Ⅱ-5. Hadron Physics (Theory)

RIKEN Accel. Prog. Rep. 52 (2019)

Proton decay matrix elements at physical quark mass
Y. Aoki,∗1,∗2 (current:∗3) Y. Kuramashi,∗4 and E. Shintani∗3

∗1
∗2
∗3
∗4

np=(1,1,1)

0.05

ts=18
ts=20

0.05

0.04

0.04

0.03

0.03

0.03

0.02

0.02

0.02

0.01

0.01

0.01

np=(0,1,2)

LL

0.04

np=(0,0,2)

0.05

W0

Proton decay is a smoking-gun signal of the physics
beyond the standard model (BSM). Grand uniﬁed theory (GUT) is the most natural origin of such an event
if observed. Despite no clear signal of the supersymmetry or any BSM phenomena at LHC, the idea ofunifying the known fundamental interactions is still
attractive. Estimate of the QCD contribution of theproton decay matrix element is needed to test GUTs
against the proton lifetime bound obtained in the experiment. Also a reliable estimate of the matrix elements is desirable for planning the future generation
proton decay detectors.
The proton decay matrix elements are obtained bynumerical computation using lattice QCD. So far, the2 + 1 ﬂavor computations have provided the matrix elements with extrapolation to the physical ud quark
mass from the results at unphysically large masses.
This procedure yields one of the largest systematic
uncertainties. Settling this systematics is important1)
and possible using current lattice gauge ﬁeld ensembles
generated at the physical point.
We use gauge ﬁeld conﬁgurations of 2 + 1 ﬂavor QCD generated with non-perturbatively O(a)improved Wilson fermions by the PACS collaboration.2) As pointed in the previous works (see e.g.3) ),computations using the three-point functions aremandatory to obtain the matrix elements of a proton decaying into a pseudoscalar (and an anti lepton).
Last year we reported on the optimization of the parameters of smearing function of the quarks for interpolating operators of proton and mesons. Using this
we are extending the computation for the three-pointfunctions. Figure 1 shows the ratio of the three andtwo-point functions for proton decay form factor W0for p → π 0 via LL operator (ϵijk (uTi CPL dj )PL uk ), forthree diﬀerent momentum values p⃗ = 2π/64∗ n⃗p . Fromthe plateau we will obtain the form factor that we areafter. Comparison of the two diﬀerent source-sink separation ts is performed, and they show consistent results in the middle. In this study we use 644 lattice atthe physical point mass. Last year we present a resultof low energy constant α computed on 644 and 964 volumes. As the results from two volumes are consistent,
we expect 644 is also good for the form factors.
We need several further steps to obtain the from factors in the physical unit (GeV2 ) and renormalized ina convenient renormalization scheme for phenomenological use. For the renormalization one needs to solve
the operator mixing due to an explicit chiral symmetry

00510t15 2000510t

15 2000510t

15 20Fig. 1. Time dependence of the ratio of three and twopoint functions for proton decay form factor W0 for
p → π 0 via LL operator (ϵijk (uTi CPL dj )PL uk ). The
middle points should converge to the form factor in theasymptotic limit of large source-sink separation ts , thus
develops a plateau.

breaking of the Wilson fermion formulation. The nonperturbative renormalization4) can be applied to solve
the mixing and at the same time to obtain the totallyrenormalized operator in the MS scheme. The use ofRI/SMOM schemes5) are under investigation. Finallythe lattice cutoﬀ squared 1/a2 needs to be multiplied.
This study has shown that we have reasonably a
good signal for the form factors and the low energy constants (last year) at the physical quark masses. Witha full statistics and a supplemental mass dependence
analysis, as well as with a completion of the operator
renormalization, this work provides results of proton
decay matrix elements with no systematic error fromchiral extrapolation for the ﬁrst time. There is another on-going project6) to calculate the same matrix
elements with domain-wall fermions with very coarse
lattice but with a better control of chiral symmetry.
These two studies are complementary to each otherand together will bring us to the ﬁnal goal of estimating the proton decay matrix elements for pseudo-scalarﬁnal states with direct simulation at the physical point.
References1) A. Martin, G. Stavenga, Phys. Rev. D 85, 095010(2012).
2) Y. Kuramashi, talk given at Lattice 2017.
3) Y. Aoki, T. Izubuchi, E. Shintani, A. Soni, Phys. Rev.
D 96, 014506 (2017).
4) Y. Aoki, C. Dawson, J. Noaki, Soni, Phys. Rev. D 75,014507 (2007).
5) C. Sturm, Y. Aoki, N. Christ, T. Izubuchi, C. Sachrajda, A. Soni, Phys. Rev. D 80, 014501 (2009).
6) J. Yoo, Y. Aoki, T. Izubuchi, S. Syritsyn, arXiv:1812.09326.

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