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Improved estimate of the cross section for inverse beta decay Artur M. Ankowski∗ Center for Neutrino Physics, Virginia Tech, Blacksburg, Virginia 24061, USA (Dated: January 26, 2016) The hypothesis of the conserved vector current, relating the vector weak and isovector electromagnetic currents, plays a fundamental role in quantitative description of neutrino interactions. Despitebeingexperimentally confirmedwith great precision,it isnotfully implementedinexisting calculations of the cross section for inverse beta decay, the dominant mechanism of antineutrino scattering at energies below a few tens of MeV. In this article, I estimate the corresponding cross section and its uncertainty, ensuring conservation of the vector current. While converging to previous calculations at energies of several MeV, the obtained result is appreciably lower and predicts moredirectionalpositronproductionnearthereactionthreshold. Thesefindingssuggestthatinthe 6 1 currentestimate of thefluxof geologically producedantineutrinos the232Th and 238U components 0 maybeunderestimatedby6.1and3.7%,respectively. Theproposedsearchforlightsterileneutrinos 2 using a 144Ce–144Pr source is predicted to collect the total event rate lower by 3% than previously estimated and to observe a spectral distortion that could be misinterpreted as an oscillation sig- n nal. In reactor-antineutrino experiments, together with a re-evaluation of the positron spectra, the a predicted event rate should be reduced by 0.9%, diminishing thesize of thereported anomaly. J 2 PACSnumbers: 13.15.+g,25.30.Pt 2 ] Interactionsoflow-energyantineutrinosprovideessen- need to be revised, with particularly large increase of h p tial information on topics as seemingly distant as super- its thorium component. In addition, the reduced esti- - nova explosions [1], the energy budget of the Earth [2], mate of the cross section is able to explain part of the p neutrinooscillations[3],andnuclearnonproliferation[4]. reactor-antineutrinoanomaly[12,13],anditsaltereden- e Of utmost importance is the process of inverse beta ergydependenceislikelytoaffectthepredictedspectraof h [ decay (IBD), produced positrons. An experimental verification of the 1 ν¯e+p→e++n, (1) pevreedntictriaotnesaonfdththiseiarretincelerg—yredgisatrrdibinugtiobno—thmtahye sroedonucbede v the cross section of which exceeds those for scattering providedbyasterileneutrinosearchusinga144Ce–144Pr 9 6 off electrons and nuclei by a few orders of magnitude at source [14, 15]. 1 energies below ∼20 MeV [5]. To obtain the cross section, recall that the matrix el- 6 At such kinematics, the estimate ofthe IBD crosssec- ement for IBD can be accurately calculated within the 0 tion by Llewellyn Smith [6]—widely employed at higher Fermi theory, . energies—isnotsuitablebecauseitneglectsthedifference 1 0 between the neutron and proton masses. As a remedy, M∝JleptJµ , (2) µ hadr 6 Vogel and Beacom[7] obtained a low-energyapproxima- 1 tion of the cross section accounting for this difference, as aninteractionbetweenthe leptonic andhadronic cur- v: and analyzed the angular distribution of the produced rents, i positrons. As this approximation becomes inaccurate at X energies above ∼20 MeV, Strumia and Vissani [8] per- Jµlept =u¯eγµ(1+γ5)uν¯, (3) r formedfully relativistic calculations andcomparedthem Jµ =u¯ (Vµ+Aµ)u , (4) a hadr n p to existing theoretical results. In this article, I point out that the vector part of the with the Dirac spinors uν¯ = uν¯(k,λ), ue = ue(k′,λ′), hadroniccurrentemployedinRefs.[7,8]isnotconserved, up =up(p,s), andun =un(p′,s′) describing the electron although its conservation is invoked to express the vec- antineutrino,positron,proton,andneutron,respectively. tor form factors by their electromagnetic counterparts. For the vectorand axialparts ofthe hadroniccurrent, I remove this theoretical inconsistency by using an ap- Vµ and Aµ, the matrix representations propriatematrixrepresentationofthecurrent. Thispro- q ceduresizablychangesthedescriptionoftheIBDprocess Vsµtd =γµF1+iσµκ2Mκ F2, (5) near the threshold, reducing the total cross section and qµ increasing the directionality of the produced positrons. Aµ =γµγ F +γ F , (6) std 5 A 5M P The importance of these findings is demonstrated on the example of geoneutrinos [9–11] and the obtained re- with M denoting the average nucleon mass, are widely sults suggest that the current estimates of the flux may adopted [6–8]. Note that second-class currents [16] are disregarded in this work because the available experimental evidence points toward their nonexistence in na- ∗ [email protected] ture [17–19]. 2 Due to the isospin invariance of strong interactions, The hypothesis of the conserved vector current is ex- the vector currents u¯ Vµu and u¯ γ Vµ†γ u together perimentally confirmed with great precision [28, 29]—at n p p 0 0 n with the isovectorelectromagnetic currentform a triplet the level of 10−4—and plays fundamental role in the es- of conserved currents [20, 21]. As a consequence, the timateoftheIBDcrosssection. However,itisimportant vector form factors F (i = 1,2) are related to the elec- to realize that the standard expression (5) violates cur- i tromagnetic form factors of proton and neutron by rent conservation when the difference between the neutron and proton masses is accounted for. In fact, F =Fp−Fn. (7) i i i q u¯ Vµ u =u¯ q γµF u =(M −M )F u¯ u , (10) µ n std p n µ 1 p n p 1 n p The numerical results presented in this article are obtained using state-of-the-art parametrization of the nu- sothe vectorcurrentisconservedonly whenthe neutron cleon form factors from Refs. [22, 23]. and proton masses, M and M , are considered to be n p Appearing in the axial part of the hadronic current, equal. the pseudoscalarformfactorFP canbe expressedby the Atthekinematicsofinterest,conservationofthevector axial form factor FA as [24, 25] current (CVC) can be ensured by applying the modified matrix representation 2M2 F = F , (8) P m2 −q2 A qµ q π Vµ = γµ− γρq F +iσµκ κ F . (11) cvc q2 ρ 1 2M 2 wherem isthepionmassandq =k−k′ =p−p′denotes (cid:18) (cid:19) π the four-momentum transfer. While this method has not been used in the previous The axial form factor, in turn, can be accurately estimates of the IBD cross sections [7, 8], it is known in parametrized as the description of resonance excitation, see e.g. [30, 31]. g The resulting differential crosssection as a function of A FA = (1−q2/M2)2, (9) q2 may be cast in the standard form [6, 8] A at the considered neutrino energies. The axial cou- dσctrvece = (GF cosθC)2 M4A+M2B(s−u)+C(s−u)2 , pling constant g = −1.2723(23) [26] is extracted from dq2 8πM2E2 A p ν neutron beta-decay measurements, and the axial mass (cid:2) (12(cid:3)) M =1.026(21)GeV is determined predominantly from where s−u = 4M E +q2−m2−2M∆, E being the A p ν ν neutrino scattering off deuteron [27]. neutrino energy, and A=4(τ +µ) (τ −µ) (F +F )2+F2 −(F −τF )2+F2 +4µF (τF −F ) 1 2 A 1 2 A P P A ∆ n (cid:2) ∆2 (cid:3) o −8µ (F +F )F + (τ +µ) (F +F )2−(1+τ)F2−F2 +4µF2 M 1 2 A M2 1 2 2 A P ∆2 (cid:2) µ2 (cid:3) + τ(F +F )2−(1+τ)F2−F2 −4µF F −2z F F M2 1 2 1 A P A τ 1 2 (cid:20) (cid:21) (13) ∆2 µ µ µ −z 1− +µ 1+ F2, M2 τ τ τ 1 h (cid:16) (cid:17) ∆ (cid:16) (cid:17)i z B =4τ(F +F )F −µ F2+(1−z)F F +2F F + F2 , 1 2 A M 2 1 2 A P τ 1 1 h i C = F2+τF2+F2 , 4 1 2 A (cid:0) (cid:1) with τ = −q2/4M2, ∆ = M −M , and µ = m2/4M2. sionsreducetothoseobtainedbyStrumiaandVissani[8]. n p In numerical calculations, the state-of-the-art values are Thekinematicallyallowedrangeofq2 extendsfromq2 − applied for the Fermi constant G = 1.1663787(6)× to q2 that can be expressed as F + 105/GeV2 and the cosine of the Cabibbo angle cosθ = C 0.97425(22)[26]. q2 = m2Mp−EνD±Eν D2−4m2Mn2 (14) The parameter z in Eq. (13) is introduced to show ± Mp+2pEν the difference between the calculations with (z =1) and with without (z = 0) the restored conservation of the vector current. Notethatinthelattercasetheobtainedexpres- D =(M +E )2−E2−M2−m2. (15) p ν ν n 3 To achieve the accuracy required by modern experi- 8 mentalapplications,theradiativecorrectionstothetree- level cross section (12) must be taken into account. De- 6 composing them into the inner and outer parts [32, 33], 2m) ascustomary,andconsideringthe leadingordertermsin c ( the fine-structure constant α, one obtains 4 vc σc q+2 dσtree 420 σ = dq2 cvc [1+δ +δ (β)]. (16) 1 2 cvc dq2 in out Zq−2 (a) For the inner correction, the value 0 0.4 δ =0.02250(38), (17) in %) 0.2 is adopted from the recent estimate [34]. The outer cor- ( rection δout(β) is a function of the positron’s speed, σcvc 0.0 / vc β = 1− m2 where E =E + q2−2M∆, (18) σ∆c −0.2 s Ee2 e ν 2Mp −0.4 (b) that can be expressed as [35, 36] 1.00 2π 3β 7 8 δout(β)= + artanhβ− artanh2β 0.98 α 2 2β β (cid:18) (cid:19) 4 4β2 x 0.96 + artanhβ−4 ln (19) σ (cid:18)β (cid:19) 1−β2 σ/cvc 0.94 8 2β M 23 p + L +3ln + , β 1+β m 4 0.92 (cid:18) (cid:19) with 0.90 (c) 1 1+x artanhx= ln , (20) 2 1−x −0.01 x ln(1−t) L(x)= dt . (21) t −0.02 Z0 θi s Note that the above expressions are equivalent to those hco −0.03 thisarticle given in Ref. [37]. Vogel–Beacom The obtained total IBD cross section and its uncer- Strumia–Vissani −0.04 tainty are presented in Figs. 1(a) and 1(b), respectively. The uncertainty is estimated by varying parameters en- (d) −0.05 tering the calculations within their uncertainties and 2 4 6 8 10 adding the corresponding cross section’s variations in E (MeV) ν quadratures. The result—not exceeding 0.35%—is over- whelmingly dominated by the uncertainty of the axial FIG.1. (Coloronline). Estimateof(a)theIBDcrosssection coupling constant, and contributions other then those σcvc and (b) its uncertainty, obtained ensuring the conser- fromthe innerradiativecorrectionsandtheCabibboan- vation of the vector current. The relevance of this effect is gle can be safely neglected. shown by (c) the ratios of σcvc to the results of Refs. [7, 8] Playing an important role only at low absolute values and (d) the comparison of the average cosine of the positron of q2, the CVC restoring procedure (11) affects the IBD production angle calculated within these approaches. crosssectioninanappreciablemannersolelyatlowneu- trino energies. This feature is illustrated in Fig. 1(c), showingthe ratioofσ tothecalculationsofVogeland that at higher energies the difference between the cross cvc Beacom [7] and those of Strumia and Vissani [8]. In all sectionsofRefs.[7]and[8]graduallybecomesvisiblebut thecases,thesametreatmentoftheradiativecorrections remains below 0.5% for E ≤13 MeV. ν has been applied. While at E = 2 MeV, the result of Asthekinematicregionoflow|q2|correspondstohigh ν this article is lower than those of Refs. [7, 8] by as much cosθ, with θ being the positron’s production angle, the as ∼6.8%, this effect reduces to ∼0.5% at 4 MeV. Note observedreduction of the cross section at low |q2| trans- 4 lates into a decrease of the averagevalue of cosθ, shown Moreover, as the antineutrino energy is closely re- in Fig. 1(d). The manifest increase of the directionality lated to the prompt energy of the produced positron, atenergiesE ∼2–3MeV,resultingpredominantlyfrom E ≃ E −0.78 MeV, the results of Fig. 1(c) cor- ν prompt ν the last term in the B factor (13), may be relevant, e.g., responding to the low-E region can be expected prompt for spatially mapping geoneutrinos [38]. to bring into better agreement the predictions and the To obtain the hcosθi dependence on antineutrino en- promptenergyspectrameasuredinneardetectorsofon- ergy presented in Fig. 1(d), the cosθ-even and cosθ-odd going reactor experiments [43, 44]. parts of the cross section has been treated separately, In summary, conservation of the vector part of the as they are subject to different outer radiative correc- weak current has important consequences for the de- tions [35, 36]. However, in agreement with the conjec- scription of inverse beta decay at the kinematics corre- ture of Ref. [7], this procedure turns out to have a small sponding to low |q2|. For energies in the vicinity of the effect on the result, affecting it by no more than 0.15% threshold,thiseffectsizablylowersthetotalcrosssection for E ≥2 MeV. and increases the directionality of positron production. ν TheIBDcrosssectionisgenerallyconsideredtobesub- These results, of particular relevance for an estimate of ject to low uncertainties and, therefore, its CVC-related the geoneutrino flux, may soon be verified by an experi- reduction may have important consequences. For exam- ment employing a 144Ce–144Pr source to search for light ple, in the context of a determination of the geoneutrino sterile neutrinos. As a consequence of the reduced total flux, I findthatσ leadsto the 232Thand238Ucompo- cross section, the size of the reactor anomaly is also di- cvc nents higher by 6.1 and 3.7%, respectively, than the es- minished, and the agreement between the predicted and timates basedon the cross sections of Refs. [7, 8]. Those observedpositron spectra can be expected to improve in values are calculated using the spectra of Ref. [39] and reactor-antineutrino experiments. refer to the KamLAND site. To facilitate analysis of other implications of the IDB In Ref. [14], it has been proposed to search for light cross section reported here—for example in the context sterile neutrinos using a 144Ce–144Pr ν¯ source. Using of big-bang nucleosynthesis—its c++ implementation, e the flux of Ref. [40], I predict an overall3% reduction of approximated expressions valid at low energies, off-shell the eventrate with respectto simulations employing the generalization,andtabulatedvaluesareprovidedinSup- crosssectionsofRefs.[7,8],andaspectraldistortionthat plemental Material [45]. maymimicanoscillationsignal. Assuchanexperimentis currently underway [15], these predictions can be tested ACKNOWLEDGMENTS within a 1-year time frame. The reevaluation of the antineutrino spectra emit- ted by nuclear reactors [41] has recently lead to the I am indebt to Makoto Sakuda for drawing my atten- conclusion that the event rates observed in past reac- tion to neutrino interactions at low energies. Special tor experiments underestimate the predicted rates by thanks are addressed to Patrick Huber and Jonathan 5.7±2.3% [12]. Combining the contributions from the Link for mostly informative discussions on the reactor- individualisotopes[41, 42]accordingto the weights[43], antineutrino anomaly and their valuable suggestions on I estimate that the CVC-related reduction of the cross the composition of this article. My work was supported section lowers the predicted rate by 0.9%, reducing the by the National Science Foundation under Grant No. reactor anomaly. PHY-1352106. [1] A.Burrows, Nature 403, 727 (2000). Phys. Rev.D 88, 033001 (2013). [2] G. Fiorentini, M. Lissia, and F. Mantovani, [11] M. Agostini et al. 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