Embed Size (px)
DESCRIPTION
Radiative corrections and enhanced power corrections for the shape function in inclusive B decays. Kazuhiro Tanaka (Juntendo U). H. Kawamura (RIKEN) J. Kodaira (KEK). OPE. Light-cone expansion. ⇒factorization formula. Bigi et al. (’94), Neubert (’94) Korchemsky, Sterman (’94) - PowerPoint PPT Presentation
Citation preview
Radiative corrections and enhanced power corrections
for the shape function in inclusive B decays
Kazuhiro Tanaka (Juntendo U)
H. Kawamura (RIKEN)
J. Kodaira (KEK)
( 224) 4 †( ) TIm ( ) (0)
u
u
iq xubu e LB X Lff
X
Vd X H B p p q d xe B J x J B- ×G - -å ò: l : m nn d
51
2LJ u b-
=n n gg
OPE ( )2
light-cone singula rity for 0Bp q- ;
max
no background
for 2
c
B
B X
mdE E
dE
nìï · ®ïïï Gíï · =ïïïîl l
l
l
;Light-cone expansion
⇒factorization formula Bigi et al. (’94), Neubert (’94)
Korchemsky, Sterman (’94)
Bauer, Pirjol, Sterwart (’02)
perturbative
nonperturbative
形状 関数unknown
s
in LO in
to all orders in bm
a
L
Shape function( )2 0 , bz tn n mº = ®¥
1
HQET-field:
+ ( ) ( )
spin symme
( tryeq. of
) (mot
) ( ) 0 ion
b
bim v xv
v v
v
Om
v
b x e h x
h x h xiv Dh x
- × æ ö÷ç ÷ç
/
÷÷çè ø=
=× =
( )2 1, 1B Bp m v v v n= = ×=m m
( ) ( ) (0) ( ) ) )( (i tv v f tB v h tn h B v d e f= =ò% ww w
0Pexp ( )
tig d n A nm
ml læ ö÷ç ÷ç ÷çè øò
6444444444447444444444448
light-cone momentum( ) d: istribution
univers l: a s
f k v
B X+ +=
®
ww
g beyond this!
Loop corrections for RG evolution
2UV UV
1 2log( ) ( ) (0)
2s F
v v
Cit h tn h
am
e ep
æ ö÷ç ÷=- +ç ÷ç ÷çè ø
IR
1( ) (0)
2s F
v v
Ch tn h
aep
=
U IRV
1( ) (0
2
1)s F
v v
Ch tn h
aeep
æ ö÷ç ÷= -ç ÷ç ÷çè ø
( ) [ ]bare 2
UV UV
( ) ( (0) ( ) (0) ) , 1 2log( ) 1
( ) 12v v v vs Fh th t ht nZn hC it
Z ta mp e e
=æ ö- ÷ç ÷= - +ç ÷ç ÷çè ø
MSEe= gm m
( ) [ ]
2
1
bareUV
0
1 3 ( ) (1 )
(1 ) 2
(( ) (01
( ))2
) (0)
( ) (1 ) ( )
,
qq F
sq q qq
P C
Zt tnn nn d Z P
xx d x
x
y y x ya
yxp
xe
x d x x
+
+= + -
-
= = - +
æ ö÷ç ÷ç ÷ç ÷è ø
ò
Grozin, Korchemsky (’96)
Bauer,Manohar (’04), Bosch et al. (’04)
4 2D e= -
Cusp divergencePolyakov (’80)
Korchemsky (’89)
{0
0
P ex
( ) (0)
( )p
P exp )
(
(
)
v v
v
t
ig d n
ig dsv A sv
h tn hA
h v
nm
mm
ml l
- ¥
æ ö÷ç ÷ç ÷ç ÷÷çè ø
æ ö÷ç ÷ç ÷è
¥
ø
-
ò
ò
P
0
v- ¥
z
[ ] ( )[ ] ( ) (0) ( ) (0)2log 0( ) 1v v v vs Fh tn h h tn hitCd
dm
am
m p- =+
0( ) ( ) ( ) ( ( 0,0) ( ) 1, 2 ) ,
NN
v vNt
dd f B v h tn h B v N
dtww w
== = =- ¥ò L
2
(0) (1)
l
og ( )
( , )
( , ) ( , ) ( , )
ˆ ( )
ˆ ( ) : ( ) , , ,
( ) (0) =
( )
i
ji v v
i
i
v v v
v vi
i
v
si
it
h t O
O h in D h h GGh h
n C t
C t C t C
qqh
h
t
m
m
m
m
m
am
mm
p
×
= + +
å
:
L
L
[ ] 2
0( ) (0) log ( ) v v
s F
th tn h
Cit
ap
m®
-:cusp anomalous dimension
double logarithmic singular behavior
hard (conta (UV) cminat omponed by ent)
" "hard softÄKorchemsky, Sterman (’94)
Bauer, Manohar (’04)
Solution of RG eq.
[ ] [ ]
1/
1/( ) ( ) (0)( ) (0)
( ) log( )
) )
( (2
v vv v
F
t
s F s
tS th tn h tn hh
S td
iC C
t
em
mm
a k akm k
kpk p
=- -æ ö÷ç ÷ç ÷çè ø
=
òIR renormalon ambiguity have to be compensated by the power corrections from nonperturbative effects:
[ ] 2 2 3 32 3
22
3
2
3
1/( ) (0)
( ) ( ) ( )
(
(
) )
1v v
v v
v v
th tn h
B v h in D h B v
B v h qqh B
t
v
tr r
r
r
+ L + L +
L
L
×
: L
:
:
L
s matching order-by-order in
exact constraints to find
independent basis of
"enhanced power corre
EO
ct
n
M
io s"
a·
·
2
0 2
1( )
logsa k
kb
=
L
1
2tL :
Tree-level matching
tre
(
e tre
0)
e ( ) (0) = ˆ ( , )( )i iv v
i
C th tn Oh m må
(0) 1 no no -depde singularitn yce: for 0iC tm· ®:
( )
1 2 1
1 2 10
( ) (0) (0) (0)!
,
N N
N N
j
v v v vN
t n n n nh tn h h D D D D h
N
D v v D D
D D igG
m m m
m n mn
m m m mm m m m
-
-
¥
^
=
= × +
é ù=-ê úë û
=åL
L
(0)N vD hm^
1(0)
N N vG hm m-+L
● Economic way: Kawamura,Kodaira,Tanaka, Prog.Theor.Phys. 113 (’05)183
Eq. of motion constraints on nonlocal operator ⇒ nonlocal op. basis
Taylor expand in the final step ⇒ independent set { }ˆ ( )iO m
response o
E
f
q. of motio
to the change of interquark separati
n constraints on nonlocal op. :
on ( ) (0)v vh x h
Light-cone expansion
2( 0)x ¹
x tn®
Nachtmann corr.
1
0( )( ) (0)( ) (0) ( ) (0)vv vvv vv duu ux v x
xgh x h h xh x v Dh hGi m n
mnm
m
¶¶
× += òsu
12
0
2
( ) (0) ( ) (0) (0
2
) 0
(
( )
)( ) (0) ( ) (0)
v v v v v v
v v v v
x tn
u
h tn h hd
tdh tnt
t du tnu x
vu
h h
nih tn uxg hG h hmm n
mn
®
æ ö¶ ÷ç ÷ç ÷çè ø
+
ì üï ïï ïï ï= í ýï ¶ ïï
-
ïï ïî þ-ò
( )2 1
4
lt
2
0( )
4( ) (0( ) ( ( )) )0 ) (0v v vv vv
x duO x
xuh h uxx h hx hh m
æ ö¶ ÷ç ÷ç ÷çè ø¶+ += ò
0
4-particle
correlation!!
( ) (0) (0) ( (( ) 0) )) ( 0v v v vv v h tn hd
td
hh htt
n h J t+ =-
( ){
}
0
12 2
0
13
0 0
2
2
( )
( ) ( )
( ) ( )
( ) (0)
( ) (0)
( ) (0)2
a
u
a
t
v v
v v
v v
J t
t q u
d
duu
du
n t nq u n
u n n s n nu dss
Dh n h
h n h
h n
i
h
g
gG gGm
m
n rmn r
t t
t t
t t
tt
t
tt
^=-
+
+
ò
ò
ò ò
su
(0) ( ) (0 ( ,= ) ˆ) ( )iv vi
ih tn C t Oh m må
( )
( ){
2
4 24
0
23
2
(0) (0)
(0
1( ) (0) (
) (0)
(0) (0)
(0
6
36
2
120
) (0)
0) (0) ( )
2
33
(0) (0)
(0) (0)
(0) (0) (0) (0)
(0) (0)
v v
a a
a
v
a
t
v v v
v v
v v
v v v v
v v
h h
i
t q t vq
t q t q
n n
h tn h h h d
t
t
t
it
J
i
h D h
h g h
h D h h g D h
h gG gG h
m
mm m
n mrmn r
l l
g
^
^ ^
= +
=
-
-
-+
-
ò
üïï +ýïïþL
1
2tL :new enhanced power corrections
independent set{ }ˆ ( )iO m
1 lÞ
1 rÞ
( ) ( )
( ) ( ) ( )
( ) ( )
( )
2
3 2
3 322 2
3
2
0
2
( ) 1
! 1
1( 1)( 2) ( 1)( 2)
( 1)
1 ( 1) 1
2 4
2( 1)( 1) ( 1)( 1)
( 1)
j
j k k
j
k
a a
j ja a
j
j
v v
v v
v v
t qt nq
t qt
it j
j j
k k j k j kj j
j
j k
n
k jj j
q
h in D D h
h in D g in D h
h in D g in D h
m
é ù-ê úê úë û
=
^
- -
- -
-+
+ + + - -
ìïïíïïîéêê+ - -
+
- - -
+ + - -
êêêë
ù
+
úúû
-
-
- + - -+
×
× ×
× ×
å
l l l l
( ) ( ) ( )
( ) ( ) ( )
4
4
42 2
2
0 0
1 ( 1)1 1
2
2 2
(0)
(0)k
k j k
k kj
j
k
k
k
v v
v vk k
j
n n
n n
h in D gG in D gG in D h
h in D gG in D gG in D h
n mrmn r
n mrmn r
- -
é ùê úê ú- ë û
=
-
=
- -æ öæ ö+ - ÷ ÷ç ç+ - -÷ ÷ç ç÷ ÷ç çè øè
éêêêêêë
´
üùïïúýïúûïþø
×
-
× ×
× × ×
å å
l
l
l
( ) -pow-th er term: correc ti on:j
j tL
2
independent operators4
j:
One-loop matching
{ }one-loop one
(0) (
l p
1)
- oo
( ) ( , ) (( ) (0) = ˆ ), ) (s
iv vi
i ih t Cn Ot th Ca
m mm
mp
+å
( )
( ) ( ) ( )( )( ) ( )( )( ) ( )
( )( )
2
2
0
2
3
4 4 2
2
3
4
,
,
:
:
,
:
:
(0),
v v
a a
a a
v v
v v
v v
v v
v v v
v
v
v v
v v
v
in
in n
h h
i
t qt
t
t
vqt
t qt q n n
i
t
n
h D h
h D h g
h D h
h g h
h D h h g D h h gG gG
h D G h
h
h D h
m
m n mrm m mn
mrm
r
r
g
^
^ ^
L
L
×L
L
×
/
×:
M
:
: L
( ) ( )
BRST
EOM operators: ,
BRST-exact operators
ˆ: 0
a av v v viv t qt qD D Gh h h hmr r
m
d
g
=
× -
Q
L L L
( ) ( )( )2
, v v v viv in ivh h h hD D D× × ×
¿
● to minimize complication due to alien operators● to maximize gauge invariance in Feynman rules Background field method:
(Q) (C)
(Q
(C) (C)
(C) (C) (C)(C)
) (C)
0,
,
v
a a
v v vh h h
A A
iv D h
D G t q qA tm m mmr r
m g+ =
+
®
× =®
Fock-Schwinger gauge
(Q) (Q) 1) (C( )
0
1( ) (0) ( ) ( P exp (
2))v
v
v
xD i
vh x gx x d A vh vv m
m lq ld×
-^
+/ æ ö÷ç ÷ç× ÷è ø= ò
1(C) (C) (C)
0( ) ( ) ( )x A x A x duux G uxm r
m m rmÞ =ò
g g gx 0
( )2For ˆ
i
v v
v vO
ih D h
h h
m^
ìïïïïíïïïïî
=
r e
2
e
2
t
( ) 5log ( ) l ( ) (0) )og(
24s F
v v
Cit it h tn h
a m pm m
p
æ ö÷ç ÷=- - +ç ÷ç ÷çè ø
ß2
1v v
v v
h h
h iD h
Z
Zm
æ ö÷ç ÷ç ÷ç ÷çè ø^
üïïïï =ýïïïïþ
Grozin, Korchemsky (’96)
Û
2
22
(1)
22
M
6
S
5log ( ) log( )
24( , )
5 log ( ) log
2
( )4
F
i
F
t
C it it
C t
C it it
pm m
mp
m m
ì æ öïï ÷ç ÷- - +ï ç ÷ï ç ÷çè øïïíï æ öï ÷çï ÷- +ç ÷ï ç ÷çï è øïî
=
[coincide with Bauer, Manohar (‘04), consistent with Bosch, Lange, Neubert, Paz (‘04)]
2For ˆ a av vi t qt vqh g hO = /
{ }3
(1) 26
36( , ) log ( ) log( )i F
itC t C it i dtm m mé ù-ê úû+ë=
● check of cancellation of IR divergence
( )
( )
22
22 2
6
7,1
3
44
( , )
( ) 51 log ( ) log( )
24
16 36
11
( )
( )
( ) (0) ˆ ( )
=
20
v
s
i
a
vi
i
a
F
v
s
s
v v v v v
v v
C t
Cit it
t it
t
O
h h i t qt vqd
h
d
tn h
h D h h g h
h D h
m
m
m
a m pm m
p
a mp
a mp
m
^
^
æ öé ù÷ç ê ú÷- - +ç ÷ç ê ú÷çè øë ûæ ö÷ç ÷ç ÷çè ø
æ ö÷ç ÷ç ÷çè
/
=
ìïï´ - -íïïî+
ïø
+ìïí+
å
27,2
7,3
21
3
3
( )
( )1
2
s
s
a av v
v v
d
d
t
n n
iqt qh g D h
h gG gG h
mm
n mrmn r
a mp
mp
g
a
^
æ ö÷ç ÷ç ÷çè ø
æ
+
+ö÷ç ÷
-ïïî
üïï +ýç ÷çè ø ïïþ- L
( ) ( )2 3 up to and ( ) corrections
model i singular UV n bde ehpend avioent: is factorizedr
s sO O ta a· ×L
·( ) ( ) ( ) (0) soft( ) "hard "v vf t B v h tn h B v= = Ä%
( )
( ) ( )0
2
2
13
2
1
1
( ) ( )
( ) ( )
( ) ( )
( ) (1 Ge
1
V) + 2G
v v
a a
v v
s
v v
C
s
B v B v
B v B
h h
i
t
v
B v Bt v vq q
h D h
h g h
m
pb
l
ra m a
^ =
/
=
ì üï ïï ï= -í ýï ïï ïî þL
Summary
beyond simple light-cone momentum distribution
``Universal'' shape function for
matching procedure for ( ) (0) " "soft
soft components
hard
e EOM
,
unravelled from xact cv v
u s
h tn h
B X B Xn g·
· = Ä
·
® ®l
hard component calculating in ``NLO'' accuracs
enhanced power corrections due to
universal double logarithms f
multi-p
rom cus
onstrai
p anoma
arton cor
lous dime
n
nt
si
relatio
ons
s
y
inter y
n
pla
·
·
IR renorm
ha
al
between a
on amigui
nd components
cancellation of
role in d
r
e c
s
a
of
y
ty
ra
t d
tes
''shape function region'' 1
( ) (0) = for 1
resummation of enhanced power corrections using EOM constraints
ˆ ( , ( )
( ) (0
)
) (
i ivi
v vv
v
t
h tn
h t
C t
n
O
h tn h
h t
dtdt
m
m
mm
·
æ ö÷ç ÷ç ÷ç ÷è ø
+
å
;
=
consistent wi
) (0) (0
th IR r
) (0
enor
( ))
malon calc us ul
v v v Jh h h t=-
