# 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 June 22, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.