Deep Inelastic Scattering and the limit of small parton

energy fraction.

M. Greco

talk based on results obtained in collaboration with B.I. Ermolaev and S.I.

Troyan

Frascati, 14 May 2007

Deep Inelastic e-p Scattering

Incoming lepton

outgoing lepton-detected

Incoming hadron

Produced hadrons - not detected

k

p

k’

X

Deeply virtual photon

q

Leptonic tensor

hadronic tensor

Hadronic tensor consists of two terms:

),( ),( qpWqpWW spindunpolarizeμνμνμν +=

Does not depend on spin

Spin-dependent

€

Wμν

symmetric antisymmetric

μ ν

p p

q q

The spin-dependent part of W is parameterized by two

structure functions:

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+= ),(),( 2

22

1 Qxgppq

SqSQxgSqi

pq

mW

spin

ρρρλμνλρμν ε

Structure functions

where m, p and S are the hadron mass, momentum and spin; q is the virtual photon momentum (Q2 = - q2 > 0). Both functions depend on Q2 and x = Q2 /2pq, 0< x < 1. At small x:

€

W μνspin

≈m

pqiεμνλρqλ Sρ

|| g1(x,Q2) + Sρ⊥(g1(x,Q2) + g2(x,Q2))[ ]

longitudinal spin-flip transverse spin -flip

Theoretical study of g1 and g2 involves both Pert and Non-Pert QCD and therefore it is not model-independent

When the total energy and Q2 are large compared to the mass scale, one

can use factorization:

k

P

q

k

P

q

k

P

q

p p

quark gluon

DIS off quarkDIS off gluon

Pert QCD

Non-pert QCD

= +

This allows to represent as a convolution of the partonic tensor and the probabilities to find a (polarized) parton (quark or gluon) in the hadron :

=spinWμν

Wquark

quark

Wgluon

gluon

+

q q

p p

μνW

DIS off quark

DIS off gluon

Probability to find a quark

Probability to find a gluon

)],(),(),(),([ 4 pkkqWpkkqWkdW gluongluon

quarkquark Φ+Φ=∫ μνμνμν

Analytically this convolution is written as follows:

Perturbative QCD

Perturbative QCD

Non-pert QCD Non-pert QCD

Pert QCD: analytical calculations of Feynman graphs

Non-perturbative QCD: no regular methods

Then DIS off quarks and gluons can be studied with perturbative QCD, by calculating the Feynman graphs involved.

The probabilities, quark and gluon involve non-perturbaive QCD. There is no regular analytic way to get them. Usually they are fitted from the experimental data at large x and Q2 , and they are called the initial quark and gluon densities and are denoted q and g . So, the conventional form of the hadronic tensor is:

gWqWW gluonquark δδ μνμνμν ⊗+⊗=

Initial quark distribution

Initial gluon distribution

DIS off the quarkDIS off the gluon

Some terminology

Contributes to nonsinglet

Contribute to singlet

Initial quark

Each structure function has both a non-singlet and a singlet components:g1 = g1

NS + g1S

The Standard Approach consists in using the perturbative Altarelli-Parisi or DGLAP Q2- Evolution Equations, together with fits for the initial parton densities.

Evolution Equations: Altarelli-Parisi, Gribov-Lipatov, Dokshitzer

),()/()2/(),( 2221 QyqyxCeQxg NS

NSqNS Δ⊗=

Evolved quark

distribution

In particular, for the non-singlet g1:

Coefficient function

The expression for the singlet g1 is similar, though more involved. It includes

coefficient functions and splitting functionsgggqqgqq PPPP , , ,

in order to evolve the quark and gluon distributions q and g

gC ,qC

NSqq

sNS

qPQd

qdΔ⊗=

Δπα2ln 2

Splitting function

where

.

€

g1NS (x,Q2) = (eq

2 /2)

dω

2πi−i∞

i∞

∫ 1

x

⎛

⎝ ⎜

⎞

⎠ ⎟ω

C(ω)δq(ω)expdk⊥

2

k⊥2γ (ω,α s(k⊥

2 ))μ 2

Q 2

∫ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

Coefficient function Anomalous dimension

Initial quark density

Using the Mellin transform, one obtains the expression for g1NS in a simpler form :

Pert QCD

Non-Pert QCD

1/x

1

Q22

x-evolution of q with coefficient function

Q2 -evolution of q with anomalous dimension

evolved quark density

q at x~1 and Q2 .>> 2

g1 at x<<1

and Q2 >> 2

q at x ~1 and Q2 ~ 2

defined from fitting exp. data

Starting point ofQ2 -evolution

Kinematics in the (1/x - Q2) plane

... )( )2/)(()( )4/)(()(

... )( )2/)((1)( )1(22)0(2

)1(2

++=

++=

ωγπαωγπαωγ

ωπαω

QQ

CQC

ss

s

LO

LO

NLO

NLO

In DGLAP, the coefficient functions and anomalous dimensions are known with LO and NLO accuracy.

LO splitting functions Ahmed-Ross, Altarelli-Parisi, Sasaki,…

NLO splitting functions

Floratos, Ross, Sachradja, Gonzale- Arroyo, Lopes, Yandurain, Kounnas, Lacaze, Curci, Furmanski, Petronzio, Zijlstra, Mertig, van Neerven, Vogelsang,…

Coefficient functions C(1)

k , C(2)k

Bardeen, Buras, Muta, Duke, Altarelli, Kodaira, Efremov, Anselmino, Leader, Zijlstra, van Neerven,…

)]/1(ln)/1([ln

)]1()1[(

xxxNq

xxNxqβα

δβα

γδγδ

+=

+−= −

There are different fits for the initial parton densities. For example,

Phenomenology of the fits for the parton densities

Altarelli-Ball-Forte-Ridolfi, Blumlein-Botcher, Leader-Sidorov- Stamenov, Hirai et al.,…

Altarelli-Ball-Forte-Ridolfi,

The parameters should be fitted from experiments. δγβα , , , ,N

This combined phenomenology works well at large and small x, though strictly speaking, DGLAP is not supposed to work in the small- x region:

Large x

1/x

1

Q2

DGLAP –region ln(1/x) are small

DGLAP accounts for logs(Q2) to all orders in s but neglects

ks

ks xx ))/1ln(( ,))/1(ln( 2 αα with

k>2

ln(1/x) are large

However, these contributions become leading at small x and should be accounted for to all orders in the QCD coupling.

Small x

1

Q22

Q2 -evolution , total resummation of

g1 at small xand large Q2

starting point

x-evolution, total resummation of

ks Q ))/ln(( 22 μα

1/x

ks

ks xx ))/1ln((,))/1(ln( 2 αα

DGLAPyes

no

DGLAP cannot perform the resummation of logs of x because of the DGLAP-ordering, a keystone of DGLAP

K3

K2

K1

DGLAP –ordering:22

32 2

2 1

2 k k k Q<<<< ⊥⊥⊥μgood approximation for large x when logs of x can be neglected. At x << 1 the ordering has to be lifted

q

p DGLAP small-x asymptotics of g1 is well-known:

1/22QCD

21 )]/(Qln [ln(1/x)ln exp ~ ΛDGLAPg

when the initial parton densities are not singular functions of x

When the DGLAP –ordering is lifted all double logarithms of x can beaccounted for, and the asymptotics is different:

2/221 )/(Q (1/x) ~ ΔΔ μDLg Bartels- Ermolaev-

Manaenkov-Ryskin

intercept

Obviously DGLAPDL gg 11 >> when x 0

,)/3(8 1/2s πα=ΔNS

singlet intercept 1/2

s )/2(3 3.45 πα=ΔS

The weakest point of this approach: the QCD coupling s is fixed at

an unknown scale.

On the contrary, DGLAP equations have always a running s

)( 2Qss αα =

Bassetto-Ciafaloni-Marchesini- Veneziano, Dokshitzer-Shirkov

Arguments in favor of the Q2- parameterization:

non- singlet intercept

DGLAP- parameterization

Intercepts of g1 in Double-Logarithmic Approximation:

K

K’

K

K’

K

K’

)( 2⊥= kss ααOrigin: in each ladder rung

DGLAP-parameterization

However, such a parameterization is good for large x only. At small x :

)( )/)(())'(( 22'22⊥⊥⊥ ≈+≈−= kxkkkk ssss αααα

Ermolaev-Greco-Troyan

When DGLAP- ordering is used and x ~1time-like argument

Contributes in the Mellin transform

Obviously, this new parameterization and the DGLAP oneconverge when x is large but they differ a lot at small x.

In this new approach for studying g1 in the small-x region, it is necessary:

1. Total resummation of logs of x2. New parametrization of the QCD coupling

How: the formula

€

α s(k2) =1/(b ln (k2/Λ2))

is valid when

Then it is necessary to introduce an infrared cut-off for k2

in the transverse space: Lipatov

€

k⊥2 > μ 2

€

k 2 >> Λ2

As the value of the cut-off is not fixed, one can evolve the structure functions

with respect to

Infra-Red Evolution Equations (IREE)

Method prevoiusly used by Gribov, Lipatov, Kirschner, Bartels-Ermolaev-Manaenkov-Ryskin, …

(name of the method)

Essence of the method

s

t

Introduce IR cut-off: μ k >⊥for all virtual particle momenta,

DL contributions, in particular, come from the region where all transverse momenta are widely different so, one can factorize the phase space into a set of separable sub-regions, in each region some virtual particle has a minimal . Let us call such a particle the softest one.

DL and SL contributions come from the integration region where

€

k⊥ < k||

both in the longitudinal and in the transverse space.

⊥k

Typical Feynman graph

DL and SL contributions of softest particles can be factorized

g g

qq

Case A: the softest particle is a non-ladder gluon. It can be factorized:

μ =+

+

+ symmetric contributions

k

k

k

⊥k

€

μis the lowest limit of integration over ),,( tskFdk ⊥⊥∫μ

Factorized softest gluons

of the softest gluon only. It does not involve other momenta

μ =+

+

+ symmetric contributions

⊥ k

⊥ k⊥ kk

k

k

New IR cut-offs for integrations over transverse

momenta of other virtual particles

⊥k

€

μ Is replaced by In the blobs with factorized gluons

Case B. The softest particle is a ladder quark or gluon. It can also be factorized:

μ = +k k

⊥ k

⊥ k

⊥ k

⊥ kDL contributions come from the region st <<− ⊥k

When

This case does not contribute when s ~ -t i.e. in the case of hard kinematics

2μ>−t such contributions disappear after applying μ∂∂

Combining case A and case B and adding Born terms leads to the IREE

quark pair

gluon pair

IREE look simpler when an integral transform (Mellin) is applied. For the Regge kinematics s >> -t, one should use the Sommerfeld-Watson transform or its simplified, Mellin-like, version :

)()(2

),( )()( ωωξπω ω

Fts

id

tsMi

i

±∞

∞−

± ⎟⎠

⎞⎜⎝

⎛=∫

where the signature factor2

1)( ±=

−±

πω

ξie

Inverse transform:

)/ln( ),,,(Im )xp(- )sin(

2)( )(

s

0

)( tstsMedF =−= ±∞

± ∫ ρωρρπω

ω

and2

),(),(),()( tsMtsMtsM

−±=±

difference with the standard Mellin transform

For singlet g1

Structure function Forward Compton amplitudewith negative signature

),(Im1 2)(

1 QxMg −=π

gq MMM +=

Compton off quark

Compton off gluon

System of IREE for the Compton amplitudes : gggqgqg

gqgqqqq

FHFHFy

FHFHFy

+=∂∂+

+=∂∂+

][

][

ω

ω

⎟⎟⎠

⎞⎜⎜⎝

⎛=

2

2

lnμ

Qy

)(ωikH Anomalous dimension matrix. Sums up DL and part of SL contributions

andwhere

IREE for the non-singlet g1 is simpler:

€

∂g1NS

∂ ln(Q2/μ 2)= (1+ω /2)HNS (ω)g1

Single Logarithms

Double Logarithms

New anomalous dimension HNSaccounts for the total resummation of DL and a part of SL of x

new non-singlet anomalous dimension, sums up DLs and SLs

Expression for the non-singlet g1 at large Q2: Q2 >> 1 GeV2

)(

2

22

1 )q( )H( -

x

1

22

ω

μωδ

ωωω

πω ω H

Qi

deg qNS

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛= ∫

Coefficient function

Anomalous dimension

Initial quark density

€

g1S =

< eq2 >

2

dω

2πi∫ 1

x

⎛

⎝ ⎜

⎞

⎠ ⎟ω

Cq(+)δq+ Cq

(+)δg( )Q2

μ 2

⎛

⎝ ⎜

⎞

⎠ ⎟

Ω ( + )

+ Cq(−)δq+ Cq

(−)δg( )Q2

μ 2

⎛

⎝ ⎜

⎞

⎠ ⎟

Ω ( − ) ⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

Expression for the singlet g1 at large Q2:

here GeV 5 ;22 ≈> μμQ

Large Q2 means

€

Ω(+) > Ω(−)

( ) ( )

( ) ( ) qQxxg

qQxg

NSNS

NSNS

NS

NS

δμμ

δμ

2/2220

2q

1

2/222q

1

/)()/(1 2

e ~

//1 2

e ~

ΔΔ

ΔΔ

++

Small –x asymptotics of g1: when x 0, the saddle-point method leads to

intercept

0.42 NS =Δ

0 >qδ

At large x, g1NS and g1

S are positive

large Q2

small Q2

As x0 > x, dependence on x is weak

In the whole range of x at any Q2

€

x0 =μ 2

2pq=μ 2

Q2x >> x

COMPASS: Q2 << 1 GeV2

€

g1NS > 0

( ) ( )

( ) ( ) 2/2220S

2q

1

2/22S

2q

1

/)()/(1 )S(2

e ~

//1 )S(2

e ~

SS

SS

Qxxg

Qxg

S

S

ΔΔ

ΔΔ

++Δ⟩⟨

Δ⟩⟨

μμ

μ22when μ>Q

22when μ<Q

g 0.08 - q -1.2 )S( S δδ=Δwhere 0.86 S =Δintercept

Asymptotics of the singlet g1 are more involved

(we assume that dq and dg are just constants)

Case A 0 g q 15 <+δδ

Case C

0 g q 15 >+δδCase B

0 g q 15 =+δδ

Sign of singlet g1:

g1 is positive at large x and goes to zero at small x

g1 is positive at large xand negative at small x

g1 is positive at large and small x

Negative and large

Negative but not large or positive

strong correlation = fine tuning

€

g1S > 0

€

g1S > 0

€

g1S > 0

Note on the singlet intercept:

A. Graphs with gluons only:

1.1 =ΔSviolates unitarity

B. All graphs0.86 =ΔS

No violation of unitarity

Values of the predictedintercepts perfectlyagree with results of several groups who have fitted theexperimental data.

Soffer-Teryaev, Kataev-Sidorov-Parente, Kotikov-Lipatov-Parente-Peshekhonov-Krivokhijine-Zotov,

Kochelev-Lipka-Vento-Novak-Vinnikov

similar to LO BFKL

non-singletintercept

singletintercept

Comparison of our results to DGLAP at finite x (no asymptotic formulae used)

Comparison depends on the assumed shape of initial parton densities.

The simplest option: use the bare quark input

(x) )( δδ =xq 1 )( =ωδq

in x- space in Mellin space

Numerical comparison shows that the impact of the total resummation of logs of x becomes quite sizable at x = 0.05 approx.

Hence, DGLAP should have failed at x < 0.05. However, it does not. Why?

Altarelli-Ball-Forte-Ridolfi

singularfactor

75.0 ,3.34 ,7.2 ,58.0 ≈≈≈≈ δγβα

])1)(x 1[( x)( - βδα γδ xNxq −+=

normalization

In order to understand what could be the reason for the success of DGLAP at small x, let us consider in more detail thestandard fits for initial parton densities.

regular factors

The parameters

are fixed from fitting experimental data at large x

In the Mellin space this fit is

)])1()(()[()( 11

1

1 −−∞

=

− −+++−++−= ∑ αωγαωαωωδ kkcNqk

k

Leading pole

Non-leading poles-k

The small-x DGLAP asymptotics of g1 is (inessential factors dropped): α(1/x) ~1

DGLAPg

Comparison with our asymptotics

( ) NSx Δ/1~g 1

shows that the singular factor in the DGLAP fit mimics the total resummation of ln(1/x) . However, the value = 0.58 differs from our non-singlet intercept =0.4

phenomenology

calculations

Although our and DGLAP asymptotics lead to an x- behavior ofRegge type, they predict different intercepts for the x- dependence and different Q2 -dependence:

( ) 2/22 1 / )/1(~g

ΔΔ μQxwhereas DGLAP predicts the steeper x-behavior and a flatter Q2 -behavior:

)(21 )(ln(1/x) ~ αγα QgDGLAP

x-asymptotics was checked by extrapolating the available exp. data to x 0. It agrees with our values of Contradicts DGLAP

our and DGLAP Q2 –asymptotics have notbeen checked yet.

our calculations

DGLAP

Common opinion: total resummation is not relevant at available values of x. Actually: the resummation has been accounted for through the fits tothe parton densities, however without realizing it.

Numerical comparison of DGLAP with our approach at small but finite x, using the same DGLAP fit for initial quark density.

R = g1 our/g1 DGLAP

Regular term in g1 our vs

regular + singular in g1 DGLAP

Only regular factors in g1

our and g1 DGLAP

x

R = g1 our/g1 DGLAP as function of Q2 at different x

X = 10-4

X = 10-3

X = 10-2

Q2 -dependence of R is flatter than the x-dependence

Q2

Q2

R = g1 our/g1 DGLAP

Structure of DGLAP fit

])1)(x 1[( x)( - βδα γδ xNxq −+=

Can be dropped when ln(x) are resummed

x-dependence is weak at x<<1 and can be dropped

ax) N(1 )( +≈xqδTherefore at x << 1

Common opinion: in DGLAP analyses the fits to the initial parton densities are related to the structure of hadrons, so they mimic effects of Non-Perturbative QCD, using the phenomenological parameters fixed from experiments .

Actually, the singular factors introduced in the fits mimic the effects of Perturbative QCD at small x and can be dropped when logarithms of x are resummed

Non-Perturbative QCD effects are included in the regular parts of DGLAP fits. Obviously, the impact of Non-Pert QCD is not strong in the region of small x. In this region, the p. densities can be approximated by an overall factors N and a linear term in x

Comparison between DGLAP and our approach at small x

DGLAP our approach

Coeff. functions and anom. dimensions are calculated with two-loop accuracy

Coeff. functions and anom.dimensions sum up DL and SL terms to all orders

To ensure the Regge behaviour,singular terms in x are used in initial partonic densities.

Regge behavior is achieved automatically, even when the initial densities are regular in x

This could be equivalent phenomenologically, but could be

Asymptotic formulae of g1 arenever used in expressions for g1

at finite xunreliable for g1 at very small x.

SUGGESTION: merging of our approach and DGLAP

1. Expand our formulae for coefficient functions and anomalous dimensions into series in the QCD coupling

2. Replace the first- and second- loop terms of the expansion by

corresponding DGLAP –expressions

DGLAP

Good at large x because includes exact two-loop calculations but bad at small x as lacks the total resummaion of ln(x)

our approach

Good at small x , includes the total resummaion of ln(x) but bad at large x because neglects some contributions essential in this region

Comparison between DGLAP and our approach at any x

New “synthetic” formulae include all advantages of both

approaches and should be equally good at large and small x.

New fits for the part. densities should not involve singular factors.

Our expressions

))(/()( ))](()[2/1()( 2/12 ωωωωωωωω HCBH −=−−=

anomalous dimension coefficient

function

First tems of the perturbative expansions in series

⎥⎦

⎤⎢⎣

⎡ +=⎥⎦

⎤⎢⎣

⎡ +=ωωπ

ωωπ

ω2

11

2

)(

2

11

2

)(211

FF CAC

CAH

New “synthetic” formulae:

DGLAPLODGLAPLO CCCcHHHh 1 1 +−=+−=

COMPASS is a high-energy physics experiment at the Super Proton Synchrotron (SPS) at CERN in Geneva, Switzerland. The purpose of this experiment is the study of hadron structure and hadron spectroscopy with high intensity muon and hadron beams.  On February 1997 the experiment was approved conditionally by  CERN and the final Memorandum of Understanding was signed in September 1998. The spectrometer was installed in 1999 - 2000 and was commissioned during a technical run in 2001. Data taking started in summer 2002 and continued until fall 2004. After one year shutdown in 2005, COMPASS will resume data taking in 2006.   Nearly 240 physicists from 11 countries and 28 institutions work in COMPASS

COMPASS COmmon Muon Proton Apparatus for Structure and Spectroscopy

Artistic view of the 60 m long COMPASS two-stage spectrometer. The two dipole magnets are indicated in red

COMPASS runs at small Q2 and small x

In order to generalize our results to the region of small Q2 , one should

recall that is the result of the integration)/ln( 22 μQ ∫⊥

⊥

2

22

2Q

k

dk

μ

However this is valid for large Q2 only. For arbitrary Q2 :

⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

+→ ∫∫

⊥

⊥

⊥

⊥2

22

022

2

2

2 Qln

22

2 μ

μ

μμ

QQ

k

dk

k

dk

Introduce a “mass” of virtual quarks and gluons to regulate infrared singularities

This is suggested by the relevant Feynman diagrams involved.

)(

2

22

0

2

1

)q( )H( -

xx

1

22ω

μμωδ

ωωω

πω

ω

H

Q

ide

g qNS

⎟⎟⎠

⎞⎜⎜⎝

⎛ +⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

+= ∫

Coefficient function

Anomalous dimension

Initial quark density

pq

Qx

pqx

22

22

0 =>=μ

weak x -dependence

weak Q2 -dependence

It leads to new expressions: Small Q2 non-singlet g1

( )

( ))(

)(

2

22)()(

2

22)()(

0

2

1

1

22

−

+

Ω

−−

Ω

++

⎟⎟⎠

⎞⎜⎜⎝

⎛ +++

⎟⎟⎠

⎞⎜⎜⎝

⎛ ++

⎟⎟⎠

⎞⎜⎜⎝

⎛

+

><= ∫

μ

μδδ

μ

μδδ

π

ωω

QgCqC

QgCqC

xxi

deg

gq

gq

qS22 μ<Q

pq

Qx

pqx

2 ,

2

22

0 ==μ

again

weak Q2-dependence

Small Q2 Singlet g1

At Q2 < 1 GeV2, the x-dependence is almost flat even at x<<1

PREDICTION:

QuickTime™ e undecompressore TIFF (LZW)

sono necessari per visualizzare quest'immagine.

C O M P A S S

(Phys.Lett.B647, 330, 2007)

Power Q2 -corrections

In practice, the power corrections are obtained phenomenologicallythrough a discrepancy between DGLAP –predictions and experimental data. However,

There exist contributions ~1/(Q2)k , with k = 1,2,.. They can be of perturbative (renormalons) and non-perturbative (higher twists) origin.

k

ka∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ +2

2

2

2

2

22

Q

Qln

Qln

μμμ

μ

when 22 μ>Q

and

k

ka∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ +2

2

2

22 QQln

μμμ

when 22 μ<Q

New power terms. TakenInto account, they can

change impact of higher twists

Conclusions

Standard Approach is based on the DGLAP evolution equations and phenomenological fits for the initial parton densities. DGLAP was originally developed, and is very successful, in the region where both x and Q2 are large, but neglects logs of (1/x), so it cannot applied safely at very small x.

In order to extend SA to the region of small x, phenomenological singular factors x-a have to be incorporated into the parton densities.These factors mimic the total resummation of leading logs of x that weperform in our approach, which leads naturally to the Regge behaviour of g1 at small x with predicted intercepts, in agreement with exps. SUGGESTION: merging of our approach and DGLAP

The region of small Q2 is also beyond the reach of SA. In our model of extension at small Q2 we predict that g1 is almost independent of x, even at x<< 1. Instead, it depends on 2pq only. In particular g1 can be pretty close to zero in the range of 2pq investigated experimentally by COMPASS. This agrees with the data.