A coordinate description of partonic processs

Transcription

1 A coordinate description of partonic processs Radcor-Loopfest UCLA, Jun 6, 205 G. Sterman, YTP, Stony Brook with Ozan Erdogan A look at perturbative long distance behavior of amplitudes and cross sections in coordinate space. Where to large corrections come from? Coordinate space leading regions for fixed-angle scattering amplitudes Approximations and factorization Wilson lines: webs and the running coupling in coordinate space Extension to cross sections Cancellation: the largest time equation

2 Coordinate space leading regions.. [Analogs in momentum space for amplitudes and cross sections are well-known and continue to be studied (Collins, GS 98; Sen, 983;... Feige & Schwartz ; Caron-Huot )] Here, look at coordinate space VEVs for all massless fields in configurations reflecting scattering. The scalar propagator, for example, with D = 2 2ε: (y x) = 4π 2 ε ( (y x) 2 + iɛ) ε Schematically, we have integrals over positions of internal vertices y k : G N (x,..., x N ) = 0 T (φ N (x N ) φ (x )) 0 d D y k numerator many terms some vertices k lines j [ ( k η jk y k + k η jl x l ) 2 + iɛ] p j Powers p j = ε (boson) or 2 ε (fermion, or derivative of boson). Once UV renormalized, G N is singular only at pinches in the complex integrals over positions of vertices, y k between incoming and outgoing propagators (on the light cone). (S. Date 983; A.O. Erdogan , PRD). 2

3 Example: d D w ( 2(y w) + (y w) (y w ) 2 + iɛ) ε ( 2(w x) + (w x) (w x ) 2 + iɛ) ε Corresponding to a pinch at w, w = 0 when x and y are lined up in the + direction: 3

4 The result: The general leading (-power) region ; as in momentum space, a physical picture : 4 J 2 S H S. H. J J... J K (a) (b) ing pinch surfaces represented by soft, jet and hard subdiagrams for (a) cusp and (b) a typical multieikonal or amplitude. Gauge lines represent arbitrary number of connections between the subdiagrams. n (b) the double s either Wilson lines or partonic propagators connected to the external vertices. General pinch surface (ρ) in coordinate space. Jets are in directions β from the position of a hard scattering. Each β 2 0 = β 2, β β =. Vertices group along the β, near the origin, or are at finite distances from these. over the positions of internal vertices considered as variables in complex coordinate space. This is the of pinches in loop momenta [7, 3, 32]. As in momentum space, at each such leading region, the diagram hysical processes with fully-consistent classical propagation for the set of lines that connect vertices tlike separated. We will refer to a manifold in coordinate space with a definite set of vertices pinched r vanishing separations as a pinch surface (PS). (We use this notation in the same sense as PSS in 4

5 A picture of where the vertices are on pinch surface ρ : x S (ρ) x K J (ρ) J (ρ) K H (ρ) FG. 3: Representation of the arrangement of vertices near a leading pinch surface ρ directly in coordinate space and their assignments to jet, J (ρ), hard, H (ρ), and soft, S (ρ) subdiagrams. For every region, the direction of the jet J (ρ) is determined by the relative position of the external point x with respect to the position of the hard scattering. Vertices in H (ρ) are near the origin can be seen by considering vertices in each of the subdiagrams associated with an arbitrary PS, ρ. For vertices x µ Vertices in J (ρ) are near rays β µ x µ for x 2 0 in the soft subdiagram, the only approximations are for denominators attached to the jets, for which jet vertices are Vertices in S (ρ) are separated from the origin and the rays. set on the lightcones, β. n neighborhood n[ρ], the x µ stay away from all of the lightcones, and the physical picture Thiscorrespondence organizes large eliminates numbers PSs involving of diagrams vertices in S related (ρ), just asby in the connecting original integral. vertices For vertices in all z µ (K) possible in jet K, the ways. integrals are unchanged, except for lines attached directly to the hard scattering, where terms that are nonleading in the scaling variable are neglected. No approximations are made for lines internal to H (ρ). Pinches of the homogeneous integral are still controlled by the distances of the external vertices x K of J (ρ) K to the relevant lightcone, and these pinches develop in the same manner in the homogeneous as in the original integral. n the homogeneous integral, defined as in Eq. (8), however, one or more of the the rescaled 5 normal variables are always order unity. Thus, the

6 n each such region (ρ), introduce an approximation operator acting on each diagram γ: (GS, Erdogan ) x S (ρ) x K J (ρ) J (ρ) K H (ρ) FG. 3: Representation of the arrangement of vertices near a leading pinch surface ρ directly in coordinate space and their assignments to jet, J (ρ), hard, H (ρ), and soft, S (ρ) subdiagrams. For every region, the direction of the jet J (ρ) is determined by the relative position of the external point x with respect to the position of the hard scattering. t ρ γ (n) = γ (n) div n[ρ] dτ () S (ρ) {µ } ({τ () }) β µ can be seen by considering vertices in each of the subdiagrams associated with an arbitrary PS, ρ. For vertices x µ in the soft subdiagram, the only approximations are for denominators attached to the jets, for which jet vertices are set on the lightcones, β. n neighborhood n[ρ], the x µ stay away from all of the lightcones, and the physical picture correspondence eliminates PSs involving vertices in S (ρ), just as in the original integral. For vertices z µ (K) in jet K, the integrals are unchanged, except for lines attached directly to the hard scattering, where terms that are nonleading in the scaling variable are neglected. No approximations are made for lines internal to H (ρ). Pinches of the homogeneous integral are still controlled by the distances of the external vertices x K of J (ρ) K to the relevant lightcone, and these pinches develop in the same manner in the homogeneous as in the original integral. n the homogeneous integral, defined as in Eq. (8), however, one or more of the the rescaled normal variables are always order unity. Thus, the pinch surfaces of the homogeneous integral will involve fewer vanishing denominators than those of the original PSs. We will use this observation in our construction of nested subtractions. We will now employ the approximations identified above to define a new set of approximation operators, denoted t ρ, one for each leading pinch surface ρ. Each operator t ρ is defined to act on any diagram γ (n) that possesses the corresponding PS and to give an expression that corresponds to the leading, singular behavior of γ (n) in the neighborhood of PS ρ. Of course, this condition defines the operator t ρ only up to a finite ambiguity. For our A µ = A µ c (x) + Aµ s ( β purposes x). (E.g., as described in Becher, Broggio, Ferroglia 204. Stewart talk here.) it will be most useful to construct subtractions similar to those employed in proofs of factorization in Ref. [7]. We define the action of the approximation operator t ρ γ (n) as the imposition of the soft-collinear and hard-collinear approximations given above in Eqs. (9), () and (2) on all lines to which they apply at PS ρ of diagram γ (n). This action can be represented schematically by β,µ dη () d D z () J (ρ)µ ν (z (), η () ) β,ν β ν d D y () H (ρ) {ν } (y() ) For gluons attaching soft function to jet in direction β, keep only the β polarization C. Approximation operators and region-by-region finiteness and the coordinate τ along the β direction: This can be a starting point for deriving Soft-Collinear EFT: in J : 6

7 The operators t ρ organize all divergences as external points approach the light cone relative to the hard scattering: R (n) P γ (n) div ˆn[ρ] = γ (n) + N N P [γ (n) ] ρ N ( t ρ ) γ (n) div ˆn[ρ] = 0 Each t ρ acts to approximate the integrand by its leading behavior near the singular surface. Following Collins and Soper (983) in axial gauge and Collins (203) in covariant gauge, the sum is over all possible nested regions, which cancels overlapping divergences. For this process formalizes a strategy of regions. (Beneke Smirnov 998, Jantzen 20.) Within each region ρ, only t ρ approximation contributes, but each approximation extends over all coordinate space. This generalizes arguments given for Sudakov-related processes. The arguments are applicable to momentum space, and the relation to NNLO arguments (c.f. talk by M. Czakon) is clear. Each subtraction corresponds to a leading region. Any application illustrates that the general elimination of double counting can be complicated even at low orders. Complicated or not, double counting can be avoided to all orders, even with jets in the final state. 7

8 As x 2 0 relative to the hard scattering (H): xµ β µ, β 2 of a factorized amplitude in coordinate space = 0, this allows the derivation Already at one loop, nesting for fixed-angle scattering becomes nontrivial because in QCD, hard scatterings can be disjoint 20 x x 3 y y 3 y 2 y 4 x 2 x 4 FG. 5: An example of disjoint hard subdiagrams. Either gluon can carry the hard scattering, with the other soft or (2 choices of) collinear (or part of the hard scattering). the vertices of H ( ) to which they attach can be in H ( ). Then, each external vertex of the hard subdiagram H ( ) either appears as an internal vertex in some jet subdiagram J ( ) L of, or is an internal vertex of the soft function, S ( ) of. We will see that when any such vertex is in a jet subdiagram of, then is not a PS for t show that when all the external vertices of H ( ) are in S ( ),then is not a leading PS of t.wewillthen. The only possibility left is that at least one vertex of H ( ) is also a vertex in H ( ), so that the hard subdiagrams are not disjoint. 8

9 The result for VEVs with fixed-angle geometry: G ({x }) = a with jet functions = dη j part (x, η β ) S ren ({β β J }) H ( {η β } ) j part[f φ] (x, η β ) = c cusp (β, β ) 0 T ( φ(x ) φ (η β )Φ [f φ] β (, η β ) ) 0, and a soft function constructed entirely from Wilson lines in the β directions. 9

10 Coordinate picture of soft radiation (cusps and polygons in QCD) (Erdogan ( , PRD), Mitov, GS, Sung, PRD,) Combinatoric exponentiation in coordinate space (t s more general, but we ll consider just the cusp.) (Gardi, Magnea see talks here) Will find an interesting geometrical interpretation directly in QCD, which becomes exact for large-n. The 2-line eikonal form factor is the exponential of a sum of two-eikonal irreducible diagrams, the webs with modified color factors: (Gatheral, Frenkel Taylor, GS, 98-83) A = exp i= w (i) Webs in the exponent, w (i). are 2-eikonal irreducible diagrams. At 2 loops: All have color factor (fundamental representation) C F C A (only no C 2 F ). 0

11 Here s how it works. Say we know the exponent w (i) to order N. Expand to N + st order, as a sum of diagrams and in terms of the exponential A (N+) = exp N+ i= w (i) (N+) A (N+) = D (N+) D (N+). This gives a formula for the highest order in the exponent: w (N+) = D (N+) D (N+) N+ m=2 m! N i m =... N i = w (i m) w (i m )... w (i ) (N+) The simplest (and most general) proofs are in coordinate spacee.

12 To the relation w (N+) = D (N+) [ N+ D (N+) m=2 m! N i m =... N i = w (i m) w (i m )... w (i ) ] (N+) Now apply the abelian graphical identity, which holds for any number, length or shape of Wilson lines, simply an expression of the path ordering of exp[ dλβ A(βλ)]: W W 2 W 2 W Figure 2: llustration of the coordinate identity, Eq. (3), where each line attaching W and W 2 to a Wilson line stands for an arbitrary number of gluons (e (a) i in the text.) The sum on the right represents the sum over all mutual orderings of the external gluons of W and W 2, preserving the orderings internal to each W along the Wilson line. The color-dependent product of the web internal factors, W and W 2 are the same on both sides of the figure. The ( shuffle ) identity allows us to interpret the products of order w n s, n =... N in terms of N + st order diagrams, so that the effect of all the lower orders is to modify color factors, since these are unaffected by the identity. where in the first equality we define τ (a) 0 0. The second form represents the integrals as a functional E, acting on the internal web function W (i) E corresponding to w(i) E. W(i) E is a function of all the τ (a) j s, and includes all color and velocity dependence associated with the gluons, including the vectors ξ µ (τ (a) j ) that are contracted with the gluon propagators. Notice that E depends only on the assignment of gluon connections, E, and is otherwise independent of the internal function W (i) E, including its order, i. We now use this property of the s to derive an identity that will serve as a lemma for our main result. 2

13 mportant: The webs automatically subtract all divergences where there is more than one subjet in the web. The entire web is either hard, soft, or collinear to one line or the other. (Erdogan, GS, and , PRD) The web acts like a single gluon (consistent with the dipole exponentiation). As ordered exponentials, the webs can be constructed in coordinate space. Care must be taken to preserve gauge invariance, or double-logarithmic exponentiation fails in general. With this in mind, the result, with an R cutoff L, is: E(L, ε) = L 0 dλ λ dσ σ w (α s (/λσ)) Here σ and λ are distances along the eikonal lines. The invariant size of the web fixes the running coupling. w = 4 A(α s) + O(ε) 3

14 Again, E(L, ε) = L 0 dλ λ dσ σ w (α s (/λσ)) A surface interpretation is tempting, viz. minimizing a 5-dimensional surface (Alday, Maldacena 2007). n QCD the coupling runs as the integral passes over the surface. The result holds to all orders in perturbation theory, and gives the structure of power corrections by Borel or related analysis. t is not limited to dressed gluon or related subsets of diagrams. 4

15 Another interesting correspondence: polygonal Wilson lines (Korchemskaya, Korchemsky (993), Drummond et al (2008)... ). Webs appear in corners, and when defined in a gaugeinvariant fashion give the leading singularities: Neglecting the running of α s, another analogy to minimal surfaces in 5-D: 4 a= W a (β a, β a ) = dy dy 2 4w conformal ( y 2 )( y2 2 ) (Again, as in Alday, Maldacena 2007) 5

16 A coordinate picture for cross section: Schematically: k d D ka ({k}) k δ + (k 2 )A({k}) Start with the cut propagator with momentum flowing out of w and into y: d D k (2π) D e ik (y w) (2π) δ + ( k 2 ) = 4π 2 ε ( (y w) 2 + iɛ(y w) 0 ) ε For w 0 > y 0 this is like a propagator in A, for w 0 > x 0, like a propagator in A. This feature leads to pinch surfaces for cut diagrams, and hence in cross sections. Then when x and y go to the + direction, w is also pinched in the plus direction. For a pinch, we can assume that w 0 > x 0. Then f y 0 > w 0, the cut propagator has a +iε, and produces a pinch just like in the amplitude for y w + > 0. f x 0 > y 0, the cut propagator has a iε, and it still produces a pinch because now y w + < 0. 6

17 All this means that cross sections have the same pinch surfaces as amplitudes, but with energy flow reversed in the complex conjugate. Just like time-ordered perturbation theory, but with the vertices everywhere in coordinate space. Because vertices of both A and A are ordered, the vertex with the largest time is always adjacent to the final state, and connected across the cut. As for example, in the simplest case of e + e annihilation: How cancellation takes place: the hermiticity of the interaction. 7

18 Cancellation of R divergences: Let w 0 be the largest time. (Veltman R. Akhoury 992, Laenen, Larsen, Rietkerk ) w 0 > x 0 and w 0 > y 0 w may be in A or in A, and we must sum over both cases: N and N with vertex at w in A and A: d D w { j (yj w w)2 + iɛ(yj w [iv ( w)] w)0 i (w x w i )2 + iɛ } + j (y w j w)2 iɛ [ iv ( w)] i (w x w i )2 + iɛ(w x w i )0 (Typically, m, n =, 2.) Vanishes because w 0 > x 0, y 0, i.e., is the largest time. 8

19 divergences at intermediate steps and the necessity to and the t Hooft coupling a sum over all final states. F (z; divergences at intermediate steps and the necessity to and the t Hooft coupling a = g 2 a) EEC () admits an equivalent representation analogous to (3) in terms of the Wightman (non-time-ordered) 2 ( z EEC N =4 = YM sum over all final states. 4z F (z; a) EEC () admits an equivalent representation analogous to (3) and in terms cut cross of thesections Wightman - all (non-time-ordered) 2 ( z), EEC N =4 = correlation function Here 0 <z<andthep 4z Directions we should go... weighted these boil down to nience. EEC is expected to top correlation function Here 0 <z<andtheprefacto E( n )E( n 2 ) q = d 4 x e iq x 0 O (x)e( n )E( n 2 )O(0) 0 nience. function EEC isof expected z, normalized to be a re E( n )E( n 2 ) q = d 4 x e iq x 0 O (x)e( n )E( n 2 )O(0) 0 function of z, normalizedas a n close analogy with 0 dz th (4) ncoupling close analogy expansion with the of F QCD (z, (4) coupling lowest expansion order term of F (z,a) comesstar fro involving the so-called energy flow operators [4 6] where lowest atorder tr[zterm 2 (x)] comes into from three-pa the involving the so-called energy flow operators [4 6] ator tr[z 2 agluonandascalarplusa (x)] into three-particle s E( n) = dτ lim agluonandascalarplusapairof E( n) = dτ r lim r2 n i T 0i (t = τ + r, r n). (5) r r2 n i T 0i (t = τ + r, r n). (5) EEC N =4 EEC N =4 = a = a z z 4z 3 4z 3 z ln Here the stress-energy tensor T µν (t, x) isplacedinfinitely Which can be defined directly Here in four the stress-energy dimensionaltensor space, T µν potentially (t, x) isplacedinfinitely avoiding an R-regulated Comparing QCD. this relation wit far from the collision region and is integrated over thecomparing this relation with (2), (... Belitsky, Hohenneger, Korchemsky, far from thesokachev, collision region Zhiboedov, and is integrated over [exact the in N=4 the redefinition the SYM]) redefinition of the of coupling the cou detector a detector working working time. time. The Theoperator (5) (5) describes describes a a QCD QCD and in and N in =4SYMhaveide N =4SYMh calorimeter and has a simplephysical physical interpretation: interpretation: it it z z. Also,. Also, both EECs both EECs exhibitex th measures the energy flux perunit unitsolid solid angle angle in ain given a given ior for z 0, but the coefficient dr directions n (with n 2 ior for z 0, but the coeffi = ). The product E( n E( n )E( n )E( n 2 ) 2 ) is different. is different. For zfor, z or equiv, o measures the correlation between energy flowing the in themeasures measures back-to-back back-to-back correlation direction of n and n 2.Then,EECisgivenbythecorre-behaviolation function (4) averagedover overthe the orientations n and n andparticles. Their contribution can behavior is governed is governed by theby emissio the n 2 kept fixed particles. Their contributio n 2,withtherelativeangleχ kept fixed semiclassical semiclassical approximation approximation and i EEC dω dω 2 δ( n n 2 cos χ) E( n )E( n 2 ) sal, independent of the choice of th q Q 2. (6) theories. On the other hand, th EEC = dω dω 2 δ( n n 2 cos χ) E( n )E( n 2 ) sal, independent of the choi q σ tot Q 2. (6) σ tot describes theories. the correlation On the other between ha For a scalar source O(x) andq µ =(Q, 0), the correlation aligned describes momenta. the t correlation is driven by b For function a scalar (4) source only depends O(x) andq on ( n µ n =(Q, 0), 2 )sothattheaverage the correlationing of aligned particles momenta. in the final t state is dr function (6) becomes (4) only trivial. depends on ( n particle content of the theory. n 2 )sothattheaverage ing of particles in final (6) Relations becomes trivial. (3), (4) and (6) rely on unitarity and the completeness of the asymptotic states, 3. Method particle content and result. of the Trying theo Relations (3), (4) and (6) rely on unitarity X X X and =. theto-leading O(a 2 )correctionto(8) completeness They hold in of a generic the asymptotic field theory, states, be it QCD 3. Method and result. or N =4 technique based on scattering amp X X X =. to-leading O(a 2 )correction They SYM. hold To make in a generic use of (3) field and 9theory, (4), we have be itto QCD specify or the N =4 the same technique complications based on as in scatter QCD.

20 Some final thoughts Could we introduce emergent degrees of freedom more naturally in coordinate space? The history of QCD jets and hadronization is there for the reading if only we can learn the language. 20