Isospin breaking in the nucleon mass and the sensitivity of β
decays to new physics
Article (Published Version)
http://sro.sussex.ac.uk
Gonzalez-Alonso, M and Camalich, J Martin (2014) Isospin breaking in the nucleon mass and the
sensitivity of β decays to new physics. Physical Review Letters (PRL), 112. 042501. ISSN 0031-
9007
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Isospin Breaking in the Nucleon Mass and the Sensitivity of β Decays to New Physics
M. González-Alonso
1,2,*
and J. Martin Camalich
3,4,5,†
1
Department of Physics, University of Wisconsin-Madison, 1150 University Avenue, Madison, Wisconsin 53706, USA
2
INFN, Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
3
Department of Physics and Astronomy, University of Sussex, BN1 9QH Brighton, United Kingdom
4
Department of Physics, University of California, San Diego, 9500 Gilman Drive, La Jolla,
California 92093-0319, USA
5
PRISMA Cluster of Excellence Institut für Kernphysik, Johannes Gutenberg-U niversität Mainz,
55128 Mainz, Germany
(Received 18 October 2013; published 28 January 2014)
We discuss the consequences of the approximate conservation of the vector and axial currents for the
hadronic matrix elements appearing in β decay if nonstandard interactions are present. In particular, the
isovector (pseudo)scalar charge g
SðPÞ
of the nucleon can be rela ted to the di fference (sum) of the nucleon
masses in the absence of electromagnetic effects. Using recent determinations of these quantities from
phenomenological and lattice QCD studies we obtain the accurate values g
S
¼ 1.02ð11Þand g
P
¼ 349ð9Þ
in the modified minimal subtraction scheme at μ ¼ 2 GeV. The consequences for searches of
nonstandard scalar i nteractions in nuclear β decays are studied, finding for the corresponding Wilson
coefficient ϵ
S
¼ 0.0012ð24Þat 90% C.L., which is significantly more stringent than current LHC bounds
and previous low-energy bounds using less precise g
S
values. We argue that our results could be rapidly
improved with updated computat ions and the direct calculation of certain ratios in lattice QCD. Final ly,
we discuss the pion-pole enhancement of g
P
, which makes β decays much more sensitive to nonstandard
pseudoscalar interactions than previously thought.
DOI: 10.1103/PhysRevLett.112.042501 PACS numbers: 23.40.Bw, 12.38.Gc, 12.60.Fr, 13.40.-f
In pure QCD, the charged d → u transitions induce
approximately conserved vector and axial currents
∂
μ
ð
¯
uγ
μ
dÞ¼−iðm
d
− m
u
Þ
¯
ud; (1)
∂
μ
ð
¯
uγ
μ
γ
5
dÞ¼iðm
d
þ m
u
Þ
¯
uγ
5
d (2)
with m
u;d
the respective light-quark masses. These
equalities are a particular case of the venerable “conserva-
tion of the vector current” (CVC) and “partial conservation
of the axial current” (PCAC) relations, which are derived
from global-symmetry considerations [1–3] and have
become a cornerstone for model-independent approaches
to the structure and interactions of hadrons [4,5].
A straightforward application of CVC and PCAC concerns
the derivation of relations between different hadronic matrix
elements of local quark bilinears and it is indeed customarily
used, e.g., in meson decays to reduce the number of
independent form factors (see, e.g., Ref. [6] for kaon decays).
A well-known application of PCAC to nucleon matrix
elements is the Golberger-Treiman relation between the
πN coupling and the nucleon axial coupling g
A
[7–9].
As shown in the next section, similar relations can be
established between the well-known isovector (axial)
vector charges g
VðAÞ
of the nucleon and their (pseudo)scalar
counterparts g
SðPÞ
[10]. The latter are needed to describe
nuclear and neutron β decays if nonstandard (pseudo)scalar
interactions are present [11–14], and they currently are
subject to intensive research mainly through lattice QCD
(LQCD) calculations [15,16]. These investigations are of
crucial importance to assess the implications of precise
β-decay measurements to constrain new physics.
To make things more interesting, it turns out that the
nucleon mass splitting in the absence of electromagnetism
δM
QCD
N
≡ ðM
n
− M
p
Þ
QCD
is a necessary input for the
calculation of the scalar charge g
S
. Actually, the isospin
corrections to the hadron masses, and in particular to the
nucleon mass, are starting to receive much attention.
While phenomenological determinations are being revised
[17], different lattice collaborations have embarked on
the ab initio computation of these effects in pure QCD
[18–20] or even including QED [21–27].
We show how this can be exploited for translating
recent calculations of δM
QCD
N
into a precise determination
of the scalar charge, which subsequently is used to extract
a stringent bound on nonstandard scalar d → u transitions
from β-decay data. Inversely, we discuss the implications
that recent LQCD calculations of the scalar charge have on
the isospin breaking effects in the nucleon mass. Finally,
we study the pion-pole enhancement of g
P
and explore its
impact on the β-decay phenomenology.
Form factors in β decay.—The theoretical description of
neutron β decay within the standard model (SM) requires
the calculation of the vector and axial hadronic matrix
elements, which can be decomposed as follows [28]:
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hpðp
p
Þj
¯
uγ
μ
djnðp
n
Þi¼
¯
u
p
ðp
p
Þ
g
V
ðq
2
Þγ
μ
þ
~
g
TðVÞ
ðq
2
Þ
2
¯
M
N
σ
μν
q
ν
þ
~
g
S
ðq
2
Þ
2
¯
M
N
q
μ
u
n
ðp
n
Þ; (3)
hpðp
p
Þj
¯
uγ
μ
γ
5
djnðp
n
Þi¼
¯
u
p
ðp
p
Þ
g
A
ðq
2
Þγ
μ
þ
~
g
TðAÞ
ðq
2
Þ
2
¯
M
N
σ
μν
q
ν
þ
~
g
P
ðq
2
Þ
2
¯
M
N
q
μ
γ
5
u
n
ðp
n
Þ; (4)
where u
p;n
are the proton and neutron spinor amplitudes,
¯
M
N
is the average nucleon mass, and q is the difference
between the neutron and the proton momenta q ¼ p
n
− p
p
.
The vector and axial charges g
V
and g
A
, respectively, are
responsible for the leading contributions to the decay rate
due to the relatively small energies (q
2
≃ 0) involved in the
process. We have g
V
¼ 1 up to second order isospin-
breaking corrections [29], whereas the axial charge has
been accurately measured in β decays, g
A
¼ 1.2701ð25Þ ×
g
V
[30]. Lastly, the subleading contributions coming from
the so-called “induced” form factors
~
g
i
are known in the limit
of isospin symmetry, a safe approximation at the current
level of experimental precision [13,31]. The description
of nuclear β decays requires the introduction of the Fermi
and Gamow-Teller nuclear matrix elements that play an
analogous role to g
V
and g
A
in the neutron decay.
If nonstandard (pseudo)scalar interactions are present we
need to introduce the following matrix elements in the
theoretical description:
hpðp
p
Þj
¯
udjnðp
n
Þi ¼ g
S
ðq
2
Þ
¯
u
p
ðp
p
Þu
n
ðp
n
Þ; (5)
hpðp
p
Þj
¯
uγ
5
djnðp
n
Þi ¼ g
P
ðq
2
Þ
¯
u
p
ðp
p
Þγ
5
u
n
ðp
n
Þ; (6)
whereas a tensor interaction introduces an additional
matrix element not relevant for our discussion [13].
Equations (5)–(6) show how the size of the (pseudo)scalar
charges modulates the sensitivity of β decay experiments to
nonstandard (pseudo)scalar interactions.
Relation between charges and the isospin breaking
contribution to the nucleon mass.—Using the CVC result
of Eq. (1) in combination with the above-defined form
factors it is straightforward to derive
g
S
ðq
2
Þ¼
δM
QCD
N
δm
q
g
V
ðq
2
Þþ
q
2
=2
¯
M
N
δm
q
~
g
S
ðq
2
Þ; (7)
where δm
q
¼ m
d
− m
u
. Note that the contribution due to
electromagnetic effects δM
QED
N
is of the same order of
magnitude as δ M
QCD
N
, and so the experimental value cannot
be used. Indeed the inclusion of QED in the analysis would
modify the CVC relation given in Eq. (1), introducing a
correction proportional to α
e:m:
.
In the limit q
2
→ 0 the expression (7) reduces to
g
S
¼
δM
QCD
N
δm
q
; (8)
up to second order isospin-breaking corrections. Notice that
the renormalization-scale and scheme dependence of g
S
and the scalar Wilson coefficient ϵ
S
[defined precisely in
Eq. (14)] is the opposite [13,32], rendering the observable
quantity ϵ
S
g
S
scale independent. Throughout this Letter we
use the modified minimal subtraction scheme at μ ¼
2 GeV for both the (pseudo)scalar charges and the light
quark masses.
Likewise, using PCAC in Eq. (2) one obtains
g
P
ðq
2
Þ¼
¯
M
N
¯
m
q
g
A
ðq
2
Þ −
q
2
=2
¯
M
N
ð2
¯
m
q
Þ
~
g
P
ðq
2
Þ; (9)
with
¯
m
q
the average light-quark mass and where we have
dropped the “QCD” subindex in the average nucleon mass,
since, in this case, all isospin breaking contributions
represent small corrections and we can just use the
experimental value of the nucleon masses. At zero momen-
tum transfer q
2
→ 0 this expression reduces to
g
P
¼
¯
M
N
¯
m
q
g
A
: (10)
Before discussing the phenomenological applications of
these relations, let us mention that an isospin-rotated
version of them has been discussed previously in the
context of electric dipole moments [33–35] and β decays
[36]. Notice that the derivation followed in this work does
not rely on the use of the isospin symmetry.
Numerical analysis.—The determination of the scalar
charge through the above-derived relation requires the
knowledge of the light-quark mass difference δm
q
, and
its contribution to the nucleon mass splitting δM
QCD
N
.
Interestingly enough, these quantities and the understand-
ing of the interplay between isospin breaking due to the
quark masses and electromagnetic effects, have become a
topic of very intensive research.
On one hand, the phenomenological determination
of the QED contributions to δM
N
¼ M
n
− M
p
using the
Cottingham’s sum rule [37,38] has been recently updated
finding δM
QED
N
¼ −1.30ð3Þð47Þ MeV [17], which com-
bined with the experimental value [30] implies the δM
QCD
N
value shown in Table I. Moreover, the error is expected to
be reduced in the future through measurements of the
isovector magnetic polarizability of the nucleon [17,39,40].
On the other hand, LQCD collaborations are starting to
implement isospin breaking effects [18–20] or even directly
simulating QED together with QCD [21–27] (for a recent
review see Ref. [42]). In addition to the determination by
the NPLQCD Collaboration in 2006 [18], we consider four
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new calculations reported in the last two years (see Table I
and Fig. 1). We do not include the results obtained in
quenched LQCD by Duncan et al. in their seminal work
[22], or the determination by Blum et al. [23] due to the
absence of an estimate of systematic errors.
The weighted average of these determinations, with their
respective errors combined in quadrature, is
½δM
QCD
N
av
¼ 2.58ð18Þ MeV (11)
with χ
2
=DOF ¼ 0.64. This average should be taken with
care, since it comes from pioneering calculations in a
rapidly developing field. The estimate of systematic errors
is a very complicated issue and future lattice studies of
isospin breaking effects are needed to confirm these first
calculations. Similar caveats apply to the recent numerical
evaluation of the Cottingham formula [17]. Nonetheless,
the average reflects the good agreement between current
determinations and we will take it as a reference number
whose robustness should improve in the future.
The other ingredient needed to calculate the scalar
charge is δm
q
. From the light quark masses results
by FLA G [41] and the PDG [30],weobtain
δm
q
¼ 2.52ð19Þ MeV and 2.55ð25Þ MeV respectiv ely.
Combining the FLAG result with the above-given average
for δM
QCD
N
, the CVC relation given in Eq. (8) yields
g
S
¼ 1.02ð8Þ
δm
q
ð7Þ
δM
N
¼ 1.02ð11Þ: (12)
It is worthwhile stressing that this result has been obtained
ignoring possible correlations between the numerator
and denominator of Eq. (8), and between the m
u
and m
d
determinations. These assumptions would be unnecessary
in a direct calculation of the ratio δM
QCD
N
=δm
q
, which
should be fairly simple to implement in future LQCD
analyses.
This determination of g
S
is significantly more precise
than direct LQCD calculations available in the literature.
The LHPC finds g
S
¼ 1.08ð32Þ [16], whereas the PNDME
Collaboration has recently published the result g
S
¼
0.66ð24Þ [15], which supersedes their original preliminary
estimate g
S
¼ 0.8ð4Þ [13].
Inversely, using again Eq. (8) these calculations provide
independent determinations of δM
QCD
N
, as shown in Table I
and Fig. 1. We see that these results are starting to have an
accuracy close to the direct calculations of δM
QCD
N
and that
the PNDME determination marginally disagrees with the
average in Eq. (11).
Likewise, the application of PCAC through Eq. (10)
yields the following result for the pseudoscalar charge:
g
P
¼ 349ð9Þ; (13)
where the error is entirely dominated by the error in
¯
m
q
¼ 3.42ð9Þ MeV, (N
f
¼ 2 þ 1 FLAG average [41]).
Notice the large enhancement experienced by this form
factor, due to a charged pion pole in the coupling of a
pseudoscalar field to the du vertex in QCD at low energies.
In fact, this result is equivalent to the Goldberger-Treiman
relation in which the pseudoscalar current serves as an
interpolator of the pion field and g
P
ðq
2
Þ is expressed as a
function with a pole at q
2
¼ M
2
π
, whose residue is defined
as the strong pion-nucleon coupling [8,9].
Implications for new-physics searches in β decays.—
Given the V-A structure of the weak interaction, the
(pseudo)scalar hadronic matrix elements of Eqs. (5)–(6)
do not appear in the SM description of β decays. However
the contribution due to new physics, like the coupling of
a heavy charged scalar to first generation fermions, would
require the calculation of these matrix elements. Such
nonstandard interactions can be described by a low-energy
effective Lagrangian for d → ueν transitions [43,44], where
the scalar and pseudoscalar interactions are described by
L
d→ueν
¼ L
SM
d→ueν
−
G
F
V
ud
ffiffiffi
2
p
½ϵ
S
¯
eð1 − γ
5
Þν
e
·
¯
ud
− ϵ
P
¯
eð1 − γ
5
Þν
e
·
¯
uγ
5
dþH:c: (14)
TABLE I. Summary of results for δM
QCD
N
, with uncertainties
shown as they were presented in the corresponding references.
We also show the results obtained using Eq. (8) (CVC) with the
FLAG value for δm
q
[41] and the g
S
calculations of LHPC and
PNDME [15,16].
Type Label δM
QCD
N
[MeV]
Pheno WLCM [17] 2.59 (03)(47)
LQCD NPLQCD [18] 2.26(57)(42) (10)
LQCD QCDSF-UKQCD [19] 3.13(15)(16)(53)
LQCD STY [20] 2.9(4)
LQCD RM123 (N
f
¼ 2) [25] 2.9(6)(2)
LQCD BMW [26] 2.28(25)(7)(9)
LQCD þ CVC LHPC [16] 2.72(83)
LQCD þ CVC PNDME [15] 1.66(62)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
(δM
N
)
QCD
[MeV]
WLCM’12
BMW’13
QCDSF-UKQCD’12
STY-PACS’12
RM123’13
NPLQCD’07
CVC-LHPC’12
CVC-PNDME’13
FIG. 1 (color online). Representation of the δM
QCD
N
results
summarized in Table I, along with our average (gray
shaded band).
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Here ν
e
, e, u, d denote the electron neutrino, electron,
and up- and down-quark mass eigenfields, whereas ϵ
S;P
are the Wilson coefficients generated by some unspecified
nonstandard dynamics. Moreover, and for the sake of
simplicity, we will assume in this work that the Wilson
coefficients ϵ
S;P
are real, corresponding to CP-conserving
interactions.
Scalar interaction—The most stringent limits on non-
standard scalar interactions arise from the contribution of
the Fierz interference term to the F t values of superallowed
pure Fermi transitions [45], namely. b
F
¼ −2g
S
ϵ
S
¼
−0.0022ð43Þ (at 90% C.L.). Alternative bounds on scalar
interactions can be obtained from the measurement of the
βν angular correlation a in pure Fermi transitions.
Although several on-going and planned experiments will
improve the current measurements of a, it seems unlikely
that they will be able to improve the above-given bound in
the near future [46]. On the other hand, the Fierz term in
neutron β decay is also sensitive to scalar interactions,
although the level of precision required to compete with the
bounds from nuclear decays looks also quite challenging, at
least for the current generation of experiments [13].
Given the experimental value of b
F
and the determi-
nation of the scalar charge derived in the previous section,
we can determine the current bound on ϵ
S
from β decays.
Following Refs. [13,44,46] we calculate the confidence
interval on ϵ
S
using the R-fit method [47], which treats
all values inside 0.91 ≤ g
S
≤ 1.13 [from Eq. (12)]onan
equal footing, whereas values outside the interval are not
permitted. Note that the bound on ϵ
S
depends only on the
lower limit of the scalar form factor, as long as b
F
is
compatible with zero at 1σ.
In this way we obtain the following limit on
CP-conserving scalar interactions
ϵ
S
¼ 0.0012ð23Þ; ð90% C:L:Þ; (15)
which, as it is shown in Fig. 2, improves significantly the
bound obtained in Ref. [13] using g
S
¼ 0.8ð4Þ. Figure 2
shows also the ϵ
S
bound that we obtain using more recent
LQCD calculations of g
S
[15,16]. It is worth mentioning
that if we abandoned the R-fit scheme and treated g
S
as a normally distributed variable we would obtain ϵ
S
¼
0.0011ð21Þ at 90% C.L., in good agreement with the
R-fit result of Eq. (15).
The LHC searches can also be used to set bounds on ϵ
S;P
.
This can be done in a model-independent way if the new
degrees of freedom that produce the effective scalar inter-
action in β decays are too heavy to be produced on shell at the
LHC, since in that case it is possible to study collider
observables using a high-energy SUð2Þ
L
× Uð1Þ
Y
-invariant
effective theory that can be connected to the low-energy
effective theory of Eq. (14).
In Fig. 2 we show the most stringent bound on ϵ
S
from
LHC searches, obtained in Ref. [46] studying the channel
pp → e þ MET þ X, where MET stands for missing
transverse energy. More specifically, a CMS search with
20 fb
−1
of data recorded at
ffiffiffi
s
p
¼ 8 TeV [48], was used to
obtain jϵ
S;P
j < 5.8 × 10
−3
at 90% C.L.
Pseudoscalar interaction—In the study of the effect of
nonstandard interactions in nuclear and neutron β decays,
it is common lore that the pseudoscalar terms can be
safely neglected in the analysis because the associated
hadronic bilinear
¯
u
p
γ
5
u
n
is of order q=
¯
M
N
, which repre-
sents a suppression of order ∼10
3
[49]. However, we
showed in the previous section how the application of
PCAC yields g
P
¼ 348ð11Þ, reducing considerably the
suppression from the pseudoscalar bilinear. This result
means that, modulo numerical factors of order 1, β decays
with a nonzero Gamow-Teller component are as sensitive to
pseudoscalar interactions as they are to scalar and tensor
couplings.
As a representative example we show here the leading
contribution of a nonzero pseudoscalar interaction to the
electron energy spectrum in the β decay of an unpolarized
neutron
dΓ
dE
e
¼
G
2
F
jV
ud
j
2
ð1 þ 3λ
2
Þ
2π
3
p
e
E
e
ðE
0
− E
e
Þ
2
ð1 þ δ
P
Þ; (16)
where E
e
and p
e
denote the electron energy and the modulus
of the three-momentum, E
0
¼ δM
N
− ðδM
2
N
− m
2
e
Þ=ð2M
n
Þ
is the electron endpoint energy, and m
e
is the electron mass.
(For subleading SM effects see Refs. [13,50]). The non-
standard contribution δ
P
coming from a nonzero effective
coupling ϵ
P
is given by
δ
P
¼ −
λ
1 þ 3λ
2
g
P
ϵ
P
E
0
− E
e
M
n
m
e
E
e
; (17)
where the factor ðE
0
− E
e
Þ=M
n
represents the above-
discussed suppression from the pseudoscalar bilinear.
For comparison we show now the (well-known) correction
stemming from a scalar coupling
-0.01 -0.005 0 0.005 0.01 0.015
ε
s
This work
PNDME’11
PNDME’13
LHPC’12
LHC’12
FIG. 2 (color online). Ninety percent C.L. bounds on ϵ
S
from the measurement of b
F
in superallowed pure Fermi
transitions [45], using different values for the scalar charge
g
S
[13,15,16]. We show also the bound obtained from
the analysis of LHC data carried out in Ref. [46].
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![FIG. 2 (color online). Ninety percent C.L. bounds on ϵS from the measurement of bF in superallowed pure Fermi transitions [45], using different values for the scalar charge gS [13,15,16]. We show also the bound obtained from the analysis of LHC data carried out in Ref. [46].](/figures/fig-2-color-online-ninety-percent-c-l-bounds-on-s-from-the-1dj6xeyx.webp)
![TABLE I. Summary of results for δMQCDN , with uncertainties shown as they were presented in the corresponding references. We also show the results obtained using Eq. (8) (CVC) with the FLAG value for δmq [41] and the gS calculations of LHPC and PNDME [15,16].](/figures/table-i-summary-of-results-for-dmqcdn-with-uncertainties-1rifefxp.webp)
