Parameter-free effective field theory calculation for the solar proton-fusion and hep processes Page: 4 of 22
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II. FORMALISM
We sketch here the basic elements of our formalism.
The explicit degrees of freedom taken into account in
our scheme are the nucleon and the pion, with all other
degrees of freedom (p- and w-mesons, A(1232), etc.) in-
tegrated out. The HBXPT Lagrangian can be written
as(3)
G= Ga=o+G1+---,
with the chiral order A defined as
d + e + -2,where d, e and n are, respectively, the numbers of deriva-
tives (the pion mass counted as one derivative), external
fields and nucleon lines belonging to a vertex. Chiral
symmetry requires A > 0. The leading-order Lagrangian
is given by
B [iv D + 2igAS A] B- CA (AFAB)2
A(4)
the two-nucleon sector is given in Ref. [27]5. With four-
fermion contact terms included, the Lagrangian takes the
form
G1 B 2mg D D + 4c3iA - iA
+ (2c4+ 2[fl, S"] [iAn, inv>]
2mN
-1i+c6 S, + B- 4id1BS-ABBB
mN y
+2id2 eabc eu svBavaaBS TbB BSot B
+ - - - , (9)
where mN ~ 939 MeV is the nucleon mass, and
fi+ =(a Lu - ttL - i [L, Lu])J t+ t( v avR -i [Rn, Rv]),
(10)
60123 1, and A z A . We have shown here only
those terms which are directly relevant to our present
study. The dimensionless low-energy-constants (LECs),
ms's mind l's_ n~r defined n~s~3 1 - 4
e3,4 = -- e3,4, d1,2
mN9A
mN dl2(11)
+ f Tr (iADiAp) + Tr(X+)
(5)
where B is the nucleon field in HBXPT; gA 1.2670 i
0.0035 is the axial-vector coupling constant [28], and
f, 92.4 MeV is the pion decay constant. Furthermore
D B (a + F )B,
F = [f, a ] - t R - L f ,A, [(, UpCJ + C'R pC
2 2
X+ = tX~t + Xt,2
with
z= z/z=zexp i j. (7)
R - z (V+ Aa) and L \ (Vz - Aa) denote ex-
ternal gauge fields, and X is proportional to the quark
mass matrix. If we neglect the small isospin-symmetry
breaking, then X m (in the absence of external scalar
and pseudo-scalar fields). For convenience, we work in
the reference frame in which the four-velocity v and the
spin operator S are
v (1, 0) and S (0, ). (8)
The NLO Lagrangian (the so-called "1/m" term) in the
one-nucleon sector is given in Ref. [26], while that inWe now consider the chiral counting of the electroweak
currents (see the Appendices for details). In the present
scheme it is sufficient to focus on "irreducible graphs"
in Weinberg's classification [21]. Irreducible graphs are
organized according the chiral index v given byv 2(A - C) + 2L + v,
(12)
where A is the number of nucleons involved in the pro-
cess, C the number of disconnected parts, and L the num-
ber of loops; v, is the chiral index A, eq.(4), of the i-th
vertex. One can show that a diagram characterized by
eq.(12) involves an nB-body transition operator, where
nB _ A - C + 1. The physical amplitude is expanded
with respect to v. As explained at length in the Ap-
pendix, the leading-order one-body GT operator belongs
to v=0. Compared with this operator, a Feynman dia-
gram with a chiral index v is suppressed by a factor of
(Q/Ay)", where Q is a typical three-momentum scale or
the pion mass, and AX ~ 1 GeV is the chiral scale.6 In
a Our definition of the pion field here is different from that
used in Ref. [27]; we have changed the sign of the pion field.
Furthermore, we employ here manifestly Lorentz-invariant
and chiral-invariant interactions.
6 For convenience, a chiral order corresponding to v is of-
ten referred to as N"LO; v=1 corresponds to NLO (next-to-
leading order), v=2 to N2LO (next-to-next-to-leading order),
and so on.
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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/m1/4/: accessed March 20, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.