: Perturbative factorization: d = f a f b ˆ F had

18 EVENT SHAPES IN PP COLLISIONS Beam thrust in pp collisions as Higgs jet veto (like DY): Fixed-order: Resummed in SCET: Berger, Marcantonini, Stewart, Tackmann, Waalewijn (2010)

19 EVENT SHAPES IN DIS: 1-jettiness in DIS measures final states with beam radiation + one additional jet 2 X Q 2 H B H J p J 1 = q J true jet axis i2x min{q B p i,q J p i } Version a: Version b: H B HJ D. Kang, CL, I. Stewart (2013) also Z. Kang, Liu, Mantry, Qiu (2012, 2013) p J q J = q + xp q B = xp q q B = xp q p B p B 1 0 d (x, Q 2 ) d a 1 Z = H(Q 2,µ) dt J dt B dk S a 1 J q (t J,µ)B q (t B,x,µ)S(k S,µ) t J Q 2 t B Q 2 k S Q 1 0 d (x, Q 2 ) d b 1 Z = H(Q 2,µ) d 2 p? dt J dt B dk S b 1 t J Q 2 J q (t J p 2?,µ)B q (t B,x,p 2?,µ)S(k S,µ) t B Q 2 k S Q Choice of axes can induce sensitivity to transverse momentum dependence of (perturbative) initial-state radiation. Resummation possible to NNLL or (for version a) NNNLL D. Kang, CL, I. Stewart (2014)

20 POWER CORRECTIONS IN PP AND DIS Universal nonperturbative shift in 3 versions of DIS 1-jettiness: dsêdt a NLO PT NNLL PT NNLL PT + NP 2 1 /Q t a Q=80 GeV x=0.2 W=0.35 GeV Using factorization theorems and boost invariance properties of soft Wilson lines, can prove that: a 1 = b 1 = c 1 D. Kang, CL, I. Stewart (2013) Surprising relation also to leading NP correction to jet mass in pp to 1 jet Boost Stewart, Tackmann, Waalewijn (2014)

21 SCET II : RAPIDITY LOGS AND RENORMALIZATION For more exclusive or for TMD distributions, collinear and soft modes have same virtuality, distinguished only by rapidity Chiu, Jain, Neill, Rothstein (2012) Need additional regulator for rapidity divergences, evolution both in virtuality and in rapidity scales. Chiu, Jain, Neill, Rothstein (2012) Echevarria, Idilbi, Scimemi

22 SCETII: EFT FOR TMD DISTRIBUTIONS pt veto in pp to Higgs + 0 jet Jet broadening TMDPDFs Stewart, Tackmann, Walsh, Zuberi (2013) Chiu, Jain, Neill, Rothstein (2012) See also: Recoil-free observables, Larkoski, Neill, Thaler (2014) Echevarria, Idilbi, Scimemi (2012)

23 SCETG: WITH GLAUBERS Add to SCET Glauber modes exchanged between jets and scattering centers in a dense medium: Idilbi, Majumder D Eramo, Liu, Rajagopal Ovanesyan, Vitev Feynman rules in covariant gauge: p Q(1, 2, ) q i Q( 2, 2, ) Ovanesyan, Vitev

24 SCET, GLAUBERS AND REGGE BEHAVIOR Much recent activity in understanding Glauber interactions and Regge behavior from SCET: S. Fleming: Glauber exchange in SCET and BFKL equation, J. Donoghue, B. Kamal El-Menoufi, G. Ovanesyan: Regge behavior in EFT, J. R. Gaunt: Glauber Gluons and Multiple Parton Interactions, I. Rothstein and I. Stewart: Glaubers, Reggeization, and Renormalization in EFT, SCET2010 and SCET2014.

25 FUTURE EVOLUTION OF SCET Jet Shapes in pp and heavy ion collisions see Y.-T. Chien and I. Vitev (2014) Non-global logarithms in EFT Jet Algorithms and Jet Substructure in SCET High-precision strong coupling from jets in DIS (EIC) Chien, Hornig, CL Hornig, CL, Stewart, Walsh, Zuberi Schwartz et al. Ellis, Hornig, CL, Vermilion, Walsh (2010) Hornig (2014) D. Kang, CL, Stewart Nuclear phenomenology from DIS and EIC Z. Kang, Mantry, Qiu TMDs and spin physics Echevarria, Idilbi, Scimemi Z. Kang, I. Vitev, Jets at LHC and the search for BSM

26 Photo credits: Background and Los Alamos, Elena Giorgi; Hotel, Inn and Spa at Loretto; Cathedral, TripAdvisor/Visit Santa Fe SCET 2015 XIIth Annual Workshop on Soft Collinear Effective Theory March 25-27, 2015 Santa Fe, New Mexico, USA The Inn and Spa at Loretto Local Organizers: Christopher Lee Contact: Yang-Ting Chien Zhongbo Kang Andrew Hornig Emanuele Mereghetti Daekyoung Kang Ivan Vitev Administrative support: Jenny Esch

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