QED Corrections to Deep Inelastic Scattering with Tagged Photons at HERA

7 Pages·0.19 MB·English

by  H. Anlauf

QED corrections to deep inelastic scattering with tagged photons at HERA
Harald Anlauf
Fachbereich Physik, Siegen University, 57068 Siegen, Germany
Andrej B. Arbuzov, Eduard A. Kuraev
Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia
Nikolaj P. Merenkov
Kharkov Institute of Physics and Technology, 310108 Kharkov, Ukraine
(February 1, 2008)
tagged photons in order to reduce the radiative correc-
9
WecalculatetheQEDcorrectionstodeepinelasticscatter-
tions and to simplify the F analysis. They did not con-
9 2
ingwith tagged photonsat HERAintheleadinglogarithmic
9 siderhigherordercorrectionsexplicitly,anditisnotclear
approximation. Due to the special experimental setup, two
1 whether their method allows an extraction of FL with-
large scales appear in the calculation that lead to two large
out running at lower beam energies. On the other hand,
n logarithms ofcomparable size. Therelation of ourformalism
a to the conventional structure function formalism is outlined. the feasibility of a quantitative measurement of FL at
J HERAusingthemethodof[2],forQ2 below5GeV2 and
1 Wepresentsomenumericalresultsandcomparewithprevious x around 10−4, has been studied in [5] and considered
calculations.
1 possible.
PACS number(s): 13.60.-r, 13.60.Hb A more general treatment of the Born cross section
2 for ep→eγX that is also valid for non-collinear photon
v emissionandintherangeofhighQ2 canbefoundinthe
3
paper by Bardin et al. [8].
3
3 I. INTRODUCTION For an accurate description of the corresponding cross
1 section one has to consider the radiative corrections. In
1 One of the major tasks of the experiments at the this letter we calculate the QED radiative corrections to
7 HERAcollideristhedeterminationofthestructurefunc- deepinelastic scatteringwithonetaggedphotontolead-
9 tions of the proton, F (x,Q2) and F (x,Q2), over a inglogarithmicaccuracyusingthestructurefunctionfor-
2 L
/
h broad range of the kinematical variables. The exten- malism[6,7]. Wewillinparticulardiscusstheappearance
p sionofthesemeasurementstotherangeofsmallBjorken oftwolargelogarithmscorrespondingtotwolargescales,
- x < 10−4 and Q2 of a few GeV2 is of particular inter- which are due to the experimental setup peculiar to the
p
est,becauseitwillhelpimproveourunderstandingofthe HERA experiments. Finally, we will show some numeri-
e
h details of the dynamics of quarks and gluons inside the calresultsandcompareourfindingswiththe calculation
: nucleon [1]. by Bardin et al. [8].
v
In order to separate F (x,Q2) from F (x,Q2), it is
i L 2
X necessaryto measurethe crosssectionfor ep→eX with
r different center-of-mass energies. However, instead of II. KINEMATICS AND LOWEST ORDER CROSS
a
running the collider at reduced beam energies, one can SECTION
employa methodsuggestedbyKrasnyetal.[2]thatuti-
lizes radiative events. This method takes advantage of a To begin with, let us start with a brief review of the
photon detector (PD) in the very forward direction, as kinematics adapted to the case of deep inelastic scatter-
seen from the incoming electron beam. Such a device ing with one exclusive hard photon,
is part of the luminosity monitors of the H1 and ZEUS
experiments. e(p )+p(P)→e(p′)+γ(k)+X(P′), (1)
e e
The idea of the method is that emission of photons in
a direction close to the incoming electron can be inter- where the polar angle ϑγ of the photon (measured with
preted as a reduction of the effective beam energy. The respect to the incident electron beam) is assumed to be
effective beam energy for each radiative event is deter- very small, ϑγ ≤ ϑ0, with ϑ0 being about 0.45 mrad in
mined from the energy of the hard photon that is ob- the case of H1.
served in the PD. In fact, radiative events were already A convenient set of invariants that takes into account
usedinameasurementofthestructurefunctionF down the energy loss from the collinearly radiated photon is
2
to Q2 >∼ 1.5 GeV2 [3]. given by [2]:
The possibility to use radiative events for structure
function analyses was already discussed by Jadach et Q2 =−(pe−p′e−k)2,
al.[4]. However,theseauthorsusedradiativeeventswith
1
Q2 III. RADIATIVE CORRECTIONS
x= ,
2P ·(p −p′ −k)
e e
P ·(p −p′ −k) There are two large sources of contributions to the
y = e e . (2)
P ·(p −k) radiative corrections from higher orders that contribute
e
terms of the order of α/π·lnQ2/m2, with m being the
f f
Sincewerestrictourselvestocollinearphotons,itissuit- mass of a light fermion. One type of contributions are
able to parameterize the energy of the radiated photon, thecorrectionstothepropagatoroftheexchangedboson
E , with the help of between the electron and the hadronic system, while the
γ
others are due to radiation of photons off the incoming
z = Ee−Eγ = Q2 , with S =2p ·P. (3) electronline,whichwewilltreatinthestructurefunction
E xyS e formalism.
e
We shall assume a calorimetric experimental setup,
The differential cross section for the process ep → eγX, whereahardphotonradiatedcollinearlytotheoutgoing
integrated over the photon emission angle 0 ≤ ϑγ ≤ ϑ0, electron line cannot be distinguished from a bare out-
reads [2] going electron, so that final state radiation can be ne-
glectedtothedesiredleadinglogarithmicaccuracyinac-
d3σBorn =σ (x,Q2;z)· α P(z,L ), (4) cordancewiththeKinoshita-Lee-Nauenbergtheorem[9].
dxdQ2dz 0 2π 0 We also assume some minimum experimental cut on the
where transverse momentum of the outgoing hadronic system,
2πα2 y2 inordertosuppressthecontributionfromQEDCompton
σ (x,Q2;z)= 2(1−y)+ F (x,Q2), (5)
0 xQ4 (cid:20) 1+R(cid:21) 2 events [10] for a leptonic measurement of the kinematic
with variables.
y =Q2/(xzS), (6)
and
1+z2 2z A. Vacuum polarization
P(z,L )= L − , (7)
0 1−z 0 1−z
The first important type of contribution to the ra-
with
E2ϑ2 diative corrections comes from the vacuum polarization,
L ≡lnζ , ζ = e 0. (8)
0 0 0 m2 which amounts to replacing the coupling constant α in
e
thehardcrosssectionσ (eq.5)bytheQEDrunningcou-
0
Here we have neglected terms of order O(ϑ2) as well as plingα(−Q2). Forthecontributionfromleptonloopswe
termsoforderO(ζ−1),andwehaveneglecte0dthecontri- use the well-known one-loop perturbative result. Since
0
butions fromZ0 exchangesince wearemainly interested weareinterestedintheregionofratherlowQ2,weusea
in the region of small Q2 (see also [5]). R is defined as parameterizationby BurkhardtandPietrzyk [11]for the
the ratio of cross sections for longitudinal to transverse hadronic contribution.
photons,
R(x,Q2)= σL = FL . (9) B. Photonic corrections
σ F −F
T 2 L
In order to illustrate the calculation of radiative cor-
The allowedrange for z follows from (3) and the restric-
rectionstothecrosssection(4)duetophotonemissionin
tion 0≤y ≤1,
the framework of the structure function formalism [6,7],
let us first note that this cross section is already a QED
Q2
≤z ≤1. (10) correction: it is that part of the inclusive first order cor-
xS rection to deep inelastic scattering (ep → eX), which
Note that the entire z-dependence of σ occurs only via is selected by requiring an exclusive hard photon seen
0
(6). in the PD. This may be exhibited by writing down the
Eq. (4) agrees with the Born cross section given in radiatively corrected inclusive cross section to DIS as a
[8] when restricted to the kinematical situation under convolutionoftheelectronnon-singletstructurefunction
consideration.1 DNS(z,Q2) with the hard cross section (5)
d2σ
RC = dz D (z,Q2)σ (x,Q2;z). (11)
dxdQ2 Z NS 0
1Weusedthesamesetofkinematicalvariablesasin[2]and
The electron non-singlet structure function,
[5]. Fordetailsontherelationbetweenthesetwosetswerefer
thereader to thediscussion in [2].
2
DNS(z,Q2)=δ(1−z)+ 2απ LP(1)(z) 2απ 2L0 (L0+L1)Pδ(1)−(L0+2L1)lnz
(cid:16) (cid:17) h i
+ α 2 L2P(2)(z)+O (αL)3 , (12) ×PΘ(1)(z)·σ0(x,Q2;z). (15)
2π 2!
(cid:16) (cid:17)
(cid:0) (cid:1)
The contribution from two hard photons in the PD,
dependsonthelargescaleQ2onlyviathelargelogarithm
with the energy fraction of each being larger than ∆, is
L = lnQ2/m2. It is known to properly sum the leading
e found to be [13]
contributions (αL)n to all orders in perturbation theory
[7]. α 2 L2
Wenowgivetherelevantcoefficientsofthepowerseries 0 P(2)(z)−2P(1)(z) P(1)−lnz
2π 2 Θ Θ δ
expansionofD . Introducingasmallauxiliaryparame- (cid:16) (cid:17) h (cid:16) (cid:17)i
NS ×σ (x,Q2;z), (16)
ter∆thatservesasaninfrared(IR)regulatortoseparate 0
virtual+softandhardphotoncontributions,thefirsttwo
withz =1−( E )/E ,where E isthetotalphoton
coefficients of the expansion of D are γ e γ
NS energy registePred in the PD. P
P(1,2)(z)=P(1,2)·δ(1−z)+P(1,2)(z)·Θ(1−∆−z), Finally,weconsiderthecontributionfromonecollinear
δ Θ photon that hits the PD, while the other one is emitted
with at an angle larger than ϑ .
1+z2 3 0
P(1)(z)= , P(1) =2ln∆+ , (13) Letusinvestigatetheregionsofphasespacewherethe
Θ 1−z δ 2 large logarithmic contributions originate. For the hard
photon that hits the PD, the major contribution comes
and similarly (see e.g. [7]),
from the region of polar angles ϑ(1) ≈ m /E ≪ ϑ .
γ e e 0
1−∆ dt z Similarly, for the other photon the biggest contribu-
PΘ(2)(z)=Zz/1(1+−∆z)2 t PΘ(1)(t)PΘ(1)(cid:16)t(cid:17)+23Pδ(1)PΘ(1)(z) tϑi(γo2n) >∼comϑ0e.sTfrhoums,ptholearleaadnignlegsccolnotsreibtuotiothnes cloowmeerelsimseint-,
=2 2ln(1−z)−lnz+ tially from the region where (ϑ(2)/ϑ(1))2 ≈ ζ ≫1. One
(cid:20) 1−z (cid:18) 2(cid:19) γ γ 0
caneasilysee thatoneobtainsthe doublelogarithmsen-
+ 1+z lnz−1+z . (14) tirelyfromthecontributionwherethephotonhittingthe
2 (cid:21) PD is emitted first, with the “lost” photon being emit-
ted second, while the reversed case is suppressed if the
Inspecting (4), it is obvious that the logarithmic piece following condition is satisfied:
of P(z) is contained in the first order correction to the
inclusive cross section; the difference in the logarithms E(2)
(L−L0) is accounted for by the remaining phase space x2·ζ0 ≫1, where x2 = Eγ . (17)
due to photons emitted with angle larger than ϑ . e
0
Let us now turn to the contributions from the second Thusweshallassumeinthefollowing∆≪1,but∆·ζ ≫
0
order. Since the maximum emission angle ϑ0 is about 1.
0.45mradfortheHERAdetectors,the“largelogarithm” Takingthis orderingofphotonemissionsinto account,
L0 appearing in (4) turns out to be moderate, the contribution from one tagged plus one undetected
photon canbe calculatedby means of the quasirealelec-
ζ ≈600, L ≈6.4 at E = 27.5 GeV,
0 0 e tron method [14]
so that the complement α 2 xm2ax dx x
L L ·P(1)(z) 2P(1) 1− 2
L1 ≡L−L0 ≈6.5...15.7 (cid:16)2π(cid:17) 0 1 Θ Z∆ z Θ (cid:16) z (cid:17)
(for e.g., Q2 = 0.1 ...1000 GeV2) ×σ˜0(x,Q2;z−x2), (18)
is of similar magnitude or even larger than L0. For this whereσ˜0 is understoodto be expressedbythe kinematic
reason we have two large logarithms L ,L entering the variables xˆ,Qˆ2, of the hard subprocess,
0 1
game.
We shall separate the contributions into vir- σ˜0(x,Q2;z−x2)≡σ0(xˆ,Qˆ2,z−x2)·J(x,Q2;x2). (19)
tual+collinear,soft+collinear,twocollinearphotons,and
one collinear plus one semicollinear photon. Note thatthis contributionexplicitly depends onthe ex-
Thesumofthecontributionsofvirtual+softcorrection perimental determination of the kinematical variables x
to collinear photon emission, with the emission angle of and Q2, since the almost collinear emission of the sec-
thesoftphotonbeingintegratedoverthe fullsolidangle, ondphotonshiftsthe“true”kinematicalvariables(xˆ,Qˆ2)
but the hard photon only over the angular range of the
PD,canbeobtainedfromtheexpressionfortheone-loop
Compton tensor. It reads [12]
3
with respect to the measured ones (x,Q2).2 The Jaco- α(−Q2) 2 αL 1−z
1+δ = 1+ 0 P(2)(z)
bianJ accountsforscalingpropertiesofthechosenkine- ho (cid:18) α(0) (cid:19) (cid:26) 4π 1+z2 Θ
matical variables under radiation of the second photon.
αL u0 σ˜ (z(1−u))
The upper limit of the x -integration in (18) is given by + 1 duP(1)(1−u) 0 −1
2 2π (cid:20)Z Θ (cid:18) σ˜ (z) (cid:19)
either some experimental cut on the maximal energy of 0 0
thesecondphoton,orbythekinematicallimit,whichalso 1
− duP(1)(1−u) . (22)
dependsonthe choiceoftheexperimentaldetermination Z (cid:21)(cid:27)
u0
of the kinematical variables, as we will discuss later.
After change of variables x = zu, u = xmax/z, the Notethatthecontributionfromtheundetectedhardpho-
2 0 2
integral in (18) may be conveniently decomposed into ton depends on the choice of kinematical variables and
IRdivergent(∆dependent)andIRconvergentcontribu- on the upper limit u .
0
tions as (suppressing the arguments x,Q2)
xm2ax dx x IV. RESULTS FOR HERA
2P(1) 1− 2 σ˜ (z−x )=
Z∆ z Θ (cid:16) z (cid:17) 0 2
u0 Inthepresenceoftheanundetected(lost)photon,the
=Z duPΘ(1)(1−u)[(σ˜0(z(1−u))−σ˜0(z))+σ˜0(z)] determinationofthekinematicalvariablesx,Q2 becomes
∆/z ambiguous, it will depend on the method chosen, and in
=σ (z)· u0duP(1)(1−u) σ˜0(z(1−u)) −1 turn the corrections (22) will depend on this choice.
0 (cid:20)Z Θ (cid:18) σ˜ (z) (cid:19) For HERA, several methods are being used to deter-
0 0
1 mine the “true” kinematical variables xˆ,Qˆ2, in order to
+2lnz−P(1)− duP(1)(1−u) , (20) reduce systematic errors or to control the influence of
δ Z Θ (cid:21)
u0 initial state radiation (see e.g., [15] for an illuminating
discussion of the latter). For the sake of brevity we will
whereinthelaststepwehaveextendedtheu-integration
discuss only the so-called electron method (E) and the
of the IR convergent piece to 0 due to the smallness of
Jacquet-Blondel method (JB).
∆, and we have used the property
In the case of the electron method, the kinematical
1 variables (2) are determined via
duP(1)(u) =0
Z0 Q2 =4EeffE′ cos2ϑ /2,
e e e e
in the simplification of the IR divergent piece. E′ Q2
y =1− e sin2ϑ /2, x = e , (23)
If we sum the contributions (15), (16), and (18), we e Eeff e e y seff
e e
see that the dependence on the auxiliary parameter ∆
with
cancels, as it should. seff =4EeffE , Eeff =E −E =zE ,
e p e e γ e
and E is the energy deposited in the PD. Radiating an
γ
C. The radiatively corrected cross section additional almost collinear photon with energy E(2) =
γ
x E leads to a shift of the “true” variables xˆ,Qˆ2, of
2 e
We may now write down our final result for the ra-
the hard subprocess with respect to the measured ones,
diative corrections. Since we restricted ourselves to the x ,Q2,
leading logarithmiccorrections,it followsthatthe radia- e e
tivelycorrectedcrosssectionmaybewritteninfactorized y +z′−1 x y z′
form, Qˆ2 =Q2e·z′, yˆ= e z′ , xˆ= y +eze′−1, (24)
e
d3σRC = d3σBorn · 1+δ (x,Q2,z) . (21) with z′ = 1−x2/z = 1−u. The kinematical limit for
dxdQ2dz dxdQ2dz ho the undetected photon follows from xˆ≤1,
(cid:0) (cid:1)
whereweretaininthecorrectionfactor(1+δho)onlythe z′ ≥ 1−ye . (25)
logarithmic terms L0,L1 from (15), (16), and (18), and 1−xeye
the contribution from vacuum polarization,
TheJacobian(19)forthischoiceofkinematicalvariables
is
′ 2
y z
2For a recent review on the influence of initial state radi- J(x ,Q2)= e . (26)
e e (cid:18)y +z′−1(cid:19)
ation on the experimental determination of the kinematical e
variables see e.g., [15], and references cited therein.
In the case of the Jacquet-Blondel method, the kine-
matical variables are determined using
4
1
y = (E −p ),
JB 2Eeff h z,h
e Xh 1
p⊥2 Q2
Q2 = h h , x = JB , (27) δho
JB P1−y JB y seff
JB JB 0.8
⊥
whereE ,p ,p aretheenergy,longitudinalandtrans-
h z,h h
verse momentum components of the final state hadrons. 0.6
The relation between the measured and “true” vari-
ables in presence of an undetected photon are now given
by 0.4
1−y y 1−y
Qˆ2 =Q2JB1−y J/Bz′, yˆ= zJ′B, xˆ=xJB1−y J/Bz′, 0.2
JB JB
(28)
0
with the Jacobian
1−y 2 -0.2
J(x ,Q2 )= JB , (29) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
JB JB (cid:18)1−yJB/z′(cid:19) ye
FIG.1. Radiativecorrectionsδho (22)inleptonicvariables
and with the kinematical limit being
fordifferentvaluesofxandataggedphotonenergyof5GeV.
y
z′ ≥ JB . (30)
1−xJB(1−yJB) Atsmallye,thecorrectionsarenegative,sincethecon-
tribution from virtual+soft corrections dominates, be-
Comparing (24) and (28), one finds that one can in cause the kinematical limit on the energy of the unde-
principle determine the energyof the undetected photon tected photon tends to zero as y →0,
e
(assuming that it is emitted almost collinearly) via y (1−x )
u =1−z′ = e e . (33)
z′ =1+y −y . (31) 0 min 1−xeye
JB e
For large y , the phase space for photon emission in-
e
This may be used to impose an experimental cut to re- creases, increasing also the shifts between “true” and
duce the size of the radiative corrections. measured variables (24), which leads to large positive
Let us now present some numerical results for the ra- corrections as y →1.
e
diative corrections (22). As input we used
E =27.5 GeV, E =820 GeV, ϑ =0.45 mrad,
e p 0
0.1
(32)
δ
ho
the GRV94-LOstructure functions [16]fromPDFLIB[17]
for x> 10−3, the BK parameterization3 [19] for GRV94
0.05
forx<10−3,andforsimplicityafixedvalueR=0.3(see
e.g.,[3]foratableofmeasuredRvalues). Figure1shows
the corrections δ for an energy of 5 GeV deposited in
ho
the PD.
0
3Wehaveverifiedthattheresults forthecorrections donot -0.05
change significantly when we use theALLM [18] parameteri-
zationforx<∼10−3,althoughthedifferenceinthepredictions
forthelow-x,low-Q2 rangeaffectsthecorrectionsalreadyfor
x around 10−4. -0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
y
JB
FIG. 2. Radiative corrections δho (22) in Jacquet-Blondel
variablesfordifferentvaluesofxandataggedphotonenergy
of 5 GeV.
5
Figure2showsthe correctionsδ forJacquet-Blondel in [8]. More details on this interference between photon
ho
variables, again for a tagged photon energy of 5 GeV. emissions can be found in [13]. Furthermore, it seems
In this case the corrections are negative for y → 1, thattheauthorsomittedthestatisticalfactor1/2!,while
JB
because in this limit the phase space for the undetected summing the contributions (iib, iic) of the semicollinear
photon tends to zero, kinematics.
Some care is needed in the comparison of numerical
u =1−z′ = (1−xJB)(1−yJB), (34) results since Bardin et al. choose kinematical variables
0 min 1−x (1−y ) different from ours. Taking into account the abovemen-
JB JB
tionedfactorof2andthedifferenceduetoourtreatment
whereas for small y the corrections remain moder-
JB oftheorderingconditionforthephotonemission,wecan
ate. Since the Jacquet-Blondel variables correspond to
qualitatively reproduce their results as presented in fig-
a “more inclusive” measurement than the leptonic vari-
ure6oftheirpaperintherangeofsmallery values(thus
ables, the corrections due to radiation of an additional
confirming the lack of the symmetry factor).
photon are generally smaller.
ACKNOWLEDGMENTS
V. DISCUSSION AND CONCLUSIONS
We are thankful to T. Ohl and E. Boos for fruitful
In this letter, we have studied the radiative correc-
discussions,toD.BardinandL.Kalinovskayafordiscus-
tions to deep inelastic scattering with taggedphotons at
sionsoftheirpaperandsendingusnumericalresultsfrom
HERA taking into account only the leading logarithms. the program HECTOR[21], to B. Bade lek for providing us
The relevant expansion parameter in the present case is with a Fortran programfor their parameterizationof F
2
not simply α/π · L, but we have two large logarithms at low x and Q2 [19]. One of us (E.K.) is grateful to the
L0,L1 = L−L0 which happen to be of similar order of Heisenberg-Landau foundation for financial support and
magnitude due to the particular choice of the geometri-
to the Technische Hochschule Darmstadt for hospitality.
cal acceptance of the photon detector. The corrections Three of us (A.A., E.K., N.M.) are thankful to the IN-
do depend on the choice of the experimental determina-
TAS foundation, grant 93–1867 (ext.) and to INFN and
tionofthekinematicalvariables,buttheyturnouttobe the Physics Department of Parma University for hospi-
oftheorderof20–40%inmostregionsofphasespacefor
tality. Finally, H.A. would like to thank Bundesminis-
leptonic variables, and below 10% for Jacquet-Blondel
terium fu¨r Bildung, Wissenschaft, Forschung und Tech-
variables. Large negative corrections occur in those re-
nologie(BMBF),Germanyforsupport,andM.Fleischer
gions of phase space that are dominated by soft photon
for drawing our attention to this problem and the mem-
emission. Theselargecorrectionsarehoweverwellunder
bers of the H1 RACO group for continued interest.
control once one takes into account the resummation of
multiple photon emission [6,7].
We have not considered the contributions from next-
to-leading order,(α/π)2×L, which may be sizable since
theindividuallogarithmsL ,L arenotverylarge,espe-
0 1
ciallyfortheexperimentallyinterestingregionoflowQ2.
However,the calculation of those terms is more involved [1] See e.g., Proceedings of the Zeuthen Workshop on Deep
and will be presented elsewhere [20]. Inelastic Scattering, J. Blu¨mlein, T. Riemann (eds.),
We have compared our results with the results given Teupitz/Brandenburg, Apr. 6-10, 1992, and references
in [8]. At the Born level, we find very good numerical cited therein.
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[7] W.Beenakker,F.A.Berends,W.L.vanNeerven,Proceed-
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6
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