8 November 2001

Physics Letters B 520 (2001) 87–91www.elsevier.com/locate/npe

Proton–proton fusion in effective field theory to fifth order

Malcolm Butlera, Jiunn-Wei Chenb

a Department of Astronomy and Physics, Saint Mary’s University, Halifax, NS B3H 3C3, Canadab Department of Physics, University of Maryland, College Park, MD 20742, USA

Received 31 January 2001; received in revised form 3 July 2001; accepted 19 September 2001Editor: W. Haxton

Abstract

The proton–proton fusion processpp→ de+νe is calculated at threshold to fifth order in pionless effective field theory.There are two unknown two-body currents contributing at the second and fourth orders. Combined with the previous resultsfor νed andνed scattering, computed to third order in the same approach, we conclude that a∼ 10% measurement of reactorνed scattering measurement could constrain thepp→ de+νe rate to∼ 7% while a∼ 3% measurement ofνed→ e−pp couldconstrain thepp rate to∼ 2%. 2001 Published by Elsevier Science B.V.

The reactionpp→ de+νe is of central importanceto stellar physics and neutrino astrophysics. Bethe andCritchfield proposed it to be the first reaction to ignitethepp chain nuclear reactions that provided the prin-cipal energy and neutrinos in the Sun [1]. Major effortshave been made to provide precise theoretical predic-tions for this reaction at zero energy [2–13]. No directexperimental constraint is available for this process,and the accuracy of the theoretical predictions mustalways be weighed against the availability of empiri-cal constraints. However, Schiavilla et al. recently cal-ibrated the matrix element with complementary cal-culations of tritium beta decay to obtain an estimateduncertainty of less than 1% in the potential model.

Alternatively, effective field theory (EFT) can pro-vide a connection betweenpp fusion and other proces-ses that might be accessible experimentally. Specifi-cally, there is a direct connection between the reactionspp→ de+νe andνed → e−pp in that they both in-volve the same matrix elements. Further, in EFT, they

E-mail addresses: mbutler@ap.stmarys.ca (M. Butler),jwchen@physics.umd.edu (J.-W. Chen).

can both be shown to depend on a single unknowncounterterm (or two-body current) at third order. Oth-erwise, the nuclear physics input is identical. Fur-ther, all fourν(ν)–d breakup channels depend on thissame counterterm, meaning that a measurement in anyone channel implies a measurement in all channels—includingpp fusion. To date, measurements are avail-able for νed breakup using reactor antineutrinos with∼ 10–20% uncertainty [14–18] and possibly∼ 5% inthe future [18]. Also, the ORLaND proposal couldprovide a measurement ofνed → e−pp to a fewpercent level [19]. It is important to understand theconstraints that these measurements could onpp→de+νe .

At the present time,νed andνed breakup processeshave been studied to the third order in pionless EFTdepending on the unknown two-body countertermL1,A. Through varyingL1,A, the four channels of po-tential model results of Refs. [20,21] are reproducedto high accuracy. This confirms that the∼ 5% differ-ence between Refs. [20,21] is largely due to differentassumptions made to short distance physics. We willdiscuss this difference later in this Letter.

0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V.PII: S0370-2693(01)01152-2

88 M. Butler, J.-W. Chen / Physics Letters B 520 (2001) 87–91

Forpp fusion, an pionless EFT calculation has beenperformed to second order by Kong and Ravndall [22],with a resulting dependence onL1,A. In this Letter,we use the same approach to push the calculation tofifth order, which introduced another unknown two-body counterterm (K1,A) that first enters at fourthorder. ConstrainingK1,A using dimensional analysis,we conclude that a measurement of reactorνed with10% uncertainty can constrainpp fusion to 7% whilea measurement ofνed→ e−pp to 3% would constrainpp fusion to 2%.

The method we use is pionless nuclear effectivefield theory, EFT(/π) [23], treating the electromag-netic interaction between protons in the manner de-veloped in [22,24]. The dynamical degrees of freedomare nucleons and non-hadronic external currents. Mas-sive hadronic excitations such as pions and the deltaare integrated out, resulting in contact interactionsbetween nucleons. The nucleons are non-relativisticbut with relativistic corrections built in systematically.Nucleon–nucleon interactions are calculated perturba-tively with the small expansion parameter

(1)Q≡ (1/app, γ,p,αMN)Λ

,

which is the ratio of the light to heavy scales. The lightscales include the inverse S-wave nucleon–nucleonscattering length 1/app (app =−7.82 fm) in the 1S0,pp channel, the deuteron binding momentumγ (=45.69 MeV) in the3S1 channel, the proton momen-tum in the center-of-mass framep, and the fine struc-ture constantα (= 1/137) times the nucleon massMN . The heavy scaleΛ, which dictates the scalesof effective range and shape parameters is set bythe pion massmπ . The Kaplan–Savage–Wise renor-malization scheme [25] is used to make the powercounting [25,26] inQ transparent. This formalism hasbeen successfully applied to many processes involv-ing the deuteron [23,27] including Compton scatter-ing [28,29], np→ dγ for big-bang nucleosynthesis[30,31],νd scattering [32] for physics at the SudburyNeutrino Observatory [33], and parity violation ob-servables [34].

There are other power counting schemes whichyield different orderings in the perturbative series andeach has certain advantages. For example, thez-pa-rametrization [35] recovers the exact deuteron wavefunction renormalization at second order; the dibaryon

pionless EFT [28] resums the effective range parame-ter contributions at first order and simplifies the calcu-lation tremendously by cleverly employing the equa-tions of motion to remove redundancies in the theory.While one of these power countings could lead to morerapid convergence in any given calculation, for a high-order calculation as we present here the distinctionsbetween different expansions are negligible.

Ignoring for the moment the weak interaction com-ponent, the relevant Lagrangian in EFT(/π) can bewritten as a derivative expansion

L=N†(i∂0+ ∇

2

2MN

)N

−C(3S1

)0

(NT PiN

)†(NT PiN

)+ C

(3S1)2

8

[(NT PiN

)†(NT←→∇ 2PiN

)+ h.c.]

− C(3S1)4

16

(NT←→∇ 2PiN

)†(NT←→∇ 2PiN

)− C

(3S1)4

32

[(NT

←→∇ 4PiN)†(NT PiN)+ h.c.

](2)+ (3

S1→ 1S0, Pi→ Pi)+ · · · ,

where←→∇ ≡ −→∇ − ←−∇ and wherePi = σ2σiτ2/

√8 and

Pi = σ2τ2τi/√

8 project out3S1 and 1S0 channels,respectively, withσ(τ) acting on spin (isospin) in-dices. The coupling constants have been fit to datain [23,32]. We only perform the calculation at thresh-old and thus the relativistic corrections and momen-tum transfer (q) effects are suppressed by factors ofγ 2/M2

N andq2/γ 2, both of which are 1%. Thus,with the goal of presenting a calculation with a preci-sion of less than 1%, the non-relativistic and zero re-coil limits are suitable approximations for us to make.

The weak interaction terms in the Lagrangian canbe written as

(3)LW =−GF√2lµ+J−µ + h.c.+ · · · ,

where GF is the Fermi decay constant andlµ+ =νγ µ(1 − γ5)e is the leptonic current. The hadroniccurrent has vector and axial vector parts,

J−µ = V −µ −A−µ.The vector current matrix element vanishes in the zerorecoil limit since it yields terms proportional to the

M. Butler, J.-W. Chen / Physics Letters B 520 (2001) 87–91 89

wave function overlapping between two orthogonalstates. The time component of the axial current givescenter of mass motion corrections which vanish inour approximation. The spatial component of the axialvector is the dominant contribution, and can be writtenin terms of one-body and two-body currents

A−k =gA

2N†τ−σkN

+L1,A[(NT PkN

)†(NT P−N)+ h.c.

]+ K1,A

8

[(NT←→∇ 2PkN

)†(NT P−N)

+ (NT PkN

)†(NT←→∇ 2P−N)+ h.c.

](4)+ · · · ,

wheregA = 1.26, τ− = (τ1 − iτ2), and the values ofL1,A andK1,A are yet to be determined by data.

The hadronic matrix element is usually parame-trized in the form

(5)∣∣〈d; j |A−k |pp〉∣∣≡ gACη

√32π

γ 3 Λ(p)δjk ,

wherej is the deuteron polarization state, and

(6)Cη =√

2πη

e2πη − 1, η= αMN

2p,

is the well-known Sommerfeld factor. In the centerof the Sun, the scale ofp is ∼ 1 MeV. Thus theSommerfeld factorC2

η changes rapidly with respectto p while Λ(p) does not. It is sufficient to keepthe p2 correction forΛ(p), since the higher ordercorrection isO((p/αMN)4). Usingε to keep track oftheQ expansion we find that, to fifth order in theQ(ε)expansion,Λ(0) can be written in the compact form

Λ(0)= 1√1− εγρd×

{eχ − γ app

[1− χeχE1(χ)

]− εγ 2app

[L1,A − ε

2γ 2

2K1,A

]}(7)+O

(ε5).

This expression can be expanded toε4, and thenεshould be set to 1.ρd = 1.764 fm is the effective range

parameter in the3S1 channel,χ = αMN/γ and

(8)E1(χ)=∞∫χ

dte−t

t.

L1,A and K1,A are the renormalization scaleµ-independent combinations of theµ-dependent para-metersL1,A, K1,A and the nucleon–nucleon contacttermsC2 andC4

L1,A = −(µ− γ )MNC

(pp)

0,−1

[L1,A

gA− MN

2

(C(pp)2,−2+C(d)2,−2

)],

(9)

K1,A = −(µ− γ )MNC

(pp)

0,−1

[K1,A

gA−MN

(C(pp)

4,−2+ 2C(d)4,−3

)].

From Ref. [32], we have

C(pp)

0,−1=4π

MN

(1

app−µ

+ αMN(

lnµ√π

αMN+ 1− 3

2γE

))−1

,

C(pp)

2,−2=MN

8πr(pp)

0 C(pp)20,−1 ,

C(pp)

4,−2=−MN

4πr(pp)

1 C(pp)20,−1 ,

C(d)2,−2=

2π

MN

ρd(µ− γ )2

,

(10)C(d)4,−3=−

π

MN

ρ2d(

µ− γ )3 ,

where the second subscripts of theC ’s denote thescaling in powers ofQ, γE= 0.577 is Euler’s constant,andr(pp)0 = 2.79 fm is thepp channel effective range.We take thepp channel shape parameter to be thesame as that in thenp channelr(pp)1 =−0.48 fm3. Anyerrors introduced by this are small numerically.

After expanding toε4 and settingµ = mπ , weobtain

(11)

Λ(0)= 2.58+ 0.011

(L1,A

1 fm3

)− 0.0003

(K1,A

1 fm5

).

At µ = mπ , dimensional analysis as developed inRefs. [23,25] would favour

|L1,A| ≈ 1

mπ(mπ − γ )2 ≈ 6 fm3,

90 M. Butler, J.-W. Chen / Physics Letters B 520 (2001) 87–91

(12)|K1,A| ≈ 1

m2π(mπ − γ )3

≈ 20 fm5.

Two observations follow. First, if we take these naivelyestimated values (with positive signs, for example),then the expansion ofΛ(0) converges rapidly

(13)Λ(0)= 2.51(1+ 0.039+ 0.029− 0.010− 0.0001).

For other sign combinations, the series also convergesrapidly and higher order effects are 1%. Second,Eqs. (11) and (12) show thatK1,A is likely to con-tribute toΛ(0) at a level less than 1%. Thus Eq. (11)is precise to 1% even withK1,A set to zero, meaningthat we can write

(14)Λ(0)= 2.58+ 0.011

(L1,A

1 fm3

)+O(1%).

This is the central result of this Letter.Ultimately the value ofL1,A must be extracted

from experimental data. Before we address this issue,let us look at what values ofL1,A are found in fitsto potential model calculations. Using the results ofRef. [32], the recent potential model results forν(ν)–d breakup of Nakamura, Sato, Gudkov, and Kubodera(NSGK) [21] are equivalent to

(15)LNSGK1,A = 5.6± 2 fm3,

while the results of Ying, Haxton and Henley (YHH)[20] are equivalent to

(16)LYHH1,A = 0.94± 2 fm3.

The uncertainties represent a conservative estimatefrom EFT of 3% from Ref. [32] (even though theNSGK results can be reproduced within 1% using thecentral value ofL1,A). Based on the size of the third-order contribution, the actual uncertainty may be assmall as 1%, but further analysis is required to ascer-tain this. The (undefinable) errors from the potentialmodels themselves are not included. As mentioned be-fore, the differences between these two calculationsare in their treatment of two-body physics. NSGK usesa model to include axial two-body meson exchangecurrents, while YHH includes vector two-body cur-rents but not the more-important axial two-body cur-rents. Given thatL1,A is representative of the dom-inant axial effects in EFT, it is not surprising to seesubstantial differences in the value inferred by eachcalculation.

Similarly, Eq. (14) translates thepp fusion rate(Λ2(0) = 7.05–7.06) calculated by Schiavilla et al.into

(17)LSchiavilla et al.1,A = 6.5± 2.4 fm3,

which is consistent with NSGK.Experimentally, it is easy to relate the EFTν(ν)–d

scattering results andpp fusion rate through the third-order results forν(ν)–d provided in [32]. The resultsfor each channel are parameterized in the form

(18)σ(Eν)= a(Eν)+ b(Eν)L1,A +O(< 3%),

wherea andb are functions of neutrino energy. Theseresults, tabulated in Ref. [32], can be easily related tothepp fusion rate through Eq. (14).

For reactorν–d scattering, one expects the rate tobe peaked around 8 MeV, which can be interpreted tomean a measurement precise to 10% can determine thepp fusion rate (∝ Λ2) to 7%. With the experimentalprecision further improved to 5%, thepp fusion ratecan be determined to 4%. Alternatively, the proposedORLaND [19] detector might measureνed → e−ppto 3%, in turn constraining thepp fusion rate to 2%.

In strong interaction processes involving externalcurrents, delicate relations between operators are re-quired to guarantee that there are no off-shell ambigu-ities in the final results. For example, in two-body sys-tems two-body currents serve to absorb the off-shelleffects from the two-body strong interactions. Thismeans that different models that reproduce the sameon-shell nucleon–nucleon scattering data might needquite distinct two-body currents to deal with off-shelleffects. In the case ofpp fusion or ν–d scattering,electromagnetic matrix elements cannot be used toconstrain weak matrix elements. Their operator struc-tures are quite different at the quark level even theymight appear the same in hadronic level. Using tri-tiumβ-decay to constrain the two-body current [12] isan excellent idea, in principle. However, contributionsfrom three-body currents that are required to absorbedthree-body off-shell effects are not yet constrained. Inlight of these facts, the need for a precise experimen-tal measurement ofνe(νe)–d breakup cannot be over-stated.

M. Butler, J.-W. Chen / Physics Letters B 520 (2001) 87–91 91

Acknowledgements

We would like to thank Ian Towner for usefuldiscussions. M.B. is supported by a grant from theNatural Sciences and Engineering Research Councilof Canada. J.-W.C. is supported, in part, by theDepartment of Energy under grant DOE/ER/40762-213.

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