Analyticity of the Dirichlet-to-Neumann map for the time ...
Analyticity and higher twists Hadron Structure’13, Tatranské Matliare , July 2, 2013
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Transcript of Analyticity and higher twists Hadron Structure’13, Tatranské Matliare , July 2, 2013
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?!