Inclusive Radiative Corrections in COMPASS and …...Jefferson Lab, May 16 – 19, 2016 B. Badelek...

Inclusive Radiative Corrections in COMPASSand Input Information for the Present and Future

RC Programs

Barbara BadelekUniversity of Warsaw

Precision Radiative Correctionsfor Next Generation Experiments

Jefferson Lab, May 16 – 19, 2016

B. Badelek (Warsaw ) Input for RC RC JLab, 2016 1 / 49

Mo & Tsai and Dubna schemes

Outline

1 Mo & Tsai and Dubna schemes

2 Extension of F2(x,Q2) down to Q2 = 0Data at low Q2

JKBBMartin-Ryskin-Stasto(Modified) saturation modelALLM97ZEUS Regge fit

3 Extension of R(x,Q2) down to Q2 = 0

4 Extension of g1(x,Q2) down to Q2 = 0

5 Outlook



Lowest order radiative processes

Mo and Tsai scheme: b) – d)

Dubna scheme b) – g) but replaces e) – g) by:

ccccccBadelek, Bardin, Kurek, Scholz, Z.Phys. C66 (1995) 591



Mo and Tsai scheme: FERRAD, model indep., non-covariant

Measured cross section:

δR(∆) is a residue of cancellation of IR divergent termsσtails processes where real photons of Eγ > ∆ are emitted



Mo and Tsai scheme: FERRAD...cont’d

Here: σmeas = vσ1γ + σtails = vσ1γ + σinel + σel + σqelv – virtual corrections + soft photon emission.

Range of kinematical variables from which the radiative tails contribute to the cross sectionmeasured at the point A(Q2, ν); parallel lines: W = const

Even if we measure at DIS, information onF1, F2 (or R, F2) needed down to Q2 =0!

Weak dependence of η(x, y) = σ1γ/σmeas on ∆(here E = 280 GeV)

Badelek, Bardin, Kurek, Scholz, Z. Phys. C66 (1995) 591



Dubna scheme: TERAD (also: POLRAD), QPM, covariant

Measured cross section: Born × { resummed collinear γ+ remnant of exponentiation+ remnant of subtraction in σin(vertex) }

inelastic radiative tailand regularization=⇒ Dubna scheme ∆ indep.

QPM calculationsof RC for hadron current

elastic radiative tailsas in MT but covariant

“Vacuum polarisation” through running of α(Q2)

O(α2) in amplitude implemented

Weak loop correction also present



FERRAD vs TERAD (µp, 280 GeV)

η(x, y) = σ1γ/σmeas ηF /ηTopen symbols = FERRAD open symbols = FERRAD without τ τ , qqclosed symbols = TERAD closed symbols = full FERRAD

Badelek, Bardin, Kurek, Scholz, Z. Phys. C66 (1995) 591



Input information for polarised/unpolarisedand inclusive/semiinclusive RC calculations

The items below should be known forxmeas < x < 1 and 0 < Q2 < Q2

max

Spin independent structure function F2(x,Q2) (nucleon, nuclei)Spin independent structure function R(x,Q2)

Spin dependent structure function g1(x,Q2)

Quasielastic suppression factors(Q2) (nuclei)Elastic form factors(Q2) (nucleon, nuclei)

All the input infoirmation is collected in a COMPASS note 2015-6,for the moment not accessible for outsiders but this may be changed




Extension of F2(x,Q2) down to Q2

= 0

Outline






5 Outlook



= 0

F2(x,Q2) and R(x,Q2) in the low Q2 region

(see e.g. Badełek, Kwiecinski, Rev. Mod. Phys. 68 (1996) 445)

F2 and R needed at: xmeas < x < 1 and 0 < Q2 < Q2max

They are either physics motivated fits or models of dynamic origin andhave to have a proper asymptotic behaviour:at Q2 →0 fulfilling the conditions for arbitrary ν

F2 = O(Q2),F1

M+F2

M

pq

q2= O(Q2).

and

R(x,Q2) =σL

σT=

(1 + 4M2x2/Q2)F2

2xF1− 1 =

FL

2xF1−→ 0 at Q2 → 0

and at Q2 →∞ joining the QCD improved parton model expressions.

Observe that:Growth of F2 with decreasing x is slower at low Q2

R(x,Q2) essentially independent of x in the low x, low Q2 region.



= 0 Data at low Q2

Outline






5 Outlook



= 0 Data at low Q2

What do the F2 data show around Q2 = 1 GeV2?

PDG 2015



= 0 Data at low Q2

What do the F2 data show around Q2 = 1 GeV2?... cont’d

10-1

1

10

10 2

10 3

10-2

10-1

1 10 Q2 (GeV2)

F

2 (x

= Q

2 /sy

, Q2 )

(× 4096)

(× 2048)

(× 1024)

(× 512)

(× 256)

(× 128)

(× 64)

(× 32)

(× 16)

(× 8)

(× 4)

(× 2)

(× 1)

(× 1024)

(× 512)

(× 256)

(× 128)

(× 64)

(× 32)

(× 16)

(× 2048)

(× 1024)

(× 512)

(× 256)

(× 128)

(× 64)

(× 32)

(× 16)

(× 8)

(× 4)

(× 1024)

(× 512)

(× 256)

(× 128)

(× 64)

(× 32)

(× 16)

(× 8)

(× 4)

(× 2)

(× 1)

y=0.8(× 4096)

y=0.7(× 2048)

y=0.6(× 1024)

y=0.5(× 512)

y=0.4(× 256)

y=0.33(× 128)

y=0.26(× 64)

y=0.2(× 32)

y=0.12(× 16)

y=0.05(× 8)

y=0.025(× 4)

y=0.015(× 2)

y=0.007(× 1)

H1 SVX 1995 H1 1994

ZEUS BPT 1997 ZEUS SVX 1995 ZEUS 1994

ZEUS QCD fitZEUS Regge fit

ZEUS 1997

hep-ex/0008069 ZEUS, Eur. Phys. J. C7 (1999) 609



= 0 JKBB

Outline






5 Outlook



= 0 JKBB

Parametrizations of F2 in the low Q2, low x regionJKBB (Kwiecinski, Badełek, Z. Phys. C43 (1989) 43; Phys. Lett. B295 (1992) 263)

The starting point is the Generalised Vector Meson Dominance (GVMD) representation of thestructure function F2(x,Q2):

F2[x = Q2/(s+Q2 −M2), Q2] =Q2

4π

∑v

M4vσv(s)

γ2v(Q2 +M2v )2

+Q2

∫ ∞Q2

0

dQ′2Φ(Q′2, s)

(Q′2 +Q2)2

≡ F(v)2 (x,Q2) + F

(p)2 (x,Q2) (1)

The function Φ(Q2, s) is expressed as follows:

Φ(Q′2, s) = −1

πIm

∫ −Q′2 dQ′′2Q′′2

FAS2 (x′, Q′′2) (2)

Asymptotic structure function FAS2 (x,Q2) assumed to be given.By construction, F2(x,Q2)→ FAS2 (x,Q2) for large Q2.The first term in (1) corresponds to the low mass vector meson dominance.Contribution of vector mesons heavier then Q0 is included in the integral in (1).This integral can be looked upon as the extrapolation of the (QCD improved) parton modelfor arbitrary Q2 (including Q2 =0).The representation (1) is written for fixed s and is expected to be valid at s� Q2, i.e. atlow x but for arbitrary Q2 – and above the resonances.



= 0 JKBB

F p2 in the low Q2, low x region...cont’d

JKBB...cont’d

Choosing the parameter Q20 > (M2

v )max where (Mv)max is the mass of the heaviestvector meson included in the sum one explicitly avoids double counting when adding twoseparate contributions to F2.

Q0 should be smaller than the mass of the lightest vector meson not included in the sum.

Representation (1) for the partonic part F (p)2 (x,Q2) may be simplified as follows:

F(p)2 (x,Q2) =

Q2

(Q2 +Q20)FAS2 (x, Q2 +Q2

0) (3)

where

x =Q2 +Q2

0

s+Q2 −M2 +Q20

≡Q2 +Q2

0

2Mν +Q20

(4)

Simplified parametrization (3) connecting F (p)2 (x,Q2) to FAS2 by an appropriate change of

the arguments posseses all the main properties of the second term in (1).

Apart from Q20, constrained by physical requirements, the representation (1) does not contain

any other free parameters except those which are implicitly present in FAS2 .



= 0 JKBB


JKBB...cont’d

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H1 Collaboration, DESY 97-042



= 0 JKBB


JKBB...cont’d

0

0.1

0.2

0.3

10-5

10-4

10-3

10-2

F 2

Q2 = 0.11 GeV2

0

0.1

0.2

0.3

10-5

10-4

10-3

10-2

Q2 = 0.15 GeV2

0

0.1

0.2

0.3

10-5

10-4

10-3

10-2

Q2 = 0.20 GeV2

0

0.1

0.2

0.3

0.4

10-5

10-4

10-3

10-2

F 2

Q2 = 0.25 GeV2

0

0.1

0.2

0.3

0.4

10-5

10-4

10-3

10-2

Q2 = 0.30 GeV2

0

0.1

0.2

0.3

0.4

10-5

10-4

10-3

10-2

Q2 = 0.40 GeV2

0

0.2

0.4

0.6

10-5

10-4

10-3

10-2

F 2

Q2 = 0.50 GeV2

0

0.2

0.4

0.6

10-5

10-4

10-3

10-2

Q2 = 0.65 GeV2

ZEUS BPC 95ZEUS 94

E665H1 94 H1 SVTX 95

DLCKMT

GRV(94)

BKABY

0

0.5

1

10-5

10-4

10-3

10-2

x

F 2

Q2 = 1.50 GeV2

0

0.5

1

10-5

10-4

10-3

10-2

x

Q2 = 3.00 GeV2

0

0.5

1

10-5

10-4

10-3

10-2

x

Q2 = 6.50 GeV2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10-6

10-5

10-4

10-3

10-2

10-1

F2 (model JK + BB)continuous - dF2 /dlogQ2

broken - dFVMD2 /DLOGQ2

dotted - dFpart2 /dlogQ2

points - ZEUS data (Vancouver 1998)

x

dF2/

dlog

Q2



= 0 Martin-Ryskin-Stasto

Outline






5 Outlook




F p2 in the low Q2, low x region

Martin-Ryskin-Stasto (Martin, Ryskin, Stasto, Eur. Phys. J. C7 (1999) 643)

Exploits further the idea of BBJK.

Perturbative and non-perturbative QCD contributions separated by the distanceconfigurations of the qq pair in the γ∗ → qq:

small distance configurations (k2T > k20) given by pQCD (unified equations, DGLAP +BFKL, unintegrated gluon distribution);

large distance configurations (k2T < k20) given by VMD (for low qq fluctuation masses,M2 < Q2

0), and additive quark model (for high qq masses, M2 > Q20).

Excellent description of the data throughout the whole Q2 region, including Q2 =0.

Fitted (at x <0.05) are 3 parameters of the gluon distribution; scales k20 and Q20 chosen as:

k20 =0.2 GeV2 (crucial) and Q20 =1.5 GeV2. Choice of k20 yields physically sensible g and

FL.

Interference between states of different qq masses is crucial for description of the data.

Importance of the perturbative contribution in the non-perturbative domain.





Martin-Ryskin-Stasto...cont’d

γ*M′2

p

z,kT

Im Aqq+p

p

M2

Fig.1

Fig.5

10 2

10 3

10 4

10-2

10-1

1 10 102

W=245 GeV (x128)

W=210 GeV (x64)

W=170 GeV (x32)

W=140 GeV (x16)

W=115 GeV (x8)

W=95 GeV (x4)

W=75 GeV (x2)

W=60 GeV (x1)

ZEUS-BPCH1-95H1-94H1ZEUSMainusch-thesis

Fit A (k02=0.2 GeV2)

Fit B (k02=0.5 GeV2)

Q2(GeV2)

σ γ*p(

µb)

≈≈



= 0 (Modified) saturation model

Outline






5 Outlook



= 0 (Modified) saturation model


(Modified saturation model, Bartels, Golec-Biernat, Kowalski, Phys. Rev. D66 (2002) 014001)

• Original saturation modelof Golec-Biernat and Wüsthoffmodified by DGLAP

• 5 parameters fitted toE665, H1 and ZEUS data,x < 0.01, 0.1< Q2 <500 GeV2

(claimed to be valid down toQ2 = 10−5 GeV2).



= 0 ALLM97

Outline






5 Outlook



= 0 ALLM97


ALLM97 (Abramowicz, Levy, hep-ph/9712415)

Parametrization of the σtot(γ∗p) at W 2 >∼ 3 GeV2 (above resonances).Valid everywhere in x and Q2 (including photoproduction).Based on Regge–type approach; extention to large Q2 compatible with QCD.Observe that it is a fit of 23 parameters to all the dataFit contains contributions of the pomeron (P ) and reggeon (R):

F2(x,Q2) =Q2

Q2 +m20

[FP2 (x,Q2) + FR2 (x,Q2)

](5)

of the form

FP2 (x,Q2) = cP (t)xα(t)P (1− x)bP (t), FR2 (x,Q2) = cR(t)x

α(t)R (1− x)bR(t) (6)

where

t = ln

(lnQ2 +Q2

0

Λ2/ ln

Q20

Λ2

)(7)

and1

xP= 1 +

W 2 −M2

Q2 +m2P

,1

xR= 1 +

W 2 −M2

Q2 +m2R

(8)

Here M is the proton mass; m20,m

2P ,m

2R, Q

20 allow a smooth transition to

photoproduction. For Q2 � m2P , Q

2 � m2R, xP → x, xR → x;

cR, aR, bR, bP ↗ Q2 ↗; cP , aP ↘ Q2 ↗.



= 0 ALLM97


ALLM97...cont’d

(σeffγ∗p ≈ σtotγ∗p at HERA energies)



= 0 ALLM97


ALLM97...cont’d

1

10

10 2

10 3

10 4

10 5

102

103

104

105

W2(GeV2)

σ totγ*

p (µb

) (sc

aled

)

0.11 (x 128)

ZEUS BPC 95 ZEUS 94 E665ZEUS, H1 γpH1 94

H1 SVTX 95 low W γpDL

ALLM

CKMT

GRV(94)

BK

W2(GeV2)

σ totγ*

p (µb

) (sc

aled

)

0.15 (x 64)

W2(GeV2)

σ totγ*

p (µb

) (sc

aled

)

0.20 (x 32)

W2(GeV2)

σ totγ*

p (µb

) (sc

aled

)

0.25

0.23 (x 16)

(x 16)

W2(GeV2)

σ totγ*

p (µb

) (sc

aled

)

0.30

0.31 (x 8)

(x 8)

W2(GeV2)

σ totγ*

p (µb

) (sc

aled

)

0.40

0.43 (x 4)

(x 4)

W2(GeV2)

σ totγ*

p (µb

) (sc

aled

)

0.50 (x 2)

W2(GeV2)

σ totγ*

p (µb

) (sc

aled

)

0.65

0.59 (x 1)

(x 1)

W2(GeV2)

σ totγ*

p (µb

) (sc

aled

)

1.50

1.50 (x 1/2)

(x 1/2)

W2(GeV2)

σ totγ*

p (µb

) (sc

aled

)

3.00

2.80 (x 1/4)

(x 1/4)

W2(GeV2)

σ totγ*

p (µb

) (sc

aled

)

6.50

7.16 (x 1/8)

(x 1/8)

Q2 (E665)

Q2 (ZEUS)

0.00 (x 256)

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1 10 100 1000 10000 10000010

-2

10-1

1

10

10 2

1 10 102

103

104

105

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]:_B. Badelek (Warsaw ) Input for RC RC JLab, 2016 28 / 49


= 0 ZEUS Regge fit

Outline






5 Outlook



= 0 ZEUS Regge fit


ZEUS Regge fit (ZEUS, Eur. Phys. J. C7 (1999) 609)

Combines the Q2 dependence of the VMD with the energydependence from the Regge model:

F2(x,Q2) =

(Q2

4π2α

)·(

M20

M2 +Q2

)·[AR · (W 2)αR−1 +AP · (W 2)αP−1

]where AR, AP , M0 are constants; αR, αP are reggeon and pomeronintercepts. Fixed: M2

0 = 0.53 GeV2, αR = 0.53.Remaining 3 parameters fitted to Q2 =0 data at W 2 >3 GeV2.

Result: αP = 1.097 ± 0.002.



= 0 ZEUS Regge fit


ZEUS Regge fit...cont’s

0

0.05

0.1

0.15

0.2

10-6

10-4

0

0.05

0.1

0.15

0.2

10-6

10-4

0

0.05

0.1

0.15

0.2

10-6

10-4

0

0.1

0.2

0.3

10-6

10-4

0

0.1

0.2

0.3

10-6

10-4

0

0.1

0.2

0.3

10-6

10-4

0

0.2

0.4

10-6

10-4

0

0.2

0.4

10-6

10-4

0

0.2

0.4

10-6

10-4

0

0.2

0.4

0.6

10-6

10-40

0.2

0.4

0.6

10-6

10-4

x

F

2 (x

,Q2 )

Q2=0.045 GeV2

x

F

2 (x

,Q2 )

Q2=0.065 GeV2

x

F

2 (x

,Q2 )

Q2=0.085 GeV2

x

F

2 (x

,Q2 )

Q2=0.11 GeV2

x

F

2 (x

,Q2 )

Q2=0.15 GeV2

x

F

2 (x

,Q2 )

Q2=0.2 GeV2

x

F

2 (x

,Q2 )

Q2=0.25 GeV2

x

F

2 (x

,Q2 )

Q2=0.3 GeV2

x

F

2 (x

,Q2 )

Q2=0.4 GeV2

x

F

2 (x

,Q2 )

Q2=0.5 GeV2

x

F

2 (x

,Q2 )

Q2=0.65 GeV2

H1 SVX 1995 ZEUS BPC 1995

ZEUS BPT 1997 ZEUS SVX 1995

E665ZEUS Regge fit

ZEUS 1997

10-1

1

10

10 2

10 3

10-2

10-1

1 10 Q2 (GeV2)

F

2 (x

= Q

2 /sy

, Q2 )

(× 4096)

(× 2048)

(× 1024)

(× 512)

(× 256)

(× 128)

(× 64)

(× 32)

(× 16)

(× 8)

(× 4)

(× 2)

(× 1)

(× 1024)

(× 512)

(× 256)

(× 128)

(× 64)

(× 32)

(× 16)

(× 2048)

(× 1024)

(× 512)

(× 256)

(× 128)

(× 64)

(× 32)

(× 16)

(× 8)

(× 4)

(× 1024)

(× 512)

(× 256)

(× 128)

(× 64)

(× 32)

(× 16)

(× 8)

(× 4)

(× 2)

(× 1)

y=0.8(× 4096)

y=0.7(× 2048)

y=0.6(× 1024)

y=0.5(× 512)

y=0.4(× 256)

y=0.33(× 128)

y=0.26(× 64)

y=0.2(× 32)

y=0.12(× 16)

y=0.05(× 8)

y=0.025(× 4)

y=0.015(× 2)

y=0.007(× 1)

H1 SVX 1995 H1 1994

ZEUS BPT 1997 ZEUS SVX 1995 ZEUS 1994

ZEUS QCD fitZEUS Regge fit

ZEUS 1997

ZEUS, Eur. Phys. J. C7 (1999) 609


Extension of R(x,Q2) down to Q2

= 0

Outline






5 Outlook



= 0

R and FL in the low Q2, low x regionBKS (Badelek, Kwiecinski, Stasto, Z. Phys. C74 (1997) 297)

A model for FL, valid at low x and low Q2;based on the photon–gluon fusion, essential at low xand extended to low Q2. Similar approach inNikolaev, Zakharov, Z. Phys. C49 (1991) 607, C53 (1992) 331

The model embodies the constraint FL ∼ Q4 at Q2 →0.

FL =

∫ 1

x

dx′

x′

∫dk0Tk0T

F 0L(x′, Q2, k0T )f(

x

x′i, k2T )

where F 0L comes from γ∗g fusion, is a longitudinal

structure function of the off-shell gluon of virtuality k2T and is calculated perturbatively; f isan unintegrated gluon distribution related to the “ordinary” g(y, µ2) by:

yg(y, µ2) =

∫ µ2dk2Tk2T

f(y, k2T )

Its evolution is controlled by (approximate) BFKL.

To extrapolate FL to low Q2 and to Q2 =0, evolution of g(y,Q2) and argument of αs(Q2)was frozen via Q2 → Q2 + 4m2

q .

HT contribution needed at moderate Q2, i.e. terms vanishing as 1/Q2 for Q2 →∞. Theywere assumed to originate from low values of the quark transverse momenta andinterpreted as coming from soft pomeron exchange (intercept = 1). Such HT has a properbehaviour both at Q2 →∞ and Q2 →0.

x / x’, kT

2

Q2

FL ^ 0 2k

f



= 0

R and FL in the low Q2, low x regionBKS (Badelek, Kwiecinski, Stasto, Z. Phys. C74 (1997) 297)...cont’d all with HT

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

10-4 10-3 10-2 10-1

R(x

,Q2 )

GRV off-shellGRV on-shellMRS(A) off-shellMRS(A) on-shell

Q2=0.1 GeV2

0.1

0.2

0.3

0.4

0.5

10-4 10-3 10-2 10-1

Q2=0.5 GeV2

0.1

0.2

0.3

0.4

0.5

10-4

10-3

10-2

10-1

Q2=1.5 GeV2

0.1

0.2

0.3

0.4

0.5

10-4

10-3

10-2

10-1

Q2=5.0 GeV2

x

0.1

0.2

0.3

0.4

0.5

10-1 1 10

R(x

,Q2 )

x=0.0001

0.1

0.2

0.3

0.4

0.5

10-1 1 10

x=0.001

0.1

0.2

0.3

0.4

0.5

10-1

1 10

x=0.01

0.1

0.2

0.3

0.4

0.5

10-1

1 10

GRV off-shell

GRV on-shell

MRS(A) off-shell

MRS(A) on-shell

x=0.1

Q2(GeV2)



= 0

R and FL in the low Q2, low x regionBKS (Badelek, Kwiecinski, Stasto, Z. Phys. C74 (1997) 297)...cont’d all with no HT

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

10-4 10-3 10-2 10-1

FL(

x,Q

2 )

Q2=0.1 GeV2

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

10-4 10-3 10-2 10-1

Q2=0.5 GeV2

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

10-4

10-3

10-2

10-1

Q2=1.5 GeV2


0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

10-4

10-3

10-2

10-1

Q2=5.0 GeV2

x

0.1

0.2

0.3

0.4

0.5

0.6

0.7

10-1 1 10

FL(

x,Q

2 )

x=0.0001

0.1

0.2

0.3

0.4

0.5

0.6

0.7

10-1 1 10

x=0.001


0.02

0.04

0.06

0.08

0.1

0.12

0.14

10-1

1 10

x=0.01

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

10-1

1 10

x=0.1

Q2(GeV2)



= 0

R and FL in the low Q2, low x regionBKS (Badelek, Kwiecinski, Stasto, Z. Phys. C74 (1997) 297)...cont’d all with HT

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

10-4 10-3 10-2 10-1

FL(

x,Q

2 )

Q2=0.1 GeV2

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

10-4 10-3 10-2 10-1

Q2=0.5 GeV2

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

10-4

10-3

10-2

10-1


Q2=1.5 GeV2

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

10-4

10-3

10-2

10-1

Q2=5.0 GeV2

x

0.1

0.2

0.3

0.4

0.5

0.6

0.7

10-1 1 10

FL(

x,Q

2 )

x=0.0001

0.1

0.2

0.3

0.4

0.5

0.6

0.7

10-1 1 10

x=0.001


0.02

0.04

0.06

0.08

0.1

0.12

0.14

10-1

1 10

x=0.01

0.02

0.04

0.06

0.08

0.1

0.12

0.14

10-1

1 10

x=0.1

Q2(GeV2)



= 0

R and FL in the low Q2, low x region

x range of points: ∼10−5 – 0.01 ICHEP04/Abstract 5–0161


Extension of g1(x,Q2) down to Q2

= 0

Outline






5 Outlook



= 0

Measurements of gp1(x) for proton

Q2 >1 (GeV/c)2 Q2 <1 (GeV/c)2

COMPASS, PLB753 (2016) 18 COMPASS DIS2016



= 0

g1 at low Q2, method IBadelek, Kiryluk, Kwiecinski, Phys. Rev. D61 (2000) 014009

The following representation of g1 was assumed:

g1(x,Q2) = gVMD1 (x,Q2) + gpart1 (x,Q2) (9)

gpart1 at low x is controlled by the ln2(1/x) terms; it was parametrised as discussed in Kwiecinski,

Ziaja, Phys. Rev. D60 (1999) 054004. gVMD1 (x,Q2) was represented as:

gVMD1 (x,Q2) =

Mν

4π

∑V=ρ,ω,φ

M4V ∆σV (W 2)

γ2V (Q2 +M2V )2

(10)

The unknown cross sections ∆σV (W 2) are combinations of the total cross sections for thescattering of polarised vector mesons and nucleons. At high W 2: ∆σV = (σ1/2 − σ3/2)/2Assume:

Mν

4π

∑V=ρ,ω

M4V ∆σV

γ2V (Q2 +M2V )2

=

C

[4

9

(∆u0val(x) + 2∆u0(x)

)+

1

9

(∆d0val(x) + 2∆d0(x)

)] M4ρ

(Q2 +M2ρ )2

, (11)

Mν

4π

M4φ∆σφp

γ2φ(Q2 +M2φ)2

= C2

9∆s0(x)

M4φ

(Q2 +M2φ)2

, (12)



= 0

g1 at low Q2, method I...cont’d

Each ∆p0j (x)→ x0 for x→0. Thus ∆σV → 1/W 2 at large W 2, i.e. zero intercept of theappropriate Regge trajectories.

Results for the spin asymmetry, A1 = g1/F1, for the proton, and for different C:

C ?? C <0 ?

-0.02

0

0.02

0.04

0.06

0.08

10-4

10-3

x

Ap1 Q2 < 1. GeV2

SMC data

C=+4C=0C=-4

Q2 = 0.02 0.06 0.1 0.2 0.3 0.6 GeV2

0

0.05

0.10

0.15

0.20

0.02 0.04 0.06 0.08 0.1x

Ap1 Q2 < 1 GeV2

E143 data,

Ebeam = 16.2 GeV2

C=0

C=-2

C=-4

Q2 = 0.5 0.6 0.7 0.8 0.9 1. GeV2



= 0

g1 at low Q2, method IIBadelek, Kwiecinski, Ziaja Eur. Phys. J. C26 (2002) 45

The following representation of g1 was assumed, valid for fixed W 2 � Q2, i.e. smallx = Q2/(Q2 +W 2 −M2):

g1(x,Q2) = gL1 (x,Q2) + gH1 (x,Q2) =Mν

4π

∑V

M4V ∆σV (W 2)

γ2V (Q2 +M2V )2

+ gAS1 (x, Q2 +Q20). (13)

The first term sums up contributions from light vector mesons, MV < Q0, Q20 ∼ 1 GeV2. The

unknown ∆σV are expressed through the combinations of nonperturbative parton distributions,evaluated at fixed Q2

0, similar to method I.

The second term, gH1 (x,Q2), represents the contribution of heavy (MV > Q0) vector mesons tog1(x,Q2) can also be treated as an extrapolation of the QCD improved parton model structurefunction, gAS1 (x,Q2), to arbitrary values of Q2: gH1 (x,Q2) = gAS1 (x, Q2 +Q2

0). The scalingvariable x is replaced by x = (Q2 +Q2

0)/(Q2 +Q20 +W 2 −M2). It follows that at large Q2,

gH1 (x,Q2)→ gAS1 (x,Q2). Thus:

g1(x,Q2) = C

[4

9(∆u0val(x) + 2∆u0(x)) +

1

9(∆d0val(x) + 2∆d0(x))

]M4ρ

(Q2 +M2ρ )2

+ C

[1

9(2∆s0(x))

]M4φ

(Q2 +M2φ)2

+ gAS1 (x, Q2 +Q20). (14)



= 0

g1 at low Q2, method II...cont’dNow fixing C in the photoproduction limit via the DHGHY sum rule.

The γ∗p scattering amplitude fulfills the dispersion relation:

S1(ν, q2) = 4

∫ ∞−q2/2M

ν′dν′G1(ν′, q2)

(ν′)2 − ν2(15)

whereG1(ν, q2) =

M

νg1(x,Q2) (16)

in the Q2, ν →∞ limit.

As a result of Low’s theorem: S1(0, 0) = −κ2p(n), G1 in the Q2 → 0 limit fulfills

the DHGHY sum rule: ∫ ∞0

dν

νG1(ν, 0) = −1

4κ2p(n). (17)



= 0

g1 at low Q2, method II...cont’d

At ν →0, eq.(15) is:

S1(0, q2) = 4M

∫ ∞Q2/2M

dν

ν2g1(x(ν), Q2). (18)

Now we define the DHGHY moment, I(Q2) as:

I(Q2) = S1(0, q2)/4 = M

∫ ∞Q2/2M

dν

ν2g1(x(ν), Q2). (19)

Before taking the Q2 → 0 limit of (18), observe that it is valid only down to some threshold valueof W , Wth

<∼ 2 GeV (above resonances). Requirement W > Wth gives the lower limit forintegration over ν in (18), where νt(Q2) = (W 2

t +Q2 −M2)/2M :

I(Q2) = Ires(Q2) +M

∫ ∞νt(Q2)

dν

ν2g1(x(ν), Q2

). (20)

Here Ires = contribution of resonances. The DHGHY sum rule now implies:

I(0) = Ires(0) +M

∫ ∞νt(0)

dν

ν2g1 (x(ν), 0) = −κ2p(n)/4. (21)



= 0

g1 at low Q2, method II...cont’d

Thus action plan for extracting C in eq.(14):

take g1(x(ν), 0), eq. (14); C is the only free parameter,

put it into eq. (21),

take Ires(0) from measurements,

extract C from eq. (21).

Taking:

Ires(0) from photoproduction, Wt=1.8 GeV GDH, Nucl. Phys. 105 (2002) 113,

gAS1 prametrized by NLO GRSV2000 Phys.Rev. D63 (2001) 094005

nonperturbative ∆p(0)j (x) at Q2 = Q2

0 = 1.2 GeV2 from

1 GRSV2000 =⇒ C = –0.302 “flat” ∆p

(0)j (x) = Ni(1− x)ηi =⇒ C = –0.24.



= 0



= 0

g1 at low Q2, method II...cont’dByproducts: g1 from eq.(14) and the DHGHY moment, I(Q2), eq.(19). Results for the proton:

-1

0

1

Q2=0.01 GeV2 x = 0.0001

-1

0

1

Q2= 0.1 GeV2 x = 0.001

-1

0

1

Q2= 1 GeV2 x = 0.01

-1

0

1

10-4

10-3

10-2

10-1

Q2= 10 GeV2

10-1

1 10

x = 0.1

x Q2 [GeV2]

g 1(x,

Q2 )

broken lines – gAS1 , dotted – gL1 ,

continuous – total g1

-0.002-0.001

00.0010.0020.0030.0040.0050.0060.0070.0080.009

0.0001 0.001 0.01

x

xg1

TOT ASYM

VMD (GRSV) SMC DATA

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 0.2 0.4 0.6 0.8 1 1.2

Q2

[GeV2]

I(Q2)

TOTASYM

VMD (GRSV)E91-023 DATA

data at Q2< 1 GeV2

points markresonancesat W < Wt(Q

2)

Figures from: Badelek, Kwiecinski, Ziaja, Eur. Phys. J. C26 (2002)45.


Outlook

Outline






5 Outlook


Outlook

Outlook

In the precision RC calculations a part of systematic uncertaintiescome from a choice of the input information.

We have collection of models (expressions) for Q2 → 0 extrapolationsfor:

form factors, suppression factorsF2(x,Q2)

R(x,Q2)

g1(x,Q2)

These expressions are valid at low x, appropriate for the EIC

but they have to be updated! =⇒ TO BE DONE


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