Gluon Spin Basics. The gluon helicity distribution in the nucleon will soon be measured... What are the foundations of this physics?

Transcription

1 Gluon Spin Basics hvordan man bruger astrologi med tibetanske medicin Winter 2004 R.L. Jaffe The gluon helicity distribution in the nucleon will soon be measured... What are the foundations of this physics? Operators? Models? Parton distributions? Evolution? RL Jaffe COMPASS Paris Meeting, March

2 Outline Gluons as partons: Distributions, total gluon spin and its operator description Helicity, twist and transversity. Operator description of the gluon helicity distribution. What is the total gluon spin contribution to the proton spin? What operator? Questions of gauge invariance, locality, and computation. Anomalies and misconceptions. Lattice computation? RL Jaffe COMPASS Paris Meeting, March

3 Outline Adding up the angular momentum In the partons In the laboratory What can we learn from models? [Not a lot!] What s the general idea? Quark Models RL Jaffe COMPASS Paris Meeting, March

4 Outline Evolution and extraction from g 1 α(q 2 ) g is constant! Characteristic evolution. Extraction from g 1 State of the data. Omitted Measurement of G in hadron-hadron collisions. Extraction of G from jet-jet or c c production in ep e jet-jet or ep e c c RL Jaffe COMPASS Paris Meeting, March

5 The Basic Basics: Degrees of Freedom Gluons are spin-1 gauge particles. AS PARTONS They have two independent degrees of freedom with helicity ±1 Confined gluons are off-shell and can have helicity 0. However these are actually composite degrees of freedom. Single parton distributions which involve only λ = ±1 helicity gluons are leading twist and scale (mod logarithms) at large Q 2. But single parton distributions involving helicity 0 gluons pay a price of one unit of twist which means one power of Q for each helicity zero gluon. λ = 1 λ = 1 RL Jaffe COMPASS Paris Meeting, March

6 The Basic Basics: Parton Distributions Gluon helicity distribution is leading twist it scales at large Q 2 ( ) ( ) 2 But gluon transversity distribution is subleading twist suppressed by 1/Q: ( ) 2 + ( ) 2 So unlike quarks, there is no transverse spin structure function for gluons. Important consequences for polarized collider physics. RL Jaffe COMPASS Paris Meeting, March

7 Helicity Distribution Function Operators I HOW IT WORKS FOR QUARKS Integral along the light-cone: q a (x)s + = 1 dξe iξx PS q(ξn)γ + γ 5 λ a I(ξn, 0)q(0) PS 4π x 0 ηµ n µ = 1 2 (1, 0, 0, 1) I(ξn, 0) is a Wilson link between 0 and ξη µ ( ξ ) Ι(ξη,ο) I(ξη, 0) = P exp ig du η 0 A(uηµ ) Moments dxx n e iξx dn dxnδ(x) So moments local operators dx q(x) a S + = 1 2 PS q(0)γ+ γ 5 λ a q(0) PS 2 qγµ 1 γ 5 q is well known to be the generator of internal (spin) rotations of quarks! x 3 RL Jaffe COMPASS Paris Meeting, March

8 Key Properties 1 2 qγµ γ 5 q Dimension 3 Spin 1 axial vector operator twist 2. Rules of the moments: dxx n 1 g(x) picks out an operator with spin n Spin of operator number of indices that can be set to + Parton distribution functions are gauge invariant Seek: Gluon operator Axial vector Dimension three Gauge invariant Local?? RL Jaffe COMPASS Paris Meeting, March

9 Helicity Distribution Function Operators HOW IT WORKS FOR GLUONS Collins & Soper, Manohar Basic gauge-covariant field strength F µν F +α F + α ɛ +αβγ F αβ A γ There is no gauge invariant, local, dimension-3, axial vector operator However, there is a perfectly good gluon helicity distribution function, g(x) = i 4π xp + dξe iξx PS F +α (ξn)i(ξn, 0) F α + (0) PS +(x x), All of the moments dxx n 1 g(x) for n =2, 3,... give local operators expected from OPE. So take n = 1 moment! And note that dx x e iξx = iπɛ(ξ) = iπ { 1 for x>0 1 for x<0 G = 1 2P + dξε(ξ) PS F +α (ξn)i(ξn, 0) F α + (0) PS RL Jaffe COMPASS Paris Meeting, March

10 The Operator for Gluon Spin G = 1 2P + dξε(ξ) PS F +α (ξn)i(ξn, 0) F α + (0) PS CONSEQUENCES Twist-two, gauge invariant, NON-LOCAL, axial vector operator Collapses to a local operator in A + =0gauge: F +α = + A α α A + g[a +,A α ] A+ =0 + A α ξ Aα and I(ξn, 0) A+ =0 1 Then integrate by parts d dξ sign(ξ) =2δ(ξ) Surface terms at ξ ± vanish G = 1 2M PS k=1,2 F + k (0)Ak (0) A + =0 PS RL Jaffe COMPASS Paris Meeting, March

11 History, Misconceptions and the Axial Anomaly G A+ =0 1 2M PS k=1,2 F + k (0)Ak (0) PS COMMENTS: Twist-two, gauge invariant, NON-LOCAL, axial vector operator This operator is coincides with the topological current operator, K µ, in this gauge, K µ = 1 2π ɛ µναβtra [F ν αβ 23 ] Aα A β. This led to a long sequence of misconceptions. Before it was realized that G involves a non-local operator, workers looked for a local, dimension three, axial vector operator. Altarelli and Ross and Teryaev and Efremov guessed that K µ was the correct operator to identify with G But it s not! Because K µ isn t gauge invariant, and anyway, we know what the correct operator is, so we don t have to guess! RL Jaffe COMPASS Paris Meeting, March

12 History, Misconceptions and the Axial Anomaly II K µ has a long and distinguished history, incuding µ K µ F F, instantons, the U(1) problem, etc. However they have nothing to do with the gluon spin in the nucleon because K µ is not the interpolating field for the gluon contribution to the nucleon spin. Lattice? G non-local operator, Or, in A + =0gauge..., And, anyway, G is a flavor singlet operator. RL Jaffe COMPASS Paris Meeting, March

13 All the Angular Momentum in the Partons The angular momentum sum rule RLJ & Manohar 1 2 = 1 2 q + G + L Q + L G Bashinsky & Jaffe Hägler & Schafer In parton form (x distributions) Harindranath & Kundu Operator description of q and G as already described L Q and L G can be written as integrals over orbital angular momentum distribution functions: L G = dxl G (x, Q 2 ) L Q = dxl Q (x, Q 2 ) And L G (x, Q 2 ) and L Q (x, Q 2 ) can be written as light-cone Fourier transforms of gauge invariant bilocal operators. However they involve the transverse coordinate x and cannot be measured in deep inelastic scattering. In fact, no one knows how to measure them! RL Jaffe COMPASS Paris Meeting, March

14 All the Angular Momentum in the Lab Another angular momentum sum rule 1 2 = 1 2 q + L Q + J G Here is a gauge invariant, local operator for L Q : L Q = d 3 xiψ ( x D) ψ X. Ji And here is a gauge invariant, local operator for J G J G = d 3 xitr x ( E B) Where D is the color gauge covariant derivative, and both expressions should look familiar! Presence of x signals need for off-forward matrix elements compare µ = 1 2 x j L Q is measurable in DVCS but J G is a stretch! RL Jaffe COMPASS Paris Meeting, March

15 What Can We Learn from Models Quark models give boundary data for evolution: Q 2 = Q GeV2. Hadrons contain ambient glue correlated with quark spin! Gluon spin effects in the nucleon are large: M M N 1 3 M N H = i j λ i λ j σ i σ j λ i σ i But model estimates depend on how the quark self-energy is treated. Eg. consider free electron. λ i σ i Exchange λ σ j j λ i σ i Self field RL Jaffe COMPASS Paris Meeting, March

16 What Can We Learn from Models II Exchange field alone gives NEGATIVE G! Gluon spin anticorrelated with nucleon spin. RLJ Bag Non-relativistic QM G 0.4 G 0.7 Self Field contribution is positive and of the same order. But should it be included? Barone, Calarco, Drago Isgur-Karl QM G 0.24 Other model estimates: Chiral Bag Model G q >0 Lee, Min, Park, Rho QCD Sum Rules G 1.0 ± 0.5 RL Jaffe COMPASS Paris Meeting, March

17 Evolution and Extraction To leading order G(Q 2 )α s (Q 2 ) = constant So G grows with Q 2 G(x, Q 2 ) and q NS (x, Q 2 ) mix under evolution: Which allows extraction of G(x, Q 2 ) from study of evolution of g 1 (x, Q 2 ). Which has been a hot topic! Original work: Altarelli, Ball, Forte & Ridolfi 1998 More recent: Blümlein & Böttcher 2002 Caveat: Note how little Q 2 dependence in the data! RL Jaffe COMPASS Paris Meeting, March

18 July 2000 F 2 2 i x = , i = 21 x = , i = 20 x = , i = 19 x = , i = 18 x = , i = 17 x = , i = 16 x = , i = 15 x = , i = 14 x = , i = 13 x = , i = 12 H1 e + p high Q H1 e + p low Q BCDMS NMC x=0.008 ( x 2048) x=0.015 ( x 1024) x=0.025 ( x 512) x=0.035 ( x 256) x=0.05 ( x 128) x = , i = 11 x = , i = 10 x = 0.013, i = 9 x = 0.020, i = 8 x = 0.032, i = 7 10 x=0.08 ( x 64) x=0.125 ( x 32) x=0.175 ( x 16) 10 x = 0.050, i = 6 x = 0.080, i = 5 x = 0.13, i = 4 g 1 p 1 x=0.25 ( x 8) x=0.35 ( x 4) x = 0.18, i = x = 0.25, i = 2 x = 0.40, i = x=0.5 ( x 2) 10-2 H1 PDF 2000 extrapolation 10-3 x = 0.65, i = Q 2 / GeV 2 H1 Collaboration 10-2 x=0.75 ( x 1) Q 2 [(GeV/c) 2 ] E155 E143 SMC HERMES EMC RL Jaffe COMPASS Paris Meeting, March

19 Early Extractions of G Altarelli, Ball, Forte, Ridolfi NLO analysis of existing polarized DIS. They use an anomalous renormalization scheme (not MS) No analysis of errors beyond variety of fits (A D) Estimates of G at 1 GeV ± 0.9 RL Jaffe COMPASS Paris Meeting, March

20 Early Extractions of G Blümlein & Böttcher NLO analysis of existing polarized DIS x Leading and next-to-leading order in MS Attempt error analysis Estimates of G at 4 GeV ± ± 0.67 RL Jaffe COMPASS Paris Meeting, March

21 Early Extractions of G J. Lichtenstadt Praha SPIN 2001 BNL EIC Workshop Preliminary NLO analysis of existing polarized DIS. Leading and next-to-leading order in anomalous scheme. Note instability to more freedom in parameterization Estimate of G at 4 GeV (stat) 0.22 (sys) 0.45 (theor) RL Jaffe COMPASS Paris Meeting, March

22 Conclusions There is a well-defined, operator based, unambiguous description of the scale dependent gluon helicity distribution, G(x, Q 2 ) And likewise for the total gluon spin contribution to the nucleon spin G(Q 2 ) Both G(x, Q 2 ) and G(Q 2 ) evolve and mix with the quark spin, allowing them to be measured indirectly in DIS. G(Q 2 ) is not directly related to the anomaly or the Kogut-Susskind current, K µ All the ingredients in the nucleon angular momentum have nice operator interpretations, but only some of them can be measured: q(x, Q 2 ) G(x, Q 2 ) L Q (Q 2 ) J G (Q 2 ) L G (Q 2 ) L Q (x, Q 2 ) L G (x, Q 2 ) RL Jaffe COMPASS Paris Meeting, March