tau polarization in (anti-)neutrino-nucleon interactions
TRANSCRIPT
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Tau Polarization in (anti-)neutrino-nucleon Interactions.
Mohammad Rafi Alam†
Luis Alvarez-Ruso# Toru Sato††
†Aligarh Muslim University, India#IFIC-CSIC, U.V.,Valencia
††RCNP, Osaka University, Japan
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Outline
1 Introduction
2 FormalismQuasi-elasticInelastic regionDeep Inelastic Scattering
3 Results
4 Summary
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Introduction
1 Introduction
2 FormalismQuasi-elasticInelastic regionDeep Inelastic Scattering
3 Results
4 Summary
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Introduction
Motivation
τ leptons were observed in experiments:Atmospheric Experiments : SuperK, IceCUBEAccelerator Experiments : DONOT, OPERA
New Experiments : SHiP, DsTau, DUNENeutrino Oscillation : appearance experiment νµ→ ντ
Observed by the ντ induced (CC)interaction(−)ν τ (k,0) + N(p,M)→ τ±(k′,ml) + X(p′)
Challenges :short lifetime of τ (10−13sec): Practically impossible to detect!
Observe the decay channels
Decay distribution has a strong dependence on τ spin polarization.
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Introduction
Motivation
Unpolarized
Polarized
0 |π/4| |π/2| |3π/4| |π|ϕ
0
1
2
3
4
5
6
7
8
Ev
ent/
bin
/4.5
×1
019p
.o.t
./1
.65
kto
n/5
yea
re, µ
πwith τ
without polarization
polarization
NPB727 (2005) 163-175
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Formalism
1 Introduction
2 FormalismQuasi-elasticInelastic regionDeep Inelastic Scattering
3 Results
4 Summary
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Formalism
(−)ν τ (k,0) + N(p,M)→ τ±(k′,ml) + X(p′)
τ± rest frame
Lab frame QE, Inel, DIS
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Formalism
(−)ν τ (k,0) + N(p,M)→ τ±(k′,ml) + X(p′)
k× k′
[k× k′]× k′
k′
Polarization Vector
~ξ = PLξL+PtξT +���HHHPt′ ξP
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Formalism
Quasi-elastic[τ±N
]Inelastic
[τ±πN, τ±KN, · · ·
]−→ (Dynamical Coupled Channel)
Deep-Inelastic[τ±+Jet]
Hadronic Tensor
Wµν(p,q) =−gµνW1(p·q,Q2) + pµpν
M2 W2(p·q,Q2)− iεµναβpαqβ
2M2 W3(p·q,Q2)
+ qµqν
M2 W4(p·q,Q2) + pµqν + qµpν
2M2 W5(p·q,Q2),+i pµqν − qµpνM2 W6(p·q,Q2)
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Formalism
Quasi-elastic[τ±N
]Inelastic
[τ±πN, τ±KN, · · ·
]−→ (Dynamical Coupled Channel)
Deep-Inelastic[τ±+Jet]
Hadronic Tensor
Wµν(p,q) =−gµνW1(p·q,Q2) + pµpν
M2 W2(p·q,Q2)− iεµναβpαqβ
2M2 W3(p·q,Q2)
+ qµqν
M2 W4(p·q,Q2) + pµqν + qµpν
2M2 W5(p·q,Q2),+i pµqν − qµpνM2 W6(p·q,Q2)
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Formalism
FPt′ =−2ml
MplW6 sinθ
FPt =−ml sinθ[±(
2W1−W2−m2l
M2W4 + 2ElM
W5
)− EνMW3
]FPl =∓
[(2W1−
m2l
M2W4
)(pl−El cosθ) +W2(pl+El cosθ)−
2m2l
McosθW5
]−W3M
(cosθ(EνEl+p2
l )−pl(Eν +El))
F =(
2W1 +m2l
M2 W4
)[El−pl cosθ
]+W2(El+pl cosθ)
±W3M
[(Eν +El)(El−pl cosθ)−m2
l
]−
2m2l
MW5
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Formalism Quasi-elastic
Quasi-elastic
ντ (k) +n(p) −→ τ−(k′) +p(p′),ντ (k) +p(p) −→ τ+(k′) +n(p′),
The hadronic current Jµ is expressed as:
Jµ = u(p′)[Vµ−Aµ
]u(p)
〈N ′(p′)|Vµ|N(p)〉 = u(p′)[γµf1(Q2) + iσµν
qν
Mp+Mnf2(Q2) +
����HHHH
2 qµMp+Mn
f3(Q2)]u(p),
〈N ′(p′)|Aµ|N(p)〉 = u(p′)[γµγ5g1(Q2) +
������
����XXXXXXXXXX
iσµνqν
Mp+Mnγ5g2(Q2) +
2 qµMp+Mn
g3(Q2)γ5
]u(p),
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Formalism Inelastic region
Dynamical Coupled Channel
Extension of Sato-Lee modelOnly πN state and works in ∆(1232) region.
The building blocks of DCCstable two-particle channels (πN,ηN,KΛ,KΣ)quasi-stable channels (ρN,σN,π∆)
The T -matrix are constructed for the meson-baryon scattering has beenobtained from the coupled-channel Lippmann-Schwinger equation:
〈α,~p ′|T (W )|β,~p〉 = 〈α,~p ′|V (W )|β,~p〉
+∑γ
∫d3k 〈α,~p ′|V (W )|γ,~k 〉G0
γ(~k,W )〈γ,~k |T (W )|β,~p〉
where α,β and γ are the meson-baryon two-body states.Satisfies two- and three- body unitarity.
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Formalism Deep Inelastic Scattering
Structure Function
Wµν =∑
q,q′=u,d,...,u,d,...
∫dξ
ξfq(ξ,Q2)Kµν(pq, q)|Vqq′ |2
Kµν = δ[(pq + q)2−m2
q′]×2{−gµνpq · q+ 2pµq pνq ∓ iεµναβpqαpqβ + (pµq qν + qµpνq )
}.
momentum fraction
Parton distributionfuction
Elements of CKMMatrix
parton momentumpq = ξp, p is nucleonmomentum
parton mass
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Formalism Deep Inelastic Scattering
ξ0 = 2x/λ1 +√
1 +4M2x2/(λQ2)
1≤Q2 ≤ 10GeV 2
00.10.20.30.40.50.60.70.80.9
1
0 0.2 0.4 0.6 0.8 1
𝜉 = 𝑥
𝜉(𝑚𝑞′=0)𝜆(𝑚𝑞′=𝑚𝑐)
𝜉(𝑚𝑞′ = 𝑚𝑐)
𝜉(𝑚𝑞′ = 0)
𝜉(𝑥,
𝜆)
𝑥 →
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Formalism Deep Inelastic Scattering
Structure Function
W1 =∑j
fj(ξ)|Vij |2 = F1(ξ) ,
W2 = M2
p · q+ ξM2
∑j
2ξfj(ξ)|Vij |2 = M2
p · q+ ξM2F2(ξ) ,
W3 = M2
p · q+ ξM2
∑j
2fj(ξ)|Vij |2
= M2
p · q+ ξM2F3(ξ) ,
W4 = M2
p · q+ ξM2
∑j
fj(ξ)|Vij |2 = M2
p · q+ ξM2F4(ξ) ,
W5 = M2
p · q+ ξM2
∑j
2fj(ξ)|Vij |2
= M2
p · q+ ξM2F5(ξ) .
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Formalism Deep Inelastic Scattering
Structure Function
F(N)1(l) = |Vud|2f
(N)d (ξl) + |Vus|2f (N)
s (ξl) +(|Vud|2 + |Vus|2)f (N)u (ξl)
F(N)1(h) = |Vcd|2f
(N)d (ξh) + |Vcs|2f (N)
s (ξh)
F(N)2(l) = 2ξl{|Vud|2f
(N)d (ξl) + |Vus|2f (N)
s (ξl) +(|Vud|2 + |Vus|2)f (N)u (ξl)}
F(N)2(h) = 2ξh{|Vcd|2f
(N)d (ξh) + |Vcs|2f (N)
s (ξh)}
F(N)3(l) = 2{|Vud|2f
(N)d (ξl) + |Vus|2f (N)
s (ξl)− (|Vud|2 + |Vus|2)f (N)u (ξl)}
F(N)3(h) = 2{|Vcd|2f
(N)d (ξh) + |Vcs|2f (N)
s (ξh)}
F(N)4(l,h) = 0
F(N)5(l,h) = 2F (N)
1(l,h)
We use CTEQ6.6 for PDFs.
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Formalism Deep Inelastic Scattering
Structure Function
ξl = 2x1+√
1+4M2x2/(Q2)
ξh = ξ0
ξN = 2x/λ1+√
1+4M2x2/(Q2)
00.10.20.30.40.50.60.70.80.9
1
0 0.2 0.4 0.6 0.8 1
𝜉 = 𝑥
𝜉(𝑚𝑞′=0)𝜆(𝑚𝑞′=𝑚𝑐)
𝜉(𝑚𝑞′ = 𝑚𝑐)
𝜉(𝑚𝑞′ = 0)
𝜉(𝑥,
𝜆)
𝑥 →
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Results
1 Introduction
2 FormalismQuasi-elasticInelastic regionDeep Inelastic Scattering
3 Results
4 Summary
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Results
Differential cross-section at Eν = 10 GeV & Wcut = 2.1GeV
00.20.40.60.8
11.21.41.61.8
2
0 2 4 6 8 10
𝜈𝜏𝑝 → 𝜏−𝑋
𝐸𝜈 =10 GeV𝜃 = 0∘
𝑑𝜎/𝑑
𝑊𝑑𝑄
2 [10
−38
𝑐𝑚2 /
𝐺𝑒𝑉
3 ]
𝑝𝜏[𝐺𝑒𝑉 ]
DIS DCC
0
0.5
1
1.5
2
2.5
3
0 2 4 6 8 10
𝜃 = 2.5∘
𝑑𝜎/𝑑
𝑊𝑑𝑄
2 [10
−38
𝑐𝑚2 /
𝐺𝑒𝑉
3 ]
𝑝𝜏[𝐺𝑒𝑉 ]
Δ
00.10.20.30.40.50.60.70.80.9
1
0 2 4 6 8 10
𝜃 = 5∘
𝑑𝜎/𝑑
𝑊𝑑𝑄
2 [10
−38
𝑐𝑚2 /
𝐺𝑒𝑉
3 ]
𝑝𝜏[𝐺𝑒𝑉 ]
0
0.05
0.1
0.15
0.2
0.25
0 2 4 6 8 10
𝜃 = 7.5∘
𝑑𝜎/𝑑
𝑊𝑑𝑄
2 [10
−38
𝑐𝑚2 /
𝐺𝑒𝑉
3 ]
𝑝𝜏[𝐺𝑒𝑉 ]
00.20.40.60.8
11.21.41.61.8
0 2 4 6 8 10
𝜈𝜏𝑛 → 𝜏+𝑋
𝐸𝜈 =10 GeV𝜃 = 0∘
𝑑𝜎/𝑑
𝑊𝑑𝑄
2 [10
−38
𝑐𝑚2 /
𝐺𝑒𝑉
3 ]
𝑝𝜏[𝐺𝑒𝑉 ]
DIS DCC
0
0.5
1
1.5
2
2.5
0 2 4 6 8 10
𝜃 = 2.5∘
𝑑𝜎/𝑑
𝑊𝑑𝑄
2 [10
−38
𝑐𝑚2 /
𝐺𝑒𝑉
3 ]
𝑝𝜏[𝐺𝑒𝑉 ]
Δ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 2 4 6 8 10
𝜃 = 5∘
𝑑𝜎/𝑑
𝑊𝑑𝑄
2 [10
−38
𝑐𝑚2 /
𝐺𝑒𝑉
3 ]
𝑝𝜏[𝐺𝑒𝑉 ]
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 2 4 6 8 10
𝜃 = 7.5∘
𝑑𝜎/𝑑
𝑊𝑑𝑄
2 [10
−38
𝑐𝑚2 /
𝐺𝑒𝑉
3 ]
𝑝𝜏[𝐺𝑒𝑉 ]
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Results
Degree of polarization
00.10.20.30.40.50.60.70.80.9
1
0 1 2 3 4 5 6 7 8 9 10
𝜈𝜏𝑝 → 𝜏−𝑋
𝐸𝜈 =10 GeV𝜃 = 0∘
|𝑃|
𝑝𝜏[𝐺𝑒𝑉 ]
DIS DCC
0.60.650.7
0.750.8
0.850.9
0.951
0 1 2 3 4 5 6 7 8 9 10
𝜃 = 2.5∘
|𝑃|
𝑝𝜏[𝐺𝑒𝑉 ]
Δ
0.80.820.840.860.880.9
0.920.940.960.98
1
0 1 2 3 4 5 6 7 8 9 10
𝜃 = 5∘
|𝑃|
𝑝𝜏[𝐺𝑒𝑉 ]
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
0 1 2 3 4 5 6 7 8 9 10
𝜃 = 7.5∘
|𝑃|
𝑝𝜏[𝐺𝑒𝑉 ]
00.10.20.30.40.50.60.70.80.9
1
0 1 2 3 4 5 6 7 8 9 10
𝜈𝜏𝑛 → 𝜏+𝑋
𝐸𝜈 =10 GeV𝜃 = 0∘
|𝑃|
𝑝𝜏[𝐺𝑒𝑉 ]
DIS DCC
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 10
𝜃 = 2.5∘
|𝑃|
𝑝𝜏[𝐺𝑒𝑉 ]
Δ
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 10
𝜃 = 5∘
|𝑃|
𝑝𝜏[𝐺𝑒𝑉 ]
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 10
𝜃 = 7.5∘
|𝑃|
𝑝𝜏[𝐺𝑒𝑉 ]
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Results
Polarization angle θP = cos−1[PLPT
]Eν = 5GeV
−1
−0.5
0
0.5
1
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
𝜈𝜏𝑛 → 𝜏−𝑋
𝐸𝜈 =05 GeV𝜃 = 0∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
DIS DCC
−0.6−0.4−0.2
00.20.40.60.8
1
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
𝜃 = 2.5∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
Δ
−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
𝜃 = 5∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
𝜃 = 7.5∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
−1
−0.5
0
0.5
1
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
𝜈𝜏𝑝 → 𝜏−𝑋
𝐸𝜈 =05 GeV𝜃 = 0∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
DIS DCC
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
𝜃 = 2.5∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
Δ
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
𝜃 = 5∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
−0.4
−0.2
0
0.2
0.4
0.6
0.8
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
𝜃 = 7.5∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
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Results
−1
−0.5
0
0.5
1
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
𝜈𝜏𝑛 → 𝜏+𝑋
𝐸𝜈 =05 GeV𝜃 = 0∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
DIS DCC
−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
𝜃 = 2.5∘co
s(𝜃 𝑝
)
𝑝𝜏[𝐺𝑒𝑉 ]
Δ
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
𝜃 = 5∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
−0.2
0
0.2
0.4
0.6
0.8
1
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
𝜃 = 7.5∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
−1
−0.5
0
0.5
1
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
𝜈𝜏𝑝 → 𝜏+𝑋
𝐸𝜈 =05 GeV𝜃 = 0∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
DIS DCC
−1−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
𝜃 = 2.5∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
Δ
−1−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
𝜃 = 5∘co
s(𝜃 𝑝
)
𝑝𝜏[𝐺𝑒𝑉 ]
−1−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
𝜃 = 7.5∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
(τ-polarisation) M.R.A. March 17, 2021 19 / 24
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Results
−1−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0 2 4 6 8 10 12 14 16 18 20
𝜈𝜏𝑛 → 𝜏−𝑋
𝐸𝜈 =20 GeV𝜃 = 0∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
DIS DCC
−1−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0 2 4 6 8 10 12 14 16 18 20
𝜃 = 2.5∘co
s(𝜃 𝑝
)
𝑝𝜏[𝐺𝑒𝑉 ]
−1−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0 2 4 6 8 10 12 14 16 18 20
𝜃 = 5∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
−1−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0 2 4 6 8 10 12 14 16 18 20
𝜃 = 7.5∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
−1−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0 2 4 6 8 10 12 14 16 18 20
𝜈𝜏𝑝 → 𝜏−𝑋
𝐸𝜈 =20 GeV𝜃 = 0∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
DIS DCC
−1−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0 2 4 6 8 10 12 14 16 18 20
𝜃 = 2.5∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
−1−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0 2 4 6 8 10 12 14 16 18 20
𝜃 = 5∘co
s(𝜃 𝑝
)
𝑝𝜏[𝐺𝑒𝑉 ]
−1−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0 2 4 6 8 10 12 14 16 18 20
𝜃 = 7.5∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
(τ-polarisation) M.R.A. March 17, 2021 20 / 24
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Results
−1−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0 2 4 6 8 10 12 14 16 18 20
𝜈𝜏𝑛 → 𝜏+𝑋
𝐸𝜈 =20 GeV𝜃 = 0∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
DIS DCC
−1−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0 2 4 6 8 10 12 14 16 18 20
𝜃 = 2.5∘co
s(𝜃 𝑝
)
𝑝𝜏[𝐺𝑒𝑉 ]
−1−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0 2 4 6 8 10 12 14 16 18 20
𝜃 = 5∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
−1−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0 2 4 6 8 10 12 14 16 18 20
𝜃 = 7.5∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
−1−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0 2 4 6 8 10 12 14 16 18 20
𝜈𝜏𝑝 → 𝜏+𝑋
𝐸𝜈 =20 GeV𝜃 = 0∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
DIS DCC
−1−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0 2 4 6 8 10 12 14 16 18 20
𝜃 = 2.5∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
−1−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0 2 4 6 8 10 12 14 16 18 20
𝜃 = 5∘co
s(𝜃 𝑝
)
𝑝𝜏[𝐺𝑒𝑉 ]
−1−0.8−0.6−0.4−0.2
00.20.40.60.8
1
0 2 4 6 8 10 12 14 16 18 20
𝜃 = 7.5∘
cos(
𝜃 𝑝)
𝑝𝜏[𝐺𝑒𝑉 ]
(τ-polarisation) M.R.A. March 17, 2021 21 / 24
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Results
2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0p × cos( )
0.0
0.5
1.0
1.5
2.0
2.5
p×
sin(
)N + X at E = 20GeV
= 0= 2.5= 5= 7.5= 10
(τ-polarisation) M.R.A. March 17, 2021 22 / 24
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Summary
1 Introduction
2 FormalismQuasi-elasticInelastic regionDeep Inelastic Scattering
3 Results
4 Summary
(τ-polarisation) M.R.A. March 17, 2021 23 / 24
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Summary
Summary
τ spin polarization is important for τ lepton detection
The use of Dynamical Couple Channel model helps us to push the invariantmass up to 2.1GeV. In this region, DCC gave an accurate description of theinelastic process.
In the DIS region, we improve the kinematic treatment (especially for low Q2)and write the structure functions in a more consistent way.
Around low invariant mass, the DIS results may need to improve for reasonablematching.
Thanks
(τ-polarisation) M.R.A. March 17, 2021 24 / 24
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Summary
Summary
τ spin polarization is important for τ lepton detection
The use of Dynamical Couple Channel model helps us to push the invariantmass up to 2.1GeV. In this region, DCC gave an accurate description of theinelastic process.
In the DIS region, we improve the kinematic treatment (especially for low Q2)and write the structure functions in a more consistent way.
Around low invariant mass, the DIS results may need to improve for reasonablematching.
Thanks
(τ-polarisation) M.R.A. March 17, 2021 24 / 24