Analyticity and higher twists

Oleg Teryaeva

aBogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia

Abstract

The representation for infinite sum of higher twists (HT) tower in DIS implied by analyticity of virtual Comptonamplitude is suggested. Its simplest realization allows to describe the Bjorken sum rule at all momentum transfers.It is stressed that TMDs accommodate the infinite tower of HT similar to non-local vacuum condensates for the caseof vacuum matrix element. The D-term in hadronic GPDs bears some similarity to vacuum cosmological constant.The negative sign of D-term may be understood as a similarity between inflation and annihilation via the gravitonexchange.

Keywords: Higher twist, dispersion relation, resummation, condensates, cosmological constant

1. Introduction

Higher twist (HT) corrections are very important forthe applications of QCD at low scales. The lower is thescale the higher twists enter into the game. Sometimesany finite number of them is insufficient, and accountfor infinite sums is necessary.

We address here several such cases. The first one cor-responds to (spin-dependent) DIS in real-photon limit,when contact with low-energy theorems (GDH sumrules) may be achieved [1]. Originally, this was real-ized by matching of the HT expansion in inverse pow-ers of Q2 (where QCD perturbative expansion in logQ2

was also included [2]) and ”chiral” expansion in pos-itive powers of Q2. Here we describe amother proce-dure, where HT series is represented in integral form,incorporating the analytic properties of virtual Comp-ton amplitude. Even in its simplest version it leads tothe rather accurate description of Bjorken sum rule atall Q2.

Another situation of all-twists relevance is repre-sented by transverse momentum dependent parton dis-tributions (TMDs). They may be considered [3] as a(partial)sum of all HTs tower, while transverse moments

Email address: teryaev@theor.jinr.ru (Oleg Teryaev)

(where Bessel moments [4] should naturally appear be-cause transverse space is 2 -dimensional) correspond todefinite twists.

The TMDs are therefore similar [5] to the non-localvacuum condensates also partially resumming the lo-cal condesates of definite dimension playing the role oftwist in the vacuum. This may be an example of moregeneral similarity/universality of vacuum and hadronicmatrix elements. It is interesting, that vacuum analogof so-called Polyakov-Weiss D-term [7] in generalizedparton distributions (GPDs) is represented by nothingless than cosmological constant. The observed definite(negative) sign of D-term may, in turn, be interpreted asa positive ”effective” cosmological constant in the an-nihilation channel of respective gravitational formfactorestablishing the relation between inflation and annihila-tion.

2. Resumming HT in spin-dependent DIS

Let us consider as a case study the lowest non-singletmoment of spin-dependent proton and neutron structurefunctions gp,n

1 defined as

Γp−n1 (Q2) =

∫ 1

0dx gp−n

1 (x,Q2) , (1)

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with x = Q2/2Mν, the energy transfer ν, and the nu-cleon mass M. We imply the elastic contribution atx = 1 to be excluded, since the low-Q2 behavior of “in-elastic” Γp−n

1 (Q2), i.e. the Bjorken Sum Rule (BSR), isconstrained by the Gerasimov-Drell-Hearn (GDH) sumrule, which allows us to investigate continuation of theBjorken integral Γp−n

1 (Q2) to low Q2 scales. As all thehigher twists contributions are divergent when Q2 → 0,only infinite sum may be matched to GDH value. Let usconsider the series

S (Q2) =∞∑1

an(M2

Q2 )n. (2)

which may correspond either to non-perturbative part(HT) of Γ1 or to I1 = 2M2Γ1/Q2 proportional to pho-toabsorption cross-sections constrained by GDHSR. Bymaking the crucial step and representing an as a mo-ments

an =

∫ ∞

−∞f (x)xn−1, (3)

the sum of HTs can be recasted as

S (Q2) =∫ ∞

−∞dx

f (x)M2

Q2 − xM2 . (4)

Such a representation in terms of moments may be com-pared to the similar one when the standard leading twistpartonic expression is derived. The later, besides thenice physical pictture, may be justified by correct ana-lytical properties of virtual Compton amplitude, havings and u cuts produced by respective poles (at LO) andcuts in the partonic subprocess.

The similar arguments may be applied for HT resum-mation. If analytical properties of S (Q2) are representedby the cut residing at Q2 ≤ 4m2 (m being the mass ofthe lightest particle in the respective channel) , the inte-gration in (3) should be limited to (−∞,−4m2/M2). Ifthe function f (x) has a definite sign it leads to the al-ternating HT series. Moreover, even for sign-changingf (x) the series will be typically alternating unless fine-tuning of f is imposed.

It is crucially important that the leading twist con-tribution and respective perturbative (logarithmic) cor-rections has the same analytic properties, leading to thesame properties of the full amplitude. This naturally se-lects the modified Analytic Perturbation Theory (APT)[8] which was successfully applied for the descriptionof BSR [9]. These dtudies manifested the duality be-tween the HT and perturbative corrrections, so that HTdecreased at NLO etc. Let me conjecture here, that the”real” HT should correspond to the piece which cannot

be absorbed to perturbative series due to is asymptoticnature.

One may now continue S to Q2 = 0 which is definedby first inverse moment

S (0) = −∫ −4m2/M2

−∞dx

f (x)x. (5)

Its sign will typically coincide with that of the first termof the series.

The derivatives of S at Q2 = 0 are defined by higherinverse momentss,say

S ′(0) = −∫ −4m2/M2

−∞dx

f (x)M2x2 . (6)

If one neglect m (which corresponds to minimal APTin the perturbative part) this integral may diverge atx ∼ 0. This divergence may be used to cancel the in-finite slope of minimal APT contribution requiring thatf (x) ∼ ρpert(s = M2x) for x ∼ 0. This divergenceis absent in the recently elaborated Massive Perturba-tion Theory (MPT [10], where Q2 → Q̃2 = Q2 + M2

gl)which together with the VDM form of HT contributionM2

HT /(Q2 + M2

HT ) lead to the reasonable description ofBSR down to rather low Q2.

Note that VDM form of HT perfectly fits to (4) withthe delta-function spectral density 1 while MPT expres-sion has also the correct analytic properties providedrelevant ”gluonic mass” Mgl ≥ ΛQCD.

At the same time, the attempt to match the MPTdescription with GDHSR fails. The reason is obvi-ous: the (average) slope of Γp−n

1 (Q2) at low Q2 is sev-eral times larger than the one following from GDHSR.This clearly supports the ”two-component” approach[1] where slope is decomposed to the sum of ”fast”rapidly decreasing component due to structure func-tion g2 and ”slow” component due to structure functiongT = g1 + g2, which for BSR provides the slope [2] en-hanced by factor μp

A/((μnA)2 − (μp

A)2) ∼ 4 determined byproton and neutron anomalous magnetic moments.

It is therefore natural to combine MPT analysis withapproach [1]. The fast component contribution is con-trolled by Butrkardt-Cottingham sum rule free from anycorrections. GDH sum rule allows to relate HT andgluon masses (appearing to be close) so that there is sin-gle free parameter remained. The one-parameter fits2

1It is interesting that similar VDM form was discussed at this con-ference also in the talks of Ya.Klopot for pion-photon transition form-factor (where it is related to axial anomaly) and A. Aleksejevs for pionformfactor.

2I am grateful to I.Gabdrakhmanov and V. Khandramai for the helpin numerical calculations.

O. Teryaev / Nuclear Physics B (Proc. Suppl.) 245 (2013) 195–198196

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.00

0.05

0.10

0.15

0.20

Q2�GeV2�

�1p�n

Figure 1: BSR at LO (dotted blue curve), NLO (solid red curve),NNLO(dashed yellow curve) compared with experimental data.

lead to the reasonable description of the data with thequality increasing with taking into account NLO MPT[10] and modifications of spectral density (4),

It is well known that modified scaling variables pro-vide the way of partial HT resummation. This corre-sponds to famous Nachtmann scaling variable and an-other one x̂ = Q̃2/2Mν, closely related to MPT andused at low x region. The advantage of the last vari-able for BSR studies is that it is larger than xB so thatg( x̂) < g1(xB) and one may hope to describe [11] the de-crease of Γp−n

1 (Q2) at small Q2 required by the finitenessof photoabsorption cross-section. However, for largerxB the spectral condition for x̂ may be violated, so thatx̂ ≥ 1. To avoid that, one may introduce another vari-able, using the representation xB = Q2/(Q2 + W2) andperforming the same change Q → Q̃:

x̃ = xB1 + a

1 + axB, a =

M2gl

Q2 . (7)

The first moment may be now presented as

Γ̃1(Q2) =∫ 1

0g1(x̃)dx = (a + 1)

∫ 1

0

g1(z)dz(1 + a(1 − z))2 .

(8)While asymptotically this result reproduces the standardpartonic expression, at smail Q2 one get

Γ̃1(Q2) → Q2

M2gl

∫ 1

0

g1(z)dz(1 − z)2 , (9)

and the slope at the origin related to GDHSR is definedby the second inverse moment. Note that the contribu-tion of large x region is enhanced. The available param-eterizations of spin-dependent parton distributions seemto provide a rather reasonable of data at low Q2 [11].

3. Hadronic vs vacuum matrix elements

3.1. Transverse Momentum Dependent Distributionsand vacuum condensates

The TMDs can be conveniently defined[5] in coordi-nate (impact parameter) space. The representative caseis Boer-Mulders function (in order to avoid consider-ation of gauge links/gluonic poles one may consider[5]Collins fragmentation function having the same Lorentzstructure) when transverse coordinate is selected by thechiral-odd Dirac structure

〈P|ψ̄(0)σμνψ(z)|P〉 = M(Pμzν − Pνzμ)I(z · P, z2) (10)

Here the z2-dependence corresponds to kT dependencein momentum space and contains all twists. The defi-nite twists may be extracted by the expansion in Taylorseries

I(z · P, z2) =∞∑

n=0

∂n

n!∂z2n I(z · P, z2)|z2=0, (11)

which in the momentum space corresponds to the trans-verse moments of TMD. Note that the lowest twist is 3,which is seen from the appearance of factor M in ther.h.s. of (10). In the momentum space this factor isshifted to the denominator, as the transverse moment istaken over dk2

T /M2 [5].

Note that the expansion in z2 may be performed onlyafter the subtraction of the singular terms in z2, whichin the collinear factorization are absorbed to coefficientfunction. For TMDs they constitute the power-like tail,after subtraction of which all the transverse momentsbecame finite, indicating the Gaussian distributions.

It is interesting that the appearance of such a tail wasrecently deduced [12] from the causality arguments, be-ing as it is well known also the origin of analytic prop-erties discussed above.

The described situation is rather similar to non-localvacuum condensates (see [6] and Ref. therein) whereone is dealing with vacuum matrix element

〈0|ψ̄(0)ψ(z)|0〉 = 〈0|ψ̄(0)ψ(0)|0〉F(z2). (12)

The Taylor expansion of F selects the local condensatesof definite dimension, corresponding to the moments ofsuitably chosen Fourier transformed function being thecomplete analog of TMD. Note that subtraction of sin-gular terms is performed here by subtraction of the per-turbative contribution corresponding to quark propaga-tor.

Generally, the hadronic matrix elements differ fromthe vacuum ones by the presence of essentially pseudo-Euclidian hadron momentum. At the same time, the Eu-clidian transverse dynamics may be more vacuum-like.

O. Teryaev / Nuclear Physics B (Proc. Suppl.) 245 (2013) 195–198 197

3.2. D-term and cosmological constantLet us discuss one more interesting example of in-

terplay between hadronic and vacuum matrix elements.It corresponds to so-called Polyakov-Weiss D-term [7](appearing in analyticity based analysis as a subtractionconstant [13]) whose moment is related to quadrupolegravitational formfactor transverse coordinate is se-lected by the chiral-odd Dirac structure

〈P+ q/2|T μν|P− q/2〉 = C(q2)(gμνq2 − qμqν)+ ... (13)

where gravitoelectric and gravitomagnetic formfactors[16] are dropped. C has definite (positive) sign in all theknown cases including hadrons [7](where also generalstability arguments are discussed), photons [14], Q-balls[15].

For vacuum matrix element one has the famous cos-mological constant

〈0|T μν|0〉 = Λgμν (14)

One may relate this matrix element in 2-dimensionaltransverse space orthogonal to P and q, so that effective2-dimensional cosmological constant is

Λ = C(q2)q2 (15)

The positive C leads to negative cosmological constantin the scattering process and to positive one in the an-nihilation process. There seems to be some relation be-tween annihilation and inflation! It may not be so sur-prising due to known similarity3 between inflation andSchwinger pair production in the electric field.

It is of course very interesting whether real cosmo-logical constant in our Universe may be understood asemerging from annihilation at extra dimensions. Qual-itatively this is similar to brane cosmology, and oneshould stress that in the suggested scenario Big Bangis due to one-graviton annihilation. The specificationof extra-dimensional states providing the cosmologicalconstant of mass dimension 4 remains to be investi-gated,

4. Conclusions

The analyticity property continues to play the ma-jor role in developing of QCD approaches to low scaleprocesses, being of most experimental interest. It al-lows one to justify the representation of infinite sums ofhigher twists contributions, providing, in particular, the

3I am indebted to A.A. Starobinsky for this comment.

accurate description of Bjorken Sum Rule data at lowQ2. Another method of higher twists partial resumma-tion is provided by modified scaling variables, some-times allowing to describe the real photon limit of DIS.

The infinite series of higher twists are required totransverse momentum dependent parton distributions,having deep similarity to non-local vacuum conden-sates.

The vacuum/hadron matrix elements similarity al-lows to describe one-graviton annihilation as effective2-dimensional cosmological constant, and this may begeberalized to describe in a similar way cosmologicalconstant in our Universe as emerging from one-gravitonannihilation at extra dimension, which is the picture ofa Big Bang in such a case.

I am grateful to Organizers for warm hospitality atHIgh Tatras and to Participants for many exciting dis-cussions. This work is supported in parst by RFBRgrants 12-02-00613 and 13-02-01060.

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