The ultrahigh energy cross section for neutrino interactions with nucleons is
reviewed, and unitarity constraints are discussed. We argue that existing QCD
extrapolations are self-consistent, and do not imply a breakdown of the
perturbative expansion in the weak coupling.Comment: 5 pages, 2 figures, RevTeX4, contribution to Snowmass 200

Reno, M. H.

Sarcevic, I.

Sterman, G.

Stratmann, M.

Vogelsang, W.

English

arXiv

The ultrahigh energy cross section for neutrino interactions with nucleons is reviewed, and unitarity constraints are discussed. We argue that existing QCD extrapolations are self-consistent, and do not imply a breakdown of the perturbative expansion in the weak coupling

Hall Reno, M.

Sarcevic, I. N.

UNT Digital Library

,“ K8RC-219.[BNL-68968Ultrahigh energy neutrinos, smallz and unitarityMary Hall Reno*Department of Physics and Astronomy, University oj Iowa, Iowa City, IA 52,%?42USAIna SarcevictDepartment of Physics, Universityof Arizona, Tucsonl AZ 85721 USAGeorge Sterman$C. N. Yang institute for Theoretical Physics, SUNY Stony Brook, Stony Brook, NY I1794 USAMarco Stratmann$Institutfiir Theoretische Physik, Universit/it Regensburg, D-93040 Regensburg, GermanyWerner VogelsangllPhysics Department and RIKEN-BNL Research Center,Brookhaven National Laboratory, Upton, NY 1197.9 USAThe ultrahigh energy cross section for neutrino interactions with nucleons is reviewed, and uni-tarity constraints are discussed. We argue that existing QCD extrapolations are self-consistent, anddo not imply a breakdown of the perturbative expansion in the weak coupling.I. INTRODUCTIONUltrahigh energy neutrinos are predicted from a nutnber of sources. One source is from cosmic ray interactionswith the microwave background radiation [1], producing charged pions which decay into neutrinos. Anotherpossible source is decaying cosmic strings or extreme] y massive relics [2], which ultimately contribute to a cosmicneutrino flux. Detection of ultrahigh energy neutrinos may shecl light on the observation of air shower eventswith energies in excess of 1011 ~Tev, revea] aspects of .granclunification or yield some insight into the sources ofthe highest energy cosmic rays.A number of detectors are able or will be able to detect neutrino induced showers. For example, the Augerexperiment should be able to detect neutrino induced horizontal air showers initiated by neutrinos with energiesabove 109 GeV. The proposed OWL/EUSO satellite experiments should be able to detect upward air showersproduced by v, + ~ just below the Earth’s surface. The event rates predictions depend on the ultrahigh energyneutrino cross section, extrapolated beyond the measured regime, as well as on the predicted neut,rino fluxes.Recent discussions by Dicus, Kretzer, Repko, and Schmidt [3] about the implications of perturbative unitarityhave refocused attention on the ultrahigh energy extrapolation of the neutrino-rlucleon cross section. In the nextsection we review the cross section evaluation including the extrapolation of the parton distribution functionto small parton momentum fraction z. We examine to what extent the cross section may he sensitive to thepresence of saturation effects in the evolution of the parton distributions. In the following section, we outline theunitarity argument and cornrnent on what can and cannot be learned by relating the neutrino-nucleon forwardscattering amplitude to the total neutrino-nuclecm cross section.The expression for the* rl)ary-hall-reno(@ uio\va.ed[]tjna,tip})ysics. ariz(,lla.ecltlIL NEUTRINO-NUCLEON CROSS SECTIONneutrino-nucleon charged current cross section, for V1(k)N(p) + 1(k’ )X (p’ ), is(1)isterrlla,l,hinsti. physics. sLltlysb.ed!J'rnarcc>.stratrrl al)ll;(!ptlysik .Llni-reger]sl} l.lrg.(ief\vvc)gcls;Lrlg, cvl>lll.g[.)v2in terms of Q2 = –(k – k’)2, z = Q2/(2J4v) and Y = vIE, for neutrino energy E and lepton energy transferv = E –“ E{ defined in the nucleon rest frame. The nucleon mass is M, and the center of mass energy squaredis S = 2ME. In Eq. (1) we have followed Refs. [3, 4] to introduce effective quark and antiquark densities thatcontain the contributions from the various flavors as well as the appropriate electroweak mixing angles. Theexpression for the neutral current reaction V1(k) N (p) -+ V1(k’)X (P’ ) can be cast into an identical form, withthe obvious replacement MW + Mz and with different effective quark densities. In what follows, we will onlyconsider the neutrino-nucleon cross section. At high energies, the antineutrino cross section is expecced to bevery similar. We will neglect perturbative QCD corrections to the cross section which were found to be smallin Refs. [3, 5]. Finally, we also neglect the contributions to the cross section arising from charm quarks in theinitial state; these can be sizable at high energies, but are unimportant for our more qualitative purposes.In Eq. (I), increasing Q2 has two effects: as Q2 rises, the cross section decreases due to the vv propagator,but the contributions of the quark and antiquark distribution functions g(z, Q) and I(z, Q) increase due toQCD evolution [6]. The propagator dominates and effectively cuts off the growth in Q2 at Q2 * fl.f~,,. As aconsequence, the typical x value as a function of incident neutrino energy E is x - J4& /(2ME), so ultrahighneutrino energies translate to small parton z. For the highest neutrino energies considered, E - 1[)11 GeV,z - 10–s. HERA measurements of structure functions [7] extend to z N 10– 6; however, such low values of ~are measured at Q2 <0.1 GeV2. For Q w MJV, the structure functions are measured down to z _ 10-3 irl theDO experiment’s analysis of inclusive jets [S]. Small x extrapolations of the parton distribution functions aretherefore necessary to extend the predictions for the neutrino-nucleon cross section above E C+107 GeV.Perturbative QCD governs the small x extrapolations. The sea quark distributions dominate the cross sectionat high energies. Sea quarks are produced by gluon splitting g + g~, so the gluon distribution g(z, Q) dictates theeventual quark and antiquark distributions at small %. The gluon distribution is parametrized as .r~(.r, QO) *x ‘~ for z << 1 at a reference scale Q(l. Approximate small-x DGLAP evolutlon [9], for A close to o,s, yieldsa gluon distribution function of the sanle form, at a larger value of Q: xg(:c, Q) ~ :r- ~ [10]. As a. practicalmatter, A was determined at Q n MW to extrapolate the parton distribution functiom [4], for example, thoseby the CTEQ collaboration [11], below z = 10-5. Cliick, Kretzer, and Reya [5] have checlcecl that the fu11DGLAP evolution of the Gliick, Reya, ancl Vogt [12] distribution functions yields only some 20% differenceat x = 10-8 compared with the power law extra.pola.tic-mof the CTEQ densities. Kwiecinski, Nlart,in, andStasto [13] have performed a BFKL-type [14] evolution, yielcling results in substantial agreement at, the highestenergies considered (1012 GeV). The resulting total neutrino-nucleon cross sections can be para.nleterizecl bypower laws for 107 C,eV< E < 1012 GeV. For example, the charged current rle~ltri?lo-~l~lcleol~CK,SS sectjorl,using the CTEQ4 parton c}istribution functions, scales as a = 5.5 x 10–36(.E/GeV)03b cn~z [4].Ultimately, the growth of the parton distribution functions – and hence that of the cross section - predictedby both DGLAP and BFKL evolution will have to slow down when the gluon densities become large enoughthat gg + g recombination processes become important [].5]. The regime in x, Q2 in which this happens isreferred to as “saturation region” and can be estinmtecl. It can be roughly characterized by the co~ldit.ioll [16](2)with A and .cOobtained from a fit, to HERA data: A = 0.288, .co = 3.04 x 10-4. Q,! is the “saturation scale” -saturation should roughly occur when Q < QS(x) at a given x. Clearlyl as x clecreases, saturation effects a,reexpected to become relevant already at larger and larger Q2.In Figs. 1 and 2 we examine whether the neutrino-nucleon cross section at ultrahigh energy, E’ = 1(:11~ ~ev,might be sensitive to such saturation effects. Figure 1 shows log10(d2cr/d loglo (z)d logl ,j(Qz)), evaluated usingthe parton distribution functions of [12], as a function of Ioglo .Vand logl” Q2. It is evident that scales Q N ~lf~},,zdominate the cross section. Note that the parton distributions of [12] can be used also down to rather smallQ’ - 1 GeV2. This is convenient because., in order to obtain the total neutrino-nucleon cross section c~neneedsto integrate the expression in Eq. ( 1) over the range O< Q2 < .rS. For Q < 1 GeV, we have frozen the scale inthe parton distribution functions to 1 GeV. Fortunately, as can be seen from Fig. 1, the region of (J2 < 1 (kV2contributes only very little to the tot,al cross secticln.The line in the xl Q2 plane corresponding to the saturation conditiou, Eq. (’2), is also shown ill Fig. L. 111Fig. 2 we follow Ref. [13] to look at a projection onto the z, Q2 plane. Tile contours are the lines at whichthe cross section d2m/clxdQ2 has falleu to 10-:~1‘osn Cmz, where n = 1, . . ., 6. From these figures, it hemmesevident that, even a.t the highest neutrino energies, (:c.)lltril~~ltic~rlsto the cross sect.ioI1 resulf.ing froth tl~eregimesensitive to gluon rec.(orlli>irlatio~leffects are marginal. For our example at E = iOl~ (;eV, flliv “s:l[{lr;~tjo]lregion” contributes far less than 1CR)to the tof.al rruss section.FIG. 1: The neutrino-nucleon cross section dza/CZZdQzat E = 1012GeV as a function of x, Q2. The “saturation region”is derived from Eq. (2).8 !-’ I I I Y1 I J765=--...@) -jE saturationregion \j., ~.]~ -lo -8 -6 -4 -2 0Ioglo(x)FIG. 2: Contour plot. of the neutrino-nucleon cross section in Fig. 1 in the x, Q2 plane.111. UNI.TARITY CONSIDERATIONSDims, Kretzer, Repko, and Schmidt in Ref. [3] have brought unitarity considerations to the fore. A rest.at.e-ment of the optical theorem relates the total ~le~ltrino-ll~lc.leorlcross section to the neutrino-nucleon forwardelastic scattering amplitude. The latter can be written in terms of the differential elastic cross section, evaluatedat. Nlanclelstam variable f = 0, which, with some approximations, yields:duellit>JJ– l(brmt~t “t=o(3)One can view this as a lower bouncl cm the forward scattering elastic cross section, or as an upper bound on thetotal cross section. In Ref. [3], the authors observe that. the inecluality is saturated at a relatively lCJWenergyby usiug the lowest order, G;, contribution for the elastic cross section on the left, and the (J? Contril’utio[lthat (’omes from the inclllsive {~r~~ss~(’t.ion (1) c)~ltl]e right. Specifically. Using the leading t,erm lrl (.~F for th(’elastic differential cross section, they conclude that4which already is violated for E ~ 2 x 108GeV. From this they deduce that at yet higher energies, where theright-hand side of Eq. (3) increases, while the left is constant (at 0 (G*)), previously neglected terms thatare higher order in the weak coupling g, in particular, g6 or gs terms, musi become important. They go onto suggest that this signals a breakdown in perturbation theory in the weak coupling, g. This is a strikingimplication indeed, especially given the small size of the cross section in Eq. (4).There is another, and we believe more natural, interpretation of the equality when E ~ 2 x 108GeV. First,we observe that the forward elastic cross section receives two qualitatively different and quantum mechanicallyincoherent contributions. The first of these describes the coherent elastic scattering of the entire nucleon throughweak vector boson exchange, which begins at tree level, that is, at G; in the cross section. The seccmd is thecontribution of high-Q2 virtual states that results from the incoherent scattering of partons. The li~tter, notthe former, is related independently by the optical theorem to the inelastic cross section on the right-hand sideof Eq. (3), and will saturate that inequality identically at order G$, regardless of its size, just. as at order G%the forward cross section is identical] y equal to the corresponding contribution from the square of the real part,which has been neglected on the right of Eq. (3).That being said, we may still ask whether the dominance of the partonic part of the cross section, higher-order by g2 compared to the elastic part, might not be a sign of large cent ributions from yet higher orders inthe weak coupling. Integrating the factorized form Eel. (1), over z and Q2, however, shows that at very highenergy the square of the total cross section behaves as G* times [g2(S/.M~, )X]2. This is to be compared toG; on the left-hand side of Eq. (3). The factor g2 is the default size of a higher-order electrowealc correction.The factor (S/Lf&)~ is due to the large number of partons of size 1/flfJv at z - ~f~v/S. For higher orders ing2 to contribute a.t a similar level, they would have to come accompanied by a similar large counting factor.At the leading power in I/.&fw, which is given by Eq. (1), this cannot happen, simply because q(x) and ~(x)already count the partons. It would still be possible if more partons are involved in the hard scattering, but thisinvolves going to higher twist, that is, to explicit suppression by additional powers of 1/11~}v, which would haveto be compensated for by higher-twist mnlti-parton matrix elements. While such contributions arc not, verywell-known even at low mornentnrn transfers, there is no experimental inclination of SLIC1llarge scales implicit,within the nucleon.The forgoing arguments, of course, assume that the unaidecl QCJ) extrapolations described above are equal tothe task of so many orders of magnitude. We have shown above the self-consistency of these extrapolate ~ons, andthat they do not, by themselves, lead to problems with unitarity, or give eviclence of a.breakdown in perturbationtheory in the weak coupling [17]. The very fact of the self-consistency of the Q~D extrapolations SIJOWSthat.ultra high energy neutrinos offer an exploration of the strong interactions, as well as of cosmic dynamics, intounprecedented length scales.AcknowledgmentsWe thank Duane Dicus, Stefan Kretzer, and Jamal ,Ja.]ilian-Marianfor cliscussions. “Thiswork was supported inpart by NSF Grants No. PHY-9802403 (M. H. R..), PHY-00!28527 (G.S. ), DOE contracts DE-FG02-9513R40906and DEF(203-93ER40792 (l.S. ). M.S. thanks SUNY Stony Brook and R.lKEN, and G.S. and M.S. thankBrookhaven National Laboratory for hospitality and support. W.V. is grateful to IUK EN, Brookhaven NationalLaboratory and the U.S. Department of Energy (contract rmmber DE-ACO2-98CH1O886) for providing thefacilities essential for the completion of his work.[1][2][3][4][5][6][7][8][9][to]R.tJ.Protheroe and P.A. Johnson, Astropart. Phys. 4, 253 (1996) [arXiv:astro-ph/9506119].For a review, see P. Bhattacharjee and C, Sigl, Phys. Rept,. 327, 109 (2000) [arXiv:astro-ph/98 110].1].D.A. Dicus, S. Kretzer, W.W. Repko, and C’. Schmi(it., Phys. Let.t. B5 14, 10:3 (‘2001) [arXiv:hep-ph/0103207].R. Gandhi, C. Quigg, M.H. Reno, and 1. Sarcevic, Phys. Rev. D58, 09:300!,1( 19!38) [ar~iv:tLer,-l]h/!~80~2G.1].M. Gliick, S. Kret.zer, and E. Reya, Astropart. Phys. 11, 327 ( 1999) [arXiv:astro-ph/980927’3].Y.M. .Andreev, V.S. Berezinsky, ancl .4.Y. 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ULTRAHIGH ENERGY NEUTRINOS, SMALL X AND UNITARITY.

Ultrahigh energy neutrinos, small x and unitarity

Ultrahigh energy neutrinos, small x and unitarity

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