Measurements of the electron-helicity dependent cross sections of deeply virtual compton scattering with CEBAF at 12 GeV

arXiv:nucl-ex/0609015v1 11 Sep 2006

of Deeply Virtual Compton S attering with CEBAF at 12 GeV Julie Ro he

∗

Rutgers, The State University of New Jersey, Pis ataway, New Jersey 08854; and Thomas Jeerson National A elerator Fa ility, Newport News, Virginia 23606

†

Charles E. Hyde-Wright

‡∗

and G. Gavalian, M. Amarian, S. Bültmann, G.E. Dodge, H. Juengst, J. La hniet, A. Radyushkin, P.E. Ulmer, L.B. Weinstein

Old Dominion University, Norfolk VA Bernard Mi hel

∗

and J. Ball, P.-Y. Bertin

§

, M. Brossard, R. De Masi, M. Garçon, F.-X. Girod, M. Guidal, M. Ma Cormi k, M. Mazouz, S. Ni olai, B. Pire, S. Pro ureur, F. Sabatié, E. Voutier, S. Wallon

LPC (Clermont) / LPSC (Grenoble) / IPNO & LPT (Orsay) / CPhT-Polyte hnique (Palaiseau) / SPhN (Sa lay)

CEA/DSM/DAPNIA & CNRS/IN2P3, Fran e Carlos Muñoz Cama ho

∗

Los Alamos National Laboratory, Los Alamos NM, 87545

A. Camsonne, J.-P.Chen, E. Chudakov, A. Deur, D. Gaskell, D. Higinbotham,C.de Jager, J. LeRose, O. Hansen, R. Mi haels, S. Nanda, A. Saha, S.Stepanyan, B. Wojtsekhowski

Thomas Jeerson National A elerator Fa ility, Newport News, Virginia 23606 P.E.C. Markowitz

Florida International University, Miami FL X. Zheng

Massa husetts Institute of Te hnology, Cambridge MA 02139

¶

R. Gilman, X. Jiang, E. Ku hina, R. Ransome

‡

Conta tPerson: hydeodu.edu

§

Rutgers, The State University of New Jersey, Pis ataway, NJ 08854 A. Deshpande

Stony Brook University, Stony Brook, NY 11794 N. Liyanage

University of Virginia, Charlottesville, VA 22904

Seonho Choi, Hyekoo Kang, Byungwuek Lee, Yumin Oh, Jongsog Song Seoul National University, Seoul 151-747, Korea

S. Sir a

Dept. of Physi s, University of Ljubljana, Slovenia (Dated: 07 July 2006 for JLab PAC30)

Weproposepre isionmeasurementsoftheheli ity-dependentandheli ity indepen-dent rossse tions for the

ep → epγ

rea tioninDeeplyVirtual ComptonS attering (DVCS) kinemati s. DVCSs aling is obtained in thelimits

Q

2

_{≫ Λ}

2

QCD

,

x

Bj

xed, and

−∆

2

= −(q − q

′

)

2

_{≪ Q}

2

. We onsider the spe i kinemati range

Q

2

_{> 2}

GeV

2

,

W > 2

GeV, and

−∆

2

_{≤ 1}

GeV

2

. We will useour su essful te hnique from the5.75 GeV Hall A DVCS experiment (E00-110). Withpolarized 6.6, 8.8, and 11 GeV beams in ident ontheliquidhydrogen target,wewill dete t thes attered ele -tron in the Hall A HRS-L spe trometer (maximum entral momentum 4.3 GeV/ ) andtheemittedphotoninaslightly expandedPbF

2

alorimeter. Ingeneral, wewill not dete t there oil proton. The H

(e, e

′

_γ)X

missingmass resolution is su ient to isolate the ex lusive hannel with

3%

systemati pre ision.

∗

Co-Spokesperson

†

PermanentAddress: OhioUniversity,AthensOH,45701

¶

Contents

I. Introdu tion 4

A. Imaging the Nu leus 4

B. Review of Hall A DVCS E00-110Methodsand Results 7

C. Physi s Goals and Proposed DVCS Kinemati s 8

II. DVCS Observables 13

A. DVCS Kinemati s and Denitions 13

B. DVCS Cross Se tion 14

C. Fourier Coe ientsand Angular Harmoni s 16

D. BH

·

DVCS Interferen e and BilinearDVCS Terms 17

III. Des ription of Experiment Apparatus 18

A. HighResolution Spe trometer 19

B. Beam Line 22

C. PbF

2

Calorimeter 23

D. Luminosity Limitsand Opti al Curingof Calorimeter 25

E. Trigger 26

IV. Proje ted Results 27

A. Systemati (Instrumental)Errors 28

B. Statisti s 29

C. Illustrationof Prospe tive Physi s 34

V. Summary 37

Referen es 38

A. Contributions to Hall A Equipment for 11 GeV 40

B. Preparation of Extensions 40

I. INTRODUCTION

A. Imaging the Nu leus

We have a quantitative understanding of the strong intera tion pro esses at extreme short distan es interms of perturbative QCD. We also understand the long distan e prop-erties of hadroni intera tions in terms of hiral perturbation theory. At the intermediate s ale: thes ale of quark onnement and the reationof[ordinary℄mass we have an under-standingofnumerous observables ataboutthe 20%levelfromlatti eQCD al ulationsand semi-phenomenologi al models. This is an extremely impressive intelle tual a hievement. However, the questions we have today about nu lear physi s, are questions at the 1%, or even 0.1

%

,levelrelativetothe onnement s ale

Λ

QCD

≈ 300

MeV/ . Forexample,the

n

-

p

masssplittingof1.3MeVandthe Deuteronbindingenergyof2.2MeVare

≤ 1%

ee tsthat are ru ialtothe evolutionofthe universe. Thepatterns ofbindingenergies ofneutron and protonri hnu leiareeven smalleree ts,andare ru ialtothesynthesisofelements

Z >

Fe in supernovae and other extreme events. It is the QCD dynami s at the distan e s ale of

1/Λ

QCD

that gives rise tothe origin of mass. To improveour understanding of onnement and of the origin of mass, we annot rely solelyon improvements intheory. We must have experimental observables of the fundamental degrees of freedom of QCDthe quarks and gluonsat the distan e s ale of onnement. The generalized parton distributions (GPDs) are pre isely the ne essary observables.

Measurements of ele tro-weak form fa tors determine the spatial stru ture of harges and urrents inside the nu leon. However, the resolution s ale

Q

2

is not independent of the distan e s ale

1/pQ

2

probed. Deep inelasti s attering of leptons (DIS) and related in lusive high

p

⊥

hadron s attering measure the distributions of quarks and gluons as a fun tion of light one momentum fra tions, but integrated over spatial oordinates. Ji [1℄, Radyushkin [2℄, and Müller et al. [3℄, dened a set of light one matrix elements, now known as GPDs,whi hrelatethe spatial andmomentumdistributionsof the partons. This allows the ex iting possibility of determining spatial distributions of quarks and gluons in the nu leon as afun tion of their wavelength.

k

k’

q’

proton

electron

p

p’

γ

e p

→

e p

=

VCS

+

+

Bethe-Heitler

Figure1: Lowest orderQED diagramsfor the pro ess

ep → epγ

,in luding theDVCSand Bethe-Heitler (BH) amplitudes. The external momentum four-ve tors are dened on the diagram. The virtualphotonmomenta are

q

µ

_{= (k − k}

′

₎

µ

intheDVCS-and

∆

µ

_{= (q − q}

′

₎

µ

intheBH-amplitudes.

Heitler (BH) pro ess (Fig. 1). In addition, the spatial resolution (

Q

2

) of the rea tion is independent of the distan e s ale

≈ 1/

√

t

min

− t

probed. Quark-Gluon operators are las-sied by their twist: the dimension minus spin of ea h operator. The handbag amplitude of Fig. 2 is the lowest twist (twist-2)

γ

∗

p → γp

operator. The fa torization proofs onrm the onne tionbetween DIS and DVCS. Theproofs thereforesuggest(butdonot establish) that, just as in DIS, higher twist terms in DVCS will be only a small ontribution to the ross se tions atthe

Q

2

and

x

Bj

range a essible with ele trons from612 GeV.

In the formalism we are using [6℄, the matrix elements of operators of twist-

n

are

Q

2

independent (ex ept for

ln(Q

2

_/Λ

2

QCD

)

evolution), but all observables of twist-

n

operators arry kinemati pre-fa tors that s ale as

[(t

min

− t)/Q

2

_]

n/2

,

[(−t)/Q

2

_]

n/2

, or

[M

2

_/Q

2

_]

n/2

. Diehl et al., [7℄ showed that the twist-2 and twist-3 DVCS-BH interferen e terms ould be independently extra ted from the azimuthal-dependen e (

φ

γγ

, ŸIIA ) of the heli ity depen-dent ross se tions. Burkardt [8℄ showed that the

t

-dependen e of the GPDs at

ξ = 0

is Fourier onjugatetothe transverse spatial distribution of quarksinthe innitemomentum frame as a fun tion of momentum fra tion. Ralston and Pire [9℄ and Diehl [10℄ extended this interpretation to the general ase of

ξ 6= 0

. Belitsky et al., [11℄ des ribe the general GPDs interms of quark and gluon Wigner fun tions.

These elegant theoreti al on epts have stimulated an intense experimental eort in DVCS. The H1 [12℄ [13℄ and ZEUS [14℄ ollaborations at HERA measured ross se tions for

x

Bj

≈ 2ξ ≈ 10

−3

. These data are integrated over

φ

γγ

and are therefore not sensitive to the BH

·

DVCS interferen e terms. The CLAS [15℄ and HERMES [16℄ ollaborations

mea-Figure2: Leadingtwist

γ

∗

_{p → γp}

amplitudeinthe DVCSlimit. Theinitialandnalquarks arry light- one momentum fra tions

x + ξ

and

x − ξ

of the light- one momenta

(1 ± ξ)(p + p

′

₎

+

(Eq. 7). The rossed diagram is also in luded in the full DVCS amplitude. The invariant momentum transfersquaredto theproton is

t = ∆

2

.

suredrelativebeamheli ityasymmetries. HERMEShasalsomeasuredrelativebeam harge asymmetries [17, 18℄and CLAS has measured longitudinaltargetrelativeasymmetries [19℄. The HERA and HERMES results integrate over nal state inelasti ities of

M

2

X

≤ 2.9 GeV

2

(or greater). Relative asymmetries are a ratio of ross se tion dieren es divided by a ross se tion sum. In general, these relative asymmetries ontain BH

·

DVCS interferen e and DVCS

†

DVCS terms in both the numerator and denominator (the beam harge asym-metry removes the DVCS

†

DVCS terms only from the numerator). Absolute ross se tion measurements are ne essary toobtain allDVCS observables.

InHallA,E00-110[20℄andE03-106[21℄,wemeasuredabsolute rossse tionsforH

(~e, e

′

_γ)

p and D

(e, e

′

_γ)pn

at

x

Bj

= 0.36

. The following se tion des ribes the methods and results of E00-110. We propose to ontinue this program using the same experimental te hniques, while taking advantage of the higher beam energy availablewith CEBAF at12 GeV.

B. Review of Hall A DVCS E00-110 Methods and Results

The rst draftpubli ationof E00-110 an befound inRef.[22℄. Usingawellunderstood experimentalapparatus,thisexperimentmeasuredboththeunpolarizedandpolarized ross-se tion of the

~ep → epγ

pro ess inthe Deeply Virtual Compton S attering (DVCS) regime at

x

Bj

= 0.36

and for Q

2

at 1.5, 1.9, and 2.3 GeV

2

. With ele trons dete ted in the HRS-L, we have an absolute a eptan e for ele tron dete tion understood to 3%, and a pre ise measurement of the s attering vertex and the dire tion of the virtual photon. DVCS is a three body nal state,but at high

Q

2

and low

t

,the nal photon ishighlyalignedwith the virtual photonand therefore highly orrelated with the s attered ele tron. Thus, even with a modest alorimeter, our oin iden e a eptan e for DVCS is essentially limited only by the ele tron spe trometer. As a onsequen e, very high values of the produ t of luminosity times oin iden ea eptan e arepossible. TheradiationhardPbF

2

alorimetergivesafast (Cerenkov) timeresponse, andea h hannel isre ordedwith a1GHzdigitizerwhi hallows o-line identi ation of the DVCS photon. The identi ation of the ex lusive hannel is illustrated in Fig. 3. In Hall A, our systemati errors are minimizedby the ombinationof theComptonpolarimeter,thewell-understoodopti sanda eptan eoftheHighResolution Spe trometer(HRS), anda ompa t, hermeti , alorimeter. Allofthose fa torsallowed the measurementsof ross-se tions. The high-pre isionele tron dete tionminimizessystemati errors on

t

and

φ

γγ

. Therefore, we exploit the pre ision

φ

γγ

-dependen e to extra t the ross se tionterms whi h havethe form ofa niteFourier series modulatedby the ele tron propagators of the BHamplitude.

The strongpointofexperimentE00-100isthatwemeasured absolute ross-se tions. For example, the

sin(φ)

term (modulated by BH propagators) of the heli ity dependent ross se tionmeasuresthe interferen e ofthe imaginarypart ofthetwist-2DVCS amplitudewith theBHamplitude,withasmall ontributionfromanadditionaltwist-3bilinearDVCSterm. The

Q

2

-dependen e of this term pla es atight limitonthe ontributionof the higher twist termstoour extra tionof thehandbag amplitude. Theunpolarized ross se tionisasum of the BH ross se tion, the real part of the BH

·

DVCS interferen e, and a twist-2 bilinear DVCS term. The E00-110 results show that the unpolarized ross se tion is not entirely dominated by the BH ross se tion, but also has a large ontribution fromDVCS. Expli it twist-3 terms were alsoextra ted fromthe

sin(2φ

γγ

)

and

cos(2φ

γγ

)

dependen e of the ross

Generate estimate of
H(e,e)Y events from measured H(e,e)Y events.

H(e,e’)X: M

X

2

kin3

Exclusive DVCS events

H(e,e’) Y

)

2

(GeV

2

X

M

0

0.5

1

1.5

2

2.5

0

1000

2000

3000

4000

cut

2

X

M

H(e,e’p) sample

H(e,e’p)

simulation,

Normalized to data

<2% in estimate of

H(e,e)N
…

below threshold M

X

2

<(M+m)

2

H(e,e’)…

H(e,e’)p

Figure 3: Missing mass squared distribution for the H

(e, e

′

_γ)X

rea tion, obtained in Hall A in E00-110. The left plot shows the raw spe trum (after a idental subtra tion) and the statisti al estimate of the ontribution of

ep → eY π

0

events of the type H

(e, e

′

_γ)γY

. This estimate was obtained from the ensemble of dete ted H

(e, e

′

_π

0

_)Y

events, with both photons from the

π

0

_{→ γγ}

de ay dete ted in the alorimeter. The right plot shows the H

(e, e

′

_γ)X

spe trum, after the

π

0

subtra tion.

se tions. These ontribute very littletothe ross se tions.

Inthe kinemati regimeofE00-100,neitherthe extra ted Twist-2orTwist-3observables show any statisti ally signi ant dependen e on Q

2

. This provides strong support to the original theoreti al predi tions that DVCS s aling is based on the same foundation as DIS s aling. We note that in the range

0.2 ≤ x

Bj

≤ 0.6

, both the higher twist terms and

ln Q

2

_/Λ

2

QCD

evolution terms are small inDIS, even for Q

2

_{≈ 2}

GeV

2

[23, 24, 25℄. Thuswe are well on our way to proving the dominan e of the leading twist term of the amplitudes ( handbag approximation) where the virtual photon s atters o a single parton. This approximation is a orner stone of the study of the nu leon stru ture interms of GPDs.

C. Physi s Goals and Proposed DVCS Kinemati s

Thepresent proposal annotfully disentanglethe spin-avorstru ture ofthe GPDs. We itemizeherethemeasurementswewillperformandthephysi alinsightsweexpe ttoobtain.

•

Measurethe

~ep → epγ

ross se tionsatxed

x

Bj

overaswidearangein

Q

2

aspossible for

k ≤ 11

GeV. This will determine with what pre ision the handbag amplitude

dominates (ornot) over the higher twist amplitudes. More generally, we onsider the virtual photon athigh

Q

2

as asuperposition of a point-likeelementary photonand a `hadroni 'photon(

qq

,ve tormesons)with atypi alhadroni transverse size. The

Q

2

dependen e ofthe DVCS rossse tionsmeasures the relativeimportan eofthese two omponentsof the photon[26℄.

•

Extra t all kinemati ally independent observables (unpolarized target) for ea h

Q

2

,

x

Bj

,

t

point. These observables are the angular harmoni superposition of Comp-ton Form Fa tors (CFFs). As fun tions of

φ

γγ

, the observable terms are

cos(nφ

γγ

)

for

n ∈ {0, 1, 2}

, and

sin(nφ

γγ

)

for

n ∈ {1, 2}

, with additional

1/[J + K cos(φ

γγ

)]

modulations from the ele tron propagators of the BH amplitude (ŸIIB ). Ea h of these ve observables isolates the

ℜ

e or

ℑ

m parts of a distin t ombination of linear (

BH · DV CS

†

) and bilinear(

DV CS · DV CS

†

) terms.

•

Measure the

t

-dependen e of ea h angular harmoni term. The

t

-dependen e of ea h CFFisFourier- onjugatetothespatialdistributionofthe orrespondingsuperposition of quark distributions in the nu leon, as a fun tion of quark light- one momentum-fra tion. In a single experiment, we annot a ess this Fourier transform dire tly, be ause we measure a superposition of terms. However, we still expe t to observe hangesin the

t

-dependen e of our observables asa fun tionof

x

Bj

. In parti ular,the

r.m.s.

impa tparameterofaquarkofmomentumfra tion

x

mustdiminishasapower of

(1 − x)

as

x → 1

. This is not a small ee t, between

x = 0.36

and

x = 0.6

, we expe t a hange in slope (as a fun tion of

t

) of afa tor two inindividual GPDs.

•

Measure the

~ep → epπ

0

ross se tion inthe same kinemati sas DVCS.

This will test the fa torization dominan e of meson ele tro-produ tion. The longi-tudinal ross se tion (

d

2

_σ

L

) is the only leading twist (twist-2) term in the ele tro-produ tion ross se tion. In this experiment, we do not propose Rosenbluth separa-tions of

d

2

_σ

L

from

d

2

_σ

T

. However, as a fun tion of

Q

2

, the ratio

d

2

_σ

T

/d

2

σ

L

falls at least

∝ 1/Q

2

. Thusthe handbag ontributionto

d

2

_σ

L

an beextra ted, within statis-ti al errors, from a

1/Q

2

expansion. The

σ

LT

,

σ

T T

, and

σ

LT

′

terms willbe obtained fromaFourierde ompositionofthe azimuthaldependen e ofthe rossse tion. These observableswillprovideadditional onstraintsonboth thelongitudinalandtransverse

Bj

x

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

)

2

(GeV

2

Q

0

5

10

2

<4 GeV

2

W

11 GeV

≤

beam

Unphysical with E

= 6.6 GeV

beam

E

= 8.8 GeV

beam

E

= 11.0 GeV

beam

E

= 5.75 GeV

beam

E

DVCS measurements in Hall A/JLab

Figure4: ProposedDVCS kinemati sfor H

(e, e

′

_γ)p

measurements inHall Awith3,4,and5pass beams of CEBAF at 12 GeV. The diamond shapestra e theapproximate a eptan eof theHRS inea h setting. The boundary of the unphysi al region orrespondsto themaximum possible

Q

2

at agiven

x

Bj

for11 GeV.This orrespondsto

180

◦

ele trons attering, equivalent to

θ

q

= 0

◦

. The pointsat

E

B

eam

= 5.75

GeV were obtained inE00-110.

The handbag amplitude of pion ele tro-produ tion is the onvolution of the axial GPDs

H

˜

and

E

˜

with the pion distribution amplitude (DA)

Φ

π

. For harged pion ele tro-produ tion, the

E

˜

term is expe ted to be dominated by the pion-polethis is theme hanismusedtomeasure thepion formfa tor inele tro-produ tiononthe pro-ton. However, neutralpionele tro-produ tionwillbedominatedbythe non-pion-pole ontributions to

H

˜

and

E

˜

.

Tostudy the DVCS pro ess with CEBAFat12GeV, welimitour denition ofDVCS to the followingkinemati range forthe entral values of our Hall A ongurations:

Q

2

≥ 2 GeV

2

W

2

_{≥ 4 GeV}

2

−t < Q

2

(1)

Our proposed kinemati s,and the physi s onstraintsof Eq. 1 are illustrated in Fig. 4. We propose a

Q

2

s an of the ross se tions forthree values of

x

B

. Our maximum entral

Q

2

of 9 GeV

2

is higherthan the

Q

2

range of the

t

-distributions of published DVCS ross se tions (integrated over

φ

γγ

) from HERA for

x

Bj

≈ 10

−3

. The beam time estimates for these high statisti smeasurementsare listedin Table I .

Within the experimental onstraints whi h we detail in se tion III, our hoi e of kine-mati s responds tothe physi s goals of high beam energy DVCS measurements.

•

At ea h

x

Bj

point, we measure the maximum possible range in

Q

2

. Although our preliminarydata fromE00-110 at

x

Bj

= 0.36

showno indi ationof

Q

2

dependen e in the observable

ℑ

m

[C

I

(F)]

, this ould result from a ompensation of terms of higher twist and QCD evolution. In this proposal, we will double the range in

Q

2

at

x

Bj

=

0.36

, and providea nearly equal

Q

2

range at

x

Bj

= 0.5

and

0.6

.

•

Simple kinemati sdi tates that asthe momentum fra tion

x + ξ

of the stru k quark goes to 1, its impa t parameter

b/(1 + ξ)

relative to the enter of momentum of the initialprotonmust shrinktowards zero. Burkardthas dis ussed the shrinkage ofthe proton as

x → 1

inthe limit

ξ = 0

. [27, 28℄. The perpendi ular momentum transfer

∆

⊥

is Fourier onjugate to the impa t parameter

b

. As a fun tion of

x

, the mean square transverse separation between the stru k quark and the enter-of-momentum of the spe tator system is

hr

2

⊥

i(x) = hb

2

i/(1 − x)

2

_,

(2) where

hb

2

i

is the mean square impa t parameter of the stru k quark. Burkardt on-sideredGPD modelsof the form

Q

2

_k

Beam Time Total (Days) (GeV

2

) (GeV)

x

Bj

= 0.36

x

Bj

= 0.50

x

Bj

= 0.6

(Days) 3.0 6.6 3 4.0 8.8 2 4.55 11.0 1 3.1 6.6 5 4.8 8.8 4 6.3 11.0 4 7.2 11.0 7 5.1 8.8 13 6.0 8.8 16 7.7 11.0 13 9.0 11.0 20 TOTAL 6 20 62 88

TableI: ProposedBeam Timeasa fun tion of

(Q

2

_{, k, x}

Bj

)

DVCSkinemati s.

Theboundthat

hr

2

⊥

i

remainniteas

x → 1

requires

n ≥ 2

. On theotherhand,

n = 1

has been used extensively for modeling GPDs. At

x = 0.6

, the hoi e of

n = 1

or

n = 2

hanges the logarithmi slope

∂ ln H/∂t

by a fa tor 2.5. This modelillustrates that without measurements, there are very large un ertainties in the behavior of the GPDs at large

x

(independent of the

t = 0

onstraints), and that measurements will improve our understanding of the transverse distan e s ales of the quarksand gluons inside the proton.

Our spe i hoi e of kinemati s in Fig. 4 and Table I maximizes our range in

x

Bj

, while maintainingmeasurements asa fun tion of

Q

2

II. DVCS OBSERVABLES

A. DVCS Kinemati s and Denitions

Fig.1 denes our kinemati four-ve tors for the

ep → epγ

rea tion. The ross se tion is a fun tionof the following invariantsdened by the ele tron s attering kinemati s:

s

e

= (k + p)

2

= M

2

+ 2kM

Q

2

= −q

2

W

2

_{= (q + p)}

2

_{= M}

2

_{+ Q}

2



1

x

Bj

− 1



x

Bj

=

Q

2

2q · p

=

Q

2

W

2

_{− M}

2

_{+ Q}

2

y = q · p/k · p.

(4)

The DVCS ross se tion also depends on variables spe i to deeply virtual ele tro-produ tion: the DVCS s aling variable

ξ

, the invariant momentum transfer squared

t

to the proton,azimuth

φ

γγ

of the nal photon around the

q

-ve tor dire tion. The DVCS s al-ingvariable

ξ

is dened interms of the symmetrized momenta:

ξ =

−q

2

q · P

= x

Bj

1 + t/Q

2

2 − x

Bj

(1 − t/Q

2

)

→

x

Bj

2 − x

Bj

for |t|/Q

2

<< 1

(5)

q

µ

= (q + q

′

₎

µ

_/2

P

µ

= (p + p

′

₎

µ

_.

P

2

_{= 4M}

2

− t

t = (p

′

− p)

2

_{= ∆}

2

_.

(6) In light one oordinates

(p

+

_{, p}

⊥

, p

−

)

,with

p

±

_{= (p}

0

_{± p}

z

_)/

√

₂

,the four-ve tors are:

p =



(1 + ξ)P

+

_,

−∆

⊥

2

,

M

2

_{+ ∆}

2

⊥

/4

2P

+

_{(1 + ξ)}



p

′

=



(1 − ξ)P

+

,

+∆

⊥

2

,

M

2

_{+ ∆}

2

⊥

/4

2P

+

_{(1 − ξ)}



P =



P

+

_{, 0,}

4M

2

− t

2P

+



.

(7)

InFig. 1,theinitial(nal)quarklight onemomentumfra tionis

x ± ξ

of thesymmetrized momentum

P

and

(x ± ξ)/(1 ± ξ)

of the initial(nal) proton momentum

p

+

(

p

′ +

).

Our onvention for

φ

γγ

is dened as the azimuthal angle in a event-by-event spheri al-polar oordinate system (inthe lab):

ˆ

z

q

= q/|q|

ˆ

y

q

= [k × k

′

] /|k × k

′

|

ˆ

x

q

= ˆ

y

q

× ˆz

q

.

(8)

tan(φ

γγ

) = (q

′

· ˆy

q

)/ (q

′

· ˆx

q

)

= [|q|q

′

· (k × k

′

]/ [q × q

′

) · (k × k

′

)] .

(9)

The quadrant of

φ

γγ

is dened by the signs of

(q

′

_{· ˆy}

q

)

and

(q

′

_{· ˆx}

q

)

We also utilize the laboratory angle

θ

γγ

between the

q

-ve tor and

q

′

-dire tions:

cos θ

γγ

= ˆ

z

q

· q

′

/|q

′

|.

(10)

The impa t parameter

b

of the light- one matrix element is Fourier onjugate to the

∆

⊥

, the momentumtransfer perpendi ular tothe light one dire tiondened by

P

+

:

∆

2

⊥

= (t − t

min

)

(1 − ξ

2

₎

(1 + ξ

2

₎

(11)

t

min

= t(θ

γγ

= 0

◦

) =

−4M

2

_ξ

2

1 − ξ

2

≈

−x

2

Bj

M

2

[1 − x

Bj

(1 − M

2

/Q

2

)]

.

(12) B. DVCS Cross Se tion

The following equations reprodu e the onsistent expansion of the DVCS ross se tion toorder twist-3of Belitsky,Müller, and Kir hner[6℄. Notethat our denition of

φ

γγ

agrees withtheTrento-Convention for

φ

[29℄,and isthe denitionusedin[16℄and[15℄. Notealso thatthisazimuth onventiondiersfrom

φ

[6℄

denedin[6℄by

φ

γγ

= π −φ

[6℄

. Inthefollowing expressions, we utilize the dierential phase spa e element

d

5

_{Φ = dQ}

2

_dx

Bj

dφ

e

dtdφ

γγ

. The heli ity-dependent(

λ = ±1

) ross se tionforaleptonof harge

±e

onanunpolarizedtarget

is:

d

5

_{σ(λ, ±e)}

d

5

_Φ

=

dσ

0

dQ

2

_dx

Bj





T

BH

(λ) ± T

DV CS

(λ)





2

/|e|

6

=

dσ

0

dQ

2

_dx

Bj

h





T

BH

(λ)





2

+





T

DV CS

(λ)





2

∓ I(λ)

i

1

e

6

(13)

dσ

0

dQ

2

_dx

Bj

=

α

3

QED

16π

2

_(s

e

− M

2

)

2

x

Bj

1

√

1 + ǫ

2

ǫ

2

_{= 4M}

2

_x

2

Bj

/Q

2

(14) The

|T

BH

_|

2

term is given in [6℄, Eq. 25, and will not be reprodu ed here, ex ept to note thatitdependsonbilinear ombinationsofthe ordinaryelasti formfa tors

F

1

(t)

and

F

2

(t)

. The interferen e term

I

is:

1

e

6

I =

1

x

Bj

y

3

P

1

(φ

γγ

)P

2

(φ

γγ

)t

(

c

I

0

+

3

X

n=1

(−1)

n

c

I

n

(λ) cos(nφ

γγ

) − λs

I

n

sin(nφ

γγ

)



)

.

(15) The

(−1)

n

and

(−λ)

fa tors are introdu ed by our onvention for

φ

γγ

, relative to[6℄. The bilinearDVCS terms have asimilar form:





T

DV CS

(λ)





2

1

e

6

=

1

y

2

_Q

2

(

c

DV CS

₀

+

2

X

n=1

(−1)

n

c

DV CS

_n

cos(nφ

γγ

) + λs

DV CS

1

sin(φ

γγ

)

)

(16) The Fourier oe ients

c

n

and

s

n

will be dened below (Eq. 2127). The

c

I

n

and

s

I

n

are linear inthe GPDs, the

c

DV CS

n

and

s

DV CS

n

are bi-linearinthe GPDs (andtheir higher twist extensions). All ofthe

φ

γγ

-dependen e ofthe ross se tion isnowexpli it inEq. 15and 16.

The

P

i

(φ

γγ

)

are the ele tron propagators of the BHamplitude:

Q

2

_P

1

(φ

γγ

) = (k − q

′

)

2

Q

2

_P

2

(φ

γγ

) = (k

′

+ q

′

)

2

.

(17)

After some algebra,the kinemati dependen e of the pre-fa tor ofEq. 15is more apparent:

1

x

Bj

y

3

P

1

(φ

γγ

)P

2

(φ

γγ

)t

=

−(1 + ǫ

2

₎

2

_(s

e

− M

2

)/t

Q

2

_{(J − 2K cos φ}

γγ

) (1 + J + (t/Q

2

) − 2K cos φ

γγ

)

(18)

J =



1 − y −

yǫ

2

2

 

1 +

t

Q

2



+ (1 − x

Bj

) (2 − y)

 −t

Q

2



→



1 − y −

yǫ

2

2



as t/Q

2

_{→ 0}

(19)

K

2

₌

t

min

− t

Q

2

(1 − x

Bj

)



1 − y −

y

2

_ǫ

2

4

 

√

1 + ǫ

2

₋

x

Bj

W

2

W

2

_{− M}

2

(t

min

− t)

Q

2



→

t

min

− t

Q

2

(1 − x

Bj

) [1 − y] as t/Q

2

→ 0

(20)

The oe ient

K

appears not onlyin the BH propagators, but alsoin the kinemati pref-a tors of the Fourier de omposition of the ross se tion (see below). For xed

Q

2

and

x

Bj

,

K

depends on

t

as

(t

min

− t)/Q

2

.

C. Fourier Coe ients and Angular Harmoni s

The Fourier oe ients

c

I

n

and

s

I

n

of the interferen e terms are:

c

I

0

= −8(2 − y)ℜ

e

 (2 − y)

2

1 − y

K

2

C

I

(F) +

t

Q

2

(1 − y)(1 − x

Bj

)

C

I

+ ∆C

I

 (F)



 c

I

1

λs

I

1



= −8K

 (2 − 2y + y

2

₎

−λy(2 − y)

  ℜ

e

ℑ

m



C

I

(F)

 c

I

2

λs

I

2



=

−16K

2

2 − x

Bj

 (2 − y)

−λy

  ℜ

e

ℑ

m



C

I

(F

eff

₎

(21)

TheFourier oe ients

c

I

3

,

s

I

3

aregluontransversityterms. Weexpe tthesetobeverysmall in our kinemati s,though itwould beex iting if they generated a measureablesignal. The

C

I

and

∆C

I

amplitudesare the angularharmoni termsdened inEqs. 69and72of [6℄(we have suppressed the subs ript unp sin e our measurements are only with an unpolarized target). These angularharmoni s depend onthe interferen e ofthe BH amplitudewith the set

F = {H, E, ˜

H, ˜

E}

of twist-2 Compton form fa tors (CFFs) or the related set

F

eff

of ee tive twist-3 CFFs:

C

I

(F) = F

1

(t)H(ξ, t) + ξG

M

(t) ˜

H(ξ, t) −

t

4M

2

F

2

(t)E(ξ, t)

(22)

C

I

(F

eff

_{) = F}

1

(t)H

eff

(ξ, t) + ξG

M

(t) ˜

H

eff

(ξ, t) −

t

4M

2

F

2

(t)E

eff

_{(ξ, t)}

(23)

C

I

+ ∆C

I

 (F) = F

1

(t)H(ξ, t) −

t

4M

2

F

2

(t)E(ξ, t) − ξ

2

G

M

(t) [H(ξ, t) + E(ξ, t)] .

(24) Note that

C

I

_{+ ∆C}

I



depends only on

H

and

E

. The usual proton elasti form fa tors,

F

1

,

F

2

and

G

M

= F

1

+ F

2

are dened to have negative argumentsin the spa e-like regime. The Comptonformfa tors are dened intermsof theve tor GPDs

H

and

E

,and the axial ve tor GPDs

H

˜

and

E

˜

. For example(

f ∈ {u, d, s}

) [6℄:

H(ξ, t) =

X

f

h

e

_f

e

i

2

(

iπ [H

f

(ξ, ξ, t) − H

f

(−ξ, ξ, t)]

+P

Z

+1

−1

dx



1

ξ − x

−

1

ξ + x



H

f

(x, ξ, t)

)

.

(25)

Twist-3CFFs ontainWandzura-Wilz ekterms, determinedby thetwist-2matrix elements, anddynami

qGq

twist-3matrixelements. Thetwist-2andtwist-3CFFsarematrixelements ofquark-gluonoperatorsandareindependentof

Q

2

(uptologarithmi QCDevolution). The kinemati suppressionofthetwist-3(andhigher)termsisexpressed inpowers of

−t/Q

2

and

(t

min

− t)/Q

2

in e.g. the

K

-fa tor. This kinemati suppression is also a onsequen e of the fa tthat thethe twist-3 terms oupletothe longitudinalpolarizationof thevirtual photon.

The bilinearDVCS Fourier oe ientsare:

c

D

V CS

0

= 2(2 − 2y + y

2

)C

D

V CS

(F, F

∗

₎







c

D

V CS

1

λs

D

V CS

1







=

8K

2 − x

Bj







2 − y

−λy













ℜ

e

ℑ

m







C

D

V CS

(F

e

f f

, F

∗

₎

(26) The

c

D

V CS

2

oe ientis a gluontransversity term. The DVCS angular harmoni s are

C

DV CS

_{(F, F}

∗

_{) =}

1

(2 − x

Bj

)

2

n

4(1 − x

Bj

)



HH

∗

_{+ ˜}

H ˜

H

∗



− x

2

Bj

2ℜ

e

h

HE

∗

_{+ ˜}

H ˜

E

∗

i

−



x

2

Bj

+ (2 − x

Bj

)

2

t

4M

2



EE

∗

− x

2

Bj

t

4M

2

E ˜

˜

E

∗



.

(27)

The twist-3term

C

DV CS

_(F

eff

, F

∗

₎

has anidenti alform,with one CFFfa tor repla edwith the set

F

eff

. The

C

DV CS

_(F

T

, F

∗

)

, appearing with a

cos(2φ

γγ

)

weighting, alsohas the same formasEq.27,butnowwithoneset

F

repla edbytheset

F

T

of(twist-2)gluontransversity Compton formfa tors.

D. BH

·

DVCS Interferen e and BilinearDVCS Terms

The BH

·

DVCS interferen e terms are not fully separable fromthe bilinear DVCS terms. Weanalyze the ross se tion (e.g. [22℄) inthe general form

d

5

_σ

d

5

_Φ

=

d

5

_σ

BH

d

5

_Φ

+

1

P

1

(φ

γγ

)P

2

(φ

γγ

)

X

n

K

cn

ℜ

e

C

I,exp

n

 cos(nφ

γγ

)

+λK

sn

ℑ

C

n

I,exp

 sin(nφ

γγ

)

(28) The fa tors

K

cn,sn

are the purely kinemati pre-fa tors dened in Eqs. 1321. The experi-mental oe ients

ℜ

e

, ℑ

m

C

I,exp

n

in lude ontributions from the bilinear DVCS terms, that mix into dierent orders in

cos(nφ)

or

sin(nφ)

due to the absen e of the BH propagators

P

1

P

2

in the DVCS

2

From Eq. 18, 15, and 16, we obtain the generi enhan ement of the interferen e terms over the DVCS

†

DVCS terms (of the same order in

sin(nφ

γγ

)

or

cos(nφ

γγ

)

):









BH · DV CS

DV CS

†

_{DV CS}









∝

y

2

_(s

e

− M

2

)

−t

.

(29) In ea h setting

(x

Bj

, Q

2

₎

setting,for ea h bin in

t

,we thereforehave the following exper-imentalTwist-2 DVCS observables

ℑ

m

[C

I,exp

_{(F)] = ℑ}

m

[C

I

(F)] + hη

s1

iℑ

m

[C

DV CS

_(F

∗

, F

eff

_)]

(30)

ℜ

e

[C + ∆C]

I,exp

(F)

= ℜ

e

[C

I

+ ∆C

I

](F) + hη

0

i ℜ

e

C

DVCS

(F

∗

, F)



(31)

ℜ

e

C

I,exp

(F)

= ℜ

e

C

I

_{(F) + hη}

c1

i ℜ

e

C

DVCS

(F

∗

, F)

.

(32)

The oe ients

hη

Λ

i

are the a eptan eaveraged ratios of the kinemati oe ients of the bilinearDVCStermstotheBH

·

DVCS terms. Inaddition,wehavethe experimentalTwist-3 DVCS observables:

ℑ

m

[C

I,exp

(F

eff

)] = ℑ

m

[C

I

(F

eff

)] + hη

s2

iℑ

m

[C

DV CS

(F

∗

, F

eff

_))]

(33)

ℜ

e

[C

I,exp

(F

eff

)] = ℜ

e

[C

I

(F

eff

)] + hη

c2

iℜ

e

[C

DV CS

(F

∗

, F

eff

_)].

(34) The values of the

η

Λ

oe ients in the E00-110 kinemati s are summarized in Table II. They are small, thoughthey growwith

|t|

. The bilinearterm inEq.30 isaTwist-3 observ-able, thereforethe oe ient

hη

s1

i

willde rease as

1/pQ

2

. Basedonthe valuesof TableII, and using our results in E00-110 to estimate

ℑ

m

[C

DV CS

_(F

∗

_{, F}

eff

_))]

, we on lude that the bilinear term likely makes less than a 10% ontribution to

ℑ

m

[C

I

e

xp

(F)]

. In any ase, any omparisonofthe experimentalresultswithmodel al ulations,ortofmodelGPDstothe observables, must in ludethe bilinear terms,with the experimental values of

hη

Λ

i

.

III. DESCRIPTION OF EXPERIMENT APPARATUS

Thisproposalisbased dire tly onthe experien e ofE00-110. We present asket h of the DVCS layout in Hall A in Fig. 5. We use the standard 15 m liquid hydrogen target. We dete t the ele trons in the HRS-L and photons (and

π

0

→ γγ

) in a PbF

2

alorimeter at beam right. We note in Fig. 5 the modied s attering hamber from E00-110 and a new modieddownstreambeampipe. Thes attering hamberis63 minradius,witha1 mAl spheri alwallfa ingthe PbF

2

alorimeterand athinwindow(16 milAl)fa ingthe HRS-L.

The HRS-Left (HRS-L) limitsthe entralvalues ofthe s attered ele tron momentum to

k

′

_{≤ 4.3GeV/c,}

θ

e

≥ 12.5

◦

.

(35)

Asdetailedinse tionIIIC ,tohandleboththeinstantaneouspile-upandintegratedradiation dose in the alorimeter,we limitthe pla ement of the alorimeter to

θ

min

PbF

2

≥ 7

◦

.

(36)

Atthe same time,to ensure adequateazimuthal overage in

φ

γγ

, we limitthe pla ementof the alorimeter

θ

q

≥ 10

◦

.

(37)

Thefollowingsubse tionsdetailourte hni alsolutions,anddemonstratethatthese te h-ni al onstraintsdo not limitthe physi s s opeof this proposal.

A. High Resolution Spe trometer

Our proposed kinemati s, and the physi s onstraints of Eq. 1 are illustrated in Fig. 4. The individual beam energies are illustrated in Fig. 6, with the HRS onstraints (Eq. 35) superimposed. WenotethefollowingpointswithregardtotheHRSand alorimeter(Eq.37) onstraints:

•

The HRS onstraints have noee t for

Q

2

_{> 2}

GeV

2

at

k = 6.6

GeV.

TableII: Weighting fa torsof bilinearDVCStermsfor BH

·

DVCS observables inE00-110.

t

(GeV

2

)

−0.37

−0.33

−0.27

−0.23

−0.17

hη

s1

i

-0.0142 -0.0120 -0.0099 -0.0080 -0.0060

hη

s2

i

-0.048 -0.042 -0.036 -0.030 -0.023

hη

c1

i

-0.050 -0.048 -0.038 -0.033 -0.026

hη

0

i

+0.015 +0.024 +0.031 +0.039 +0.045

hη

c2

i

-0.038 -0.030 -0.022 -0.014 -0.010

LH 2

Q1

e'

BEAM DUMP

6

_˚

₇

_˚

Ca lori meter

at 1.50 M

calorimeter

at 3.00 M

Figure5: HallAlayoutfor DVCSwithCEBAFat 12GeV.Thes attering hamberisidenti alto theE00-110 hamber, withamidplane 63 mradius spheri alse tion with1 mAlwall thi kness, and a 16 mil Al window fa ing the HRS L. We propose a oni al downstream beam pipe, with half-opening angle of 6 degrees on beam-right, and length 3m. The drawing shows the expanded PbF

2

alorimeter at beam-right, inthe losest and farthest ongurations: front fa e150 mand 300 m, respe tively, from target enter. The alorimeter is shown in its smallest angle setting (inner edgeat

7

◦

).

•

At

k = 8.8

and 11 GeV, primarily the

k < 4.3

GeV onstraint removes roughly the lower half of the

Q

2

range at ea h

x

Bj

. This

Q

2

range for ea h

x

Bj

is overed by the lower beam energies.

•

A ording to the Base Equipment plan for CEBAF, the three Halls A, B, C, when runningsimultaneously, must operate at dierent multiples of 2.2GeV.

•

Thesmallangle alorimeter utof

θ

q

> 10

◦

removestheveryhighestphysi allyallowed

Q

2

TableIII: DetailedDVCSKinemati s. TherstlineisfromE00-110,andisin ludedfor omparison purposes. Theangle

θ

q

isthe entral angleof thevirtual photon dire tion

q = (k − k

′

₎

.

Q

2

_x

Bj

k

k

′

θ

e

θ

q

q

′

(0

◦

)

W

2

(GeV

2

) (GeV) (GeV)

(

◦

₎

₍

◦

₎

(GeV) (GeV

2

) 1.90 0.36 5.75 2.94 19.3 18.1 2.73 4.2 3.00 0.36 6.60 2.15 26.5 11.7 4.35 6.2 4.00 0.36 8.80 2.88 22.9 10.3 5.83 8.0 4.55 0.36 11.00 4.26 17.9 10.8 6.65 9.0 3.10 0.50 6.60 3.20 22.5 18.5 3.11 4.1 4.80 0.50 8.80 3.68 22.2 14.5 4.91 5.7 6.30 0.50 11.00 4.29 21.1 12.4 6.50 7.2 7.20 0.50 11.00 3.32 25.6 10.2 7.46 8.1 5.10 0.60 8.80 4.27 21.2 17.8 4.18 4.3 6.00 0.60 8.80 3.47 25.6 14.1 4.97 4.9 7.70 0.60 11.00 4.16 23.6 13.1 6.47 6.0 9.00 0.60 11.00 3.00 30.2 10.2 7.62 6.9

•

All of the onstraintsprevent us fromkinemati s for

x

Bj

≤ 0.2

.

•

Kinemati s at

x

Bj

= 0.7

are allowed at

k = 8.8

and 11 GeV. At this time, it is very di ulttomake reliableestimatesof the DVCS signalatthis extreme

x

Bj

, andwe do not in ludethese kinemati s inour proposal.

The detailedkinemati s are summarized inTable III.

The HRS performan e is entral to this experiment. On e we make a sele tion of ex- lusive H

(e, e

′

_γ)p

events, the resolution in

t = (q − q

′

₎

2

is determined by approximately equal ontributions from the HRS momentum resolution and the angular pre ision of the dire tion

ˆ

q

′

. The resolution onthe dire tionof

q

′

has equal ontributions fromthe position resolution in the alorimeter and the vertex resolution, as obtained from the spe trometer. The ombinationof pre isea eptan e, kinemati ,and vertex resolution in the HRS makes

Bj

x

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

)

2

(GeV

2

Q

0

2

4

6

8

10

k=6.6 GeV

Bj

x

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

)

2

(GeV

2

Q

0

2

4

6

8

10

k=8.8 GeV

Bj

x

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

)

2

(GeV

2

Q

0

2

4

6

8

10

k=11.0 GeV

2

<4.0 GeV

2

W

kp> 4.3 GeV

<12.5 deg

kp

θ

=6.0 deg

q

θ

=10.0 deg

q

θ

=14.0 deg

q

θ

proposed measurements

2

<4.0 GeV

2

W

kp> 4.3 GeV

<12.5 deg

kp

θ

=6.0 deg

q

θ

=10.0 deg

q

θ

=14.0 deg

q

θ

proposed measurements

2

<4.0 GeV

2

W

kp> 4.3 GeV

<12.5 deg

kp

θ

=6.0 deg

q

θ

=10.0 deg

q

θ

=14.0 deg

q

θ

proposed measurements

Figure 6: Proposed DVCS entral kinemati s for H

(e, e

′

_γ)p

measurements in Hall A with 6.6, 8.8, and 11 GeV in ident beam. The experimental onstraints of Eq. 35,37 are indi ated. Ea h experimentalpoint isthe enterof one ofthe'diamond' regionsof Fig.4.

B. Beam Line

InE00-110and E03-106,with thefrontfa eofthe alorimeter1.10mfromtarget enter, wehada6 diameter ylindri albeam pipe,weldeddire tlytothes attering hamber. This aperture orresponds to roughly a

6

◦

half-opening angle from target enter. We require pla ingthe alorimeterat distan es from 1.5mto 3.0m fromthe target (

§

IIIC ). To avoid ex ess alorimeterba kgroundgenerated by se ondariesstrikingthe beam pipe,wepropose a thin walled (1/8 Al) oni al beam pipe, welded to the same aperture. The one must be slightly o-axis, to prevent an interferen e with HRS-Q1 atthe minimum spe trometer angle setting of

18

◦

(Table III), while at the same time preserving a full

6

◦

aperture on Beam-right. The one should be 3.0 m long, to ontinue downstream of the alorimeter.

C. PbF

2

Calorimeter

Table IV: Properties ofPbF

2

.

Density 7.77g/ m

3

RadiationLength 0.93 m MolièreRadius 2.20 m Indexof Refra tion (

λ = 180

nm) 2.05 (

λ = 400

nm) 1.82

Criti al Energy 9.04MeV

We will dete t the s attered photon in a

13 × 16

element PbF

2

alorimeter. This is the existing

11 × 12

E00-110 alorimeter, with 76 additional elements. Ea h blo k is

3 × 3

m

2

_{× 20X}

0

. The additionalblo ks willadd two morerows onthe topand bottom,and two olumns onthe wide angle side. The properties of PbF

2

are summarized in Table IV. The importantdesign onsiderationsfor DVCS are as follows.

•

PbF

2

is aradiationhard pure Cerenkov rystal medium [30℄;

•

Withnos intillationlight[31℄,the alorimetersignalisinsensitivetolowenergynu lear parti les, and the pulse rise and fall time is determined only by geometry and the response of the PMT. This allows us to use the 1GHz Analog Ring Sampler (ARS) digitizer[32℄to minimizepileup.

•

The high luminosity of this proposal requires fast response PMTs operated at low gain and apa itively oupledto apre-amplier. The lowgain redu es the DC anode urrent. The apa itive ouplingremovesthe average pile-upfrom lowenergy

γ

-rays.

•

ThesmallMolièreradius (2.2 m)allowsustoseparate losely spa e showers from

π

0

de ay, and minimizeshower leakage atthe boundary.

•

The shortradiationlengthminimizesu tuationsin light olle tionfromu tuations inthe longitudinalprole of the shower.

bremsstrahlung energy loss ex eeds ionization loss for ele trons) also improves the resolution e.g. relativetoPb-Glass.

•

In E00-110, we obtained a signal of 1photo-ele tron per MeV of deposited energy in the E.M. shower, and a energy resolution of 2.4% from elasti H

(e, e

′

C

alo

p

H

RS

)

ele -tron of 4.2 GeV. For our simulations, we proje t a resolution of

σ

E

/E = 2.0% ⊕

(3.2%)p(1 GeV)/q

′

. We alsoa hieved a spatialresolution of 2 mmat4.2GeV. From the ombination of energy and spatial resolution, we obtained a

π

0

→ γγ

mass reso-lution of 9MeV.

The size, granularity,and position of the alorimetermust a ommodatethe following on-straints:

1. Nearly

2π

azimuthal photon a eptan e for

|∆

⊥

| < 0.6

GeV/ . independent of the entral kinemati s. The angularsize required thereforeshrinks as

∆

max

⊥

/q

′

(0

◦

)

. 2. Good separation of the two lusters from the

π

0

→ γγ

de ay, in order to measure the H

(e, e

′

_π

0

_)p

rea tion. For

π

0

_{→ γγ}

re onstru tion, we require a enter to enter separation of 3 PbF

2

blo ks, or 9 m. In high energy DVCS or deep virtual

π

0

kine-mati s,

q

′

DVCS

≈ E

π

. In this limit, the minimum half opening angle of the

π

0

→ γγ

de ay is

θ

πγ

≥ m

π

/q

′

. The minimum distan e of the alorimeter from the target, based on the luster separation requirement is given as the

D

parameter in Table I. Furthermore,50% of the

π

0

de ay events yield both photons withina halfangle one of

θ

πγ

≤ (

p3/2)(m

π

/q

′

₎

.

3. Maximum distan e from target to minimize pile-up and radiation dose per blo k at xed luminosity.

Item 2 requires us to in rease

D

in proportion to

q

′

. However, from item 1 we see that the a eptan e is

∆

⊥

remains invariant. The solid angle per PbF

2

blo k at the distan e

D

determinesthe maximum feasible luminosityforea h setting. Therefore item3 allows usto in rease the luminosity inproportionto

D

2

∝ q

′ 2

.

We will alibrate the entral blo ks of the alorimeter via elasti H

(e, e

′

C

alo

p

H

RS

)

mea-surements We anti ipate three sequen es of elasti measurements of one day ea h at the

alibrateallof the blo ks and maintaina ontinuousmonitorof the alibration with the

π

0

mass re onstru tion from H

(e, e

′

_π

0

_)X

events. We anti ipate su ient statisti s to obtain anindependent alibration from ea h day of running. The elasti alibrations also serve to verify the geometri al surveys of the spe trometer and alorimeter.

D. Luminosity Limits and Opti al Curingof Calorimeter

In E03-106, we took

D(e, e

′

_γ)X

data at a maximum luminosity (per nu leon) of

4 ·

10

37

_/

m

2

/s, with the front fa e of the PbF

2

alorimeter at 110 m from the target enter. We did not obtain any degradation of resolution in the alorimeter from pileup of signals. During the entire 80 day run of E00-110 and E03-106, we delivered a total of 12 C to the 15 m liquid Hydrogen and Deuterium targets. We performed absolute alibrations of the alorimeterwith elasti H

(e, e

′

Calo

p

HRS

)

events atthe beginningand end ofdata taking. We observed up to 20% de rease in signal amplitude in individual blo ks, without observable lossinmissingmass resolution afterre alibration. Custompre-amplierswere used tokeep the PMT anode urrentsmall. Therefore the loss ofamplitude is attributedto degradation of the transmission properties of the blo ks, and not to degradation of the photo- athodes of the PMTs. Independent numeri al[33℄ and analyti simulations indi ate that below 10 degrees, the radiationdose tothe alorimeterisdominated by Møllerele trons (and related bremsstrahlung). Beyond 20 degrees, the dominant ba kground arises from de ay photons from in lusive

π

0

photo-produ tion. The simulations also indi ate that from 7.5 to 11.5 degrees, theradiationdosediminishesby afa tor5. Onthisbasis,we on ludethatroughly

50%

oftheradiationdosere eivedbythesmallangleblo ksofthe alorimetero urredwhen the smallangle edge of the alorimeter was at

7

◦

. This is the same minimum angle we will use in the present proposal.

A henba h et al. studied the radiationdamage and opti al uring of PbF

2

rystals[34℄. They found the radiationdamage tobe linear for doses up to8 KGy (froma

60

Co sour e). For a dose of 1 KGy, they observed a loss of

25%

in transmission for blue lightof

λ = 400

nm. They also obtained good results for uring the radiationdamage by exposure to blue light. The front fa e of the 1000 element A4 array was exposed for 17 hours to a Hg(Ar) pen il lamp (ltered to pass only

λ > 365

nm) at a distan e of 50 m (intensity on the alorimeter surfa e of

2 µ

W/ m

2

to100% and 97%, respe tively, of its initialvalue.

At this stage, it is not possible to ompare the absolute dose in our simulations with the dose re orded in the MAMI-A4 trials. The A4 dose is re orded as a volumetri dose (1Gy = 1Joule/kg) yet the gamma rays from

60

Co are predominantly absorbed in a layer of thi kness 1/10 the transverse size of the rystals. The radiation dose in Hall A during a DVCS experiment isprimarily fromphotons and ele trons 501000 MeV. Thuswe onsider theabsolutes aleofdose omparisontobeun ertainbyafa torof10. Instead,wenormalize thefutureradiationdamageofthe alorimetertothemaximumvalueforthefra tionalsignal attenuation per integrated luminosity obtained in onditions during E00-110 and E03-106 thatwerenearlyidenti alourproposed onguration. WeutilizethestudiesfromMAMI-A4 to determine that a radiation-dose indu ed attenuation of up to25% (

λ = 400

nm) an be ured (to within

1%

) with a 17 hour exposure to blue light. This uring must be followed by several dark hoursto allowthe phosphores en e of the PMT photo- athodes to de ay.

Pile-up within the 20nsanalysis window of the pulse shape analysis ofthe PbF

2

signals will limit our instantaneous luminosity. Based on our previous experien e we an operate atan instantaneous luminosity times a eptan eperPbF

2

blo kof

L(D) =

 (4.0 · 10

37

₎

cm

2

_{· s}

 

D

(110 cm)



2

.

(38)

with

D

the distan e from targetto alorimeter. Our proje ted ount rates and beam times are based onthis luminosity asa fun tion of the kinemati setting.

We plan to use blue light uring of the blo ks every time the signal attenuation rea hes

20%

. For those settings with the minimum edge of the alorimeter

θ

min

Calo

equal to

7

◦

, this orresponds to5days ofrunningatthe luminosityof Eq.38. Wewilluse the

π

0

→ γγ

mass resolution from both single arm H

(e, π

0

_)Y

(pres aled) and oin iden e H

(e, e

′

_π

0

_)X

events tomonitorthe light yieldsinthe PbF

2

array. Atlarger alorimeterangle settings,the time between uring will be orrespondingly longer. We estimate a total of 1012 uring days during the experiment (inaddition tothe running time).

E. Trigger

Theele tron dete tor sta k inthe HRS willbethe standard ongurationofVDC I and II,segmented S1andS2, gasCerenkov, andPb-Glass alorimeter. The Cerenkov (

Cer

ˇ

)and

Pb-Glass provide redundant ele tron identi ation in the o-lineanalysis. The main HRS trigger is

HRS = [∪

i,j

(S1R

i

∩ S1L

i

) ∩ (S2R

j

∩ S2L

j

)] ∩ ˇ

Cer.

(39)

This requires a oin iden e between the PMTs at the two ends of at least one s intillator paddle in ea h of

S1

and

S2

. In addition, we require a oin iden e with the Cerenkov ounter. Supplementary pres aled triggers with either

S1

,

S2

, or

Cer

ˇ

removed from the oin iden e willmonitorthe e ien y of ea htrigger dete tor.

In addition to the HRS ele tron trigger, we will upgrade our present oin iden e validation/fast- lear logi . The alorimeter signals are ontinuously re orded by the 128 sample

⊗

1GHzARS[20,32℄array. Thisarrayisdigitized,ataslowerrate,followinga trig-ger validation. For ea h HRStrigger, westop the ARS sampling,and triggera Sampleand Hold(SH) ir uit, that is oupledtothe alorimeter signalsvia ahigh impedan e input (in thefuture, wemayrepla ethiswithapipelineADC).TheSHsignalsare digitizedandthen in a eld programmable gate array (FPGA), we will form the following trigger validation signals.

1. A validationofthe HRS-ele tron trigger,basedon dete tingatleast one

2 × 2

luster abovea programmablethreshold

E

Th

γ

. 2. A

π

0

triggerbasedondete tingatleasttwoseparated

2 × 2

lusters,with ea h luster above a programmable threshold

E

1

and the sum of the two lusters above a pro-grammable threshold

E

π

. This triggerwill sele t andidate H

(e, e

′

_π

0

_)X

events. If a validsignal is found, the ARS array is digitized and re orded (together with the HRS signals). If the HRS trigger is not validated by the alorimeter signals, a fast lear is issued to the ARS array, no digitization o urs, and a quisitionresumes. During E00-110, the SH/Fast- lear y le took 500 ns. With upgrades to the FPGA, we an shorten this deadtime by a fa tor of two. In addition, upgrades to the VME standard will allow us to in rease the total bandwidthof data a quisition for this proposal.

IV. PROJECTED RESULTS

The detailed

(e, e

′

₎

kinemati s, alorimeter onguration, H

(e, e

′

_γ)X

missing mass reso-lution in

M

2

TableV: ExperimentalConditions forDVCS.Forea h

(Q

2

_{, k, x}

Bj

)

setting, we present: Maximum photonenergy

q

′

₍₀

◦

₎

;Calorimeterdistan e

D

;Virtualphoton dire tion

θ

q

;Angle

θ

min

calo

of theedge of the alorimeter, relative to the beam line; Kinemati minimum

|t

min

|

and upper bound

|t

max

|

for approximately full a eptan ein

φ

γγ

; Resolution

σ(M

2

X

)

in H

(e, e

′

_γ)X

missingmass squared; Luminosity

L

; H

(e, e)X

in lusive trigger rate, H

(e, e

′

_γ)p

ex lusive DVCS ount rate. The beam time is al ulated to obtain an estimated 250K events at ea h setting, or at least 40,000 events per bin in

t

min

− t

. The distan e

D

of the front fa e of the alorimeter from the target enter is optimized for the separation of the lusters from

π

0

_{→ γγ}

de ay (se tion IIIC). The intrinsi missing massresolution in E00-110 is

σ(M

2

X

) = 0.20

GeV

2

. The luminosity is determined by the maximumrateallowedbypileupinthe alorimeter. Thisluminosityisproportionalto

D

2

(

§

IIID).

Q

2

_k

_x

Bj

q

′

(0

◦

) D

θ

q

θ

min

calo

t

min

t

max

σ(M

X

2

) L/10

38

HRS DVCS Time (GeV

2

) (GeV) (GeV) (m) (deg) (deg) (GeV

2

) (GeV

2

) (GeV

2

) ( m

−2

/s) (Hz) (Hz) (days) 3.0 6.6 0.36 4.35 1.5 11.7 7.1 -0.16 -0.42 0.23 0.75 479 1.16 3 4.0 8.8 0.36 5.83 2.0 10.3 7.0 -0.17 -0.42 0.26 1.3 842 1.74 2 4.55 11.0 0.36 6.65 2.5 10.8 7.0 -0.17 -0.42 0.27 2 2460 4.63 1 3.1 6.6 0.5 3.11 1.5 18.5 11.0 -0.37 -0.64 0.17 0.75 873 0.77 5 4.8 8.8 0.5 4.91 2.0 14.5 8.9 -0.39 -0.70 0.20 1.3 716 0.82 4 6.3 11.0 0.5 6.50 2.5 12.4 7.9 -0.40 -0.72 0.20 2. 778 0.99 4 7.2 11.0 0.5 7.46 2.5 10.2 7.0 -0.40 -0.75 0.25 2. 331 0.53 7 5.1 8.8 0.6 4.18 1.5 17.8 10.4 -0.65 -1.06 0.16 0.75 338 0.27 13 6.0 8.8 0.6 4.97 2.0 14.8 9.2 -0.67 -1.05 0.18 1.3 227 0.22 16 7.7 11.0 0.6 6.47 2.5 13.1 8.6 -0.69 -1.10 0.20 2. 274 0.28 13 9.0 11.0 0.6 7.62 3.0 10.2 7.3 -0.71 -1.14 0.22 3. 117 0.17 20

hosen to give atotal statisti s of 250K events per

(x

Bj

, Q

2

₎

setting.

A. Systemati (Instrumental)Errors

Table VI shows the main systemati errors on the ross-se tions extra tion during esti-mation of those errors for the proposed experiment. The main improvements of the errors