Parameter-free effective field theory calculation for the solar proton-fusion and hep processes Page: 3 of 22
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operators in SNPA and EFT are identical, and that their
matrix elements can be reliably estimated with the use
of realistic SNPA wave functions for the initial and final
nuclear states. Next, we note that in EFT the oper-
ators representing two-body corrections2 to the leading-
order one-body term can be controlled by systematic chi-
ral expansion in heavy-baryon chiral perturbation theory
(HBXPT) . Then, since the ratio of a two-body ma-
trix element to the leading-order one-body matrix ele-
ment can be evaluated with sufficient accuracy with the
use of the realistic SNPA wave functions3, we are in a
position to obtain a reliable estimate of the total (one-
body + two-body) contribution. This approach takes full
advantage of the extreme high accuracy of the wave func-
tions achieved in SNPA while securing a good control of
the transition operators via systematic chiral expansion.
For convenience, we will refer to this method, which ex-
ploits the powers of both SNPA and EFT, as "MEEFT"
(short for more effective EFT). MEEFT which is close
in spirit to Weinberg's original scheme  based on the
chiral expansion of "irreducible terms" has been found
to have an amazing predictive power for the n+p -> d+'
process [14,22] and several other processes . An al-
ternative approach, which however is in line with our
reasoning, has been discussed by Ananyan, Serot and
An early HBXPT study of the pp process was made in
Ref.  (hereafter referred to as PKMR98) by four of the
authors. The calculation in PKMR98 was carried out up
to next-to-next-to-next-to-leading order (N3LO) in chi-
ral counting (see below). At N3LO, two-body meson-
exchange currents (MEC) begin to contribute, and there
appears one unknown parameter in the chiral Lagrangian
contributing to the MEC. This unknown constant, called
dR in Ref. , represents the strength of a four-nucleon-
axial-current contact interaction. In Ref. , since no
method was known to fix the value of aR, the dR-term
was simply ignored by invoking a qualitative argument
that the short-range repulsive core would strongly sup-
press its contribution. Due to uncertainties associated
with this argument, Ref.  was unable to corroborate
or exclude the result of the latest SNPA calculation ,
652B 0.5 ~ 0.8 %, where 32B is the ratio of the con-
tribution of the two-body MEC to that of the one-body
current (see below).
The situation can be greatly improved by using
MEEFT. As first discussed in Refs. [7,8] and as will be ex-
2The argument made here should apply generally to n-body
currents (n > 2) but since the 2-body terms are dominant,
we shall continue to restrict our discussion to the latter.
This statement holds only for the finite-range part of two-
body operators, with the zero-range part requiring a regular-
ization to be specified below.
pounded here, the crucial point is that exactly the same
combination of counter terms that defines the constant
dR enters into the Gamow-Teller (GT) matrix elements
that feature in pp fusion, tritium /3-decay, the hep pro-
cess, p-capture on a deuteron, and v d scattering and
that the short-range interaction involving the constant
dR is expected to be "universal," that is, A-independent.
Therefore, assuming that three- and four-body currents
can be ignored (which we will justify a posteriori), if the
value of JR can be fixed using one of the above pro-
cesses, then we can make a totally parameter-free predic-
tion for the GT matrix elements of the other processes.
Indeed, the existence of accurate experimental data for
the tritium /-decay rate, F , and the availability of ex-
tremely well tested realistic wave functions for the A=3
nuclear systems allow us to carry out this program. In
the present work we determine the value of JR from F
and perform parameter-free EFT-based calculations of
SP(0) and SheP(0).
As described below, our scheme has a cutoff parame-
ter A, which defines the energy/momentum cutoff scale
of EFT below which reside the chosen explicit degrees
of freedom4. Obviously, in order for our result to be
physically acceptable, its cutoff dependence must be un-
der control. In our scheme, for a given value of A in a
physically reasonable range (to be discussed later), dR
is determined to reproduce F; thus JR is a function
of A. According to the premise of EFT, even if JR it-
self is A-dependent, physical observables (in our case the
S-factors) should be independent of A as required by
renormalization-group invariance. We shall show that
our results meet this requirement to a satisfactory de-
gree. The robustness of our calculational results against
changes in A allows us to make predictions on SP(0)
and SheP(0) with much higher precision than hitherto
achieved. Thus we predict: SP(0) 3.94x (1 0.004)x
10-25 MeV-b and SheP(0) (8.6 1.3) x 10-z0 keV-b.
The remainder of this article is organized as follows.
In Section II we briefly explain our formalism; in par-
ticular, we describe the relevant transition operators de-
rived in HBXPT. Section III presents the calculation of
SP (0), while Section IV is concerned with the estimation
of SheP(0). Section V is devoted to discussion and conclu-
sions. We have made the explanation of the formalism in
the text as brief and focused as possible, relegating most
technical details to the Appendices.
4The cutoff specifies not just the relevant degrees of freedom
but also their momentum/energy content. This should be un-
derstood in what follows although we do not always mention
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Park, T.S.; Marcucci, L.E.; Schiavilla, R.; Viviani, M.; Kievsky, A.; Rosati, S. et al. Parameter-free effective field theory calculation for the solar proton-fusion and hep processes, article, August 1, 2002; Newport News, Virginia. (digital.library.unt.edu/ark:/67531/metadc742856/m1/3/: accessed May 22, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.