Two-loop QED Corrections to Bhabha Scattering

0.19 MB·English

by  R. Bonciani

Two-loop QED Corrections to Bhabha Scattering
6
0 R. Bonciania∗ and A. Ferrogliab
0
2 aDepartament de F´ısica Teo`rica, IFIC, CSIC – Universitat de Val`encia, E-46071 Val`encia, Spain

n
bPhysikalisches Institut, Albert-Ludwigs-Universit¨at Freiburg, D-79104 Freiburg, Germany
a
J
0 Recent developments in the calculation of the NNLO corrections to the Bhabha scattering differential cross
3 section in pureQED are briefly reviewed and discussed.

1
v
6
1. Status of the NNLO corrections beam effects that lead to a non monochromatic
4
2 Bhabha scattering, e+e− → e+e−, is a crucial luminosity spectrum [3].
1 Experimentally, Bhabha scattering is measur-
0 processinthe phenomenologyofparticlephysics. able with a very high accuracy. At LEP, the ex-
6 Its relevance is mainly due to the fact that it is
perimental error on the luminosity measurement
0 the process employed to determine the luminos- was 4·10−4 [4]. At ILC it is expected to be of
h/ ityLate+e−colliders: infact,L=NBhabha/σth, the same order of magnitude or better (the goal
p where NBhabha is the rate of Bhabha events and of the TESLA forward calorimeter collaboration
p- σth is the Bhabha scattering cross section calcu- istoreach,inthe firstyearofrun,anexperimen-
e lated from theory. tal error of 1·10−4 [5]). This remarkable accu-
h Twokinematicregionsareofspecialinterestfor
racyrequires,asacounterpart,anequallyprecise
: the luminosity measurements, since in these re-
v theoretical calculation of the Bhabha scattering
i gionsthe Bhabhascattering crosssectionis com- cross section, in order to keep the luminosity er-
X
paratively large and QED dominated. At collid-
ror small. Therefore, radiative corrections to the
r ers operating at c. m. energies of O(100GeV),
a basic process have to be under control.
the relevant kinematic region is the one in which
In order to match the detector geometry and
the angle between the outgoing particles and the
experimental cuts of any particular machine, a
beam line is of few degrees. This, for instance,
Monte Carlo event generator is needed. In the
was the case at LEP, where the luminometers
recent past, several groups have been working
were located between 50 and 100 mrad, and it
on Monte Carlo generators for Bhabha scatter-
will be also the case at the future ILC (lumi-
ing, inboththe large-angleandsmall-anglekine-
nometers between 25 and 80 mrad [1]). At ma-
matic regions. The LEP theoretical simulations
chines operating at c. m. energies of the order
of Bhabha events were based on BHLUMI [6],
of 1 − 10GeV, the region of interest is instead
whose theoretical error, mainly due to missing
the one in which the scattering angle is large; as
higher order corrections, is estimated to be 4.5·
an example, at KLOE, the luminosity measure- 10−4 (see for instance [7]). The KLOE collab-
mentinvolvesBhabhascatteringeventsthattake
oration employs the Monte Carlo event genera-
◦ ◦
place at angles between 55 and 125 [2]. More-
tors BABAYAGA [8] and BHAGENF [9], which
over, the large angle Bhabha scattering will be have an estimated theoretical error of 5 · 10−3;
employed at the ILC in order to study the beam
within the error claimed, they are in agreement
with each other. Moreover, BHWIDE [10] and
∗This workwas supported bytheEuropean Unionunder MCGPJ [11] provided valuable checks. All the
the contract HPRN-CT2002-00311 (EURIDICE) and by mentioned Monte Carlo programs for Bhabha
MCYT(Spain)underGrantFPA2004-00996,byGeneral-
scattering employ the mass of the electron as a
itatValenciana(GrantsGRUPOS03/013andGV05/015).
1
2 R. Bonciani and A. Ferroglia
cut-offforcollineardivergences;thisistobetaken 2e-05
into account when calculating NLO (O(α3)) and
NNLO (O(α4)) corrections to the cross section. 1.5e-05 DNF=1(θ,E)
The complete O(α3) corrections to Bhabha
scattering, in the full Electroweak Standard 1e-05
Model,havebeenknownforalongtime[12]. The
◦
10
corrections of O(α4) to the differential cross sec- 5e-06 90◦
tion in the Standard Model are not yet known.
In recent years, several papers were devoted to 0
the study of NNLO corrections in pure QED. 0.02 0.04 0.06 0.08 0.1 0.12 0.14
In [13,14,15] the second order radiative correc- E (GeV)
tions, both virtual and real, enhanced by factors
of lnn(s/m2) (with n = 1,2, s the c. m. energy Figure1. D◦ NF=1 as a function of th◦ebeam energy,
forθ =10 (solidline)andθ =90 (dashedline).
squared, and m the mass of the electron) where
The soft-photon energy cut-off is set equal to E.
studied. The complete set of these corrections
wasfinallyobtainedin[16]. Thiswasachievedby
employing the QED virtual corrections for mass-
less electron and positrons of [17] and the results
of [18], as well as by using the known structure Bhabha scattering cross section, as well as the
oftheIRpolesindimensionalregularization[19]. calculation of the subset of radiative corrections
In [20], the complete set of photonic O(α4) cor- due to the interference of one-loop diagrams (al-
rections to the differential cross section that are ready considered in [31]) was completed. A few
not suppressed by positive powers of the ratio papers discuss the leading NNLO weak correc-
m2/s was calculated. The virtual corrections of tions to Bhabha scattering cross section [32].
O(α4) involving a closed fermion loop, together In summary, the complete NNLO corrections
with the correspondingsoft-photon emission cor- to Bhabha scattering in the full Standard Model
rections,wereobtainedin[21];nomassexpansion are still far from being completely known. Con-
or approximation was employed, and the result cerning the pure QED contributions, from a
retains the full dependence on the electron mass phenomenological point of view, all the numer-
m. The calculation was performed by means of ically relevant corrections to the NNLO differen-
the Laporta-Remiddi algorithm [22] which takes tial cross section are known, with the only ex-
advantage of the integration by parts [23] and ception of the ones arising from the production
Lorentz-invariance [24] identities in order to re- of soft pairs, for which only the terms enhanced
ducetheproblemtothecalculationofasmallset by lnn(s/m2) with n = 1,2 are present. If we
of master integrals. The master integrals were considerinsteadtheexactfixedordercalculation,
calculatedusingthedifferentialequationsmethod the situation is less satisfactory. The unapproxi-
[25]; their expression in terms of harmonic poly- mated two-loopQEDphotonic box contributions
logarithms [26] is given in [27]. Several papers are still missing. Moreover, it would be interest-
deal with the unapproximated calculation of the ing to evaluate the corrections arising from dia-
master integrals necessary for the evaluation of grams with closed loops of heavier fermions (like
the photonic NNLO corrections [28]; in partic- muons or taus).
ular, the reduction to master integrals of these
correctionsiscomplete,andonlythemasterinte-
2. The Small Mass Limit
grals related to the two-loop boxes have not yet
been all evaluated. The status of the calculation At the colliders mentioned above, the mass of
of these master integrals is discussed in [29]. In the electron is very small in comparisonwith the
[30], finally, the unapproximated calculation of c. m. energy. Therefore, it is reasonable to ex-
the photonic vertex contributions to the O(α4) pect that it can be safely ignored except in the
Two-loopQED Corrections to Bhabha Scattering 3
2e-05
4e-06
DBox Box(θ,E)
1.5e-05 DVertices(θ,E) 3e-06
◦
1e-05 2e-06 10◦
90
◦ 1e-06
10
5e-06 ◦
90
0
0
-1e-06
0.04 0.06 0.08 0.1 0.12 0.14 0.04 0.06 0.08 0.1 0.12 0.14
E (GeV) E (GeV)
Figure 2. DVertices as a function of the beam en- Figure3. DBox Box as afunctionoftheenergy, for
◦ ◦ ◦ ◦
ergy, for θ =10 (solid line) and θ =90 (dashed θ = 10 (solid line) and θ = 90 (dashed line).
line). The soft-photon energy cut-off is set equal The soft-photon energy cut-off is set equal to E.
to E.
the same quantity as dσ(i)/dΩ, aside for the fact
terms where it acts as a cut-off for collinear di- 2
that the terms proportionalto positive powersof
vergencies,asitwasdoneinobtainingtheresults
the ratio m2/s are neglected.
of[20]. For the setof correctionsobtainedin [21]
and [30], for which unapproximated analytic re- In Figs. 1, 2 and 3 the functions Di, evalu-
◦ ◦
atedfortwodifferentsampleangles(10 and90 ),
sults are available, it is possible to determine the
are shown. It is clear that the approximation in
numerical relevance of the terms suppressed by
positivepowersofthe ratiom2/sasafunctionof which terms proportional to positive powers of
the ratio m2/s are neglected is extremely good
the beam energy.
already at energies that are significantly smaller
This analysis is performed in [30]. The three
than the ones encountered in e+e− experiments.
contributionstakenintoaccountaretheonearis-
The NNLO corrections arising from the two-
ingfromgraphswithaclosedfermionloop(NF =
loop photonic boxes are not taken into account
1), the photonic corrections involving at least a
vertex graph (Vertices),and the interference of in this analysis, since unapproximated result for
one-loop box diagrams (Box Box). For each con- that set of corrections are not available. Never-
theless,itisreasonabletoexpectinthem2/s→0
tribution, the quantity
limit a behavior similar to the one of the other
α 2 dσ(i) dσ(i) NNLO corrections.
Di = 2 − 2
π (cid:12) dΩ dΩ (cid:12) !(cid:12)
(cid:16) (cid:17) (cid:12) (cid:12)L (cid:12)
(cid:12) −(cid:12)1 (cid:12) 3. Results
dσ(cid:12)0 α dσ1 (cid:12) (cid:12)
× (cid:12) + (cid:12) , (cid:12) (1)
dΩ π dΩ In Fig. 4 all the QED contributions to the
(cid:18) (cid:16) (cid:17) (cid:19) O(α4) Bhabha scattering cross section known at
withi=(NF =1),Vertices,Box Box,isplotted the momentareplotted as afunction ofthe scat-
as a function of the beam energy E. In Eq. (1),
tering angle. Terms suppressed by positive pow-
dσ2(i)/dΩistheunapproximatedO(α4)correction ers of the ratio m2/s were neglected.
to the cross section taken into account. It in- Thedottedlinerepresentsthephotoniccorrec-
cludes the contribution of the virtual diagrams tions [16,20]. The corrections of O(α4(NF = 1))
and the one of the corresponding diagrams with [21] (dashed line) have, for this choice of ω (ω =
the emission of up to two soft photons (with en- E), anoppositesignwithrespecttothephotonic
ergy smaller than the cut-off ω). dσ2(i)/dΩ|L is ones. Moreover they include large terms propor-
4 R. Bonciani and A. Ferroglia
0.01 neglected, and should be included in the Monte
Penin
0.008 dσ2(NF=1)/dσ0 Carlo event generators.
dσ(NF=1+P)/dσ
2 0
0.006 tot
dσ2 /dσ0 4. Acknowledgment
0.004
The authors wish to thank the participants of
0.002 RADCOR05 in particular A. Penin, C. Carloni
Calame, S. Jadach and L. Trentadue for useful
0
discussions. Special thanks to the organizers.
-0.002
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