Neutrinoless double beta decay and chiral $SU(3)$

0.4 MB·

by  V. Cirigliano

LA-UR-17-20043
Neutrinoless double beta decay and chiral SU(3)
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V. Ciriglianoa, W. Dekensa,b, M. Graessera, and E. Mereghettia
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a Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
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] b New Mexico Consortium, Los Alamos Research Park, Los Alamos, NM 87544, USA
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1 Abstract
v
TeV-scale lepton number violation can affect neutrinoless double beta decay through
3
4 dimension-9 ∆L = ∆I = 2 operators involving two electrons and four quarks. Since the
4 dominant effects within a nucleus are expected to arise from pion exchange, the π− →π+ee
1 matrix elements of the dimension-9 operators are a key hadronic input. In this letter we
0
provide estimates for the π− →π+ matrix elements of all Lorentz scalar ∆I =2 four-quark
.
1 operators relevant to the study of TeV-scale lepton number violation. The analysis is based
0 on chiral SU(3) symmetry, which relates the π− → π+ matrix elements of the ∆I = 2
7 operators to the K0 → K¯0 and K → ππ matrix elements of their ∆S = 2 and ∆S = 1
1
: chiral partners, for which lattice QCD input is available. The inclusion of next-to-leading
v
orderchiralloopcorrectionstoallsymmetryrelationsusedintheanalysismakesourresults
i
X robust at the 30% level or better, depending on the operator.
r
a
Introduction – Neutrinoless double beta decay (0νββ) is a rare nuclear process in which
two neutrons inside a nucleus convert into two protons with emission of two electrons and no
neutrinos,thuschangingthenumberofleptonsbytwounits. Sinceleptonnumberisconservedin
the Standard Model (SM) at the classical level, observation of 0νββ would be direct evidence of
new physics, with far reaching implications: it would demonstrate that neutrinos are Majorana
fermions[1], shedlightonthemechanismofneutrinomassgeneration, andprobeleptonnumber
violation (LNV), a key ingredient needed to generate the matter-antimatter asymmetry in the
universe via “leptogenesis” [2]. The current experimental limits on the half-lives are already
impressive [3–9], at the level of T > 2.1×1025 y for 76Ge [6] and T > 1.07×1026 y for
1/2 1/2
136Xe [3], with next generation ton-scale experiments aiming at a sensitivity of T ∼ 1027−28 y.
1/2
By itself, the observation of 0νββ would not immediately point to the underlying physical
originofLNV.While0νββ searchesarecommonlyinterpretedintermsoftheexchangeofalight
Majorana neutrino, other new physics mechanisms deserve careful evaluation. In an effective
theory approach to new physics, the light Majorana neutrino exchange dominates whenever the
scale of lepton number violation, Λ , is very high compared to the electroweak scale: as long
LNV
as Λ (cid:29) TeV, the only low-energy manifestation of this new physics is a Majorana mass for
LNV
lightneutrinos,encodedinasinglegauge-invariantdimension-5operator[10]. However,asΛ
LNV
is lowered, new contributions to 0νββ are possible, which typically involve the exchange of new
TeV-mass Majorana fermions, for example R-handed neutrinos in left-right symmetric models
or neutralinos in certain supersymmetric models (for recent reviews see Refs. [11–14]). At low
energy, the effects of this TeV scale LNV dynamics can be encoded in a set of local dimension-9
operators (involving two leptons and four quarks) that change lepton number by two units. The
operators have been classified both according to SU(3) ×U(1) gauge invariance [15,16],
C EM
directly relevant at low-energy, and SU(3) ×SU(2) ×U(1) gauge invariance [17], important
C W Y
to connect 0νββ to possible LNV signals at the Large Hadron Collider.
To interpret positive or null 0νββ results in the context of TeV-scale LNV dynamics, it is
essential to quantify the hadronic and nuclear matrix elements involving the ∆L = 2 dimension-
9operators. Thisisconvenientlytackledbyfirstmatchingthedimension-9quarkleveloperators
onto appropriate operators at the pion-nucleon level, and subsequently computing the nuclear
matrix elements. As illustrated in Fig. 1, the dimension-9 operators induce a variety of effective
vertices at the pion-nucleon level. The use of chiral power counting has led to the identification
of the two-pion exchange in Fig. 1 as the dominant contribution [16,18]. It is therefore very
importanttoestimateasaccuratelyaspossibletheπ−π− → eematrixelementsofthedimension-
9operators. Currentknowledgeofthesematrixelementsisbasedonvacuumsaturationornaive
dimensional analysis, with the exception of two operators for which chiral symmetry was used
to relate the two-pion matrix elements to the K± → π±π0 amplitude [19].
In this paper we generalize the chiral symmetry analysis to all Lorentz scalar ∆I = 2 four-
quark operators, O defined in Eq. (2) below. We provide estimates for the π− → π+
1,...,5
matrix elements of O by relating them to the K0 → K¯0 matrix elements of their ∆S = 2
2,...,5
chiral partners, which have been computed by several lattice QCD groups [20–24]. By including
the leading chiral loop corrections, we are able to estimate the uncertainty on the symmetry
relations, finding that it does not exceed 30%.
Operator basis and chiral transformation properties – At the hadronic scale, short-
distance contributions to 0νββ can be parameterized by a number of dimension-9 operators [16,
1
Figure 1: Feynman diagrams representing the insertion of dimension-9 operators of Eq. (1) –
denoted by a black square – at the hadronic level. In this paper we study the π− → π+ee vertex
appearing in the leftmost diagram, which is enhanced in the chiral power counting.
17,25]
(cid:34) (cid:35)
L = 1 (cid:88) (cid:0)c e¯ec+c(cid:48) e¯γ ec(cid:1) O + e¯γ γ ec (cid:88) c Oµ , (1)
eff Λ5 i,S i,S 5 i µ 5 i,V i
LNV i=scalar i=vector
where O and Oµ denote scalar and vector four-quark operators, respectively. In this letter we
i j
focus on the scalar operators and in order to discuss their properties under the chiral SU(3) ×
L
SU(3) group we chose to work in the following basis 1
R
O = q¯αγ τ+qα q¯βγµτ+qβ (2a)
1 L µ L L L
O = q¯ατ+qα q¯βτ+qβ (2b)
2 R L R L
O = q¯ατ+qβ q¯βτ+qα (2c)
3 R L R L
O = q¯αγ τ+qα q¯βγµτ+qβ (2d)
4 L µ L R R
O = q¯αγ τ+qβ q¯βγµτ+qα , (2e)
5 L µ L R R
where qT = (u,d,s), q = (1/2)(1∓γ )q, α, β denote color indices, and τ+ = T1 +iT2 in
L,R 5
terms of the SU(3) generators Ta. Three additional operators O(cid:48) are obtained from O
1,2,3 1,2,3
by the interchange L ↔ R everywhere. Parity invariance of QCD implies (cid:104)π+|O(cid:48) |π−(cid:105) =
1,2,3
(cid:104)π+|O |π−(cid:105).
1,2,3
TheoperatorsO belongtoirreduciblerepresentationsofthechiralsymmetrygroupSU(3) ×
i L
SU(3) (q → U q with U ∈ SU(3) ). O transforms as 27 ×1 , O transform
R L,R L,R L,R L,R L,R 1 L R 2,3
as 6 ׯ6 , and finally O transform as 8 ×8 . The transformation properties of O were
L R 4,5 L R 1
exploited in Ref. [19] to relate the matrix element of (cid:104)π+|O |π−(cid:105) to the ∆I = 3/2 K+ → π−π0
1
amplitude. Here we exploit the transformation properties of O to relate their two-pion
2,3,4,5
matrix elements to the matrix elements of their chiral partners between a K0 and a K¯0 meson,
which have been computed with lattice QCD by several groups [20–24]. Strictly speaking the
symmetry relation is valid only to leading order in the chiral expansion, and is expected to
receive O(30%) corrections. To make our analysis more robust, we also estimate the size of
next-to-leading order (NLO) quark-mass corrections by computing the leading chiral loops.
1 ThisisconsistentwiththebasesusedinRefs.[26]and[27]forthe∆S =2effectiveHamiltonianbeyondthe
Standard Model. Compared to the basis presented in Ref. [17], we are able to eliminate the operator involving
tensor densities σ ⊗σµν.
µν
2
Figure2: FeynmandiagramscontributingtotheNLOcorrectionstoMππ andMKK¯. Solidlines
represent π,K,η. The black squares represent an insertion of the lowest-order chiral Lagrangian
(see Eq. (3)), while the open square represents an insertion from the NLO effective Lagrangian.
Determination of (cid:104)π+|O |π−(cid:105) – The argument proceeds as follows. O and O
2,3,4,5 2,3 4,5
can be written as linear combinations of operators transforming according to the 6 ׯ6 and
L R
8 ×8 representationsofthechiralgroup, respectively. Theseoperatorsinturnadmitaunique
L R
hadronic realization to leading order in the chiral expansion
F4 (cid:16) (cid:17)
Oa,b = q¯ Taq q¯ Tbq → g 0 Tr TaUTbU (3a)
6ׯ6 R L R L 6ׯ6 4
F4 (cid:16) (cid:17)
Oa,b = q¯ Taγ q q¯ Tbγµq → g 0 Tr TaUTbU† , (3b)
8 L µ L R R 8×8 4
wherethetraceisoverflavorandU istheusualmatrixofpseudo-Nambu-Goldstonebosonfields
transforming as U → U U U† under SU(3) ×SU(3) ,
L R L R
 
(cid:32)√ (cid:33) √π32 + √π86 π+ K+
2iπ  
U = exp F0 , π =  π− −√π32 + √π86 K0  , (4)
K− K¯0 −√2 π8
6
and F is the pseudoscalar decay constant in the chiral limit (in our normalization F (cid:39)
0 π
92.4 MeV). The non-perturbative dynamics is encoded in the low-energy constants g , g ,
6ׯ6 8×8
andforeachrepresentationtherearetwoindependentconstants,correspondingtodifferentcolor
contractions (e.g. O and O ).
2 3
The operators in Eqs. (2) are obtained by setting Ta = Tb → T1 + iT2 in Eq. (3). The
same representations, however, contain ∆S = 2 operators that contribute to K0-K¯0 mixing in
extensions of the Standard Model (Ta = Tb → T6−iT7 in Eq. (3)). The relevant K0-K¯0 matrix
elements have been computed in lattice QCD, thus providing the couplings g and g to
6ׯ6 8×8
leading order in the chiral SU(3) expansion. Note that the g couplings can be independently
8×8
extracted through their contributions to K0 → (ππ) amplitudes via the ∆S = 1 electroweak
I=2
penguin operators, that transform as 8 ×8 .
L R
We first focus on the relation to K0-K¯0 mixing. A straightforward calculation based on the
leading chiral realization of Eq. (3) leads to:
Mππ ≡ (cid:104)π+|O1+i2,1+i2|π−(cid:105) = (cid:104)K¯0|O6−i7,6−i7|K0(cid:105) ≡ MKK¯ (5a)
6ׯ6 6ׯ6 6ׯ6 6ׯ6
Mππ ≡ (cid:104)π+|O1+i2,1+i2|π−(cid:105) = (cid:104)K¯0|O6−i7,6−i7|K0(cid:105) ≡ MKK¯ . (5b)
8×8 8×8 8×8 8×8
3
NLO chiral corrections arising from the one-loop diagrams of Fig. 2 and O(p2) counterterms
could alter the above relation (see for example Refs. [28] and [29] for the analogous discussion
of K0-K¯0 mixing and K± → π±π0 amplitudes in the Standard Model). For the relations of
interest here, we find to NLO at zero momentum transfer (defining L ≡ logµ2/m2 )
π,K,η χ π,K,η
(cid:26) 1 (cid:18) m2 3 (cid:19)(cid:27)
MKK¯ = g F2 1+ m2 (−1+2L )− πL − m2L +δKK¯ (6)
8×8 8×8 K (4πF )2 K K 4 π 4 η η 8×8
0
(cid:26) (cid:18) (cid:19)(cid:27)
1
Mππ = g F2 1+ m2 (−1+L )+δππ . (7)
8×8 8×8 π (4πF )2 π π 8×8
0
To identify the finite parts of the loops we have followed the modified MS scheme commonly
used chiral perturbation theory [30,31]. Moreover, δKK¯ and δππ denote linear combinations
8×8 8×8
of O(p2) counterterms, that reabsorb the µ dependence of L and contain additional finite
χ π,K,η
corrections. Using the NLO effective Lagrangian [32], we find
(cid:18) (cid:19) (cid:18) (cid:19)
1 1
δKK¯ = a m2 +b m2 + m2 δππ = a m2 +b m2 + m2 , (8)
8×8 8×8 K 8×8 K 2 π 8×8 8×8 π 8×8 K 2 π
with a and b dimensionless constants. Similarly, for the 6 ׯ6 . representation we find
8×8 8×8 L R
(cid:26) 1 (cid:18) m2 7 (cid:19)(cid:27)
MKK¯ =−g F2 1+ m2 (−1+2L )+ πL − m2L +δKK¯ (9)
6ׯ6 6ׯ6 K (4πF )2 K K 4 π 12 η η 6ׯ6
0
(cid:26) (cid:18) (cid:19)(cid:27)
1 2
Mππ =−g F2 1+ m2(−1+L )+ m2L +δππ , (10)
6ׯ6 6ׯ6 π (4πF )2 π π 3 η η 6ׯ6
0
with counterterm contributions analogous to the ones in Eq. (8). The loop corrections to
MKK¯ have been calculated in Ref. [33] and we agree with them. Eqs. (6)-(10) lead to
8×8,6ׯ6
the central result of our work, namely a relation between Mππ and MKK¯ valid to NLO in the
chiral expansion
F2
Mππ = MKK¯ × π ×(1+∆ ) = MKK¯ ×R (11a)
8×8 8×8 F2 8×8 8×8 8×8
K
Mππ = MKK¯ × Fπ2 ×(cid:0)1+∆ (cid:1) = MKK¯ ×R , (11b)
6ׯ6 6ׯ6 F2 6ׯ6 6ׯ6 6ׯ6
K
with
∆ = 1 (cid:20)m2π(−4+5L )−m2 (−1+2L )+ 3m2L −a (cid:0)m2 −m2(cid:1)(cid:21) (12a)
8×8 (4πF )2 4 π K K 4 η η 8×8 K π
0
∆ = 1 (cid:20)−m2π(4−3L )−m2 (−1+2L )+ 5m2L −a (cid:0)m2 −m2(cid:1)(cid:21).(12b)
6ׯ6 (4πF )2 4 π K K 4 η η 6ׯ6 K π
0
Eq. (8) implies that the low-energy constants a could be extracted from lattice QCD
8×8,6ׯ6
calculations of K0-K¯0 mixing at different values of both m = m , and m . Moreover, at
u d s
NLO one can derive counterterm-free relations that connect Mππ, MKK¯, and K → ππ matrix
elements (MK→ππ). For the 8 ×8 case, using the NLO Lagrangian of Ref. [32], we obtain
L R
(2m2 −m2 )F2Mππ = m2 F2MKK¯
π K π 8×8 K K 8×8
4
4iF2F (cid:20) (cid:21)
+ √π K 2(m2 −m2 )MK→π+π−−(m2 +2m2)MK→π0π0 +∆loop , (13)
π K 8×8 K π 8×8
2
where ∆loop is a calculable loop correction. In practice, at the moment neither of these two
approachesisfeasible, duetomissingornotsufficientlypreciselatticeinput. Soinourestimates
we adopt the following strategy: to obtain central values for ∆ , we evaluate the chiral
8×8,6ׯ6
loops at the scale µ = m and set the counterterms to zero. We then assess the counterterm
χ ρ
uncertainty in two ways: first, assuming naive dimensional analysis (NDA), namely |a | ∼
8×8,6ׯ6
O(1), we find ∆ = 0.02(20) and ∆ = 0.07(20). Second, requiring that the counterterms
8×8 6ׯ6
(ct) (loops)
be of comparable size to their beta-functions, namely ∆ = ±|d∆ /d(logµ )| (n = 8×8
n n χ
or 6 × ¯6), we find ∆ = 0.02(36) and ∆ = 0.07(16). To account for the strong scale
8×8 6ׯ6
(ct)
dependence of loops in ∆ , we enlarge the NDA estimate to ∆ = ±0.3 and use in the
8×8 8×8
subsequent analysis ∆ = 0.02(30) and ∆ = 0.07(20). The above results point to the fact
8×8 6ׯ6
that the dominant SU(3) × SU(3) correction to the relations (5) is captured by the ratio
L R
(F /F )2 in (11). Putting together the effect of chiral loops and F /F = 1.19 [34], the total
π K K π
chiral corrections in (11) amount to R = 0.72(21) (∼ 30% uncertainty) and R = 0.76(14)
8×8 6ׯ6
(∼ 20% uncertainty). Given that the chiral expansion is well behaved for these quantities, we
expect residual higher order corrections not to exceed 10%, well within the assigned ranges.
Using Eqs. (11) and the matrix elements of the ∆S = 2 operators calculated in Refs. [20–24]
we find for the two-pion matrix elements renormalized in the MS scheme at the scale µ = 3 GeV
5 2F2 m4
(cid:104)π+|O |π−(cid:105) = − B K ×R K = K K (14a)
2 12 2 6ׯ6 (m +m )2
d s
1
(cid:104)π+|O |π−(cid:105) = B K ×R (14b)
3 12 3 6ׯ6
1
(cid:104)π+|O |π−(cid:105) = − B K ×R (14c)
4 5 8×8
3
(cid:104)π+|O |π−(cid:105) = −B K ×R , (14d)
5 4 8×8
where the dimensionless scale- and scheme-dependent B are reviewed in [34]. 2 To obtain
2,3,4,5
the central value estimates for the matrix elements we use F = 110 MeV, m (3GeV) =
K d
4.3 MeV, m (3GeV) = 87.5 MeV [35]. For the B we take the midpoint of a conservative range
s i
encompassing the maximum and minimum values of the N = 2+1 [22], and N = 2+1+1 [20]
f f
results summarized in Ref. [34]. The uncertainty associated with this treatment of the B is at
i
the level of 10% for B , 20% for B , and 30% for B . For the matrix elements of O the
2,3 4 5 2,3
uncertainty due to the quark masses is non negligible, but subdominant compared to the effect
ofNLOchiralcorrections. Wesummarizeourcurrentbestestimatesforthematrixelementsand
their uncertainties in Table 1. The fractional uncertainty is at the 20% level for (cid:104)π+|O |π−(cid:105)
2,3
(dominated by chiral corrections), at the 35% for (cid:104)π+|O |π−(cid:105) (dominated by chiral corrections),
5
andatthe40%levelfor(cid:104)π+|O |π−(cid:105)(equallysharedbychiralcorrectionandlatticeQCDinput).
4
The effective coupling g can also be extracted from the electroweak penguin matrix ele-
8×8
ments (cid:104)(ππ) |Q |K0(cid:105) [36,37]. This extraction was recently updated in Ref. [38] to LO in the
I=2 7,8
(i)
chiral expansion (in [38] the notation g → −A was used). Using the value of g from
8×8 iLR 8×8
Ref. [38] in Eq. (7) and neglecting chiral corrections leads to (cid:104)π+|O |π−(cid:105) = −1.9×10−2 GeV4
4
2Our operators O are related to the Q of Ref. [34] as follows: O =(1/4)Q , O =−(1/2)Q .
i i 1,2,3 1,2,3 4,5 5,4
5
(cid:104)π+|O |π−(cid:105) = (1.0±0.1±0.2)×10−4 GeV4
1
(cid:104)π+|O |π−(cid:105) = −(2.7±0.3±0.5)×10−2 GeV4
2
(cid:104)π+|O |π−(cid:105) = (0.9±0.1±0.2)×10−2 GeV4
3
(cid:104)π+|O |π−(cid:105) = −(2.6±0.8±0.8)×10−2 GeV4
4
(cid:104)π+|O |π−(cid:105) = −(11±2±3)×10−2 GeV4
5
Table 1: Pionic matrix elements of the operators in Eqs. (2) in the MS scheme at the scale µ=3 GeV.
The first uncertainty refers to the lattice QCD input on kaon matrix elements. The second uncertainty
is associated to the size of partially known NLO chiral corrections (only loops are taken into account)
and possible higher order effects. See text for discussion.
and (cid:104)π+|O |π−(cid:105) = −8.5×10−2 GeV4, in reasonable agreement with the estimate of these ma-
5
trix elements based on K0-K¯0 mixing given in Eq. (14) and Table 1. NLO chiral effects in
K0 → (ππ) change the extracted low-energy constant as follows, g → g /(1+∆ ), with
I=2 8×8 8×8 2
∆ = −0.30±0.20 [39], where the central value stems from chiral loop and known countert-
2
erms, while the error encompasses an estimate of the unknown counterterms. Taking this into
account and keeping the chiral logs in Eq. (7) leads to (cid:104)π+|O |π−(cid:105) = −2.7×10−2 GeV4 and
4
(cid:104)π+|O |π−(cid:105) = −12.7×10−2 GeV4, in excellent agreement with the results of Table 1.
5
Determination of (cid:104)π+|O |π−(cid:105) – For completeness, we also update the analysis of Ref. [19].
1
First, note that O belongs to the 27 ×1 representation of SU(3) ×SU(3) , along with
1 L R L R
O = s¯γ (1−γ )ds¯γµ(1−γ )d and the component Q(27×1) of the ∆S = 1 operator Q =
∆S=2 µ 5 5 2 2
s¯γ (1−γ )uu¯γµ(1−γ )d = Q(27×1)+Q(8×1). Using the normalization conventions of Ref. [31],
µ 5 5 2 2
the leading order chiral realization of these operators is 3
(cid:18) (cid:19)
2
Q(27×1) → g F4 L Lµ + L Lµ (15a)
2 27×1 0 µ32 11 3 µ31 12
5
O → g F4 L Lµ (15b)
∆S=2 3 27×1 0 µ32 32
5
4O → g F4 L Lµ , (15c)
1 3 27×1 0 µ12 12
with Lµ = i(U†∂µU) . The factor of 4 multiplying O in (15) accounts for the different
ij ij 1
normalization of O compared to Q and O . In principle, Eqs. (15) allow one to relate
1 2 ∆S=2
(cid:104)π+|O |π−(cid:105) to both K0-K¯0 mixing (as done for (cid:104)π+|O |π−(cid:105)) and (cid:104)π+π0|Q |K+(cid:105). However,
1 2,...,5 2
it turns out that the determination in terms of K0-K¯0 mixing suffers from potentially large
chiral corrections. The NLO analysis leads to
1 m2F2
(cid:104)π+|O |π−(cid:105) = π π (cid:104)K¯0|O |K0(cid:105) (1+∆ ) (16)
1 4m2 F2 ∆S=2 27×1
K K
(loops)
with a strongly scale-dependent one-loop correction given by ∆ = {+0.24,−0.11,−0.39}
27×1
at µ = {m ,m ,1 GeV}, respectively. This is not unexpected, as large chiral corrections to
χ η ρ
3 Explicitly the projection reads Q(27×1) = 2/5 [s¯γ (1−γ )uu¯γµ(1−γ )d+s¯γ (1−γ )du¯γµ(1−γ )u)]−
2 µ 5 5 µ 5 5
1/5 [s¯γ (1−γ )dd¯γµ(1−γ )d+s¯γ (1−γ )ds¯γµ(1−γ )s].
µ 5 5 µ 5 5
6
(cid:104)K¯0|O |K0(cid:105)werealreadyfoundinRefs.[29]and[33]. TakingforB themidpointofarange
∆S=2 K
thatincludestheN = 2+1andN = 2+1+1latticeresults[34],namelyB (3GeV) = 0.52(3),
f f K
from (16) at µ = m we obtain (cid:104)π+|O |π−(cid:105) = 0.97×10−4 GeV4. The uncertainty from chiral
χ ρ 1
corrections is at least 40-50%, given the strong scale dependence of the one-loop effects. In light
of this, we focus next on the relation of (cid:104)π+|O |π−(cid:105) to K+ → π+π0 [19].
1
At one loop in chiral perturbation theory we find
5 (cid:26) m2 (cid:27)
(cid:104)π+|O |π−(cid:105) = g m2F2 1+ π (−1+3L )+δππ . (17)
1 3 27×1 π π (4πF )2 π 27×1
0
(cid:104)π+π0|iQ |K+(cid:105) = 5 g F (cid:0)m2 −m2(cid:1) (cid:110)1+∆K+π+π0(cid:111). (18)
2 3 27×1 π K π 27
where δππ denotes a linear combination of counterterms. ∆K+π+π0 receives both loop and
27×1 27
counterterms contributions: the loops have been computed in Ref [31] and they are small while
displaying a very mild scale dependence. The counterterms have been estimated in the large-
N approximation and are negligible [31]. Overall, one finds |1+∆K+π+π0| = 0.98±0.05 [31].
C 27
Comparing Eq. (18) to the lattice QCD results for (cid:104)π+π0|Q |K+(cid:105) [36,37] (in the MS scheme
2
at µ = 3 GeV) we obtain g = 0.34(3) (2) , where the first error is from the lattice
27×1 LQCD χ
input and the second from the chiral corrections in Eq. (18) 4. Using this value in Eq. (17),
and assigning a conservative 20% error due to the unknown counterterms in δππ , we obtain
27×1
the result reported in Table 1. Finally, note that the determination of (cid:104)π+|O |π−(cid:105) from K0-K¯0
1
mixing, though plagued by larger uncertainty, is quite consistent with the result of Table 1.
Discussion and conclusion – In this letter we have provided estimates for the π− → π+
matrix elements of all Lorentz scalar ∆I = 2 four-quark operators relevant to the study of
TeV-scale lepton number violation. The analysis is based on (i) chiral SU(3) symmetry, which
relates the π− → π+ matrix elements of O defined in Eq. (2) to the K0 → K¯0 and K → ππ
1,...,5
matrix elements of their ∆S = 2 and ∆S = 1 chiral partners; (ii) lattice QCD input for the
relevant kaon matrix elements. Our main results are summarized in Eqs. (14) and Table 1.
A preliminary lattice QCD calculation of the matrix elements considered in this letter has
appeared in Ref [40] 5. A complete comparison is not yet possible because Ref. [40] presents
results for bare matrix elements. Nonetheless, already at this level, we find the hierarchy of bare
matrix elements in [40] to be in qualitative agreement with our results.
For all the symmetry relations used here, we have included the NLO chiral loop corrections,
showing that the chiral expansion is well behaved and the relations are robust at the 20-30%
level, depending on the operator under consideration. The remaining uncertainty can be fur-
ther reduced as the precision on K0-K¯0 matrix elements improves. Our results provide a first
controlled estimate of the hadronic matrix elements needed to assess the sensitivity of 0νββ
to TeV-scale sources of lepton number violation, and can be used as input in nuclear structure
calculations of the leading pion-exchange operators [14].
4This result is in good agreement with g (cid:39) 0.29 found by a fit to the K → ππ decay rates [31]. The
27
identification g = g , however, neglects mixing of Q with other operators and its scale dependence. Our
27 27×1 2
result is also in good agreement with Ref [19], once we take into account that the coupling g(27) of Ref. [19] is
related to our g by g(27) =(5/12)g .
27×1 27×1
5There is a slight difference between the operators used here (O ) used here and the ones used in Ref. [40]
j
(O++). Using parity-invariance of QCD, for the π− → π+ matrix elements we have the following relations:
i+
(cid:104)O++(cid:105)=(cid:104)O (cid:105), (cid:104)O(cid:48)++(cid:105)=(cid:104)O (cid:105), (cid:104)O++(cid:105)=2 (cid:104)O (cid:105), (cid:104)O(cid:48)++(cid:105)=2 (cid:104)O (cid:105), (cid:104)O++(cid:105)=2 (cid:104)O (cid:105).
1+ 4 1+ 5 2+ 2 2+ 3 3+ 1
7
Acknowledgements
VC, MG and EM acknowledge support by the LDRD program at Los Alamos National Labo-
ratory. WD acknowledges support by the Dutch Organization for Scientific Research (NWO)
through a RUBICON grant. We thank Brian Tiburzi for discussions on the operator basis.
References
[1] J. Schechter and J. W. F. Valle, Phys. Rev. D25, 2951 (1982).
[2] S. Davidson, E. Nardi, and Y. Nir, Phys. Rept. 466, 105 (2008), 0802.2962.
[3] KamLAND-Zen, A. Gando et al., Phys. Rev. Lett. 117, 082503 (2016), 1605.02889, [Ad-
dendum: Phys. Rev. Lett.117,no.10,109903(2016)].
[4] CUORE, K. Alfonso et al., Phys. Rev. Lett. 115, 102502 (2015), 1504.02454.
[5] EXO-200, J. B. Albert et al., Nature 510, 229 (2014), 1402.6956.
[6] GERDA, M. Agostini et al., Phys. Rev. Lett. 111, 122503 (2013), 1307.4720.
[7] KamLAND-Zen, A. Gando et al., Phys. Rev. Lett. 110, 062502 (2013), 1211.3863.
[8] S. R. Elliott et al., Initial Results from the MAJORANA DEMONSTRATOR, 2016,
1610.01210.
[9] SNO+, S. Andringa et al., Adv. High Energy Phys. 2016, 6194250 (2016), 1508.05759.
[10] S. Weinberg, Phys. Rev. Lett. 43, 1566 (1979).
[11] W. Rodejohann, Int. J. Mod. Phys. E20, 1833 (2011), 1106.1334.
[12] A. de Gouvˆea and P. Vogel, Prog. Part. Nucl. Phys. 71, 75 (2013), 1303.4097.
[13] S. Dell’Oro, S. Marcocci, M. Viel, and F. Vissani, Adv. High Energy Phys. 2016, 2162659
(2016), 1601.07512.
[14] J. Engel and J. Men´endez, (2016), 1610.06548.
[15] H. P¨as, M. Hirsch, H. V. Klapdor-Kleingrothaus, and S. G. Kovalenko, Phys. Lett. B498,
35 (2001), hep-ph/0008182.
[16] G. Pr´ezeau, M. Ramsey-Musolf, and P. Vogel, Phys. Rev. D68, 034016 (2003), hep-
ph/0303205.
[17] M. L. Graesser, (2016), 1606.04549.
[18] A. Faessler, S. Kovalenko, F. Simkovic, and J. Schwieger, Phys. Rev. Lett. 78, 183 (1997),
hep-ph/9612357.
[19] M. J. Savage, Phys. Rev. C59, 2293 (1999), nucl-th/9811087.
[20] ETM, N. Carrasco et al., Phys. Rev. D92, 034516 (2015), 1505.06639.
8
[21] ETM, V. Bertone et al., JHEP 03, 089 (2013), 1207.1287, [Erratum: JHEP07,143(2013)].
[22] SWME, B. J. Choi et al., Phys. Rev. D93, 014511 (2016), 1509.00592.
[23] RBC, UKQCD, P. A. Boyle, N. Garron, and R. J. Hudspith, Phys. Rev. D86, 054028
(2012), 1206.5737.
[24] RBC/UKQCD, N. Garron, R. J. Hudspith, and A. T. Lytle, JHEP 11, 001 (2016),
1609.03334.
[25] M. Gonz´alez, M. Hirsch, and S. G. Kovalenko, Phys. Rev. D93, 013017 (2016), 1511.03945.
[26] F. Gabbiani, E. Gabrielli, A. Masiero, and L. Silvestrini, Nucl. Phys. B477, 321 (1996),
hep-ph/9604387.
[27] A. J. Buras, M. Misiak, and J. Urban, Nucl. Phys. B586, 397 (2000), hep-ph/0005183.
[28] J. F. Donoghue, E. Golowich, and B. R. Holstein, Phys. Lett. B119, 412 (1982).
[29] J. Bijnens, H. Sonoda, and M. B. Wise, Phys. Rev. Lett. 53, 2367 (1984).
[30] J. Gasser and H. Leutwyler, Nucl. Phys. B250, 465 (1985).
[31] V. Cirigliano, G. Ecker, H. Neufeld, and A. Pich, Eur. Phys. J. C33, 369 (2004), hep-
ph/0310351.
[32] V. Cirigliano and E. Golowich, Phys. Lett. B475, 351 (2000), hep-ph/9912513.
[33] D. Be´cirevi´c and G. Villadoro, Phys. Rev. D70, 094036 (2004), hep-lat/0408029.
[34] S. Aoki et al., (2016), 1607.00299.
[35] Particle Data Group, K. A. Olive et al., Chin. Phys. C38, 090001 (2014).
[36] T. Blum et al., Phys. Rev. D 86, 074513 (2012), 1206.5142.
[37] T. Blum et al., Phys. Rev. D 91, 074502 (2015), 1502.00263.
[38] V. Cirigliano, W. Dekens, J. de Vries, and E. Mereghetti, (2016), 1612.03914.
[39] V. Cirigliano and E. Golowich, Phys. Rev. D65, 054014 (2002), hep-ph/0109265.
[40] A. Nicholson et al., Neutrinoless double beta decay from lattice QCD, in Proceedings, 34th
International Symposium on Lattice Field Theory (Lattice 2016): Southampton, UK, July
24-30, 2016, 2016, 1608.04793.
9