The all-order structure of long-distance singularities in massless gauge theories

Transcription

1 The all-order structure of long-distance singularities in massless gauge theories Lorenzo Magnea CERN Università di Torino INFN, Torino Lausanne EPFL - March 22, 2010

2 1 Introduction History and motivation Tools 2 Planar amplitudes and form factors Factorization and evolution Results 3 Beyond the planar limit Soft anomalous dimensions Factorization constraints The dipole formula 4 Beyond the dipole formula Further constraints Three-loop analysis 5 Perspective Outline

3 Introduction

4 A remarkable achievement, before quantum field theory was born. Ancient History

5 Theory Singularities may arise only when internal propagators go on shell. 2p k = 2p 0 k 0 (1 cos θ pk ) = 0, k 0 = 0 (IR); cos θ pk = 1 (C). Note: p 0 = 0 singularity will be integrable. Emission is not suppressed at long distances. Emission of a massless gauge boson Isolated charged particles are not true asymptotic states of unbroken gauge theories. A serious problem: the S matrix does not exist in the usual Fock space. Possible solutions: construct finite transition probabilities (KLN theorem); construct better asymptotic states (coherent states). Long-distance singularities obey a pattern of exponentiation i h i M = M 0 h1 κ α 1 π ɛ +... M = M 0 exp κ α 1 π ɛ +...

6 Practice Just a formal issue in quantum field theory? Are there practical applications? Higher order calculations at colliders cross hinge upon cancellation of divergences between virtual corrections and real emission contributions. Cancellation must be performed analytically before numerical integrations. Need local counterterms for matrix elements in all singular regions. State of the art: NLO multileg. NNLO available only for e + e annihilation. Cancellations leave behind large logarithms: they must be resummed. 1 ɛ {z} virtual Z m 2 + (Q 2 ) ɛ dk 2 0 (k 2 ) 1+ɛ {z } real = ln(m 2 /Q 2 ), For inclusive observables: analytic resummation to high logarithmic accuracy. For exclusive final states: parton shower event generators, (N)LL accuracy. Resummation probes the all-order structure of perturbation theory. Power-suppressed corrections to QCD cross sections can be studied Links to the strong coupling regime can be established for SUSY gauge theories.

7 Modern History Factorization M {ri } «pi µ, αs(µ2 ), ɛ = X L «pi M L µ, αs(µ2 ), ɛ = S LK ρ ij, α s(µ 2 ), ɛ H K Progress ny i=1 «pi M L µ, αs(µ2 ), ɛ (c L ) {ri } (2p i n i ) 2 J i n 2, α s(µ 2 ), ɛ i µ2! 2p i p j µ 2, (2p i n i ) 2 n 2, α s(µ 2 ), ɛ i µ2! Exponentiation applies to non-abelian gauge theories. Exponentiation extends to collinear divergences. Exponentiation is performed at the amplitude level. An optimal regularization scheme is used.

8 At Tevatron Z boson spectrum at Tevatron (A. Kulesza et al., hep-ph/ ) 66 < Q < 116 GeV CDF CDF data on Z production compared with QCD predictions at fixed order (dotted), with resummation (dashed), and with the inclusion of power corrections (solid).

9 At LHC Higgs boson spectrum at LHC (M. Grazzini, hep-ph/ ) Predictions for the q T spectrum of Higgs bosons produced via gluon fusion at the LHC, with and without resummation.

10 For theory Understanding long-distance singularities to all orders provides a window into non-perturbative effects. The structure of long-distance singularities is universal for all massless gauge theories. A very special theory has emerged as a theoretical laboratory: N = 4 Super Yang-Mills. It is conformal invariant: β N =4 (α s) = 0. Exponentiation of IR/C poles in scattering amplitudes simplifies. AdS/CFT suggests a simple description at strong coupling, in the planar limit. Exponentiation has been observed for MHV amplitudes up to five legs. Higher-point amplitudes are strongly constrained by (super)conformal symmetry. A string calculation at strong coupling matches the perturbative result. Amplitudes admit a dual description in terms of polygonal Wilson loops. Integrability leads to possibly exact expressions for anomalous dimensions.

11 For theory Understanding long-distance singularities to all orders provides a window into non-perturbative effects. The structure of long-distance singularities is universal for all massless gauge theories. A very special theory has emerged as a theoretical laboratory: N = 4 Super Yang-Mills. It is conformal invariant: β N =4 (α s) = 0. Exponentiation of IR/C poles in scattering amplitudes simplifies. AdS/CFT suggests a simple description at strong coupling, in the planar limit. Exponentiation has been observed for MHV amplitudes up to five legs. Higher-point amplitudes are strongly constrained by (super)conformal symmetry. A string calculation at strong coupling matches the perturbative result. Amplitudes admit a dual description in terms of polygonal Wilson loops. Integrability leads to possibly exact expressions for anomalous dimensions. (Anastasiou, Bern, Dixon, Kosower, Smirnov; Alday, Maldacena; Brandhuber, Heslop, Spence, Travaglini; Drummond, Ferro, Henn, Korchemsky, Sokatchev; Beisert, Eden, Staudacher;...)

12 Tools: dimensional regularization Nonabelian exponentiation of IR/C poles requires d-dimensional evolution equations. The running coupling in d = 4 2ɛ obeys µ α µ β(ɛ, α) = 2 ɛ α + ˆβ(α), ˆβ(α) = α2 2π The one-loop solution is " α µ 2, ɛ = α s(µ 2 0 )! ɛ µ 2 µ ɛ X «α n b n. π n=0! ɛ! # 1 1 µ2 b 0 µ 2 0 4π αs(µ2 0 ). The β function develops an IR free fixed point, so that α(0, ɛ) = 0 for ɛ < 0. The Landau pole is at µ 2 = Λ 2 Q πɛ «1/ɛ b 0 α s(q 2. ) Integrations over the scale of the coupling can be analytically performed. All infrared and collinear poles arise by integration of α s(µ 2, ɛ).

13 Tools: factorization All factorizations separating dynamics at different energy scales lead to resummation of logarithms of the ratio of scales. Renormalization group logarithms. Renormalization factorizes cutoff dependence G (n) 0 (p i, Λ, g 0 ) = ny i=1 Z 1/2 i (Λ/µ, g(µ)) G (n) R (p i, µ, g(µ)), dg (n) 0 dµ = 0 d log G (n) nx R d log µ = γ i (g(µ)). i=1 RG evolution resums α n s (µ2 ) log n `Q 2 /µ 2 into α s(q 2 ). Note: Factorization is the difficult step. It requires a diagrammatic analysis all-order power counting (UV, IR, collinear...); implementation of gauge invariance via Ward identities.

14 Collinear factorization logarithms. Tools: factorization Mellin moments of partonic DIS structure functions factorize «««ef 2 N, Q2 m 2, αs = ec N, Q2 µ 2, α s ef N, µ2 F F m 2, αs def 2 dµ F = 0 d log ef d log µ F = γ N (α s). Altarelli-Parisi evolution resums collinear logarithms into evolved parton distributions (or fragmentation functions). Note: Sudakov (double) logarithms are more difficult. A double factorization is required: hard vs. collinear vs. soft. Gauge invariance plays a key role in the decoupling. After identification of the relevant modes, effective field theory can be used (SCET).

15 Sudakov factorization Divergences arise in fixed-angle amplitudes from leading regions in loop momentum space. Soft gluons factorize both form hard (easy) and from collinear (intricate) virtual exchanges. Jet functions J represent color singlet evolution of external hard partons. The soft function S is a matrix mixing the available color representations. In the planar limit soft exchanges are confined to wedges: S I. In the planar limit S can be reabsorbed defining jets J as square roots of elementary form factors. Beyond the planar limit S is determined by an anomalous dimension matrix Γ S. Leading regions for Sudakov factorization. Phenomenological applications to jet and heavy quark production at hadron colliders.

16 Form Factors and Planar Amplitudes (with Lance Dixon and George Sterman)

17 Gauge theory form factors Consider as an example the quark form factor Q Γ µ(p 1, p 2 ; µ 2 2 «, ɛ) 0 J µ(0) p 1, p 2 = v(p 2 )γ µu(p 1 ) Γ µ 2, αs(µ2 ), ɛ. The form factor obeys the evolution equation» Q 2 Q 2 «Q 2 log Γ µ 2, αs(µ2 ), ɛ = 1» Q K ɛ, α s(µ 2 2 «) + G 2 µ 2, αs(µ2 ), ɛ, Renormalization group invariance requires µ dg dk = µ dµ dµ = γ K α s(µ 2 ). γ K (α s) is the cusp anomalous dimension (G. Korchemsky and A. Radyushkin;...). Dimensional regularization provides a trivial initial condition for evolution if ɛ < 0 (for IR regularization). α(µ 2 = 0, ɛ < 0) = 0 Γ 0, α s(µ 2 ), ɛ = Γ (1, α (0, ɛ), ɛ) = 1.

18 Detailed factorization Operator factorization for the Sudakov form factor, with subtractions.

19 The functional form of this graphical factorization is Q 2 «Γ µ 2, αs(µ2 ), ɛ Operator definitions! = H Q2 µ 2, (p i n i ) 2 n 2, α s(µ 2 ), ɛ S β 1 β 2, α s(µ 2 ), ɛ i µ2 2 «3 (p 2Y J i n i ) 2 n 6 2, i µ2 αs(µ2 ), ɛ 4 «7 i=1 (β J i n i ) 2 5. n 2, α s(µ 2 ), ɛ i We introduced factorization vectors n µ i, with n2 i 0, to define the jets, (p n) 2 «J n 2 µ 2, αs(µ2 ), ɛ u(p) = 0 Φ n(, 0) ψ(0) p. where Φ n is the Wilson line operator along the direction n µ. " # Φ n(λ 2, λ 1 ) = P exp ig Z λ2 The jet J has collinear divergences only along p. λ 1 dλ n A(λn).

20 Operator definitions The soft function S is the eikonal limit of the massless form factor S β 1 β 2, α s(µ 2 ), ɛ = 0 Φ β2 (, 0) Φ β1 (0, ) 0. Soft-collinear regions are subtracted dividing by eikonal jets J.! J (β 1 n 1 ) 2 n 2, α s(µ 2 ), ɛ = 0 Φ n1 (, 0) Φ β1 (0, ) 0, 1 S and J are pure counterterms in dimensional regularization. β i -dependence of S and J violates rescaling invariance of Wilson lines. It arises from double poles, associated with γ K. A single pole function where the cusp anomaly cancels is S ρ 12, α s(µ 2 ), ɛ It can only depend on the scaling variable S `β 1 β 2, α s(µ 2 ), ɛ «Q 2 i=1 J (β i n i ) 2, α s(µ 2 ), ɛ ρ 12 (β 1 β 2 ) 2 n 2 1 n2 2 (β 1 n 1 ) 2 (β 2 n 2 ) 2. n 2 i

21 Jet evolution The full form factor does not depend on the factorization vectors n µ i. Defining x i (β i n i ) 2 /n 2 i, Q 2 «x i log Γ x i µ 2, αs(µ2 ), ɛ = 0. This dictates the evolution of the jet J, through a K + G equation x i log J i = x i log H + x i x i x i 1 2 hg i x i, α s(µ 2 ), ɛ Imposing RG invariance of the form factor leads to the final evolution equation log J i x i i + K α s(µ 2 ), ɛ, γ S (ρ 12, α s) + γ H (ρ 12, α s) + 2γ J (α s) = 0. Q log Γ = β(ɛ, αs) log H γ Q α S 2 γ J + s 2X (G i + K). i=1

22 Results for Sudakov form factors The counterterm function K is determined by γ K. µ d dµ K(ɛ, αs) = γ K(α s) = K ɛ, α s(µ 2 ) = 1 Z µ 2 dλ λ 2 γ K ᾱ(λ 2, ɛ). The form factor can be written in terms of just G and γ K, ( Z Γ Q 2 1 Q 2 dξ 2 h, ɛ = exp 2 0 ξ 2 G 1, α ξ 2, ɛ, ɛ 1 Q 2 γ K α ξ 2 2 «ff, ɛ log ξ 2. In general, poles up to α n s /ɛ n+1 appear in the exponent. The ratio of the timelike to the spacelike form factor is» Γ(Q 2, ɛ) log Γ( Q 2 = i π, ɛ) 2 K(ɛ)+ i Z π h G α e iθ Q 2, ɛ i Z θ dφ γ K α e iφ Q 2 i Infinities are confined to a phase given by γ K. The modulus of the ratio is finite, and physically relevant.

23 Form factors in N = 4 SYM In d = 4 2ɛ conformal invariance is broken and β(α s ) = 2 ɛ α s. All integrations are trivial. The exponent has only double and single poles to all orders (Z. Bern, L. Dixon, A. Smirnov).» Q 2 «log Γ µ 2, αs(µ2 ), ɛ = 1 X αs(µ 2 «n ) µ 2 2 π Q 2 n=1 = 1 X αs(q 2 ) 2 π n=1 «nɛ " (n) γ K 2n 2 ɛ 2 + G(n) (ɛ) nɛ # «n " (n) γ e iπnɛ K 2n 2 ɛ 2 + G(n) (ɛ) nɛ In the planar limit this captures all singularities of fixed-angle amplitudes in N = 4 SYM. The structure remains valid at strong coupling, in the planar limit (F. Alday, J. Maldacena). The analytic continuation yields a finite result in four dimensions, arguably exact. Γ(Q 2 2» ) π 2 Γ( Q 2 ) = exp 4 γ K α s(q ) 2., #

24 Characterizing G(α s, ɛ) The single-pole function G(α s, ɛ) is a sum of anomalous dimensions G(α s, ɛ) = β(ɛ, α s) α s log H γ S 2γ J + 2X G i, In d = 4 2ɛ finite remainders can be neatly exponentiated i=1 C α s(q 2 ), ɛ 2 Z Q 2 dξ 2 = exp 4 0 ξ 2 8 < : d log C α ξ 2, ɛ, ɛ d ln ξ " = Z 1 Q 2 dξ 2 5 exp ; 2 0 ξ 2 G C α ξ 2 #, ɛ, ɛ The soft function exponentiates like the full form factor ( S α s(µ 2 1 ), ɛ = exp 2 Z µ 2 dξ 2 " 0 ξ 2 G eik α ξ 2, ɛ 1 2 γ K α ξ 2, ɛ log µ 2! #) ξ 2. G(α s, ɛ) is then simply related to collinear splitting functions and to the eikonal approximation G(α s, ɛ) = 2 B δ (α s) + G eik (α s) + G H (α s, ɛ), G H does not generate poles; it vanishes in N = 4 SYM. Checked at strong coupling, in the planar limit (F. Alday).

25 Beyond the Planar Limit (with Einan Gardi)

26 Factorization at fixed angle Fixed-angle scattering amplitudes in any massless gauge theory can also be factorized into hard, jet and soft functions. M L p i /µ, α s(µ 2 ), ɛ The soft function is now a matrix, mixing the available color tensors. (c L ) {αk } S LK = X {η k } 0 β i β j, α s(µ 2 ), ɛ ny i=1 = S LK β i β j, α s(µ 2 ), ɛ H K "!, ny (p i n i ) 2 J i n 2, α s(µ 2 ), ɛ i=1 i µ2 hφ βi (, 0) αk,η k i 0 (c K ) {ηk },! p i p j µ 2, (p i n i ) 2 n 2, α s(µ 2 ) i µ2 (β i n i ) 2 J i n 2, α s(µ 2 ), ɛ i! #, Soft exchanges mix color structures.

27 Soft anomalous dimensions The soft function S obeys a matrix RG evolution equation µ d dµ S IK β i β j, α s(µ 2 ), ɛ = Γ S IJ β i β j, α s(µ 2 ), ɛ S JK β i β j, α s(µ 2 ), ɛ, Note: Γ S is singular due to overlapping UV and collinear poles. As before, S is a pure counterterm. In dimensional regularization, then " S β i β j, α s(µ 2 ), ɛ = P exp 1 Z µ 2 dξ 2 # 2 0 ξ 2 ΓS β i β j, α s(ξ 2, ɛ), ɛ. Double poles cancel in the reduced soft function S LK ρ ij, α s(µ 2 ), ɛ = S LK `βi β j, α s(µ 2 ), ɛ! ny (β i n i ) 2 J i n 2, α s(µ 2 ), ɛ i=1 i S must depend on rescaling invariant variables, ρ ij n2 i n 2 j (β i β j ) 2 (β i n i ) 2 (β j n j ) 2. The anomalous dimension Γ S (ρ ij, α s) for the evolution of S is finite.

28 Surprising simplicity Γ S can be computed from UV poles of S Non-abelian eikonal exponentiation selects the relevant diagrams: webs Γ S appears highly complex at high orders. A web contributing to Γ S. The two-loop calculation (M. Aybat, L. Dixon, G. Sterman) leads to a surprising result: for any number of light-like eikonal lines Γ (2) S = κ «67 2 Γ(1) S κ = 18 ζ(2) C A 10 9 T FC F. No new kinematic dependence; no new matrix structure. κ is the two-loop coefficient of γ K, rescaled by the appropriate Casimir, γ (i) K (αs) = C(i) h 2 αs π + κ ` α s 2i + O `α π 3 s.

29 Factorization constraints The classical rescaling symmetry of Wilson line correlators under β i κβ i is violated only through the cusp anomaly. For eikonal jets, no β i dependence is possible at all except through the cusp. In the reduced soft function S the cusp anomaly cancels. S must depend on β i only through rescaling-invarant combinations such as ρ ij, or, for n 4 legs, the cross ratios ρ ijkl (β i β j )(β k β l )/(β i β k )(β j β l ) Consider then the anomalous dimension for the reduced soft function Γ S IJ ρ ij, α s(µ 2 ) = Γ S IJ β i β j, α s(µ 2 ), ɛ δ IJ nx This poses strong constraints on the soft matrix. Indeed k=1 Singular terms in Γ S must be diagonal and proportional to γ K.! (β k n k ) 2 γ Jk n 2, α s(µ 2 ), ɛ. k Finite diagonal terms must conspire to construct ρ ij s combining β i β j with x i. Off-diagonal terms in Γ S must be finite, and must depend only on the cross-ratios ρ ijkl.

30 Factorization constraints The constraints can be formalized simply by using the chain rule. Γ S depends on x i in a simple way. x i Γ S IJ x (ρ ij, α s) = δ IJ x i γ J (x i, α s, ɛ) = 1 i x i 4 γ(i) K (αs) δ IJ. This leads to a linear equation for the dependence of Γ S on ρ ij X j i ln(ρ ij ) ΓS MN (ρ ij, α s) = 1 4 γ(i) K (αs) δ MN i, The equation relates Γ S to γ K to all orders in perturbation theory and should remain true at strong coupling as well. It correlates color and kinematics for any number of hard partons. It admits a unique solution for amplitudes with up to three hard partons. For n 4 hard partons, functions of ρ ijkl solve the homogeneous equation.

31 The dipole formula The cusp anomalous dimension exhibits Casimir scaling up to three loops. γ (i) K (αs) = C i bγ K (α s) with C i the quadratic Casimir and bγ K (α s) universal. Denoting with eγ (i) K X j i possible terms violating Casimir scaling, we write ln(ρ ij ) ΓS (ρ ij, α s) = 1» C i bγ K (α s) + eγ (i) K 4 (αs) By linearity, using the color generator notation, the scaling term yields X j i ln(ρ ij ) ΓS Q.C. (ρ ij, α s) = 1 4 T i T i bγ K (α s), An all-order solution is the dipole formula (E. Gardi, LM; T. Becher, M. Neubert) i i, Γ S dip (ρ ij, α s) = 1 8 bγ K (α s) X j i ln(ρ ij ) T i T j + 1 bδ 2 S (α X s) i T i T i, as easily checked using color conservation, P i T i = 0. Note: all known results for massless gauge theories are of this form.

32 The full amplitude It is possible to construct a dipole formula for the full amplitude enforcing the cancellation of the dependence on the factorization vectors n i through ln! (2p i n i ) 2 n 2 + ln i! (2p j n j ) 2 n 2 + ln j ( β i β j ) 2 n 2! i n2 j 2(β i n i ) 2 2(β j n j ) 2 = 2 ln ( 2p i p j ). Soft and collinear singularities can then be collected in a matrix Z ««pi M µ, pi αs(µ2 ), ɛ = Z, α s(µ 2 f µ ), ɛ pi H f µ, µ «f µ, αs(µ2 ), ɛ, satisfying a matrix evolution equation «««d pi Z, α s(µ 2 f d ln µ f µ ), ɛ pi = Γ, α s(µ 2 f f µ ) pi Z, α s(µ 2 f f µ ), ɛ. f The dipole structure of Γ S is inherited by Γ, which reads (T. Becher, M. Neubert) pi Γ dip λ, αs(λ2 ) = 1 4 bγ X K α s(λ 2 ) ln j i 2 pi p j λ 2 «T i T j + nx i=1 γ Ji α s(λ 2 ).

33 Beyond the dipole formula (with Lance Dixon and Einan Gardi)

34 Beyond the minimal solution The cusp anomalous dimension may violate Casimir scaling starting at four loops. This would add a contribution Γ S H.C. satisfying X ln(ρ j, j i ij ) ΓS H.C. (ρ ij, α s) = 1 4 eγ(i) K (αs), i. For n 4 the constraints do not uniquely determine Γ S : one may write Γ S (ρ ij, α s) = Γ S dip (ρ ij, α s) + S (ρ ij, α s), where S solves the homogeneous equation X j i ln(ρ ij ) S (ρ ij, α s) = 0 S = S `ρ ijkl, α s. By eikonal exponentiation S must directly correlate four partons. A nontrivial function of ρ ijkl cannot appear in Γ S at two loops. eh [f] = X X i f abc T a j Tb k Tc l ln `ρ `ρiklj `ρiljk ijkl ln ln. j,k,l a,b,c The minimal solution holds for matter loop diagrams at three loops (L. Dixon).

35 Collinear constraints Factorization of fixed-angle amplitudes breaks down in collinear limits, as p i p j 0. New singularities are captured by a universal splitting function M n (p 1, p 2, p j ; µ, ɛ) 1 2 Sp (p 1, p 2 ; µ, ɛ) M n 1 (P, p j ; µ, ɛ). Infrared poles of the splitting function are generated by a splitting anomalous dimension " Sp(p 1, p 2 ; µ, ɛ) = Sp (0) H (p 1, p 2 ; µ, ɛ) exp 1 Z # µ 2 dλ λ 2 Γ Sp(p 1, p 2 ; λ), related to the soft anomalous dimensions of the two amplitudes: Γ Sp (p 1, p 2 ; µ f ) Γ n `p1, p 2, p j ; µ f Γn 1 `P, pj ; µ f. If the dipole formula receives corrections, so does the splitting amplitude Γ Sp (p 1, p 2 ; λ) = Γ Sp, dip (p 1, p 2 ; λ) + n `ρijkl ; λ n 1 `ρijkl ; λ. Universality of Γ Sp constrains n n 1 : it must depend only on the collinear parton pair (T. Becher, M. Neubert).

36 Bose symmetry, transcendentality Contributions to n(ρ ijkl ) arise from gluon subdiagrams of eikonal correlators. They must be Bose symmetric. With four hard partons, 4 (ρ ijkl ) = X i h (i) abcd Ta i Tb j Tc k Td l (i) 4, kin (ρ ijkl), the symmetries of (i) 4, kin must match those of h(i) abcd. For polynomials in L ijkl log ρ ijkl one easily matches symmetries of available color tensors h 4 (ρ ijkl ) = T a 1 Tb 2 Tc 3 Td e 4 f ade fcb Lh i L h Lh ( 1)h 1+h 2 +h 3 L h Lh cycl., Transcendentality constrains the powers of the logarithms. At L loops h tot h 1 + h 2 + h 3 τ 2L 1 For N = 4 SYM, and for any massless gauge theory at three loops the bound is expected to be saturated. Collinear consistency requires h i 1 in any monomial.

37 Three loops n can first appear at three loops. A general n is a sum over quadrupoles. Relevant webs are the same in N = 4 SYM. The only available color tensors are f ade f e cb Polynomials in L ijkl are severely constrained. Using Jacobi identities for color and L L L 1342 = 0 for kinematics, only one structure polynomial in L ijkl survives. j k b c i l a d Three-loop web contributing to Γ S.

38 Survivors Just one maximal transcendentality, Bose symmetric, collinear safe polynomial in the logarithms survives. h (122) 4 (ρ ijkl ) = T a 1 Tb 2 Tc 3 Td e 4 f ade fcb L 1234 (L 1423 L 1342 ) 2 i e + f cae fdb L 1423 (L 1234 L 1342 ) 2 e + f bae fcd L 1342 (L 1423 L 1234 ) 2. Allowing for polylogarithms, structures mimicking the simple symmetries of L ijkl must be constructed. Two examples are» (122, Li 2) 4 (ρ ijkl ) = T a 1 Tb 2 Tc 3 Td e 4 f ade fcb L 1234 Li 2 (1 ρ 1342 ) Li 2 (1 1/ρ 1342 ) Li 2 (1 ρ 1423 ) Li 2 (1 1/ρ 1423 ) + cycl..» (311, Li 3) 4 (ρ ijkl ) = T a 1 Tb 2 Tc 3 Td e 4 f ade fcb Li 3 (1 ρ 1342 ) Li 3 (1 1/ρ 1342 ) L 1423 L cycl.. Higher-order polylogarithms are ruled out by their trancendentality combined with collinear constraints.

39 Perspective After O `10 2 years, soft and collinear singularities in massless gauge theories are still a fertile field of study. We are probing the all-order structure of the nonabelian exponent. All-order results constrain, test and help fixed order calculations. Understanding singularities has phenomenological applications through resummation. Factorization theorems Evolution equations Exponentiation. Dimensional continuation is the simplest and most elegant regulator. Transparent mapping UV IR for pure counterterm functions. Remarkable simplifications in N = 4 SYM point to exact results. Only three functions, γ K, G eik and B δ determine all singularities in the planar limit, and possibly beyond. Factorization and classical rescaling invariance severely constrain soft anomalous dimensions to all orders and for any number of legs. A simple dipole formula may encode all infrared singularites for any massless gauge theory. The study of possible corrections to the dipole formula is under way.