+ −
χcJ → e e
decay within the NRQCD
effective theory framework
Nikolay Kivel
11th European Research Conference on
“Electromagnetic Interactions with Nucleons and Nuclei”
1-7 November, 2015, Cyprus
Motivation and existing results
C-odd
+ −
e+
Br[J/Ψ(1S) → e e ] = 5.971%
Br[Υ(1S) → e+ e− ] = 2.38%
C-even
Γ[χc1 → e+ e− ] � 0.1 − 0.46 eV
+ −
Γ[χc2 → e e ] � 0.014 eV
γ∗
e−
γ∗
γ∗
χcJ , X(3872)
Γ[X(3872) → e+ e− ] � 0.03 eV
VMD model
Kühn, Kaplan, Safiani 1979
Denig, Guo, Hahnhart, Nefediev 2014
might be detected
at high-luminosity
accelerator BES III
NRQCD factorisation framework
k
� ∼ mv
∆
v2 � 1
expansion with respect to small
integrate out hard modes k0 ∼ m �k ∼ m
QCD
QED
Bodwin, Braaten,Lepage
1994
QCD
Non-relativistic
QED
NRQCD factorisation: hard configuration
soft part is described within the NRQCD
# of the non-perturbative constants
are reduced to one due to the
heavy-quark spin symmetry
hard contributions are computed in pQCD
�
TJ=1
2
α
∼ 3 ln
mc
m2c
�2
∆
�
TJ=2
2
α
∼ 3
mc
�
ln
m2c
�2
∆
1
+ (ln 2 − 1 + iπ)
3
IR-sensitive !
☛
�
Kühn, Kaplan, Safiani 1979
factorisation formula is not complete!
smth must be subtracted to cancel soft scale
NRQCD factorisation: ultrasoft configuration
additional region:
is hard
is ultrasoft
hard k0 ∼ m �k ∼ m
soft
� ∼ mv
∆
ultrasoft
soft non-perturbative part, NRQCD & pNRQED
pQCD
soft-collinear QED
NRQCD factorisation: ultrasoft configuration
additional region:
is hard
is ultrasoft
hard k0 ∼ m �k ∼ m
soft
ultrasoft
soft photons are resummed into the soft Wilson lines
+
+
+...
� ∼ mv
∆
Complete NRQCD factorisation
NK, Vanderhaeghen 2015
?
Low energy effective theory
effective degrees
of freedom:
soft photons & cc-mesons:
k
n
χcJ
n̄
includes exact and approximates symmetries of NRQCD
in a systematic way using 1/m expansion
Casalbuoni et al, 1993
Low energy effective theory
effective degrees
of freedom:
soft photons & cc-mesons:
k
n
χcJ
n̄
Operator matching
Complete NRQCD factorisation formula
Γ[χc1
Γ[χc2
√ 2
� 3
�
1
α
(1)
→e e ]=
Mχ |Cγγ (µ0 ) O( P0 ) + Cγ ec S(µ0 )/ 2 |
12π
π
+ −
� 3
�
1
α
(2)
→e e ]=
Mχ |Cγγ (µ0 ) O( P0 ) + Cγ ec S(µ0 ) |2
20π
π
+ −
!
matching scale
Eichten, Quigg 1995
Numerical estimates
Γ[χc1
Γ[χc2
√ 2
� 3
�
1
α
(1)
→e e ]=
Mχ |Cγγ (µ0 ) O( P0 ) + Cγ ec S(µ0 )/ 2 |
12π
π
+ −
� 3
�
1
α
(2)
→e e ]=
Mχ |Cγγ (µ0 ) O( P0 ) + Cγ ec S(µ0 ) |2
20π
π
+ −
matching scale
Kühn, Kaplan, Safiani 1979
Γ[χc2 → e+ e− ] � 0.014 eV
eV
usoft 82% , hard 13%
eV
usoft 53% , hard 16%
Denig, Guo, Hahnhart, Nefediev 2014
Summary
The decay rate Γ[χcJ → e+ e− ] is computed using effective filed
theory framework
There are two different contributions: hard (both photons
are hard) and ultrasoft (one photon is ultrasoft)
Ultrasoft contribution can be computed within the low energy
effective theory framework (em sector of HHχPT)
Nonperturbative contributions are presented in terms of matrix
elements in NRQCD. All these matrix elements are known and
related to the radial wave functions at zero.
Higher order corections are of order v2 and can not be very
large numerically
!ank y"!