One-loop chiral amplitudes of Moller scattering process

0.16 MB·English

by  A. I. Ahmadov

One-loop chiral amplitudes of Møller scattering process
A. I. Ahmadov∗
Institute of Physics, Azerbaijan National Academy of Science, Baku, Azerbaijan and
Joint Institute for Nuclear Research, Dubna, Russia
Yu. M. Bystritskiy† and E. A. Kuraev‡
Joint Institute for Nuclear Research, Dubna, Russia
A. N. Ilyichev§
2
National Center of Particle and High Energy Physics
1
0
2 of Belarussian State University, Minsk, Belarus

n
a
V. A. Zykunov¶
J

2
Belarussian State University of Transport, Gomel, Belarus

]
h
p The high energy amplitudes of the large angles Møller scattering are calculated
-
p
in frame of chiral basis in Born and 1-loop QED level. Taking into account as well
e
h
the contribution from emission of soft real photons the compact relations free from
[

1 infrared divergences are obtained. The expressions for separate chiral amplitudes
v
0
contribution to the cross section are in agreement with renormalization group pre-
6
4
dictions.
0
.
1
0
2
1
: I. INTRODUCTION
v
i
X
r Main prediction of Standard Model (SM) — Parity Violation (PV) — can be experimen-
a
tally measured in Møller scattering process (quasi-elastic electron-electron scattering) due
to possible contributions from mechanism with massive SM boson exchange between the
∗Electronic address: [email protected]
†Electronic address: [email protected]
‡Electronic address: [email protected]
§Electronic address: [email protected]
¶Electronic address: [email protected]
2
electrons. The first PV observation in Møller scattering was made by E-158 experiment at
SLAC [1–3], which studied scattering of 45- to 48-GeV polarized electrons on the unpolar-
ized electrons of a hydrogen target. Its result at low Q2 = 0.026 GeV2 for PV asymmetry
A = (1.31 0.14 (stat.) 0.10 (syst.)) 10−7 [4] allowed one of the most important
LR
± ± ×
parameters in the Standard Model – the sine of the Weinberg angle – to be determined
with unprecedented accuracy. The next-generation experiment to study e e scattering,
−
MOLLER [5], planned at JLab following the 11 GeV upgrade, will offer a new level of sensi-
tivity and measure the PV asymmetry in the scattering of longitudinally polarized electrons
off unpolarized target to a precision of 0.73 ppb. That would allow a determination of
the weak mixing angle with an uncertainty of about 0.1%, a factor of five improvement in
fractional precision over the measurement by E-158.
The polarized Møller scattering has allowed the high-precision determination of the
electron-beam polarization at many experimental programs, such as SLC [6], SLAC [7, 8],
JLab [9] and MIT-Bates [10]. A Møller polarimeter may also be useful in future experiments
planned at the ILC [11].
Since the polarized Møller scattering is a very clean process with a well-known initial
energy and extremely suppressed backgrounds, any inconsistency with the Standard Model
will signal new physics. Some examples of physics beyond the Standard Model to which
Møller scattering measurement extends sensitivity include new neutral bosons (Z′), electron
compositeness, supersymmetry and doubly charged scalars [5]. Thus Møller scattering ex-
periments can provide indirect access to physics at multi-TeV scales and play an important
complementary role to the LHC research program [12, 13].
Moreover the accuracy of calculations in Born approximation (sometimes even on one-
loop level) seems to be insufficient to compare with the results of modern experiments. A
significant theoretical effort has been dedicated to one-loop radiative corrections already.
A short review of the literature to date on that topic is done in [14]. The motivation of
our paper is to calculate radiative corrections (on this stage within the QED framework
but planning in the future to add weak contribution) in one-loop approximation, in order
to show up the correct renormalization group behavior of leading logarithms contribution.
This will allow us to extend the applicability of this calculation to higher order corrections.
The paper is organized as follows. After a brief review of results in Born approximation
and formalism of chiral amplitudes in Section II, the virtual corrections associated with the
3
p1 p′1 p1 p′2
e− e− e− e−
γ q1 γ q2
e− e− e− e−
p2 p′2 p2 p′1
(a) (b)
Figure 1: Born approximation diagrams.
known expressions of vertex functions aswell as ones arising frompolarizationof vacuum are
considered in Section III. As well the radiative corrections from emission of real soft photon
are presented there. In Section IV the contribution from box type Feynman diagrams
are considered. In Section V the total result for the chiral amplitudes and the relevant
contributions to the cross sections are presented. In Section VI we make some concluding
remarks about the accuracy of our result. Finally, in Appendix A we present some useful
integrals for box-diagrams calculation.
II. BORN LEVEL CHIRAL AMPLITUDES OF MOLLER SCATTERING
PROCESS
We consider the process of electron–electron scattering (so called Møller process [15])
e−(p ,λ )+e−(p ,λ ) e−(p′,λ′)+e−(p′,λ′), (1)
1 1 2 2 → 1 1 2 2
where λ = and λ′ = are the chiral states of initial and final particles. Matrix
1,2 ± 1,2 ±
element of this process within the QED has two terms (see Fig. 1):
M(0) = M(0) M(0), (2)
γ1 − γ2
(0)
where M is the contribution which goes from photon exchange in t-channel (Fig. 1(a)),
γ1
(0)
M is also the contribution of the photon in intermediate state but with the interchange
γ2
of final leptons (p′ p′) which should be done in order to take into account the Pauli
1 ↔ 2
principle (see Fig. 1(b)). The explicit form of these terms are the following (we use the
t’Hooft–Feynman gauge, i.e. ξ = 1):
M(0)λ1λ2λ′1λ′2 = e2 u¯(λ′1)(p′)γµu(λ1)(p ) u¯(λ′2)(p′)γ u(λ2)(p ) ,
γ1 q2 1 1 2 µ 2
1 h ih i
M(0)λ1λ2λ′1λ′2 = e2 u¯(λ′2)(p′)γµu(λ1)(p ) u¯(λ′1)(p′)γ u(λ2)(p ) ,
γ2 q2 2 1 1 µ 2
2 h ih i
4
where q = p p′ = p′ p and q = p p′ = p′ p are the transferred momenta in
1 1 − 1 2 − 2 2 1 − 2 1 − 2
direct and exchanged diagram, e is the positron electric charge.
We will consider the process using chiral amplitudes approach. In this approach you
select the specific chiral spin state of initial and final particles:
u(λ) = ω u, u¯(λ) = u¯ω , λ = 1 = R,L, (3)
λ −λ
±
where projective operators ω has a form:
λ
1
ω = (1+λγ ), ω2 = ω , ω ω = 0, ω +ω = 1. (4)
λ 2 5 λ λ + − + −
For the case of massless fermions the completeness condition take place
u(λ)(p)u¯(λ)(p) = ωλ/p. (5)
Keeping in mind the parity conservation in QED Mλ = M−λ we have for the summed on
polarization matrix element square
M 2 = 2[ M++++ 2 + M+−+− 2 + M+−−+ 2]. (6)
| | | | | | | |
sXpins
Let us calculate the QED contribution of RR RR chiral amplitude, with all fermions
→
of right hand ones uR = ω u:
+
e2
M(0)++++ = [u¯(p′)ω γµω u(p )][u¯(p′)ω γ ω u(p )],
γ1 t 1 − + 1 2 − µ + 2
e2
M(0)++++ = [u¯(p′)ω γµω u(p )][u¯(p′)ω γ ω u(p )],
γ2 u 2 − + 1 1 − µ + 2
where we have used the kinematics Mandelstam invariants in electron mass vanishing limit
(m 0):
e
→
s = (p +p )2 = 2(p p ) = 2(p′p′),
1 2 1 2 1 2
t = (p p′)2 = 2(p p′) = 2(p p′), (7)
1 − 1 − 1 1 − 2 2
u = (p p′)2 = 2(p p′) = 2(p p′),
1 − 2 − 1 2 − 2 1
s+t+u = 0.
In order to transform this amplitudes into calculable traces we multiply the terms which
contain factor (1/t) by the following quantity:
ab
= 1, (8)
ab
5
and the terms with factor (1/u) by
cd
= 1, (9)
cd
where
a = u¯(p1)ω−/p2ω+u(p′2), c = u¯(p1)ω−p/2ω+u(p′1), (10)
b = u¯(p2)ω−/p1ω+u(p′1), d = u¯(p2)ω−p/1ω+u(p′2).
After this we obtain trace in numerator and can calculate it immediately:
e2 1 e2 1
Mγ(10)++++ = t abSp p/′1γµω+/p1/p2ω+p/′2γµω+/p2/p1ω+ = t ab2s2t, (11)
h i
e2 1 e2 1
Mγ(20)++++ = u cdSp p/′2γµω+/p1/p2ω+p/′1γµω+/p2/p1ω+ = u cd2s2u.
h i
Thus QED amplitude in Born approximation have a form:
1 1
M(0)++++ = M(0)++++ M(0)++++ = 2(4πα)is2M0, M0 = ;
γ γ1 − γ2 γ γ ab − cd
u2
M(0)+−+− = 2(4πα)i , (12)
γ tc d
1 1
t2
M(0)+−−+ = 2(4πα)i ,
γ − uc d
2 2
where c and d are the modified factors similar to (8) or (9) which in this case have a form:
1 1
c = u¯(p )ω u(p′), d = u¯(p )ω u(p′),
1 1 − 2 1 2 + 1
c = u¯(p )ω u(p′), d = u¯(p )ω u(p′). (13)
2 1 − 1 2 2 + 2
Using the relations
a 2 = b 2 = st, c 2 = d 2 = su, abc∗d∗ = s2tu,
| | | | − | | | | − −
c 2 = d 2 = u, c 2 = d 2 = t,
1 1 2 2
| | | | − | | | | −
we obtain for the sum of squares of all six amplitudes the known result [16, 17]
s2 s2 2s2 t2 u2 s4 +t4 +u4
M(0)λ 2 = 8(4πα)2 + + + + = 8(4πα)2 . (14)
γ (cid:20)(cid:18)t2 u2 tu (cid:19) (cid:18)u2 t2(cid:19)(cid:21) t2u2
X(λ) (cid:12) (cid:12)
(cid:12) (cid:12)
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γ
γ
γ l
γ
(a) (b)
Figure 2: Diagrams of vacuum polarization and vertex correction types.
III. VERTEX, POLARIZATION OF VACUUM AND SOFT REAL PHOTON
EMISSION CONTRIBUTIONS
Born level and virtual photons emission radiative corrections to chiral amplitudes can be
put in form
Mλ = M(0)λ +Mλ , (15)
VΠB
where λ denotes different chiral states. In this section we will consider the contribution from
Feynman diagrams of vertex and photon vacuum polarization types (see Fig. 2(a), 2(b)):
1 1
M++++ = 8παis2 (1+Π +2Γ ) (1+Π +2Γ ) ;
t t u u
(cid:20)ab − cd (cid:21)
8παiu2
M+−+− = (1+Π +2Γ ); (16)
t t
tc d
1 1
8παit2
M+−−+ = (1+Π +2Γ ),
u u
uc d
2 2
where vertex and vacuum polarization operators are well known (see [17]):
α 3 1 π2
Γ = l (l 1)+ l l2 2+ζ , ζ = ,
t 2π (cid:20)− λ t − 2 t − 2 t − 2(cid:21) 2 6
α 5
Π = l ,
t t
3π (cid:18) − 3(cid:19)
t m2
Γ = Γ (t u), Π = Π (t u), l = ln − , l = ln ,
u t u t t λ
→ → m2 λ2
and λ is a fictitious ”photon mass”.
7
Corrections to chiral amplitudes squared from polarization operators are
1 1
∆ M++++ 2 = 2(8πα)2s3 Π + Π ,
Π| | − (cid:18)t2u t u2t u(cid:19)
u2
∆ M+−+− 2 = 2(8πα)2 Π ; (17)
Π t
| | t2
t2
∆ M+−−+ 2 = 2(8πα)2 Π .
Π| | u2 u
And the similar expression for the vertex operators corrections
1 1
∆ M++++ 2 = 4(8πα)2s3 Γ + Γ ,
V t u
| | − (cid:18)t2u u2t (cid:19)
u2
∆ M+−+− 2 = 4(8πα)2 Γ ;
V t
| | t2
t2
∆ M+−−+ 2 = 2(8πα)2 Γ . (18)
V u
| | u2
Soft real photons emission contribution to the squares of chiral amplitudes are [17]:
Mλ 2 = δ M(0)λ 2,
| soft| soft| |
4πα ′ d3k p p p′ p′ 2
δ = 1 + 2 1 2 , (19)
soft −16π3 Z ω (cid:18)p k p k − p′k − p′k(cid:19)
1 2 1 2
where ′ means that photon energy ω = k2 +λ2 do not exceed some small value ω <
R p
∆E E = √s/2, where ∆E and E are defined in center of mass system. Standard
≪
calculation [18] leads to
2α 1
δ = (l 1+L)(l +2l )+ l2 +Ll +K ,
soft π (cid:20) s − λ ǫ 2 s s soft(cid:21)
s ∆E t u
l = ln , l = ln , L = l +l , l = ln − , l = ln − ,
s ǫ ts us ts us
m2 E s s
1 1
K = l2 + l2 2ξ +Li cos2(θ/2) +Li sin2(θ/2) , (20)
soft 2 ts 2 us − 2 2 2
(cid:0) (cid:1) (cid:0) (cid:1)
and θ is the angle (center of mass frame implied) between the directions of motion of one of
initial and the scattered electrons.
Combining the vertex and soft parts of the corrections we obtain
2α s4 3 3
∆ M++++ 2 = (8πα)2 (l 1) 2l + +2l L+ +K +
VS s ǫ ǫ soft
| | π (cid:20)t2u2 (cid:20) − (cid:18) 2(cid:19) 2 (cid:21)
s3 1 3 s3 1 3
+ l2 l +2 ζ + l2 l +2 ζ
t2u (cid:20)2 ts − 2 ts − 2(cid:21) u2t (cid:20)2 us − 2 us − 2(cid:21)(cid:21)
2αu2 3
∆ M+−+− 2 = (8πα)2 (l 1) 2l + +2l L+l (l +l )
VS s ǫ ǫ us λ s
| | π t2 (cid:20) − (cid:18) 2(cid:19)
8
γ γ
γ γ
(a) (b)
Figure 3: Diagrams of vacuum polarization and vertex correction types.
1 3 1
+K + l l2 +ζ ; (21)
− 2 soft 2 ts − 2 ts 2(cid:21)
2α t2 3
∆ M+−−+ 2 = (8πα)2 (l 1) 2l + +2l L+l (l +l )
VS s ǫ ǫ ts λ s
| | π u2 (cid:20) − (cid:18) 2(cid:19)
1 3 1
+K + l l2 +ζ .
− 2 soft 2 us − 2 us 2(cid:21)
IV. BOX DIAGRAMS CONTRIBUTION
Let us consider box-type diagrams (see Fig. 3). Using the symmetry reasons we can
restrict ourself by considering only two chiral amplitudes
1
M++++ = iα2(1 ) m++++, m++++ = M++++ +M++++;
B −P ab 1 2
M+−−+ = M+−+−;
B −P B
1
M+−+− = iα2 m+−+−, m+−+− = M+−+− +M+−+−, (22)
B c d 1 2
1 1
where M and M correspond to diagram on Fig. 3(a) and 3(b) respectively. Operator
1 2
P
interchanges the momenta of the scattered electrons
F (p′,p′) = F (p′,p′), (23)
P 1 2 2 1
and
dk dk d4k
M++++ = N ; M++++ = N ; dk =
1 Z (0)(1)(2)(q) 1 2 Z (0)(1)(2′)(q) 2 iπ2
dk dk
M+−+− = M ; M+−+− = M .
1 Z (0)(1)(2)(q) 1 2 Z (0)(1)(2′)(q) 2
Here the denominators are:
(0) = k2 λ2, (1) = (p k)2 m2 +i0; (q) = (q k)2 λ2;
1 1
− − − − −
(2) = (p +k)2 m2 +i0; (2′) = (p′ k)2 m2 +i0, (24)
2 − 2 − −
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and the numerators are
1
N1 = Sp /p′γµ /p k/ γνp/ /p /p′γµ /p +k/ γνp/ /p ;
2 1 1 − 1 2 2 2 2 1
h (cid:16) (cid:17) (cid:16) (cid:17) i
1
N2 = Sp /p′γµ /p k/ γνp/ /p /p′γν /p′ k/ γµ/p p/ ;
2 1 1 − 1 2 2 2 − 2 1
h (cid:16) (cid:17) (cid:16) (cid:17) i
1
M1 = Sp /p′γµ /p k/ γνp/ /p′γµ /p +k/ γν/p ;
2 1 1 − 1 2 2 2
h (cid:16) (cid:17) (cid:16) (cid:17) i
1
M2 = Sp /p′γµ /p k/ γνp/ /p′γν /p′ k/ γµ/p . (25)
2 1 1 − 1 2 2 − 2
h (cid:16) (cid:17) (cid:16) (cid:17) i
Calculating the traces and using the table of relevant integrals, listed in Appendix A we
obtain:
m++++(s,t,u) = 4s3tI012q 2tuI012′q +4t2(s u)I01q +4ut(s u)I012′
− − − −
4stl +4stl , (26)
u t
−
m+−+−(s,t,u) = 2s(t2 2su)I012q 4u3I012′q +4t(s u)I01q +4s(s u)I012
− − − − −
4ul +4ul .
t s
−
Inserting the explicit expressions for scalar integrals with 3 and 4 denominators (see Ap-
pendix A) we obtain
m++++(s,t,u) = 8s2(l +l )l +2u2l2 2s2L2 +4tsl ;
λ s su tu − tu
8s2
m+−+−(s,t,u) = l l +4ul +2(s u)l2 2s2L2 +4tsl
t λ su su − ts − tu−
8u2 8u2
l l l l +12(u s)ζ .
s us us ts 2
− t − t −
The lowest order contribution arises from the interference term of Born amplitude with the
box type amplitude
∗
1 1 1 1
∆ M++++ 2 = 2α2(8παs)2 m++++(s,t,u) m++++(s,u,t) ,
B
| | (cid:18)ab − cd(cid:19) (cid:18)ab − cd (cid:19)
∗
1 1
∆ M+−+− 2 = 2α2(8παu)2 m+−+−(s,t,u), (27)
B
| | (cid:18)c d (cid:19) c d
1 1 1 1
∗
1 1
∆ M+−−+ 2 = 2α2(8παt)2 m+−−+(s,t,u).
B
| | (cid:18)c d (cid:19) c d
2 2 2 2
The simple calculation of (27) gives the total result (the contributions of vertex and soft
10
photons (21) are added):
8s4 3
∆( M++++ 2 + M−−−− 2) = 32πα3 (l 1) +2l +(l +l +2l )(l +l ) +
s ǫ λ s ǫ ts us
| | | | (cid:26)t2u2 (cid:20) − (cid:18)2 (cid:19) (cid:21)
8s4 3 s
+ +K 2u2l2 2s2(l +l )2 +4tsl
t2u2 (cid:20)2 soft(cid:21)− t2u tu − ts us tu −
(cid:2) (cid:3)
s
2t2l2 2s2(l +l )2 +4usl
− u2t tu − ts us ut −
(cid:2) (cid:3)
8s3 3 1
(l +l )l + l l2 2+ζ
− t2u (cid:20)− λ s ts 2 ts − 2 ts − 2(cid:21)−
8s3 3 1
(l +l )l + l l2 2+ζ . (28)
−u2t (cid:20)− λ s us 2 us − 2 us − 2(cid:21)(cid:27)
We see the cancelation of terms containing the quantity l + l . That is the result of well
λ s
known Bloch–Nordsieck theorem of infrared divergencies cancelation [19]. Similar cancela-
tion takes place for corrections to two remaining chiral amplitudes. As a final result we have
for all contributions except vacuum polarization:
8s4 3
∆( M++++ 2 + M−−−− 2) = 32πα3 (l 1) +2l +2l (l +l ) +
s ǫ ǫ ts us
| | | | (cid:26)t2u2 (cid:20) − (cid:18)2 (cid:19) (cid:21)
8s4 3 s
+ +K 2u2l2 2s2(l +l )2 +4tsl
t2u2 (cid:20)2 soft(cid:21)− t2u tu − ts us tu −
(cid:2) (cid:3)
s
2t2l2 2s2(l +l )2 +4usl
− u2t tu − ts us ut −
(cid:2) (cid:3)
8s3 3 1 8s3 3 1
l l2 2+ζ l l2 2+ζ ;
−t2u (cid:20)2 ts − 2 ts − 2(cid:21)− u2t (cid:20)2 us − 2 us − 2(cid:21)(cid:27)
(29)
and
8u2 3
∆( M+−+− 2 + M−+−+ 2) = 32πα3 (l 1) +2l +2l (l +l ) +
s ǫ ǫ ts us
| | | | (cid:26) t2 (cid:20) − (cid:18)2 (cid:19) (cid:21)
8u2 1 1 3
+ ζ l2 + l +K +
t2 (cid:20) 2 − 2 − 2 ts 2 ts soft(cid:21)
1 8u2
+ 4ul +12(u s)ζ l l +2(s u)l2 . (30)
t (cid:20) su − 2 − t us ts − ts(cid:21)(cid:27)
At least
∆( M+−−+ 2 + M−++− 2) = ∆( M+−+− 2 + M−+−+ 2). (31)
| | | | P | | | |