Y and ψ leptonic decays as probes of solutions to the R(D (*)) puzzle
Received: March
Published for SISSA by Springer
Open Access 0 1
c The Authors. 0 1
0 decays necessarily modi es
1 Rehovot , 7610001 Israel
2 Department of Physics, LEPP, Cornell University
3 Department of Particle Physics and Astrophysics, Weizmann Institute of Science
Experimental measurements of the ratios R(D( )) deviation from the Standard Model prediction. In the absence of light righthanded neutrinos, a new physics contribution to b ! c
Beyond Standard Model; Heavy Quark Physics; Precision QED

((BB!!DD(( ))` )) (` = e; )
transitions. These contributions lead to violation
of lepton
and/or cc !
avor universality in, respectively,
leptonic decays. We analyze the
(1S; 2S; 3S) !
in the upcoming Belle II experiment.
Contents
1 Introduction V
! `` decay rate
The e ective
eld theory
Numerical results
(1; 3)0
(3; 1)+2=3
(3; 3)+2=3
(3; 1) 1=3
(3; 3) 1=3
(1; 2)+1=2
(3; 2)+7=6
(3; 2) 5=6
Discussion and future prospects
Summary and conclusions
A The leptonic width in the SM
B Other EFT operators
B.1 Z mediated operators
B.2 Dipole operator
Introduction
R(D( ))
; (` = e; ):
R(D) = 0:403
0:047; R(D ) = 0:310
result, R(D ) = 0:270
0:044 [6], is not included in the HFAG average [8].) The SM
predictions are [9, 10]
R(D) = 0:300
0:008; R(D ) = 0:252
works analyze the deviation in terms of e ective
eld theory (EFT), construct explicit
and/or cc !
purely lefthanded, NP contributions to the b ! c
in ref. [47] which analyze the high PT distribution of the
are governed by the cc !
and bb !
quarkonia. Speci cally, we study the ratios
transition imply that NP contributions
transitions are unavoidable. This point was discussed
signature at the LHC,
transitions: nonuniversality in leptonic decays
RV=`
; (V = ; ; ` = e; );
making two assumptions:
e ective operators of dimension six.
R(D( )) is modi ed by new physics that a ects only the B ! D( )
decays (and
a ected (and not V ! ``).
decays of
vectormesons.
anomaly and those modifying the leptonic decays of
is that all processes occur at
mixing, as well as its compatibility with other relevant observables.
Tests of LFU have been carried out for
(2S) and
(3S) [60, 61]. We collect
RV=` ' (1 + 2x2)(1
4x2)1=2
BR( (2S) !
BR( (2S) !
) = (3:1
) = (7:9
BR( (2S) ! e+e ) = (7:89
threshold, respectively.
lepton pairs obey [63]
is below the
We also do not consider
(3770) and
(4S) which have
SM prediction
Exp. value
m (1S) = 9:46030
m (2S) = 10:02326
m (3S) = 10:3552
m (2S) = 3:686097
0:00026 GeV;
0:00031 GeV;
0:0005 GeV;
0:000025 GeV;
0:00012 GeV:
as is evident from table 1.
the formalism is similar.
! `` decay rate
The most general V ! `+` decay amplitude can be written as
M(V ! `+` ) =
v(p2; s2) (p);
where Aq`; BVq`; CVq`; DVq` are dimensionless parameters which depend on the Wilson
coef
V
cients of the operators controlling the V ! `+`
decays at the perturbative level, and on
width and RV=` are given by
[V ! ``] =
1 4x`2 jAqV`j2 1+2x`2 +jBVq`j2 1 4x`2 +
1 4x`2 + 2Re hAqV`CVq`i x` 1
RV=` '
q 2 1 + 2x2 + jBV j
4x2 + 2Re A
Within the SM, A`;SM
Qq and B`;SM; C`;SM; D`;SM
V V V
' 0. For the SM calculations
q + CSq`LeReLqRqL + CSq`ReReLqLqR + h:c:i :
CVq`LL + CVq`RR + CVq`LR + CVq`RL
L`q = CVq`RReR
We nd:
AqV` =
BVq` =
parametrization:
q jV (p)i = fV mV
q jV (p)i = ifVT [ (p)p
(p)p ] ;
the heavy quark limit fV = f T . This is an excellent approximation for the
V
5q jV (p)i = 0. The
meson. For
The e ective eld theory
and Lorentz invariant operators:
LLQQ; eLuQ; eLQd; LLuu; LLdd; eeuu; eedd; eeQQ;
new (scalar or vector) boson. Following ref. [27], we specify the NP
eld content which
in the quark interaction basis):
L) (uL uL)+( L
L) dL dL +(eL eL) (uL uL)+(eL eL) dL dL ;
(eL eL)(uL uL)+(eL eL) dL dL
OV1L = ( L
OV2L =
OV3L = ( L
OV4L = ( L
OSL =
OT = (eR
( L eL) dL uL + (eL
eR) (uL
eR) (uR
L) (uR uR) + (eL eL) (uR uR) ;
uL) + (eR eR) dL dL ;
(eReL) (uRuL) + (eR L) (uRdL) ;
eL) (uR
OD = HL
lL;R lL;R and the dipole operators
modify the Z !
Field content
Fierz identities
(eL) (uQ)
(uL) (eQ)
Lc Q (euc)
Qd (eL)
Lu (uL)
e) Q e
Qe (eQ)
(euc) (uce)
(edc) dce
2 OVL + 2OV2L
136 OV1L + 14 OV2L
OVL
8 OT
OT
OVL
(3; 1)+2=3
(3; 3)+2=3
(3; 1) 1=3
(3; 3) 1=3
(1; 2)+1=2
(3; 2)+7=6
(3; 1) 1=3
(1; 2)+1=2
(3; 1)+2=3
(3; 2) 5=6
(3; 2)+1=6
(3; 2)+7=6
(3; 2) 5=6
(3; 2)+1=6
(3; 2) 5=6
(3; 2)+7=6
(3; 2)+5=3
(3; 1) 1=3
(3; 1)+2=3
(3; 1) 4=3
arising from this operator is given by
RV=` ' RV=;S`M
of Zpole observables [68] constrain this e ect to be smaller than 10 5
. For more details
by the mixing with four fermion operators, in which case
netuning cancelation may be
needed to ensure that the Zpole constraints are not violated.
our analysis.
Numerical results
have the right quantum numbers to modify the b ! c
transition. For each case, we
and (2S) states. Some of the simpli ed models we
2 = 9 (corresponding to 3
with one degree of freedom) are included. Note that in some
cases only one of the decay modes, i.e.
, is modi ed, due to the speci c
(1; 2)+1=2
UV eld content
S (3; 1) 1=3
(3; 1)+2=3
(3; 2) 5=6
Current measurement
Achievable uncertainty (with current data)
R =(`1S)
1:005 0:025 0:39 0:05
R =(`2S)
0.3890.390 {
Predicted modi cation to R =(`1S)
Decrease by 0:2%{0:4%
Decrease by 0:3%{4:0%
Decrease by 0:5%{1:6%
V =
for more details.
and NC e ective operators by O (
level splitting is typically of order
nal results
As concerns the
avor structure of the fundamental couplings, we impose a global
transitions in the
rst and second generation.1 To determine uniquely the
avor
strucQ3 =
operators (generated by the S and D
elds) one could choose alignment to the upmass
basis. In this case neither cLcR !
Nevertheless, other NC operators which result in bLbL !
transitions are generated.
and cRcR !
once upalignment is taken.
We assume no signi cant mixing between the NP and SM
elds (this is crucial for
at the
TeV scale. We stress again that, once an anomaly is found in LFU of
decays, these
be necessary in some cases [71].
elds is absent due
real and denote Xij
XiXj .
evident from the accurate SM predictions. (See also
gures 1{2 in ref. [50].) As for the
it to the SM point. The latter gives 2
As concerns kinematical observables, the q2 distribution of
] is hardly
as analysed in ref. [49] are expected to be small.
(1; 3)0
interaction Lagrangian:
We introduce a vectorboson, colorsinglet, SU(2)Ltriplet W 0
(1; 3)0 with the following
Integrating out W 0 , we obtain the following EFT Lagrangian:
EFT =
L3 =
The relevant CC interactions are given by
The best tpoint (BFP) and 95% C.L. intervals are given by
LCC =
bL) + h:c::
g1B2FP = 7:7
g12 = [4:4; 10:9]
2 = 0:5;
@ 95% C:L::
where g12
has the same Lorentz structure as in the SM.
The relevant NC interactions are given by
LNC =
nd the following nonuniversalities
R =(`1S) = 0:989
R =(`2S) = 0:990
R =(`3S) = 0:990
R =(`2S) = 0:390:
(3; 1)+2=3
ing interaction Lagrangian:
We introduce a vectorboson, colortriplet, SU(2)Lsinglet U
(3; 1)+2=3 with the
followLU = g1Q3U= L3 + g2d3U= e3 + h:c::
d3) + h:c:
2Vcbg1g2 ( R L) (cLbR) + h:c::
Integrating out U , we obtain the following EFT Lagrangian:
. Given the 95% C.L. intervals quoted above, we nd the following
2Mg1Ug22 ( R L) bLbR + h:c: :
EFT =
LCC =
LNC =
They induce only
nonuniversalities
R =(`1S) = 0:952
R =(`2S) = 0:949
R =(`3S) = 0:946
{ 10 {
The relevant CC interactions are given by
The BFP which explains R(D( )) is given by g1g2 < 0 and
= (3:3; 0:4)
2 = 0:
The 95% C.L. intervals are presented in
gure 1. The q2 distribution of [B ! D
modi ed compared to the SM one. Yet, as is evident from
gure 3b, this change is not
very signi cant given the current uncertainties.
The relevant NC interactions are given by
of R =(`1S).
(3; 3)+2=3
We introduce a vectorboson, colortriplet, SU(2)Ltriplet X
(3; 3)+2=3 with the
following interaction Lagrangian:
LX = gQ3 aX= L3 + h:c::
Integrating out X , we obtain the following EFT Lagrangian:
LX
EFT =
Q3 =
The relevant CC interactions are given by
The BFP and 95% C.L. interval are given by
LCC =
bL) + h:c::
g 2 BFP = 0
jgj2 = [0; 1:5]
2 = 20:4;
@ 95% C:L::
Lorentz structure as in the SM.
The relevant NC interactions are given by
LNC =
nd the following nonuniversalities
R =(`1S) = 0:992;
R =(`2S) = 0:993
R =(`3S) = 0:994
R =(`2S) = 0:390:
(3; 1) 1=3
interaction Lagrangian:
We introduce a scalarboson, colortriplet, SU(2)Lsinglet S
(3; 1) 1=3 with the following
LNC = 2jM1jS2 Vc2b ( L
2MS2 Vcb ( R L) (cRcL) +
cL) + h:c: :
They induce only
nonuniversalities
. Given the 95% C.L. intervals quoted above, we nd the following
We impose a global 3B
L symmetry, which prevent an additional Yukawa couplings of
1 2 Lc3 Q3 (e3uc2) + h:c:
= j 1j
The relevant CC interactions are given by
R =(`2S) = 0:389
LCC =
The BFP which explains R(D( )) is given by 1 2 < 0 and
( R L)(cRbL)
bL) + h:c::
2 BFP; j 2j
= (0:3; 0:9)
2 = 0:
signi cant given the current uncertainties.
The relevant NC interactions are given by
(3; 3) 1=3
interaction Lagrangian:
We introduce a scalarboson, colortriplet, SU(2)Ltriplet S
(3; 3) 1=3 with the following
LT =
We impose global 3B
L symmetry to forbid T QQ terms. Integrating out T , we obtain
the following EFT Lagrangian:
EFT =
2 L3 a
3 aL3 =
The relevant CC interactions are given by
The BFP and 95% C.L. interval are given by
2 BFP = 0
j j2 = [0; 3:0]
2 = 20:4;
@ 95% C:L::
Lorentz structure as in the SM.
The relevant NC interactions are given by
nd the following nonuniversalities
R =(`1S) = 0:992
R =(`2S) = 0:994
R =(`3S) = 0:995
R =(`2S) = 0:390:
(1; 2)+1=2
A scalarboson, colorsinglet, SU(2)Ldoublet
generates only scalar couplings.
(3; 2)+7=6
ing interaction Lagrangian:
We introduce a scalarboson, colortriplet, SU(2)Ldoublet D
(3; 2)+7=6 with the
followIntegrating out D, we obtain the following EFT Lagrangian:
The relevant CC interactions are given by
The BFP and 95% C.L. interval are given by
EFT = jM1D2j jQ3e3j2
+ h:c: :
LCC =
2M D2 ( R L) (cRbL) +
bL) + h:c::
MD
2 = 17;
12 = [0; 0:6]
@ 95% C:L::
The q2 distribution of [B ! D
] is modi ed compared to the SM one. Yet, as is evident
The relevant NC interactions are given by
LNC =
2M D2 Vcb ( R L) (cRcL) +
cL) + h:c: :
nd the following nonuniversalities
R =(`1S) = 0:889{0:992;
R =(`2S) = 0:879{0:994;
R =(`3S) = 0:873{0:995;
R =(`2S) = 0:386{0:390:
(3; 2) 5=6
ing interaction Lagrangian:
We introduce a vectorboson, colortriplet, SU(2)Ldoublet V
(3; 2) 5=6 with the
follow
LV = g1Q3V= ec3 + g2L3V= dc3 + h:c::
Integrating out V , we obtain the following EFT Lagrangian:
The relevant CC interactions are given by
The BFP and 95% C.L. interval are given by
2Vcbg1g2 ( R L) (cLbR) + h:c::
g12 = [1:9; 5:9]
@ 95% C:L::
MV
2 = 8:2;
LCC =
g1B2FP = 4:0
L3 + h:c:
EFT =
LNC =
They induce only
nonuniversalities
2Mg1Vg22 ( R L) bLbR + h:c: :
. Given the 95% C.L. intervals quoted above, we nd the following
The q2 distribution of [B ! D
] is modi ed compared to the SM one. Yet, as is evident
The relevant NC interactions are given by
R =(`1S) = 0:976
R =(`2S) = 0:976
R =(`3S) = 0:976
from ref. [2].
Discussion and future prospects
]. Data points and error bars are taken
{ 16 {
Nonuniversality in leptonic
decays was tested by CLEO [60] for the 1S; 2S and 3S
states, and by BaBar [61] for the 1S state. These measurements read
R =(1S); BaBar = 1:005
R =(1S); CLEO = 1:02
R =(2S); CLEO = 1:04
R =(3S); CLEO = 1:05
0:013stat
0:022syst;
0:02stat
0:04stat
0:08stat
0:05syst;
0:05syst;
0:05syst:
states, respectively. The
(2S) onresonance sample collected by BaBar (Belle) is about
10 (16) times larger than CLEO's sample. The
(3S) onresonance sample collected by
BaBar (Belle) is about
20 (2) times larger than CLEO's sample. Analyzing these existing
2 percent.
Additional improvement can be achieved by using the
cascade chains, which will render
this error negligible.
on the di erent
total e ciencies and event shapes. The larger statistics can
to study the
(3S) into one test of universality. Eq. (1.6) as a function of m (nS),
R =`(m (nS)) = 1 + 2x2;1S
2#1=2
where x ;1S
the di erent
this part of the error with a combined analysis.
{ 17 {
BESII [73] measurement reads BR( (2S) !
10 3 using 14M
the statistical error by a factor of
3. (KEDR also measures this tauonic branching
ratio [74] but it is not used by the PDG
t.) The relative systematic uncertainty on
) (as measured by BaBar) is 10% [75], while the relative systematic
BR( (2S) !
be achieved in Bess III to start probing the relevant parameter space.
Summary and conclusions
There is a 3:9 evidence that the ratio R(D( ))
)= (B ! D( )` ), where
` =
new bosons which mediate the B
decay at tree level. There are seven such
or two SU(2)Ldoublet quark
elds. Consequently, a variety of processes, in addition to
, are a ected. Some of these, such as t ! c +
decay, Bc !
decay [24, 41],
decay [51, 59], and bb =cc !
scattering [47], have been previously
of observables: lepton nonuniversality in leptonic decays of the
vectormesons,
parameterized by the ratio RV=`
)= (V ! ``).
hand, for
could be explained by operators which do not a ect V !
accuracy of about 1.5%. If Belle II operates below the
(4S) resonance, it can contribute
{ 18 {
Acknowledgments
The leptonic width in the SM
0 [V ! ``] = 4
4x`2)1=2 1 + 2x`2 :
0 [V ! ``] 1 + Ztree + QCD + EM :
The treelevel Z mediated correction is
4Qqc2W s2W
8Qqc2W s2W
1+2x`2
exchange. The corrections to
namely evaluating ( ) at
a ect RV=`.
= m2V =m2Z and gVq = T
2Qqs2W . This is an O(10 4) correction, well below the
be absorbed in the de nition of the vector meson form factor, fV .
absent: the LandauYang theorem [78, 79] implies M(V
) = 0 for massive vector
are taken into account by using the running couplings,
{ 19 {
we quote here the inclusive [V ! `+`
depend on the experimental setup. It reads [80]
+ ] at oneloop order in QED, which does not
EM =
4x`2] 1 + log[x`2]
16x`2(2 + 3x`2)
+ 16x`2 1 + 2x`2 log 4 +
3 + 16x`2 (2
log 4) ' 0:002 + 0:006x`2:
We further estimate the twoloop nonuniversality e ect to be
2 loop = O
We therefore consider, for all practical purposes,
[V ! ``] = 4
Qq2(1 4xl2)1=2 1 + 2xl2
log 4) ;
RV=` = (1
4x2)1=2 1 + 2x2
log[4]) :
Other EFT operators
Z mediated operators
Here we consider the following set of dimension six operators
LHl = iCH1` Hy D!
+ iCHe Hy D!
decays are given by
AqV` =
BVq` =
CH1` +CH3`=4+CHe
CH1` + CH3`=4
2Qqs2W , and for the Z
vertex corrections we
. This e ect
is negligible given the current and future experimental sensitivity.
vertex. Their contributions to the
Dipole operator
Here we consider the dimension six dipole operator
Its contribution to the
leptonic decays is given by
LD =
eRHF
AqV` =
CVq` = 8
DV` = 8
Qq + 16
= 4
constraints read
0:0011 GeV 1
0:0023 GeV 1
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