QCD Phenomenology at High Energy Bryan Webber CERN Academic Training Lectures 2008 Lecture 3: DIS and Evolution Equations ● Deep Inelastic Scattering ❖ Parton model ❖ Scaling violation and DGLAP equation ❖ Quark and gluon distributions ❖ Solution by moments ❖ Small x ● Parton Showers ❖ Sudakov form factor ❖ Infrared cutoff ● Soft Gluon Coherence ❖ Angular ordering ❖ Coherent branching
Deep Inelastic Scattering ● Consider lepton-proton scattering via exchange of virtual photon: ● Standard variables are: − q 2 Q 2 x = 2 p · q = 2 M ( E − E ′ ) k · p = 1 − E ′ q · p y = E where Q 2 = − q 2 > 0 , M 2 = p 2 and energies refer to target rest frame. ● Elastic scattering has ( p + q ) 2 = M 2 , i.e. x = 1 . Hence deep inelastic scattering (DIS) means Q 2 ≫ M 2 and x < 1 . 1
● Structure functions F i ( x, Q 2 ) parametrise target structure as ‘seen’ by virtual photon. Defined in terms of cross section d 2 σ 8 πα 2 ME »„ 1 + (1 − y ) 2 « = 2 xF 1 Q 4 dxdy 2 – +(1 − y )( F 2 − 2 xF 1 ) − ( M/ 2 E ) xyF 2 . ● Bjorken limit is Q 2 , p · q → ∞ with x fixed. In this limit structure functions obey approximate Bjorken scaling law, i.e. depend only on dimensionless variable x : F i ( x, Q 2 ) − → F i ( x ) . 2
● Figure shows F 2 structure function for proton target. Although Q 2 varies by two orders of magnitude, in first approximation data lie on universal curve. ● Bjorken scaling implies that virtual photon is scattered by pointlike constituents (partons) — otherwise structure functions would depend on ratio Q/Q 0 , with 1 /Q 0 a length scale characterizing size of constituents. 3
● Parton model of DIS is formulated in a frame where target proton is moving very fast — infinite momentum frame . ❖ Suppose that, in this frame, photon scatters from pointlike quark with fraction ξ of proton’s momentum. Since ( ξp + q ) 2 = m 2 q ≪ Q 2 , we must have ξ = Q 2 / 2 p · q = x . s, ˆ ❖ In terms of Mandelstam variables ˆ t, ˆ u , spin-averaged matrix element squared for massless eq → eq scattering (related by crossing to e + e − → q ¯ q ) is s 2 + ˆ u 2 q e 4 ˆ |M| 2 = 2 e 2 X ˆ t 2 where P denotes average (sum) over initial (final) colours and spins. ❖ In terms of DIS variables, ˆ t = − Q 2 , ˆ s = Q 2 /xy . Differential cross u = ˆ s ( y − 1) and ˆ section is then dxdQ 2 = 4 πα 2 d 2 ˆ σ Q 4 [1 + (1 − y ) 2 ]1 2 e 2 q δ ( x − ξ ) . ❖ From structure function definition (neglecting M ) dxdQ 2 = 4 πα 2 d 2 σ [1 + (1 − y ) 2 ] F 1 + (1 − y ) ff ( F 2 − 2 xF 1 ) . Q 4 x ❖ Hence structure functions for scattering from parton with momentum fraction ξ is F 2 = xe 2 ˆ q δ ( x − ξ ) = 2 x ˆ F 1 . 4
❖ Suppose probability that quark q carries momentum fraction between ξ and ξ + dξ is q ( ξ ) dξ . Then Z 1 X dξ q ( ξ ) xe 2 F 2 ( x ) = q δ ( x − ξ ) 0 q X e 2 = q xq ( x ) = 2 xF 1 ( x ) . q ❖ Relationship F 2 = 2 xF 1 (Callan-Gross relation) follows from spin- 1 2 property of quarks ( F 1 = 0 for spin-0). ● Proton consists of three valence quarks (uud), which carry its electric charge and baryon number, and infinite sea of light q ¯ q pairs. Probed at scale Q , sea contains all quark flavours with m q ≪ Q . Thus at Q ∼ 1 GeV expect ( x ) ≃ 4 u ( x )] + 1 F em 9 x [ d ( x ) + ¯ 9 x [ u ( x ) + ¯ d ( x ) + s ( x ) + ¯ s ( x )] 2 where u ( x ) = u V ( x ) + ¯ u ( x ) d V ( x ) + ¯ d ( x ) = d ( x ) s ( x ) = s ( x ) ¯ 5
with sum rules Z 1 Z 1 dx u V ( x ) = 2 , dx d V ( x ) = 1 . 0 0 R 1 ● Experimentally one finds P 0 dx x [ q ( x )+ ¯ q ( x )] ≃ 0 . 5 .. Thus quarks only carry about q 50% of proton’s momentum. Rest is carried by gluons . Although not directly measured in DIS, gluons participate in other hard scattering processes such as large- p T jet and prompt photon production. ● Figure shows typical set of parton distributions extracted from fits to DIS data, at Q 2 = 10 GeV 2 . 6
Scaling Violation and DGLAP Equation ● Bjorken scaling is not exact. This is due to enhancement of higher-order contributions from small-angle parton branching, discussed earlier. ● Incoming quark from target hadron, initially with low virtual mass-squared − t 0 and carrying a fraction x 0 of hadron’s momentum, moves to more virtual masses and lower momentum fractions by successive small-angle emissions, and is finally struck by photon of virtual mass-squared q 2 = − Q 2 . ● Cross section will depend on Q 2 and on momentum fraction distribution of partons seen by virtual photon at this scale, D ( x, Q 2 ) . ● To derive evolution equation for Q 2 -dependence of D ( x, Q 2 ) , first introduce pictorial representation of evolution, also useful for Monte Carlo simulation. 7
● Represent sequence of branchings by path in ( t, x ) -space. Each branching is a step downwards in x , at a value of t equal to (minus) the virtual mass-squared after the branching. ● At t = t 0 , paths have distribution of starting points D ( x 0 , t 0 ) characteristic of target hadron at that scale. Then distribution D ( x, t ) of partons at scale t is just the x -distribution of paths at that scale. ● Consider change in the parton distribution D ( x, t ) when t is increased to t + δt . This is number of paths arriving in element ( δt, δx ) minus number leaving that element, divided by δx . 8
● Number arriving is branching probability times parton density integrated over all higher momenta x ′ = x/z , Z 1 δt dx ′ dz α S P ( z ) D ( x ′ , t ) δ ( x − zx ′ ) ˆ δD in ( x, t ) = t 2 π x Z 1 δt dz α S ˆ = P ( z ) D ( x/z, t ) t z 2 π 0 ● For the number leaving element, must integrate over lower momenta x ′ = zx : Z x δt dx ′ dz α S P ( z ) δ ( x ′ − zx ) ˆ δD out ( x, t ) = t D ( x, t ) 2 π 0 Z 1 δt dz α S ˆ = t D ( x, t ) P ( z ) 2 π 0 ● Change in population of element is δD ( x, t ) = δD in − δD out Z 1 δt dz α S » 1 – ˆ = P ( z ) zD ( x/z, t ) − D ( x, t ) . t 2 π 0 9
● Introduce plus-prescription with definition Z 1 Z 1 dz f ( z ) g ( z ) + = dz [ f ( z ) − f (1)] g ( z ) . 0 0 Using this we can define regularized splitting function P ( z ) = ˆ P ( z ) + , and obtain Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equation: Z 1 t ∂ dz α S ∂tD ( x, t ) = 2 πP ( z ) D ( x/z, t ) . z x Beware! Note that Z 1 Z 1 dz f ( z ) g ( z ) + = dz Θ( z − x ) f ( z ) g ( z ) + x 0 Z 1 Z x = dz [ f ( z ) − f (1)] g ( z ) − f (1) dz g ( z ) x 0 ● Here D ( x, t ) represents parton momentum fraction distribution inside incoming hadron probed at scale t . In timelike branching, it represents instead hadron momentum fraction distribution produced by an outgoing parton. Boundary conditions and direction of evolution are different, but evolution equation remains the same. 10
Quark and Gluon Distributions ● For several different types of partons, must take into account different processes by which parton of type i can enter or leave the element ( δt, δx ) . This leads to coupled DGLAP evolution equations of form Z 1 t ∂ dz 2 πP ij ( z ) D j ( x/z, t ) ≡ α S α S X ∂tD i ( x, t ) = 2 πP ij ⊗ D j z x j ● Quark ( i = q ) can enter element via either q → qg or g → q ¯ q , but can only leave via q → qg . Thus plus-prescription applies only to q → qg part, giving ! 1 + z 2 ˆ P qq ( z ) = P qq ( z ) + = C F 1 − z + P qg ( z ) = T R [ z 2 + (1 − z ) 2 ] ˆ P qg ( z ) = ● Gluon can arrive either from g → gg (2 contributions) or from q → qg (or ¯ q → ¯ qg ). Thus number arriving is 11
Z 1 ( " # δt dz α S D g ( x/z, t ) + D g ( x/ (1 − z ) , t ) ˆ δD g, in = P gg ( z ) t 2 π z 1 − z 0 ˆ " «#) P qq ( z ) „ x « „ x + D q 1 − z, t + D ¯ 1 − z, t q 1 − z Z 1 ( «–) δt dz α S „ x « » „ x « „ x 2 ˆ + ˆ = P gg ( z ) D g z, t P qq (1 − z ) D q z, t + D ¯ z, t , q t z 2 π 0 ● Gluon can leave by splitting into either gg or q ¯ q , so that Z 1 δD g, out = δt dz α S h i P gg ( z ) + N f ˆ ˆ t D g ( x, t ) P qg ( z ) dz . 2 π 0 ● After some manipulation we find "„ z + 1 − z « 1 − z + 1 P gg ( z ) = 2 C A 2 z (1 − z ) z + # − 2 + 1 2 z (1 − z ) 3 N f T R δ (1 − z ) , 12
1 + (1 − z ) 2 q ( z ) = ˆ P gq ( z ) = P g ¯ P qq (1 − z ) = C F . z ● Using definition of the plus-prescription, can check that „ z « z 2 z (1 − z ) + 11 1 − z + 1 + 1 2 z (1 − z ) = 12 δ (1 − z ) (1 − z ) + + ! 1 + z 2 1 + z 2 + 3 = 2 δ (1 − z ) , 1 − z (1 − z ) + + so P qq and P gg can be written in more common forms " # 1 + z 2 + 3 P qq ( z ) = C F 2 δ (1 − z ) (1 − z ) + z + 1 − z » – P gg ( z ) = 2 C A + z (1 − z ) (1 − z ) + z + 1 6(11 C A − 4 N f T R ) δ (1 − z ) . 13
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