What is the twist of TMDs? Como, June 12, 2013
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What is the twist of TMDs?
Como, June 12, 2013
Oleg TeryaevJINR, Dubna
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Outline Definitions of twist TMDs as infinite towers of twists Quarks in vacuum and inside the
hadrons: TMDs vs non-local condensates
HT resummation and analyticity in DIS
HT resummation and scaling variables: DIS vs SIDIS
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HT resummation in DIS
Higher Twists in spin-dependent DIS: GDH sum rule – finite sum of infinite number of divergent terms
Resummation of HT and analyticity
Comparing modified scaling variables
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What is twist? Power corrections ~1/Q2
------//------------- ~ M2
DIS – it’s the same (~ M2 /Q2)
TMD – usually ~1/Q2 - (M2 /kT
2 )i attributed to Leading Twist
However – tracing the powers of M is helpful for studying HT in coordinate (~impact parameter) space
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Collins FF and twist 3 x(T) –space : qq correlator ~ M - twist 3
Cf to momentum space (kT/M ) – M in denominator – “LT”
x <-> kT spaces Moment – twist 3 (for Sivers – Boer,
Mulders, Pijlman) Higher (2D-> Bessel) moments – infinite
tower of twists (for Sivers - Ratcliffe,OT)
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Resummation in x-space (DY) Full x/kT – dependence
DY weighted cross-section
Similarity with non-local quark condensate: quarks in vacuum ~ transverse d.o.f. of quarks in hadrons (Euclidian!) ?! –cf with Radyushkin et al
Universal hadron(type-dependent)/vacuum functions?!
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Hadronic vs vacuum matrix elements
Hadron-> (LC) momentum; dimension-> twist; quark virtuality -> TM; (Euclidian) space separation -> impact parameter
D-term ~ Cosmological constant in vacuum; Negative D-> negative CC in space-like/positive in time-like regions: Annihilation~Inflation!
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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
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Generalized GDH sum rule Define the integral – scales
asymptotically as
At real photon limit (elastic contribution subtracted) – - Gerasimov-Drell-Hearn SR
Proton- dramatic sign change at low Q2!
...1142 QQ
Q2
1
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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: Lorentz invariance Summed by
representing
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Summation and analyticity? Justification (in addition to nice parton
picture) - analyticity! Correct analytic properties of virtual
Compton amlitude Defines the region of x Require: Analyticity of first moment in Q2
Strictly speaking – another integration variable (Robaschik et al, Solovtsov et al)
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Summation and analyticity! 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 Analytic properties proper integration region
(positive x, two-pion threshold) Finite value for Q2 =0: -< f(x)/x> - inverse
moment!
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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 Analyticity of HT analyticity of pQCD series
– (F)APT Finite linit -> series starts from 1/Q2 unless the
density oscillates Annihilation – (unitarity - no oscillations)
justification of “short strings”?
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Short strings Confinement term in the heavy
quarks potential – dimension 2 (GI OPE – 4!) scale ~ tachyonic gluon mass
Effective modification of gluon propagator
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Decomposition of (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
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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
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The model for transition to small Q (Soffer, OT ’04)
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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
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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
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Bjorken Sum Rule – most clean test
Strongly differs from smooth interpolation for g1
(Ioffe,Lipatov,Khoze) Scaling
down to 1 GeV
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New option: Analytic Perturbation Theory Shirkov, Solovtsov: Effective coupling – analytic in Q2
Generic processes: FAPT (BMS) 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
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Bjorken Sum Rule-APT Accurate data + IR stable coupling ->
low Q region
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PT/HT duality
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Matching in PT and APT
Duality of Q and 1/Q expansions
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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!?
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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.
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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 (cf Ioffe e.a. interpolation)!
)1( x
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Account for transverse polarization -> descripyion in the whole Q region (Khandramai, OT, in progress)
1-st order – LO coupling with (P) gluon mass + (NP) “VDM”
GDH – relation between P and NP masses
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NP vs P masses
Non-monotonic!
“Phase diagram”
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P/NP masses
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Data at LO
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NLO
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Data vs NLO
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Modification of spectral function for HT Add const × Q2/(M22+Q2) 2 -> First of second
derivative of delta-function appear – double and triple poles (single – almost cancelled)
Masses: P= 0.68 NP=0.76 Expansion at
low Q2
Real scale – pion mass?!
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HT – modifications of scaling variables (L-T relations) Various options since Nachtmann ~ Gluon mass
-//- new (spectrality respecting) modification
JLD representation
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Resummed twists: Q->0 (D. Kotlorz, OT)
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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)?!
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Conclusions/Discussion TMD – infinite towers of twists Similar to non-local quark condensates –
vacuum/hadrons universality?! Infinite sums of twists – important for DIS at Q-
>0 Representation for HT similar to parton model:
preserves analyticity changing the poles to cuts Modified scaling variables – models for twists
towers at DIS and (TMD) SIDIS Good description of the data at all Q2 with the
single scale parameter