A. Integral cross section of the l − τ conversion
Now using effective four-fermion operators we calculate
the integral cross sections for the ETC and MTC. The total
cross section of the l − τ conversion on a nucleus (1) can
be approximated by the sum over the corresponding cross
section on its constituent nucleons
σðl þðA; ZÞ → τ þ XÞ¼Zσðl þ p → τ þ XÞ
þðA − ZÞσðl þ n → τ þ XÞ:
ð12Þ
Here nucleon N ¼ p, n cross section is
σðl þ N → τ þ XÞ
¼
X
if
Z
1
0
dx
Z
1
0
dy
d
2
ˆσ
dxdy
ðl þ q
i
→ τ þ q
f
Þq
N
i
ðx; Q
2
Þ
þ
d
2
ˆσ
dxdy
ðl þ
¯
q
f
→ τ þ
¯
q
i
Þ
¯
q
N
f
ðx; Q
2
Þ
; ð13Þ
where q
N
i
ðx; Q
2
Þ and
¯
q
N
i
ðx; Q
2
Þ are quark and antiquark
PDFs, respectively. We will consider two nuclear targets:
Fe with A ¼ 56 and Z ¼ 26 and Pb with A ¼ 207 and
Z ¼ 82. Quark/antiquark PDFs depend on the resolution
scale set by the square momentum transferred to the
nucleon
q
2
¼ −Q
2
¼ −ðs − m
2
N
− m
2
l
Þxy ≃−sxy; ð14Þ
where m
N
is the nucleon mass, x ¼ Q
2
=ðq · PÞ is Bjorken
variable, y ¼ðq · PÞ=ðk · PÞ is inelasticity. Therefore, we
should substitute Q
2
by sxy in Eq. (13). In the present paper
we use quark PDFs from the CT10 next-to-next-to-leading
order global analysis of QCD [26]. In fact, PDF fits using
the standard CTEQ PDF evolution [27] but using the
HOPPET α
s
running solution.
The elementary differential cross sections corresponding
to the contact 4-fermion interactions in Eq. (4) are given by
d
2
ˆσ
dxdy
ðl þ q
i
→ τ þ q
f
Þ¼
X
I;XY
1
ðΛ
lτ
I
if;XY
Þ
4
ˆ
sf
I;XY
ðyÞ
64π
; ð15Þ
d
2
ˆσ
dxdy
ðl þ
¯
q
f
→ τ þ
¯
q
i
Þ¼
X
I;XY
1
ðΛ
lτ
I
if;XY
Þ
4
ˆ
sg
I;XY
ðyÞ
64π
: ð16Þ
Here f
I;XY
ðyÞ and g
I;XY
ðyÞ are functions related to the
matrix elements of the effective operators (5)–(7). They are
given in Appendix.
Substituting (15), (16) into (13) and (12) we find
σðl þðA; ZÞ → τ þ XÞ¼
X
I;if;XY
Q
A
I
if;XY
Λ
4
I
if;XY
ð17Þ
with
Q
A
I
if;XY
¼
s
64π
Z
1
0
dx
Z
1
0
dy½xf
I;XY
ðyÞq
A
i
ðx; sxyÞ
þ xg
I;XY
ðyÞ
¯
q
A
f
ðx; sxyÞ; ð18Þ
where
u
A
ðx; Q
2
Þ¼Zu
p
ðx; Q
2
ÞþðA − ZÞd
p
ðx; Q
2
Þ;
d
A
ðx; Q
2
Þ¼Zd
p
ðx; Q
2
ÞþðA − ZÞu
p
ðx; Q
2
Þ;
u
A
ðx; Q
2
Þþd
A
ðx; Q
2
Þ¼Aðu
p
ðx; Q
2
Þþd
p
ðx; Q
2
ÞÞ;
¯
u
A
ðx; Q
2
Þ¼A
¯
u
p
ðx; Q
2
Þ;
¯
d
A
ðx; Q
2
Þ¼A
¯
d
p
ðx; Q
2
Þ;
s
A
ðx; Q
2
Þ¼
¯
s
A
ðx; Q
2
Þ¼As
p
ðx; Q
2
Þ;
c
A
ðx; Q
2
Þ¼
¯
c
A
ðx; Q
2
Þ¼Ac
p
ðx; Q
2
Þ;
b
A
ðx; Q
2
Þ¼
¯
b
A
ðx; Q
2
Þ¼Ab
p
ðx; Q
2
Þð19Þ
are the quark and antiquark PDFs in a nucleus A. Numerical
results for the double moments Q
A
I
if;XY
are shown in
Tables I–IV for Fe and Pb nuclear targets and for the
electron and muon beams.
The dominant contribution to the inclusive l þ A cross
section is due to the bremsstrahlung of leptons on nuclei,
given by the formula [28,29]
σ
BS
ðl þðA; ZÞ → l þ XÞ¼4αr
2
l
Z
2
7
9
log
183
Z
1=3
m
l
m
e
ð20Þ
TABLE I. Double moments of quark PDF Q
A
I
if;XY
(in GeV
2
)
with f ¼ u, d, s, c, b and i specified in the Table. The case of a
Fe target and an electron beam with E
e
¼ 100 GeV.
(IiXY)
Q
A
I
if;XY
(IiXY )
Q
A
I
if;XY
S operators
(SuXY) 3.82 (SdXY) 4.07
(SsXY) 0.74 (ScXY) 0.21
(SbXY) 0.006
V operators
(VuLL=RR) 43.83 (VuLR=RL) 20.51
(VdLL=RR) 46.23 (VdLR=RL) 22.46
(VsLL=RR) 5.85 (VsLR=RL) 5.85
(VcLL=RR) 1.41 (VcLR=RL) 1.41
(VbLL=RR) 0.02 (VbLR=RL) 0.02
T operators
(TuLL=RR) 453.52 (TdLL=RR) 484.37
(TsLL=RR) 81.84 (TcLL=RR) 19.21
(TbLL=RR) 0.23
DEEP INELASTIC e − τ AND μ − τ … PHYS. REV. D 98, 015007 (2018)
015007-3