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Isovector EMC Effect and the NuTeV Anomaly

I. C. Cloet

W. Bentz

A. W. Thomas

A. W. Thomas William & Mary

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Recommended Citation Cloet, I. C., Bentz, W., & Thomas, A. W. (2009). Isovector EMC effect and the NuTeV anomaly. Physical review letters, 102(25), 252301.

This Article is brought to you for free and open access by the Arts and Sciences at W&M ScholarWorks. It has been accepted for inclusion in Arts & Sciences Articles by an authorized administrator of W&M ScholarWorks. For more information, please contact [email protected]. week ending PRL 102, 252301 (2009) PHYSICAL REVIEW LETTERS 26 JUNE 2009

Isovector EMC Effect and the NuTeVAnomaly

I. C. Cloe¨t,1 W. Bentz,2 and A. W. Thomas3 1Department of Physics, University of Washington, Seattle, Washington 98195-1560, USA 2Department of Physics, School of Science, Tokai University, Hiratsuka-shi, Kanagawa 259-1292, Japan 3Jefferson Lab, 12000 Jefferson Avenue, Newport News, Virginia 23606, USA and College of William and Mary, Williamsburg, Virginia 23187, USA (Received 28 January 2009; published 26 June 2009) A neutron or proton excess in nuclei leads to an isovector-vector mean field which, through its coupling to the quarks in a bound nucleon, implies a shift in the quark distributions with respect to the Bjorken scaling variable. We show that this result leads to an additional correction to the NuTeV measurement of 2 sin W . The sign of this correction is largely model independent and acts to reduce their result. Explicit calculation in nuclear matter within a covariant and confining Nambu–Jona-Lasinio model predicts that this vector field correction may account for a substantial fraction of the NuTeV anomaly. We are therefore led to offer a new interpretation of the NuTeV measurement, namely, that it provides further evidence for the medium modification of the bound nucleon wave function.

DOI: 10.1103/PhysRevLett.102.252301 PACS numbers: 24.85.+p, 11.80.Jy, 13.60.Hb, 21.65.Cd Within relativistic, quark-level models of nuclear struc- becomes ture, the mean scalar and vector fields in the medium ð1 4 sin2 Þhx uiþð1 2 sin2 Þhx di generate fundamental changes in the internal structure of R ¼ 6 9 W A A 6 9 W A A ; PW hx di1 hx ui bound hadrons. These modifications lead to a good de- A A 3 A A scription of the EMC effect in finite nuclei and predict a (2) more dramatic modification of the bound nucleon spin x structure function [1–3]. We show that in nuclei with N Þ where A is the Bjorken scaling variable of the nucleus A h...i x Z this approach leads to interesting and hitherto unex- multiplied by , implies integration over A, and q q q plored effects connected with the isovector-vector mean A A A are the nonsinglet quark distributions of field, which is usually represented by the 0, and is in part the target. responsible for the symmetry energy. In a nucleus such as Ignoring quark mass differences and possible electro- u d 56Fe or 208Pb where N>Z, the 0 field will cause the weak corrections the - and -quark distributions of an u quark to feel a small additional vector attraction and the isoscalar target will be identical, and in this limit Eq. (2) d quark to feel additional repulsion. becomes In this Letter we explore the way in which this additional N¼Z 1 R ! sin2 : vector field modifies the traditional EMC effect. However, PW 2 W (3) there is an even more important issue which is our main focus. Even though the 0 mean field is completely con- If corrections to Eq. (3) are small the PW ratio provides a sistent with charge symmetry, the familiar assumption that unique way to measure the Weinberg angle. u ðxÞ¼d ðxÞ d ðxÞ¼u ðxÞ Motivated by Eq. (3) the NuTeV Collaboration extracted p n and p n will clearly fail for a 2 a value of sin W from neutrino and antineutrino DIS on an nucleon bound in a nucleus with N Þ Z. Therefore cor- 2 0 iron target [6], finding sin W ¼ 0:2277 0:0013ðstatÞ recting for the field is absolutely critical in a situation 0:0009ðsystÞ. The three-sigma discrepancy between this where symmetry arguments are essential, such as the use of 2 result and the world average [7], namely sin W ¼ N Þ Z nuclear data from and deep inelastic scattering 2 0:2227 0:0004, is the NuTeV anomaly. Some authors (DIS) to extract sin W via the Paschos-Wolfenstein (PW) have speculated that the NuTeV anomaly supports the relation [4]. Indeed, we show that the deviation from the existence of physics beyond the standard model [8]. naive application of charge symmetry to the and data 56 Standard model corrections to the NuTeV result have on Fe may naturally explain the famous NuTeVanomaly. largely been focused on nucleon charge symmetry violat- The PW ratio is defined by [5] ing effects [9] and a nonperturbative strange quark sea [8]. A A Charge symmetry violation, arising from the u- and R ¼ NC NC ; PW A A (1) d-quark mass differences, is probably the best understood CC CC and constrained correction and can explain approximately where A represents the target, NC indicates weak neutral one-third of the NuTeV anomaly [10]. Standard nuclear current, and CC weak charged current interaction. corrections such as Fermi motion, binding, and off-shell Expressing the cross sections in terms of quark distribu- effects are found to be small [11]. However, effects from tions and ignoring heavy flavor contributions, the PW ratio the medium modification of the bound nucleon, in particu-

0031-9007=09=102(25)=252301(4) 252301-1 Ó 2009 The American Physical Society week ending PRL 102, 252301 (2009) PHYSICAL REVIEW LETTERS 26 JUNE 2009 lar, the impact of the 0 field, have hitherto not been characterized by a 4-fermion contact interaction between explored in relation to the NuTeV anomaly. These effects the quarks. The NJL model has a long history of success in are potentially important because they are now widely describing mesons as qq bound states [18] and more accepted as an essential ingredient in explaining the recently as a self-consistent model for free and in-medium EMC effect [12]. baryons [2,3,13,19]. The original 4-fermion interaction In our approach, presented in Refs. [2,3,13], the scalar term in the NJL Lagrangian can be decomposed into and vector mean fields inside a nucleus couple to the various qq and qq interaction channels via Fierz trans- quarks in the bound nucleons and self-consistently modify formations [20], where the relevant terms to this discussion their internal structure. The influence of the vector fields on are given in Ref. [2]. the quark distributions arises from the nonlocal nature of The scalar qq interaction term generates the scalar field, the quark bilinear in their definition [13]. This leads to a which dynamically generates a constituent quark mass via largely model independent result for the modification of the gap equation. The vector qq interaction terms are used the in-medium parton distributions of a bound nucleon by to generate the isoscalar-vector, !0, and isovector-vector, the vector mean fields [13–15], namely 0, mean fields in-medium. The qq interaction terms give t pþ pþ Vþ the diquark matrices whose poles correspond to the scalar qðxÞ¼ q x q : and axial-vector diquark masses. The nucleon vertex func- pþ Vþ 0 pþ Vþ pþ Vþ (4) tion and mass are obtained by solving the homogeneous The subscript 0 indicates the absence of vector fields and Faddeev equation for a quark and a diquark, where the pþ is the nucleon light cone plus component of momen- static approximation is used to truncate the quark exchange þ þ kernel [19]. To regularize the NJL model we choose the tum. The quantities V and Vq are the light cone plus component of the net vector field felt by the nucleon and a proper-time scheme, which enables the removal of unphys- quark of flavor q, respectively. ical thresholds for nucleon decay into quarks, and hence Before embarking on explicit calculations, we first ex- simulates an important aspect of confinement [21,22]. plore the model independent consequences of Eq. (4) for To self-consistently determine the strength of the mean the PW ratio and the subsequent NuTeV measurement of scalar and vector fields, an equation of state for nuclear 2 matter is derived from the NJL Lagrangian, using hadro- sin W. The NuTeV experiment was performed on a pre- 56 nization techniques [22]. In a mean field approximation the dominately Fe target, and therefore isoscalarity correc- 2 2 !0 0 result for the energy density is [22] E ¼ EV þ tions need to be applied to the PW ratio before extracting 4G! 4G sin2 W. Isoscalarity corrections to Eq. (3) for small isospin Ep þ En, where G! and G are the qq couplings in the asymmetry have the general form isoscalar-vector and isovector-vector channels, respec- 7 hx u x di tively. The vacuum energy EV has the familiar Mexican R ’ 1 sin2 A A A A ; PW 3 W hx u þ x di (5) hat shape and the energies of the protons and neutrons A A A A moving through the mean scalar and vector fields are 2 where the Q dependence of this correction resides com- labeled by Ep and En, respectively. The corresponding 2 " ¼ E þ V ¼ pletely with sin W. NuTeV perform what we term naive qprotonffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and neutron Fermi energies are F F isoscalarity corrections, where the neutron excess correc- 2 2 MN þ pF þ 3!0 0, where ¼ p or n, the plus tion is determined by assuming that the target is composed sign refers to the proton, MN is the in-medium nucleon of free nucleons [16]. However, there are also isoscalarity p corrections from medium effects, in particular, from the mass, and F the nucleon Fermi momentum. Minimizing the effective potential with respect to each vector field medium modification of the structure functions of every ! ¼ 6G ð þ Þ nucleon in the nucleus, arising from the isovector 0 field. gives the following useful relations: 0 ! p n 0 ¼ 2G ð Þ For nuclei with N>Z the field develops a nonzero and 0 p n , where p is the proton and n expectation value that results in Vu

Z/N Z/N Z/N Z/N Z/N Z/N Z/N Z/N Z/N Z/N Z/N

FIG. 1 (color online). Isospin dependence of the EMC effect FIG. 2 (color online). Isospin dependence of the EMC effect for proton-neutron ratios greater than one. The data are from for proton-neutron ratios less than one. Ref. [31] and correspond to N ¼ Z nuclear matter. less modified while medium modification of the d-quark masses in the expressions for the free parton distributions distribution is enhanced. Since the EMC effect is domi- discussed in Ref. [19]. To include the nucleon Fermi nated by the u quarks it decreases. The isospin dependence motion, the quark distributions modified by the scalar field of the EMC effect for nuclear matter with Z=N < 1 is are convoluted with the appropriate Fermi smearing func- given in Fig. 2. Here the medium modification of the tion, namely u-quark distribution is enhanced, while the d-quark distri-   bution is modified less by the medium. Since the EMC N 3 M^ 3 p 2 E 2 f ðy Þ¼ N F F y ; ratio is initially dominated by the u quarks the EMC effect 0 A A 4 p ^ ^ A (6) F MN MN first increases as Z=N decreases from one. However, even- d N ¼ Z N ¼ N M^ ¼ Z E þ N E tually the -quark distribution dominates the ratio and at where p , n , and N A Fp A Fn. this stage the EMC effect begins to decrease in the valence Vector field effects can be included in Eq. (6) by the quark region. We find a maximal EMC effect for Z=N ’ E ! " M^ ! M ¼ Z " þ substitutions F F and N N A Fp 0:6, which is slightly less than the proton-neutron ratio in N " A Fn. Our final result for the infinite asymmetric nuclear Pb. This isospin dependence is clearly an important factor matter quark distributions, which includes vector field in understanding the A dependence of the EMC effect, even effects on both the quark distributions in the bound nucleon after standard neutron excess corrections are applied. and on the nucleon smearing functions, is given by Now we turn to the consequences of the isospin depen- M M V dence of the EMC effect for the NuTeV measurement of q ðx Þ¼ N q N x q : sin2 A A ^ A0 ^ A ^ (7) W. The NuTeV experiment was performed on an iron MN MN MN target, which, because of impurities had a neutron excess Z=N The subscript A0 indicates a distribution which includes of 5.74% [6]. Choosing our ratio to give the same effects from Fermi motion and the scalar mean field. The neutron excess, we use our medium modified quark distri- distributions calculated in this way are then evolved [23] butions and Eq. (5) to determine the full isoscalarity cor- 2 2 rection to the isoscalar PW ratio, given by Eq. (3). Using from the model scale, Q0 ¼ 0:16 GeV , to an appropriate 2 the standard model value for the Weinberg angle we obtain Q for comparison with experimental data. RPW ¼0:0139. If we break this result into the three The EMC effect is defined by the ratio separate isoscalarity corrections, by using Eq. (5) and the F F 4u þ d R ¼ 2A ¼ 2A ’ A A ; various stages of modification of the in-medium quark naive (8) F2A ZF2p þ NF2n 4uf þ df distributions, we find

0 qA qf naive Fermi where are the quark distributions of the target and are RPW ¼ RPW þ RPW þ RPW the distributions of the target if it was composed of free ¼ð0:0107 þ 0:0004 þ 0:0028Þ: nucleons. Results for the isospin dependence of the EMC (9) effect are given in Figs. 1 and 2. Figure 1 illustrates the EMC effect for proton rich Higher order corrections to Eq. (5) do not change this matter, where we find a decreasing effect as Z=N increases. result. The NuTeV analysis includes the naive isoscalarity An intuitive understanding of this result may be obtained correction [16] but is missing the medium corrections. The 0 by realizing that it is a consequence of binding effects at new correction of RPW ¼0:0028 would account for the quark level. For Z=N > 1 the 0 field is positive, which almost two-thirds of the NuTeV anomaly. means Vu >Vd and hence the u quarks are less bound than To estimate the effect on the NuTeV experiment, we use the d quarks. Therefore the u-quark distribution becomes the standard classical nuclear matter approximation for an 252301-3 week ending PRL 102, 252301 (2009) PHYSICAL REVIEW LETTERS 26 JUNE 2009 iron nucleus, based on the quasielastic electron scattering [4] E. A. Paschos and L. Wolfenstein, Phys. Rev. D 7,91 results of Ref. [24]. In practice this means that we rescale (1973). the nuclear matter density by 0.89. The 0 field depends [5] The cross sections in Eq. (1) have been integrated over the Bjorken scaling variable and energy transfer. The Q2 linearly on the density, and therefore a first order estimate sin2 of the isovector correction for an ironlike nucleus, that is, a dependence of the PW ratio resides with W . [6] G. P. Zeller et al., Phys. Rev. Lett. 88, 091802 (2002); 90, finite nucleus with the same neutron excess as the NuTeV 0 239902(E) (2003). experiment, would be RPW !0:89 0:0028 ¼ [7] D. Abbaneo et al., arXiv:hep-ex/0112021. 0:0025. Another approach is to take our medium modi- [8] S. Davidson, S. Forte, P. Gambino, N. Rius, and A. fied nucleon distributions and use the NuTeV CSV func- Strumia, J. High Energy Phys. 02 (2002) 037. 0 274 tional given in Ref. [25], this gives R !0:0021. [9] E. Sather, Phys. Lett. B , 433 (1992); E. N. Rodionov, PW A. W. Thomas, and J. T. Londergan, Mod. Phys. Lett. A 9, Therefore we conclude that medium effects, in particular, 0 1:5 1799 (1994). a nonzero field, can explain approximately of the [10] J. T. Londergan and A. W. Thomas, Phys. Lett. B 558, 132 NuTeV anomaly. (2003). If we also include the well constrained charge symmetry [11] S. A. Kulagin, Phys. Rev. D 67, 091301 (2003); S. A. CSV violation (CSV) correction, RPW ¼0:0017 [10], Kulagin and R. Petti, Phys. Rev. D 76, 094023 (2007). which originates from the quark mass differences, we [12] J. R. Smith and G. A. Miller, Phys. Rev. C 65, 055206 medium CSV have a total correction of RPW þ RPW ’0:0045. (2002); D. F. Geesaman, K. Saito, and A. W. Thomas, The combined correction largely accounts for the NuTeV Annu. Rev. Nucl. Part. Sci. 45, 337 (1995). [13] H. Mineo, W. Bentz, N. Ishii, A. W. Thomas, and K. anomaly [26]. Corrections to the NuTeV result from A735 s quarks may also be important; however, current experi- Yazaki, Nucl. Phys. , 482 (2004). [14] F. M. Steffens, K. Tsushima, A. W. Thomas, and K. Saito, mental uncertainties are too large to draw any firm con- Phys. Lett. B 447, 233 (1999). clusions [8,27]. [15] W. Detmold, G. A. Miller, and J. R. Smith, Phys. Rev. C Since our nuclear matter calculation suggests that CSV 73, 015204 (2006). and medium modification corrections largely explain the [16] NuTeV do not directly utilize Eq. (5) for their naive discrepancy between the NuTeV result and the standard isoscalarity correction, because in their case, details of model, we propose that this NuTeV measurement provides this correction depend explicitly on the Monte Carlo rou- strong evidence that the nucleon is modified by the nuclear tine used to analyze their data. medium, and should not be interpreted as an indication of [17] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 physics beyond the standard model. In our opinion this (1961); Y. Nambu and G. Jona-Lasinio, Phys. Rev. 124, 246 (1961). conclusion is equally profound since it may have funda- 27 mental consequences for our understanding of traditional [18] U. Vogl and W. Weise, Prog. Part. Nucl. Phys. , 195 (1991); T. Hatsuda and T. Kunihiro, Phys. Rep. 247, 221 nuclear physics. We stress that the physics presented in this 0 (1994). Letter, in particular, the effects of the mean field, are [19] I. C. Cloe¨t, W. Bentz, and A. W. Thomas, Phys. Lett. B consistent with existing data [28–30], but can strongly 621 0 , 246 (2005). influence other observables. For example, the field gives [20] N. Ishii, W. Bentz, and K. Yazaki, Nucl. Phys. A587, 617 rise to a strong flavor dependence of the EMC effect, and (1995). there is an excellent chance this effect will be measured in [21] D. Ebert, T. Feldmann, and H. Reinhardt, Phys. Lett. B future experiments at, for example, Jefferson Lab. 388, 154 (1996); G. Hellstern, R. Alkofer, and H. I. C. thanks Jerry Miller for helpful discussions. This Reinhardt, Nucl. Phys. A625, 697 (1997). work was supported by the U.S. Department of Energy [22] W. Bentz and A. W. Thomas, Nucl. Phys. A696, 138 (2001). under Grant No. DEFG03-97ER4014 and by Contract 94 No. DE-AC05-06OR23177, under which Jefferson [23] M. Miyama and S. Kumano, Comput. Phys. Commun. , 185 (1996). Science Associates, LLC operates Jefferson Laboratory [24] E. J. Moniz, I. Sick, R. R. Whitney, J. R. Ficenec, R. D. and by the Grant in Aid for Scientific Research of the Kephart, and W. P. Trower, Phys. Rev. Lett. 26, 445 Japanese Ministry of Education, Culture, Sports, Science (1971). and Technology, Project No. C-19540306. [25] G. P. Zeller et al., Phys. Rev. D 65, 111103 (2002). [26] The NuTeV result for RPW after naive isoscalarity correc- 1 2 tion is 0.2723 which differs from 2 sin W ¼ 0:2773 by the amount 0:005. [1] K. Saito, A. Michels, and A. W. Thomas, Phys. Rev. C 46, [27] D. Mason et al., Phys. Rev. Lett. 99, 192001 (2007). R2149 (1992); J. R. Smith and G. A. Miller, Phys. Rev. C [28] S. Kumano, Phys. Rev. D 66, 111301 (2002). 72, 022203 (2005). [29] M. Hirai, S. Kumano, and T. H. Nagai, Phys. Rev. D 71, [2] I. C. Cloe¨t, W. Bentz, and A. W. Thomas, Phys. Lett. B 113007 (2005). 642, 210 (2006). [30] K. J. Eskola and H. Paukkunen, J. High Energy Phys. 06 [3] I. C. Cloe¨t, W. Bentz, and A. W. Thomas, Phys. Rev. Lett. (2006) 008. 95, 052302 (2005). [31] I. Sick and D. Day, Phys. Lett. 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