FACTORIZATION SIMPLIFIED Loopfest XIII June 19, 2014 Matthew - PowerPoint PPT Presentation

June 19, 2014 Matthew Schwartz FACTORIZATION SIMPLIFIED Loopfest XIII June 19, 2014 Matthew Schwartz Harvard University Based on arXiv:1306.6341 with Ilya Feige and arXiv:1403.6472 with Ilya Feige

June 19, 2014 Matthew Schwartz Main result: · · · h X N | W † = C ( S ij ) h X 1 | � ? W 1 | 0 i N � | 0 i h X | O | 0 i ⇠ h X s | Y † 1 · · · Y N | 0 i h 0 | Y † h 0 | W † 1 W 1 | 0 i N Y N | 0 i p i · p j • Two different amplitudes in QCD are equal at leading power in Q 2 • We prove this rigorously to all orders in perturbation theory C ( S C ( S M {±} ⇠ X C I, {±} ( S ij ) = I ⇥ · · · h X i | ¯ ψ i W i | 0 i ± h i · · · h X j | A µ W j | 0 i ± a j · · · h X k | W † k ψ k | 0 i ± h k QCD: · · · tr h 0 | Y † tr h 0 | Y † tr h 0 | W † i W i | 0 i j W j | 0 i k Y k | 0 i I ) h i l i · · · ( Y † I ) l j − 1 a j l j +1 · · · ( T k I Y k ) l k h k · · · | 0 i ⇥ h X s | · · · ( Y † j T j i T i

June 19, 2014 Matthew Schwartz Perturbative QCD • Why is perturbative QCD useful at all? Factorization Universal Asymptotic freedom Perturbative • α s is small at high energy d σ =[PDFs] • Perturbation theory works x [hard process] x [soft/collinear radiation] β = µ d dµ α s < 0 x [hadronization] (Re)summable Small Determined by UV properties of QCD Determined by IR properties of QCD

June 19, 2014 Matthew Schwartz Why is proving factorization so hard? 1. Non-perturbative effects m P Λ QCD • To show factorization up to or Q Q • No access to non-perturbative scales in perturbation theory 2. Perturbative effects • Infrared singularities (pinch surfaces) complicated • Gauge dependence subtle • Off-shell modes (Glauber gluons) 3. Hard even to formulate theorem • Precisely what is supposed to hold? • Gauge-invariant and regulator-independent formulation?

June 19, 2014 Matthew Schwartz Historically, four approaches 1. OPE approach 2. Pinch surface approach 3. Amplitude approach 4. Effective field theory approach

June 19, 2014 Matthew Schwartz Approach 1: Operator Products Deep inelastic scattering • Use OPE around ω =0 to expand at large Q 2 • Physical region has ω >1 k ′ k Im( ω ) ω q ↓ analytic P analytic ge Q . In mom Re( ω ) Since ω = 2 P · q − 1 1 Photon momentum Q 2 analytic analytic (i.e. µ ν is determined by re The hadronic tensor • OPE is possible because we can analytically continue • We know analytic structure because 1. Inclusive over final states 2. Analytic structure of two-point function known exactly { J µ ( x ) J ν (0) }| • Analytic structure for more complicated processes not known exactly

404 J.C. Collins, D.E. Soper / Back-to-back jets S \ A_.~" ,¢~.;_B A A d Fig. 5.10. Adding tulip B to the garden produces a cancellation. the finger of TA enters S. There the boundary of TB follows the boundary of S. Of course it is supposed that the finger of which we have been speaking has no .L C. Collins, D.E. Soper / Back-to-back jets subfingers interior to it; if it does, we apply the argument to the smallest subfinger. 401 If the garden contains several tulip fingers extending from S into J, there is an independent cancellation for each finger. There are also (independent) cancellations when the exterior of a tulip has a finger that extends from J into S. The cancellations for a three tulip garden of reasonably complicated geometry are illustrated in fig. 5.11. This argument shows that GR-- 0 when soft and collinear integration regions are taken into account. We now extend this argument to cover ultraviolet momenta, June 19, 2014 Matthew Schwartz proceeding in two stages. In the first stage, imagine that ultraviolet loop momenta in vertex and self-energy subgraphs are allowed provided that no soft gluon from Approach 2: Pinch surfaces the upper soft subgraph S in fig. 4.3a ends on an ultraviolet line in J. Upon rereading the original argument for GR--0, one discovers that no change in the argument is required to cover this case. Fig. 5.6. Eikonal rotation matrix next to a collinear three-gluon vertex. The "'soft" q~' and q~' are neglected in the expression for the vertex. (The 8 here signifies that the octet SU(3) representation matrices appear on the eikonal line.) Collins, Soper, Sterman: Thus we can adopt the convention that k"+q~+q~ is replaced by k" at the I three-gluon vertex in the "soft" approximation. In coordinate space with the vertex pinch surfaces factorize at x ", this amounts to saying that the O/Ox" at the vertex does not differentiate the eikonal rotation matrix U(x ; A) that appears at the end of the collinear gluon line. 5.5. A BOTANICAL CONSTRUCTION fig. 5.7 and summing over attachments of the four gluons leaving the larger tulip, one obtains this Fig. 6.1. The fate of fig. 5.7. After making the soft approximation on the gluons leaving each tulip in contributing to (OldJ(k')a+(P)a(P) × Consider a cut Feynman diagram G T{A(q)d~(k)}[O), as illustrated in fig. 5.7. When the gluon A(q) is coupled to the 21 I 2 k I ~ ~ ~ k ~ 2 I x ./ q :5 Fig. 5.11. Cancellations for a complicated garden. The shaded area is the soft subgraph. The solid lines are tulip boundaries. Addition of tulips with new boundary portions along one or more of the dashed Collins & Soper, 1981 I or dotted lines produces cancellations. diagram, as well as several others. all "soft" insertions of Ta into G. From the discussion of subsect. 5.3 it is evident Fig. 5.7. A two-tulip garden. a particular insertion of TR into (~ in the soft approximation. We now sum over subgraph T and all of the gluons leaving T are erased. The T term of eq. (6.1) is Let us denote by t~ the graph that is left over from the complete graph G if its (6.2) J be considered to be a Feynman graph in its own right, and we can define a subtracted Consider now a particular term in the sum over largest tulips T. The tulip T can (6.1) It is useful to rewrite eq. (5.18) by grouping together all gardens that have the . initial quark with a square vertex of fig. 4.2. Each such graph can be decomposed T ) 5.7, where now we imagine that the gluon with momentum q" is coupled to the 1 shown in fig. 4.1. A typical diagram contributing to fig. 4.1 is the one shown in fig. (-1) N iS(T1)''" S(Tn-1)]G. - n T 407 ( I /" S (-1)N-IS(T1) "' • m T,,=T;N>I k t n s e n l e a s d v t r e Tn=T;N>I i a • u j g q t k e n s c n e n a l i e a b d Y v r - i TR = T+ o a u g q t - e k n c i a according to the garden formula (5.18). B G = GR q-~, S(T) { 1 + / r e p o S . version of T by E T . D same largest tulip. Thus , s n i l l o C . C . J

June 19, 2014 Matthew Schwartz Approach 2: Pinch surfaces p µ = 0 Soft region : all particles have Hard region (drawn as points) Jet regions : all particles have p µ i = c i p µ • All momenta zero or exactly proportional to some external momentum • Sidesteps soft/collinear overlap region (zero bin) • More work needed to factorize finite-momentum amplitudes • Factorizes hard from jet/soft – does not factorize jet from soft • Do not provide operator definitions

June 19, 2014 Matthew Schwartz Approach 3: Amplitudes Collinear Primary goal is practical formulas (e.g. for subtractions): mitting a photon, so all that changes is a group factor ij gets Tree-level One-loop p i p i + k = − → = p i p 1 p 1 p 1 q k , , q M → M × P ab q eikonal factor is now . As in QED, this factor is ind p 2 p 2 p 2 q q DGLAP splitting functions (1977) p 1 p 1 q p 1 � � 1 + z 2 � � � 1 + 3 ⌘ , G ↑ k � P qq ( z ) = C F 2 δ (1 − z ) ↑ k ⌘ 1 − z + p 2 p 2 ion is known as a DGLAP splitting function , afte • Leading order splitting functions universal p 2 (process independent) • IR divergent at 1-loop • Splitting functions for PDF evolution defined to all orders • Relevant diagrams are gauge and process-dependent • Bern and Chalmers (1995): collinear universality proven at 1-loop • Kosower (1999): universality proven to all orders at leading color (large N) • No all-orders proof in QCD (until now)

June 19, 2014 Matthew Schwartz Approach 3: Amplitudes Soft 0 i h k 1 · · · k ` | Y † 1 · · · Y N | 0 i x = = Z ∞ ✓ ◆ ds n j · A ( x µ + s n µ Y † j ) e − " s j ( x ) = exp ig Soft gluons see hard particles as classical sources 0 Wilson lines • Wilson line picture does not disentangle soft from collinear • Universal soft current conjecture (Catani & Grassini 2000) 1 + O ( g 4 � a | M ( q, p 1 , . . . , p m ) � ≃ ε µ ( q ) J a � � µ ( q, � ) | M ( p 1 , . . . , p m ) � S ) , Computed in dim reg at 1-loop (Catani & Grassini 2000) q q q q q i i i i i i k+q + + + + k k q -k j j j j j j (a) (b) (c) q (a) (b) (c) q q i i i l i J (0) (q) + - J (0) (q) - i k j j j j j (d) (e) (f) (d) (e) • Soft current computed in dim reg at 2-loop (Duhr & Gehrmann 2013, Zhu & Li 2013) • Required for NNLO subtractions and automation • No operator definition of J • all orders universality unproven (until now)