The Long and Short of High-Energy Jets Some methods; a few principles FRIF Workshop on First Principles Non-Perturbative QCD of Hadron Jets Paris, Jan 12, 26 George Sterman, YITP, Stony Brook Motivation: QCD, the protean theory Power corrections: the OPE and massive gluons Massive gluons and the dispersive coupling Resummation & shape functions From resummation back to dispersion Toward the far infrared
MOTIVATION QCD: what s all the fuss about? Why not leave a true theory alone? It should be good for popping heavy new states from the vacuum. Quantum chromodynamics, in the words of the Duke of Gloucester, can add colors to the chamelion/change shapes with Proteus for advantages. [Henry VI, part III]. Simple laws to complex behavior is a central theme of twenty-first century physics. In QCD: the transition of the theory of Yang and Mills to the theory of Yukawa.
The Beginning of this era and today... it is clear that the increasing ability of experimental physicists to study quantitatively many-particle final states in high energy collisions is going to lead to a data explosion... On the other hand, in the midst of enormous complexities there are very striking characteristics exhibited by high energy collisions... A number of ideas, models etc. been introduced to accommodate such characteristics... The ideas behind these models are not always mutually exclusive, especially since none of them is completely precisely defined. C.N. Yang, 1969 Now we have a theory, QCD, and we understand regularities as keys to the protean transition from weak- to strong-coupling. Our challenge: to use these characteristics as guides in a transition from models to a principles.
POWER CORRECTIONS: THE OPE & A MASSIVE GLUON From PT to Operator Expansions Mueller (1985,92): σ tot, Unitarity and IR Safety σ e+ e tot = (4πα)2 Q 2 Im π(q 2 ) π(q 2 ) = ( i 3Q 2 ) d 4 x e iq x Tj µ ()j µ (x),
m = Im ( m 2 = Im + +... ) The OPE: j µ ()j µ (x) = 1 x 6 C (x 2 µ 2 ),α s (µ)) + 1 x 2 C F 2(x 2 µ 2,α s (µ)) F µν F µν () +.... Infrared Safety: π(q) C (Q 2 /µ 2,α s (µ)) = n= c (n) (Q2 /µ 2 ) α n s(µ)
π PT C only: IR and CO finite for m q = in D = 4 So what about C F 2...? The series for C doesn t converge! Generic pinched integration region: k = only q k ~ k 2 like a gluon mass (come back to this later) C (1/Q 4 ) Q 2 dk 2 k 2 ln n (k 2 /µ 2 ) (1/2)(1/2) n n!.
Think about it as: RG + Gauge inv. C = C (reg) +C (pinch) + O(Q 6 ) κ 2 C pinch = H(Q) dk 2 k 2 α s (k 2 ) κ 2 = H(Q) dk 2 k 2 α s (Q 2 ) ) 1 + b 2 ln(k 2 /Q 2 ) ( αs (Q 2 ) 4π κ 2 = H(Q) dk 2 k 2 4π b 2 ln(k 2 /Λ 2 ) Call this: Internal Resummation: all logarithmic behavior in f(α s (k 2 ))
Now here s the OPE: Soft integrals in C (pinch) and for F 2 (κ) identical! k ~ Axiom of substitution, (also known as matching ): (Behavior of true C ) F 2 C PT Q 4 Creg PT ( Q 2 µ 2, κ ) Q,α s(q) + C F 2(Q,κ) α s F 2 () (κ) Q 4
Nonconvergence of PT need for new (IR) regularization Cost: new NP parameter ( F 2 ) implicit in PT Benefit: new NP parameter ( F 2 ) implicit in PT Possible whenever there is an internal resummation: f(αs (k 2 ))dk 2 to all logs, for some f When starting from OPE, a cancellation (see Borel plane) But the value of NP parameters depends on the definition of PT Method of effective charges (Maxwell) shows how flexible they can be!
Borel interpetation Change variables: b 2α s (Q 2 )ln(q 2 /k 2 ), and C becomes: C pinch (α s (Q)) = 1 2α s (Q) H(Q)Q4 db e b/α s(q) 1 1 b 8π b Pole (= ambiguity) at b = 8π/b 2 Q 4 The b-borel plane: b 8π/b π(b) analytic
Applications of these ideas to semininclusive cross sections... Manohar, Wise, Akhoury,Zakharov; Banfi, Lucenti, Beneke; Beneke,Braun; Braun Contopanagos, Dokshitzer,Marchesini,Webber; Dasgupta,Webber; Dasgupta,Salam Grunberg; Gardi; Rathsmann, Korchemsky; Tafat,,Vogelsang, Forte, Ridolfi... What was new in 94+ was applications to semi-inclusive cross sections Beyond the OPE: Because in all infrared-safe cross sections we can find f(αs (k 2 ))dk 2 for some f (Is it always internal resummation? Is it always all logs?)
MASSIVE GLUONS AND DISPERSIVE COUPLINGS Particle masses in event shapes give generic m/q corrections (in thrust: Basu 1984) 94: Webber: gluon mass (Bigi et al) in thrust µ/q Can be given precise realization by large-β /dressed gluons with model self-energies. Widely useful: Akhoury & Zakharov; Beneke & Braun... Gardi, Berger & Magnea The apotheosis of the massive gluon: the dispersive coupling...
Conjecture α s (k 2 ) = dµ 2 µ 2 + k 2 ρ s(µ 2 ), ρ s (µ 2 ) = d d lnµ 2α eff(µ 2 ) The template for observable F, with mass µ, after an integration by parts F(x,Q) = dµ 2 µ 2 α eff(µ) df(x,q,µ) dµ 2 Isolate the NP part δα eff, and universal terms like A 2p dµ 2 µ 2 µ2p ln(µ 2 /µ 2 )δα eff (µ 2 ) Linear α s requires MC scheme: α s = αs MS (1 + (α s /2π)K)
Neat, but the dispersion relation requires an inclusive sum over final states. Event shapes weight different parts of phase space differently by definition. (Nason, Seymour) So applications to event shapes require a bit more analysis: The MC scheme doesn t get the entire α s 2 part right. Can compensate with a Milan factor : α s,eff Mα s,eff. It s computable.
Angularities e + e RESUMMATION & SHAPE FUNCTIONS Flexible event shapes (C.F. Berger, Kúcs, GS (23), Berger, Magnea (24)) τ a = i in N θ i angle to thrust (a = ) axis E i Q (sin θ i) a (1 cos θ i ) 1 a = broadening: a = 1; inclusive limit a NLL resummed cross section σ (τ a,q,a) = σ tot C i in N dν e ν τ a [ J i (ν,p Ji )] 2 p T,i Q e (1 a) η i
The jet in transform space J i (ν,p Ji ) = dτ a e ντ Ji J i (τ Ji,p Ji ) = e 1 2 E(ν,Q,a) The resummed form again an integral over α s (µ): E(ν,Q,a) = 2 1 [ uq 2 du u u 2 Q 2 dp 2 T p 2 T A(α s (p T )) ( ) e u1 a ν(p T /Q) a 1 + 1 2 B ( α s ( uq) ) ( ) ] e u(ν/2)2/(2 a) 1 Argument of α s vanishes for u. Yet expansion in α s (Q) is finite at all orders.
Isolating the shape function p T > κ, PT p T < κ, expand exponentials Low p T replaced by f NP shape function E(ν,Q,a) = E PT (ν,q,κ,a) + 2 1 a n=1 1 nn! ( ν Q ) κ 2 n dp 2 T p 2 T p n T A(α s (p T )) +... E PT (ν,q,κ,a) + ln f a,np ( ν Q,κ ) +...
Shape function factorizes in moments convolution, σ(τ a,q) = dξf a,np (ξ) σ PT (τ a ξ,q). Fit at Q = M Z predictions for all Q All such IR safe event shapes related to correlations of nonlocal energy flow operators.
Shape function phenomenology for ρ H thrust (Korchemsky,GS, Belitsky; Gardi Rathsman,Magnea) First pass: f,np (ρ) = const ρ a 1 e bρ2 : a : no. particles/ unit rapidity b : related to energy flow between hemispheres
Scaling property for τ a event shapes (C.F. Berger & GS (23) Berger and Magnea (24)) Test of rapidity-independence of NP dynamics ln f a,np ( ν Q,κ ) = 1 1 a n=1 ( λ n (κ) ν ) n Q f a ( ν Q,κ ) = [ f ( ν Q,κ ) ] 1 1 a
What PYTHIA gives 1. R PY (ν,a)/r PT (ν,a).8.6.4.2 a = a = -.5 a = -.25. 1 2 3 4 5 6 7 8 9 ν Most event shapes were invented for jet physics of the late 7 s Address existing data with new analysis New observables to analyze final states; aid in searches for new physics (Tkachov (1995), C.F. Berger et al. (Snowmass, 21))
RESUMMATION & THE DISPERSIVE COUPLING (Aybat, GS, Vogelsang) Relation to effective theories (Korchemsky, GS; Lee, Manohar, Wise) k m+1 k m...... k N k 1 S C Eikonal lines on + and - lightcones; k the total final-state momentum Interactions on + line instantaneous at x =
Do all loop-l + integrals at fixed total final-state momentum k. All k + = (k 2 + k 2 )/2k dependence is in a factor F(k) = { N N c=1 = i N n=1 n=c+1 1 k + D n iǫ 2πδ(k+ B D c) 1 N k + D n iǫ + i n=1 c 1 n=1 1 k + D n + iǫ 1 k + B D n + iǫ } D n = i n k2 i /2k i k + -independent; dispersive structure manifest
Dispersive structure is respected in the exponent: webs. Leading operators in an effective theory ds ea ({β i }) de a = n δ (e a e a (n)) W n n W ( 2 ) W = T Φ βj (,) ( j=1 Φ βi (λ,x) = P exp ig λ ) dλ β i A(λ β + x)
β β β β k k β Typical (virtual) diagram
GS; Gatheral, Frenkel and Taylor Resummation for e a in this formalism dσ de a = n= 1 n! de δ(e e s ) P e i =e a n i=1 E(e i ) E(e) = states n W n (e) = M C(M n ) M 2 n(e) M 2 n: momentum parts of cut graphs irreducible by cutting two eikonal lines
C(M n ): modified color factors for M n Examples: all C(M)=C F CA Exponentiation under transform S e (N) Ne ds de e de = exp [ ] de e Ne E(e )
Double logarithmic behavior encoded in properties of W s Boost invariance in the eikonal annihilation cross section ln ˆσ (eik) (N, Q) = N dps N θ(q 2 k 2 ) M (eik) 2 e Nk /Q = Q 2 ρ(α s (u,ε)) + ln N u 2 Q 2 [ ( ) 2Nu K Q du 2 u 2 A(α s(u,ε)) + ln u Q ] Factorization requires an MC-scheme ρ(α s (u,ε)) = A(α s (u,ε)) + D ln µ 2 with D(α s,) the NNLL D-term
CONCLUSION: LOOKING TOWARD THE FAR INFRARED Dispersion for A(α s ) + D(α s ) related to sum over real final states From (N/Q) p to (N/Q 2 ) p (Analog Bloom-Gilman duality?) Resummation and power corrections: it s gluons mostly, but eventually: it all becomes valence quarks. The hadronization of event generators: clusters, string breaking When we learn how to describe this we ll be there!