Analyticity and higher twists

Hadron Structure’13, Tatranské Matliare, July 2, 2013

Oleg TeryaevJINR, Dubna

QCD without confinement problem solved – description and classification of NP inputs (~biology before Darwin)

Complicated processes – complicated classification – UGDFs, TMDs, GPDs, HTs (~multiparton pdfs)

Are such classifications compatible with symmetries ( -> low-energy theorems),

CAUSALITY AND ANALYTICITY ?

Outline HT resummation and analyticity in

(spin-dependent) DIS HT resummation and scaling

variables: DIS vs SIDIS TMDs as infinite towers of twists Quarks in vacuum and inside the

hadrons: TMDs vs non-local condensates

Spin dependent DIS Two invariant tensors

Only the one proportional to contributes for transverse (appears in Born approximation of PT)

Both contribute for longitudinal Apperance of only for longitudinal case –result

of the definition for coefficients to match the helicity formalism

g1

gggT 21

Generalized GDH sum rule Define the integral – scales asymptotically as

At real photon limit (elastic contribution subtracted) – - Gerasimov-Drell-Hearn SR

Finite limit of infinite sum of inverse powers!

Proton- dramatic sign change at low Q2!

...1142 QQ

Q2

1

Finite limit of infinite sum of inverse powers?!

How to sum ci (- M2 /Q2 )i ?!

May be compared to standard twist 2 factorization

Light cone:

Moments and partonic expression

Lorentz invariance:

Summed by representing

Summation and analyticity Justification (in addition to nice parton

picture) - analyticity! Correct analytic properties of virtual

Compton amplitude Defines the region of x Generalized Parton Distributions –

compatibility of analyticity with factorization – non-trivial: Radon transform technique (OT’05; Anikin,OT’07,D. Muller,Kumericky’09-13)

Summation and analyticity-HT Parton model with |x| < 1 – transforms

poles to cuts! – justifies the representation in terms of moments

For HT series ci = <f(x) xi> - moments of HT “density”- geometric series rather than exponent: Σ ci (- M2

/Q2 ) = < M2 f(x)/(x M2

+ Q2 )>

Like in parton model: pole -> cut

Summation and analyticity-HT Analytic properties proper

integration region (positive x, two-pion threshold)

Finite value for Q2 =0: -< f(x)/x> - inverse moment!

Relation to matrix elements unclear (probably – Wilson lines: transverse momentum)

Summation and analyticity

“Chiral” expansion: - (- Q2/M2 )i <f(x)/x i+1>

“Duality” of chiral and HT expansions: analyticity allows for EITHER positive OR negative powers (no complete series!)

Analyticity – (typically)alternating series

Summation and analyticity

Analyticity of HT analyticity of pQCD series – (F)APT

Finite limit -> series starts from 1/Q2 unless the density oscillates

Annihilation – (unitarity - no oscillations) extra justification of “short strings”?

Short strings Confinement term in the heavy

quarks potential – dimension 2 (GI OPE – 4!) scale ~ tachyonic gluon mass

Effective modification of gluon propagator

Back to DIS: (J. Soffer, OT ‘92) Supported by the fact

that

Linear in , quadratic term from

Natural candidate for NP, like QCD SR analysis – hope to get low energy theorem via WI (C.f. pion F.F. – Radyushkin) - smooth model

For -strong Q – dependence due to Burkhardt-Cottingham SR

gggT 21

g2

g2

Models for :proton Simplest - linear

extrapolation – PREDICTION (10 years prior to the data) of low (0.2 GeV) crossing point

Accurate JLAB data – require model account for PQCD/HT correction – matching of chiral and HT expansion

HT – values predicted from QCD SR (Balitsky, Braun, Kolesnichenko)

Rather close to the data

gT

For Proton

The model for transition to small Q (Soffer, OT ’04)

Models for :neutron and deuteron Access to the

neutron – via the (p-n) difference – linear in ->

Deuteron – refining the model eliminates the structures

gT

for neutron and deuteron

Duality for GDH – resonance approach Textbook (Ioffe, Lipatov. Khoze)

explanation of proton GGDH structure –contribution of dominant magnetic transition form factor

Is it compatible with explanation?! Yes!– magnetic transition contributes

entirely to and as a result to

)1232(

g2

g2

gggT 21

Bjorken Sum Rule – most clean test

Strongly differs from smooth interpolation for g1

Scaling down to 1 GeV

New option: Analytic Perturbation Theory Shirkov, Solovtsov: Effective coupling –

analytic in Q2 Landau pole automatically removed Generic processes: F(ractional)APT Does not include full NPQCD dynamics

(appears at ~ 1GeV where coupling is still small) –> Higher Twist

Depend on (A)PT Low Q – very accurate data from JLAB

Bjorken Sum Rule-APT Accurate data + IR stable coupling ->

low Q region

PT/HT duality

Matching in PT and APT

Duality of Q and 1/Q expansions

4-loop corrections included V.L.

Khandramai, R.S. Pasechnik, D.V. Shirkov, O.P. Solovtsova, O.V. Teryaev. Jun 2011. 6

pp. e-Print: arXiv:1106.6352 [hep-ph]

HT decrease with PT order and becomes compatible to zero (V.I. Zakharov’s duality)

Analog for TMD – intrinsic/extrinsic TM duality!?

Asymptotic series and HT Duality: HT can be eliminated at all (?!)

May reappear for asymptotic series - the contribution which cannot be described by series due to its asymptotic nature.

Another version of IR stable coupling – “gluon mass” – Cornwall,.. Simonov,.. Shirkov(NLO) arXiv:1208.2103v2 [hep-th] 23 Nov 2012

HT – in the “VDM” form M2/(M2

+ Q2 ) Corresponds to f(x) ~ Possible in principle to

go to arbitrarily small Q BUT NO matching with

GDH achieved Too large average slope

– signal for transverse polarization role !

)1( x

Account for transverse polarization -> description in the whole Q region (Khandramai, Shirkov, OT, in progress)

1-st order – LO coupling with (P) gluon mass + (NP) “VDM”

GDH – relation between P and NP masses

P/NP

Data vs LO/NLO/NNLO

Smaller Q

HT – modifications of scaling variables Various options since Nachtmann ~ Gluon mass

-//- new (spectrality respecting) modification

JLD representation

Resummed twists: Q->0 (D. Kotlorz, OT)

Modified scaling variable for TMD First appeared in P. Zavada model

XZ =

Suggestion – also (partial) HT resummation(M goes from denominator to numerator in cordinate/impact parameter space)?!

HT for TMDs - case study: Collins FF and twist 3 x(T) –space : qq correlator ~ M - twist 3

Cf to momentum space (kT/M ) – M in denominator – “Leading Twist”

x <-> kT spaces Moment – twist 3 (for Sivers – Boer,

Mulders, Pijlman) Higher (2D-> Bessel) moments – infinite

tower of twists (for Sivers - Ratcliffe,OT)

Resummation in x-space (DY) Full x/kT – dependence and its expansion

Singularities (-> power/log tail – cf Efremov, Vladimirov – causality -  arXiv:1306.3929 ) should be subtracted to get exponential falloff (required to have all moments in TM finite)

DY weighted cross-section

What happens in vacuum? Suggestion : similarity with non-local

quark condensates (Radyushkin et al) : quarks in vacuum ~ transverse d.o.f. of quarks in hadrons (Euclidian!) ?!

Universality in hadron( type-dependent) TMD and vacuum NLC functions?!

Hadronic vs vacuum matrix elements

Hadron -> (LC) momentum; dimension-> twist

Transverse dynamics – looks more similar to vacuum: quark virtuality -> TM(squared)

(Euclidian) space separation -> impact parameter

More complcated objects Modification of BFKL kernel - talk of E. Levin:

May be a way to modify NP part (impact factor): exponential falloff in coordinate space -> finiteness in momentum space (cf GDH)

D-term for GPDs (~ quadrupole gravitational FF) ~ Cosmological Constant in vacuum; Negative D-term -> negative CC in space-like/positive in time-like regions: Annihilation~Inflation

Conclusions

Representation for HT similar to parton model: preserves analyticity changing the poles to cuts

Infinite sums of twists – important for DIS at Q->0

Good description of the data at all Q2 with the single scale parameter

Discussion

TMD – infinite towers of twists

Modified scaling variables – models for infinite twists towers in DIS and SIDIS

Similar to non-local quark condensates – vacuum/hadrons universality?!