Generalized parton distributions: what to measure, what to ... · What can we learn? What can we measure? QCD as the theory of strong interactions I a quantum eld theory with a quite

Generalized parton distributions: what tomeasure, what to learn?

M. Diehl

Deutsches Elektronen-Synchroton DESY

Schloss Waldthausen, 31 March 2009

DESY

What can we learn? What can we measure?

1. Usual and generalized parton densities in QCD

2. From cross sections to generalized parton distributions

M. Diehl Generalized parton distributions: what to measure, what to learn? 2


QCD as the theory of strong interactions

I a quantum field theory with a quite simple Lagrangiangiving rise to a wide array of phenomena

I chiral symmetry breakingI strong bindingI confinementI asymptotic freedom

I interplay of these aspects can be studied in processes withhigh momentum transfer (high Q2, p2

T , etc.)



WARNINGI we are entering an area of

I relativistic physicsI quantum effects impossible to probe system without disturbing it,

interference can play tricks on classical intuitionI quantum field theory renormalization, anomalies ↔ breakdown of

scale invariance and chiral symmetryI gauge theories some degrees of freedom are unphysical,

which ones depends on the chosen gauge

I we should not expect that physics is always similar to the oneof Kepler orbits, spinning tennis balls, . . .

compare:if you go up a high mountain, you should (i) bring sunscreen and warm clothing (thesun is strong even if it doesn’t feel, and it can suddenly get cold), (ii) expect effectsfrom lack of oxygen above 3000 m (take time to acclimatize and move cautiously)

despite all this, the experience can be very rewarding



Parton distributions (including generalized ones)

I have intuitive interpretation as densities or correlationsin the parton model

• visualize hadron structure in quantitative way• can “derive” interpretation from QCD under simplifications

quantize theory on the light-cone, put aside complicationsfrom gluons with zero light-cone momentum (= zero modes)

• a useful guide, as long as remember its limitations

I are defined in full QCD

• non-perturbative definition, suitable e.g. for lattice calculations→ talks C. Alexandrou, M. Gockeler

• isolate hadron properties from full reaction dynamicsas far as this is possible

factorization theorems give the details



Generalized parton distributions (GPDs)

I defined from matrix elements

〈h′(p′, s′)|O|h(p, s)〉h = hadron of your choice

+ξx −ξx

p,s p’,s’

t

O = operator involving quark and/or gluon fields

I two dist’s for unpolarized partons in spin 1/2 hadron:

• H for conserved hadron helicity• hadron helicity flip from E

I interpretation: longitudinal momentum fractions x and ξξ−x−ξ− x

x−ξ ξ0 1−1

+ξxxξ− x+ξ x−ξ

I for |x| < ξ coherent emission of qq pairI for |x| > ξ similar to parton densities:

correlation ψ∗x−ξ ψx+ξ instead of probability |ψx|2use light-cone quantization and ignore complications

I still know little about interplay of x and ξ: major challengefor describing/parameterizing/understanding GPDs

I no simple way known to get GPD(x, 0, t) from GPD(x, ξ, t)



Generalized parton distributions (GPDs)

I interpretation: longitudinal momentum fractions x and ξξ−x−ξ− x

x−ξ ξ0 1−1

+ξxxξ− x+ξ x−ξ

I for |x| < ξ coherent emission of qq pairI for |x| > ξ similar to parton densities:

correlation ψ∗x−ξ ψx+ξ instead of probability |ψx|2use light-cone quantization and ignore complications

I still know little about interplay of x and ξ: major challengefor describing/parameterizing/understanding GPDs

I no simple way known to get GPD(x, 0, t) from GPD(x, ξ, t)



Some fine print about parton distributions (generalized or not)

I to have well defined operator O must renormalize it→ factorization scale µphysically: resolution of structure in given process

I can calculate µ dependence perturbatively for large enough µ(DGLAP/ERBL equations)

I renormalization → choice of scheme (MS, DIS, ABJ, . . . )

• at quantum level no unique separation of “object” and “probe”• parton densities are not directly observable

but neither are αs or quark masses



Some fine print about parton distributions (generalized or not)

I to have well defined operator O must renormalize it→ factorization scale µphysically: resolution of structure in given process

I can calculate µ dependence perturbatively for large enough µ(DGLAP/ERBL equations)

I renormalization → choice of scheme (MS, DIS, ABJ, . . . )

• at quantum level no unique separation of “object” and “probe”• parton densities are not directly observable

but neither are αs or quark masses



More fine print about parton distributions (generalized or not)

I O contains a Wilson line, e.g. q(z1)PeigR z2z1

dx−A+(z)q(z2)

• simple parton-model interpretation only in A+ = 0 gaugewhich involves tricky issues concerning zero modes

I O follows from how distribution appears in physical process

* how a proton “looks like” depends on how you look at it *I concept of “a single parton” too naive

• already in QED a “bare electron” is not observable, need toinclude soft photons

• in QCD additional complications from collinear quarks andgluons

parton densities and factorization theorems provide concretesetting to investigate this physics



Nucleon tomography: impact parameter

I at fixed longitudinal momentum fractions x, ξ:t dependence of GPD → transverse mom. transfer ∆→ Fourier transf. to transverse position b of struck parton

I fully relativistic, no ambiguities ∼ Compton wavelength

I different from interpretation in three-dimensional spacefor form factors: Sachs ’62

for GPDs: Polyakov, Shuvaev ’02, Belitsky, Ji, Yuan ’03





I for ξ = 0 have joint density in parton-model sense:long. momentum fraction x and transv. position b

q(x, b2) = (2π)−2∫d2∆ e−ib∆Hq(x, ξ = 0, t = −∆2)

D. Soper ’77; M. Burkardt ’00, ’02

I∫dxxn gives interpretation of form factorsfrom experiment and lattice → talks C. Carlson, M. Gockeler





I for ξ = 0 have joint density in parton-model sense:long. momentum fraction x and transv. position b

q(x, b2) = (2π)−2∫d2∆ e−ib∆Hq(x, ξ = 0, t = −∆2)

D. Soper ’77; M. Burkardt ’00, ’02

I q(x, b2) not Fourier conjugate to q(x,k2)transverse-momentum dependent density

both generated by higher-level function

q(x,k2)∆=0← H(x, ξ = 0,∆,k)

Rd2k d2∆ e−ib∆

→ q(x, b2)

complementary information about transverse structure




in exclusive reactions do not measure at ξ = 0I for ξ 6= 0 get distance of quark to average positions of initial

and final proton M. Diehl ’02

I especially simple for x = ξ

∆→ b with t = ζ2m2p+∆2

1−ζ ξ = ζ2−ζ

distance of struck parton from spectator systemM. Burkardt ’07

1

1−ζ

x

b

0=ζ

1

1−ζ

b

0

ζ= x



Nucleon tomography and polarization

I impact parameter transform of E(x, ξ, t) spin-orbit correlations of partons

I parton distribution in nucleon polarized along x-axisis shifted in y direction

qX(x, b) = q(x, b2)− by

m

∂

∂b2eq(x, b2)

where eq(x, b) is Fourier transform of Eq(x, ξ = 0, t)I semi-classical picture: rotating matter distribution

• gives alternative view of Ji’s sum rule: Lx = by pz

• connection with Sivers effect via “chromodynamic lensing”

M. Burkardt ’02, ’04, ’05



Another interpretation of E

I Eq,g ↔ ∆L3 = 1 from helicity imbalance

qE

I similar to Sivers distributionI E(x, ξ, t) b dependence of ∆L3

I can formalize using light-cone wave functionsM. Burkardt, G. Schnell ’05



Spin sum rules

I Ji’s sum rule: 12 = Jg +

∑qJq with

Jq = 12

∫dxx(Hq + Eq)

∣∣t=0ξ=0

Jg = 12

∫dx (Hg + Eg)

∣∣t=0ξ=0

I further decompose Jq = Lq + 12Σ

with Σ from ordinary parton densities

I in analogous decomposition Jg = Lg + ∆gno obvious operator interpretation of Lg

I Jaffe-Bashinski decomposition: 12 = J g +

∑qJ q

with J g = Lg + ∆g and J q = Lq + 12Σ

I Jq 6= J q, Lq 6= Lq and Jg 6= J gI no known connection of Lq and Lg with observables



Spin sum rules

I ambiguities in decomposition reflects difficulty to separate

• “quarks” from “gluons” in a gauge theory• “intrinsic” from orbital angular momentum for spin-1 particles

similar issues already in QED

I there need not be a unique choice

I remember: axial anomaly limits simple parton-modelinterpretation of Σ and ∆g

does not affect Ji’s Jq and Jg, whichare derived from energy-momentum tensor

I striving to understand these issues better may eventually shedfurther light on inner workings of QCD



Spin sum rules

I ambiguities in decomposition reflects difficulty to separate

• “quarks” from “gluons” in a gauge theory• “intrinsic” from orbital angular momentum for spin-1 particles

similar issues already in QED

I there need not be a unique choice

I remember: axial anomaly limits simple parton-modelinterpretation of Σ and ∆g

does not affect Ji’s Jq and Jg, whichare derived from energy-momentum tensor

I striving to understand these issues better may eventually shedfurther light on inner workings of QCD



Summary Part 1

generalized parton distributionsI share properties of usual parton densities

I intuitive, pictorial interpretation of hadron structurewithin the limitations of a gauge theoryexploring these limits is relevant in itself

I well-defined in full QCD, suitable for the use of full-fledgedperturbative and non-perturbative calculations

I reveal characteristic aspects of hadron structure that areinaccessible in usual parton densities or form factors

I correlation of (transverse) spatial and (longitudinal)momentum structure → three-dimensional information

I spin-orbit correlations, orbital angular momentumI particularly simple interpretations for ξ = 0 and for

∫dxxn

but already get physics from GPD(ξ, ξ, t), GPD(x, ξ, t)



From cross sections to amplitudes

I have all-order factorization theorems forI Compton scattering:

DVCS: `N → `N γ (γ∗N → γN)Timelike Compton Scattering : γN → `+`−N (γN → γ∗N)

I meson electroproduction and πN → `+`−N

note: do not have this for pp→ jets +X or γp→ charm +X

I available calculationsI full NLO and partial NNLO for Compton scatt.I full NLO for meson productionI twist three (LO and partial NLO) for Compton scatt.

otherwise no systematic theory for power suppressed terms

I from differential cross sections can extractI linear combinations of amplitudes from Compton scatt.

(using interference with Bethe-Heitler process )polarization and beams of both charges highly valuable

I quadratic combinations of amplitudes from meson production



From amplitudes to GPDs: leading order in αs

amplitudes for DVCS and vector meson production at LO in αs :

ImH(ξ, t, Q2) = πˆH(ξ, ξ, t;Q2)−H(−ξ, ξ, t;Q2)

˜• compare with inclusive DIS: F2(xB , Q

2) = xBPq

ˆq(xB ;Q2) + q(xB ;Q2)

˜

ReH(ξ, t, Q2) = PV

Z 1

−1

dxH(x, ξ, t;Q2)

»1

ξ − x −1

ξ + x

–

= PV

Z 1

−1

dxH(x, x, t;Q2)

»1

ξ − x −1

ξ + x

–+D(t;Q2)

access to x values different from measured ξ = xB/(2− xB)

D(t) related with Polyakov-Weiss D-term

O.V. Teryaev ’05, I.V. Anikin and O.V. Teryaev ’07

quark flavor labels suppressed for brevityhave analogous eq’s for mesons with other quantum numbers



From amplitudes to GPDs: leading order in αs

amplitudes for DVCS and vector meson production at LO in αs :

ImH(ξ, t, Q2) = πˆH(ξ, ξ, t;Q2)−H(−ξ, ξ, t;Q2)

˜• compare with inclusive DIS: F2(xB , Q

2) = xBPq

ˆq(xB ;Q2) + q(xB ;Q2)

˜

ReH(ξ, t, Q2) = PV

Z 1

−1

dxH(x, ξ, t;Q2)

»1

ξ − x −1

ξ + x

–= PV

Z 1

−1

dxH(x, x, t;Q2)

»1

ξ − x −1

ξ + x

–+D(t;Q2)

access to x values different from measured ξ = xB/(2− xB)

D(t) related with Polyakov-Weiss D-term

O.V. Teryaev ’05, I.V. Anikin and O.V. Teryaev ’07

quark flavor labels suppressed for brevityhave analogous eq’s for mesons with other quantum numbers



From amplitudes to GPDs: leading order and beyond

I at leading order:

amplitude ↔ GPD(x, x, t;Q2) and subtraction constant D(t;Q2)

• greatly simplified phenomenology• limited accuracy and no sensitivity to x ≥ ξ

I Q2 dependence from evolution:

d

d lnQ2H(ξ, ξ, t;Q2) = kernel⊗

{H(x, ξ, t;Q2) for |x| ≥ ξ

} need GPD in DGLAP region |x| ≥ ξ

I from DGLAP region can reconstruct missing region |x| ≤ ξup to ambiguity corresponding to D(t;Q2)

K. Passek-Kumericki, D. Muller, K. Kumericki ’08

I note: measure parton distributions at Q2>∼ 1 GeV2

need evolution to make contact with low-energy descriptions



From amplitudes to GPDs: leading order and beyond

I at leading order:

amplitude ↔ GPD(x, x, t;Q2) and subtraction constant D(t;Q2)

• greatly simplified phenomenology• limited accuracy and no sensitivity to x ≥ ξ

I Q2 dependence from evolution:

d

d lnQ2H(ξ, ξ, t;Q2) = kernel⊗

{H(x, ξ, t;Q2) for |x| ≥ ξ

} need GPD in DGLAP region |x| ≥ ξ

I from DGLAP region can reconstruct missing region |x| ≤ ξup to ambiguity corresponding to D(t;Q2)

K. Passek-Kumericki, D. Muller, K. Kumericki ’08

I note: measure parton distributions at Q2>∼ 1 GeV2

need evolution to make contact with low-energy descriptions



From amplitudes to GPDs: beyond leading order

I at NLO and higher:

H(ξ, t;Q2) =1∫

−1

dxC(x, ξ)H(x, ξ, t;Q2)

with process dependent kernels C

for DVCS and meson prod’n can limit to DGLAP region

+ process dependent D(t;Q2) M.D. and D.Yu. Ivanov’07

I compare with inclusive processes:

dσ/dx ∝1∫x

dz C(x, z)PDF(z;Q2)

• for any process at NLO• for pp→ jet +X, γp→ c +X, . . . already at LO

I access to x 6= ξ in GPDs only via loop effects:

• Q2 dependence compare: g(x), ∆g(x) from DIS struct. fcts.• compare different processes



Summary Part 2

I theory to go from cross sections to integrals over GPDsamong the most solid cases of factorization in QCD

I higher-order results available

I DVCS and TCS: many observables with factorized descriptionmeson production: valuable complementary information

but need higher Q2 for theoretical accuracy

I at LO and without evolution simple description limited to x = ξfor sensitivity to x 6= ξ need loop effects:

I deviations from Bjorken scalingI comparing different processes, e.g.

• DVCS vs. TCS• DVCS vs. meson production• `N → `πN vs. πN → `+`−N


Documents

Generalized parton distributions: what to measure, what to ... · What can we learn? What can we measure? QCD as the theory of strong interactions I a quantum eld theory with a quite