Particles and Deep Inelastic Scattering

Transcription

1 Particles and Deep Inelastic Scattering University HUGS - JLab - June 2010 June 2010 HUGS 1

2 Sum rules You can integrate the structure functions and recover quantities like the net number of quarks. Momentum Sum Rule x tot = F 2 (x)dx x(q(x) + q(x))dx which yields the total fraction of the proton momentum carried by quarks. (After adjustment for the charges) We find that x tot for quarks is about 50%. Half of whatever is in the proton is not quarks - it turns out to be gluons but you can t detect a gluon directly with an electron beam. We can also measure the momentum fraction of anti quarks from xf 3 (x)dx and x tot and get 10-15%. June 2010 HUGS 2

3 Sum rules that don t say much The numerical sum rule N quark = dx x F 2(x) i (q i (x) + q i (x))dx can reach thousands as Q 2 decreases. as the parton distributions actually diverge. June 2010 HUGS 3

4 Adler Sum rule N u N d 1 0 dx 1 νp (F2 F νp 2 x ) 2 0 dx [u(x) + d(x) + s(x) d(x) u(x) c(x)] 2 x This has small corrections if one needs to worry about charm production thresholds. It is hard to measure as neutrino-proton scattering data is very scarce. June 2010 HUGS 4

5 Gross-Llewellyn-Smith sum rule N q N q (F νp νp 3 +F3 ) dx[u(x)+d(x)+s(x) u(x) d(x) s(x)] u v (x)+d v (x) 3 0 June 2010 HUGS 5

6 Gottfried sum rule N u N d 1 0 dx ep (F2 F2 en x 1 0 dx x [4 9 (u(x) + u(x) + 1 (d(x) + d(x)) (u(x) + u(x)) 4 (d(x) + d(x))] dx [[u(x) d(x) + u(x) d(x)] x [u(x) d(x)] 3 June 2010 HUGS 6

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8 Zeus detector 30 GeV e GeV p June 2010 HUGS 8

9 E665 detector 470 GeV µ + p, d, C, Ca, Pb at rest June 2010 HUGS 9

10 F 2 (x,q 2 ) * 2 i x H1 ZEUS BCDMS E665 NMC SLAC Q 2 (GeV 2 ) June 2010 HUGS 10

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12 As you can see from these data, the structure functions F 2 and xf 3 do seem to have some Q 2 dependence. This can be explained by higher order QCD corrections to the basic scattering diagrams. The Leading Order QCD corrections involve the emission of gluons from quarks and pair production of quarks from gluons. e e γ zx x g x(1-z) June 2010 HUGS 12

13 e e γ zx x g x(1-z) A quark which originally has momentum fraction x has a probability, P qqg α s (Q 2 )P qq (z), emit a gluon with momentum fraction x(1 z), leaving a quark carrying a lower momentum fraction xz. Our lepton then scatters off of that quark. Since we find the x value from the photon quark vertex, we measure xz instead of x as the momentum fraction. June 2010 HUGS 13

14 e e γ u x u xz g x(1-z) A gluon which originally has momentum fraction x has a probability, P Gqq α s (Q 2 )P Gq (z), emit an antiquark with momentum fraction x(1 z), leaving a quark carrying a lower momentum fraction xz. Our lepton then scatters off of that quark. Since we find the x value from the photon quark vertex, we measure xz instead of x as the momentum fraction (and see a gluon as a quark.) June 2010 HUGS 14

15 g zx x g (1-z) x A gluon can also split into 2 gluons with momentum fraction xz and x(1 x) with probability α s (Q 2 )P + GG(z) June 2010 HUGS 15

16 You can calculate these splitting probabilities using QCD: P qq (z) = z 2 1 z P qg (z) = 1 2 [z2 + (1 z) 2 ] P Gq (z) = (1 z) 2 3 z z P GG (z) = 6[ (1 z) + 1 z z + z(1 z)] for z < 1. The PDF s we measure are thus produced by PDF s at higher x radiating down to the value we measure. June 2010 HUGS 16

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18 They lead to the following evolution equations a for the parton distribution functions: dq i (x, Q 2 ) d ln Q 2 = α s(q 2 ) 2π dg(x, Q 2 ) d ln Q 2 = α s(q 2 ) 2π y x y x dy y dy y q j (y, Q 2 )P q i,q j (x y ) + G(y, Q2 )P q i,g( x y ) j q j (y, Q 2 )P G,q j ( x y ) + G(y, Q2 )P G,G ( x y ) j a DGLAP for Yu. L. Dokshitzer, V.N. Gribov, L.N. Lipatov, G. Altarelli and G. Parisi June 2010 HUGS 18

19 Show evolution slides June 2010 HUGS 19

20 Longitudinal Structure Function F L The general structure function formulation in terms of 2xF 1, F 2 and xf 3 allows for a longitudinal structure function F L = F 2 2xF 1. F L can be extracted by measuring the y dependence of the cross section. June 2010 HUGS 20

21 2 2 June 2010 HUGS 21

22 ε = 2(1-y)/(1 + (1-y) 2 ) F L ~ the slope June 2010 HUGS 22

23 R = F L / 2xF 1 June 2010 HUGS 23

24 The state of the art is now NNLO or NNNLO for the parton distribution functions so the equations become more complicated. One can no longer interpret the structure functions as sums over quarks once one gets above LO, so the mathematics gets complicated very quickly. But you can still define parton distribution functions and use them. June 2010 HUGS 24

25 QCD measurements with deep inelastic scattering The Q 2 behavior depends on the strong coupling constant and you can measure α S (Q 2 ) by measuring the Q 2 dependence. This requires good calibration and understanding of your detector!. June 2010 HUGS 25

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27 α S (Q 2 ) from HERA s QCD s (M Z ZEUS ZEUS ZEUS ) = ! (S Bethke, hep-ex/ ) 10 HERA H1 H1 (inclusive-jet NC DIS) (inclusive-jet p) (norm. dijet NC DIS) (norm. inclusive-jet NC DIS) (event shapes NC DIS) 100 μ = Q or E T jet (GeV) HERA s working group June 2010 HUGS 27