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8/6/2019 Lecture 18, Y23 distribution
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Slide 1 of 11
Today’s lecture
Y 23 distribution
Literature:
G. Dissertori, M. Schmelling, Phys. Lett. B 361 (1995) 167-178
A. Banfi, G. Salam, G. Zanderighi, JHEP 01 (2002) 018
A.Heister et al., Eur.Phys.J. C35 (2004) 457
G.Abbiendi at al., Eur. Phys. J. C40 (2005) 287
B. Adeva et al., Z. Phys C55 (1992) 39
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8/6/2019 Lecture 18, Y23 distribution
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Slide 2 of 11
Y 23 distribution. Shifting NLL result
f Hu 0L = 0, u 0 =32
u = ln1 y
f Hu L = - y „ R I y M
„ y ,or alternatively one can introduce the logarithmic distribution:
but let's consider the tree-level kinematic constrain: y = H1 - x 1L x 2
x 3for odered energy fractions: x i =
2 E i
t , x 1 > x 2 > x 3
for the symmetric point x i =23 , y =
13 thus u min = ln3 º 1.1 < u 0
Let's simply shift the distribution in order to have the correct zero position: f èHu L = f u - ln3 +
32
.
Usual prescription to take into account kinematic constrans is slightly different. Let's
imagine we have an event shape "y" andthe NLL cumulative distribution R(a,L), where L=ln1/y is the "large log", but y < y max (for
example, thrust 1-T<1/2), then:
R I y M = ADq I y ME2 = expC-2 C F ‡ yt
t d m2
m2
as I m2M2 p
lnt
m2 -32
G -„ R I M
„ y = C F
as I y t Mp y
ln1 y
-32
R I y M,the differential distribution is
L Ø ln1 y
-1
y max+ 1 so that all radiative corrections disappear at y = y max
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Slide 3 of 11
Y 23 distribution. NLL accuracy
2 4 6 8 10
0.00
0.05
0.10
0.15
0.20
-lnH y 23L
y 2 3
d s ê d y 2 3
BSZ result
CKKW numeric
CKKW analytic
Analytic shifted
as in "Z pole scheme"
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Slide 4 of 11
Y 23 distribution. First nontrivial corrections
2 p
as C F
ds
sH0L dy = -7 + 12 b -
3 y + 16 y -
I1 - y M I12 y - 12 b y - 5 y 2MI2 - 2 b - y M2 +
6 - 8 y + 2 y 2
1 - b + y
+b 2 + 2 b y + 4 y 2
1 - y -
4 y
1 - y LogC1 + y
1 - y G -
1
y
1 + y
1 - y LogC2 - 2 b - y
2 - 2 b + y G
+
2
y LogCy - 1 + b
y + 1 - b G + 2
1 + y
1 - y LogC1 - y
2 G +
2 + y
y LogC21 - y
2 - b G
+6 - 10 y + 8 y 2
1 - y LogC2 2 - 2 b - y
1 - y G -
4 y I1 - y M LogC y
1 - b G
-1 + y
1- y
LogA2 b y - y E + I8 - 4 y MLogC 2 y
1-
b +
y
G,
b = 1 + y
4-
y
2+
y 2
16
ds
sH0L dy=
as C F
2 p‡ dx1 dx2
x 12
+ x 22
H1 - x 1L H1 - x 2L dI y - y H x i LM, y = H1 - x 1L x 2
x 3for odered energy fractions: x i =
2 E i
t , x 1 > x 2 > x 3
six sectors
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Slide 5 of 11
Y 23 distribution. Numerical test
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Slide 6 of 11
Y 23 distribution. log-R matching
R I y M = I1 + C m am M S a, ln
1 y
+ D Ia, y M,
lnR = ln S + R H1L -IG 12 aL2+ G 11 aLM,
SHa, L L = exp ‚n =1
¶
‚m =1
n +1
G nm an L m
D Ia, y M = ‚i
ai D i I y M, so that D i H0L = 0.
resummation factor
perturbative (power) correction
lnR = ln S - ‚n =1
2
‚m =1
n +1
G nm an L m
+ R H1L + R H2L -12@R H1LD2,
hence R = 1 + R H1LI y M + O Ia2M
hence R = 1 + R H1LI y M + R H2LI y M + O Ia3M
Instead of finding C m and D, one can use the following log-R matching
or alternatively (R-matching): R I y M = A1 + C m am
+ D Ia, y ME S a, ln1 y
=A1 + R H1L - IG 12 aL2+ G 11 aLME S a, ln
1 y
,
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Slide 7 of 11
Y 23 distribution. log-R matching
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Slide 8 of 11
L3 and OPAL data at Z pole
0.00 0.05 0.10 0.15 0.20 0.25 0.300.01
0.1
1
10
100
y 23
d s ê d y 2 3
L3 1992
OPAL 2005
NLO+NLL
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Slide 9 of 11
ALEPH data in Z-pole
2 4 6 8 10-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
-lnH y 23L
y 2 3
d s ê d y 2 3
BSZ result
logHRL-matching as =0.1176
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Slide 10 of 11
ALEPH data all energies
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