CERN-TH-2017-013 Single top polarisation as a window to new physics J. A. Aguilar–Saavedraa, C. Degrandeb, S. Khatibic a Departamento de F´ısica Te´orica y del Cosmos, Universidad de Granada, E-18071 Granada, Spain b CERN, Theory Department, Geneva 23 CH-1211, Switzerland c School of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM) P.O. Box 19395-5531, Tehran, Iran 7 1 Abstract 0 2 We discuss the effect of heavy new physics, parameterised in terms of four- n a fermion operators, in the polarisation of single top (anti-)quarks in the t-channel J process at the LHC. It is found that for operators involving a right-handed top 0 2 quarkfieldtherelativeeffectonthelongitudinalpolarisationistwicelargerthanthe relativeeffectonthetotalcrosssection. Thisenhanceddependenceonpossiblefour- ] h fermion contributions makes the polarisation measurements specially interesting, in p - particular at high momenta. p e h [ Single top quark production is the sub-dominant source of top quarks in high-energy 1 collisions at the Large Hadron Collider (LHC). The three production modes, t-channel, v 0 s-channel and tW associated production, are sensitive to various manifestations of new 0 physics beyond the Standard Model (SM), such as new particles exchanged in the s or t 9 5 channels [1–4], a modified tbW interaction [5–8], top flavour-changing neutral currents [9– 0 . 15] or effective four-fermion interactions [8,16]. Most of the phenomenological studies 1 0 adressing the effect of new physics in single top production have concentrated on the total 7 1 cross section, whereas the top quark polarisation, which can be measured with similar or : v better precision, deserves a more detailed investigation. i X In the t-channel process, single top quarks are produced with a large longitudinal r a polarisationP (cid:39) 0.9inthedirectionzˆofthespectatorjetinthetopquarkrestframe[17], z and a slightly smaller polarisation P (cid:39) 0.8 for anti-quarks. (In the t-channel process z qg → q(cid:48)t¯b the spectator jet j is the light quark q(cid:48).) The top quark polarisation can also be measured along two additional orthogonal axes, the so-called “transverse” (xˆ, within the production plane) and “normal” (yˆ, orthogonal to it). The definitions of the three axes are [18] p(cid:126) p(cid:126) ×p(cid:126) j j q zˆ= , yˆ= , xˆ = yˆ×zˆ, (1) |p(cid:126) | |p(cid:126) ×p(cid:126) | j j q 1 with the direction of the initial quark q taken as the direction of the proton closest to the spectator jet. All momenta are taken in the top rest frame. In the SM, P (cid:39) 0 for x quarks and P (cid:39) 0.1 for anti-quarks at the tree level, and in both cases P = 0 because x y a normal polarisation requires a non-trivial complex phase in the amplitude. The effect in the top polarisation of new physics at the loop level, which manifests in the form of anomalous tbW couplings, has been studied before [18,19]. In this Letter we will fill the gap by studying the effect in the top polarisation of new heavy resonances, described in terms of four-fermion effective interactions.1 We work in a framework of the SM extended with an effective Lagrangian containing dimension-six gauge invariant effective operators O , i (cid:88) C i L = O +h.c., (2) eff Λ2 i i where Λ is the new physics scale and C are dimensionless coefficients. Conventionally, we i will add the Hermitian conjugate even if the operators O are Hermitian. As aforemen- i tioned, in this work we restrict ourselves to operators involving four quark fields. There is a huge number (185) of such operators contributing to single top production [16], so we need some guiding principle to select among the possible four-fermion contributions and still obtain results that are representative of the phenomenology induced by such operators. u d u d W b t b t Figure 1: Left: sample 2 → 2 diagram for t-channel single top production. Right: four- fermion contribution. Neglecting the charged-current mixing, the amplitudes for the leading SM t-channel single top processes involve the exchange of a W boson between two fermion lines, one with u and d quarks and the second one with b and t quarks, such as the 2 → 2 diagram depicted in Fig. 1 (left), plus diagrams related by crossing symmetry. There are sixteen independent four-fermion operators with these quark fields [16], depicted symbolically 1On-shell particles exchanged in the s channel obviously produce large effects in the top polarisation as well [20], if they are within kinematical reach. 2 in the right panel. All of them can contribute to these leading single top production processes. But not all of them are expected to give qualitatively different effects on the cross section and polarisation. There are six independent operators that involve the same colour flow as the SM ub → dt amplitude, namely the diagram in Fig. 1 (left), which can be taken as O(3,3311) = (q¯ γµτIq )(q¯ γ τIq ) ⊃ 2(t¯ γµb )(d¯ γ u ), qq L3 L3 L1 µ L1 L L L µ L O(1133) = (d¯ γµu )(t¯ γ b ), du R R R µ R O(1133) = (q¯ u )(t¯ q ) ⊃ (d¯ u )(t¯ b ), qu L1 R R L3 L R R L O(3311) = (q¯ b )(d¯ q ) ⊃ (t¯ b )(d¯ u ), qd L3 R R L1 L R R L O(1,3311) = (q¯ t )(cid:15)(q¯ d )T ⊃ −(¯b t )(u¯ d ), qud L3 R L1 R L R L R O(1,1133) = (q¯ u )(cid:15)(q¯ b )T ⊃ −(d¯ u )(t¯ b ). (3) qud L1 R L3 R L R L R We use standard notation with q = (u d )T the left-handed doublet of the ith genera- Li Li Li tion, τI the Pauli matrices, and (cid:15) = iτ2. Six additional operators O(1,3113) = (q¯ γµq )(q¯ γ q ) ⊃ 2(t¯ γµu )(d¯ γ b ), qq L3 L1 L1 µ L3 L L L µ L O(3113) = (t¯ γµu )(d¯ γ b ), ud R R R µ R O(3113) = (t¯ γµu )(q¯ γ q ) ⊃ (t¯ γµu )(d¯ γ b ), uq R R L1 µ L3 R R L µ L O(1331) = (d¯ γµb )(q¯ γ q ) ⊃ (d¯ γµb )(t¯ γ u ), dq R R L3 µ L1 R R L µ L O(1,1331) = (q¯ t )(cid:15)(q¯ d )T ⊃ (u¯ t )(¯b d ), qud L1 R L3 R L R L R O(1,3113) = (q¯ u )(cid:15)(q¯ b )T ⊃ (t¯ u )(d¯ b ), (4) qud L3 R L1 R L R L R givesimilartermsbutwiththecolourindicescontractedbetweent,uandd,b,respectively. For the operators with two or more right-handed fields this difference has no effect in the colour-averaged differential cross sections at the partonic level, because the diagrams including four-fermion operators do not interfere with the SM amplitudes for massless u and d quarks. On the other hand, for the two operators with four left-handed fields, the interference of O(1,3113) is damped by a colour factor of 1/3 while for O(3,3311) it is not; qq qq however, neither of these operators change the top polarisation significantly, therefore the former operator does not lead to a qualitatively different phenomenology either. The remaining four operators O(8,3311) = (q¯ λat )(cid:15)(q¯ λad )T ⊃ −(¯b λat )(u¯ λad ), qud L3 R L1 R L R L R O(8,1133) = (q¯ λau )(cid:15)(q¯ λab )T ⊃ −(d¯ λau )(t¯ λab ), qud L1 R L3 R L R L R O(8,1331) = (q¯ λat )(cid:15)(q¯ λad )T ⊃ (u¯ λat )(¯b λad ), qud L1 R L3 R L R L R O(8,3113) = (q¯ λau )(cid:15)(q¯ λab )T ⊃ (t¯ λau )(d¯ λab ) (5) qud L3 R L1 R L R L R 3 are also expected to yield similar results up to rescaling factors, because they do not interfere with the SM amplitude. Wedonotconsiderherethe80additionalfour-fermionoperatorsobtainedbyreplacing u → c and/or d → s,b in the former set. Either their effect is smaller than the ones considered, because of parton distribution functions (PDFs), if the second generation or b quarks are in the initial state, or the effect in the cross section and polarisation is similar, if they are in the final state. By the same reckoning, the effect of flavour-changing neutral four-fermion operators (involving three charge-2/3 light quarks plus a top quark) is expected to be quite similar. Therefore, we restrict our study to the six operators in (3), which generate the effective Lagrangian 1 L = (cid:2)C (t¯ γµb )(d¯ γ u )+C (t¯ γµb )(d¯ γ u )+C (t¯ b )(d¯ u ) eff Λ2 qq L L L µ L du R R R µ R qu R L L R +C (t¯ b )(d¯ u )−C (t¯ b )(d¯ u )−C (t¯ b )(d¯ u )(cid:3)+h.c. (6) qd L R R L qudR R L R L qudL L R L R In order to ease the notation, we have dropped flavour superindices in the effective oper- ator coefficients and introduced a chirality label to distinguish the coefficients of the two O operators, C = C(1,3311)∗, C = C(1,1133). A factor of 4 has been absorbed in qud qudR qud qudL qud the definition of C . qq This Lagrangian is implemented in Feynrules [21] in order to perform our calcula- tions with MadGraph5 aMC@NLO [22]. We use a centre-of-mass energy of 13 TeV. ¯ ¯ ¯ ¯ ¯ Our results include ug → tbd, dg → tbu¯ and also ud → tbg, which is s-channel pro- duction with the radiation of an extra gluon, and the charge conjugate processes for top anti-quark production. The latter process has a collinear enhancement for low transverse momentum (p ) of the spectator jet, in this case the gluon; Since experimental measure- T ments require the presence of the spectator jet we set a lower cut p ≥ 20 GeV. We use T the CTEQ6L1 PDFs [23] and the default factorisation and renormalisation scales in the generator. Processes with initial s, c quarks are included as well but without four-fermion operator contributions, which as seen in Eq. (6) only involve u, d, t and b fields. We only study the effect of one effective operator at a time, because their interference vanishes in most cases, or is suppressed by the b quark mass otherwise. Therefore, the simultaneous inclusion of two or more operators does not lead to qualitatively new effects. The operator coefficients are always taken real. (C is always real due to the Hermiticity qq of the effective operator.) We have not found significant variations in the transverse polarisation P nor, for complex operator coefficients, in the normal polarisation P . x y The dependence of the total cross section and longitudinal polarisation on the effective 4 operator coefficients is given by C (cid:18) C (cid:19)2 σ = A +A +A , 0 int Λ2 2 Λ2 (cid:34) (cid:35) C (cid:18) C (cid:19)2 P = B +B +B /σ, (7) z 0 int Λ2 2 Λ2 with A = 93.7 pb (51.6 pb) the SM cross section for top quarks (anti-quarks) and 0 B = 81.6 pb (41.8 pb). The values of the remaining coefficients are collected in Table 1. 0 They are extracted from quadratic fits to σ and σP , numerically obtained with the z Monte Carlo calculations for several values of each coefficient. The dependence of the cross section and polarisation is depicted in Fig. 2. For the operators involving right- handed fields t , it is seen that the effect in the top quark polarisation is twice larger R than in the total cross section, the reason being that for these operators A ∼ −B . 2 2 t t¯ B B A A B B A A int 2 int 2 int 2 int 2 C −11.8 9.3 −11.9 10.8 −4.1 2.3 −4.8 3.7 qq C 0 −9.2 0 10.8 0 −3.0 0 3.7 du C , C 0 −5.4 0 6.0 0 −2.1 0 2.3 qu qudR C , C 0 5.3 0 6.0 0 1.8 0 2.3 qd qudL Table 1: Numerical values of the coefficients in Eqs. (7) for the different four-fermion operators. A , B are given in units of pb TeV2, and A , B in units of pb TeV4. int int 2 2 The relative effect of four-fermion contributions on the single top cross section and polarisation is compactly presented in Fig. 3. As for the total cross section we take a 10% relative error, assuming some improvement over current measurements at 13 TeV, σ = 156± 5 (stat)± 27 (syst) pb, σ = 91 ± 4 (stat) ± 18 (syst) pb [24], σ = 232± t t¯ t+t¯ 13 (stat)±28 (syst) pb [25]. Polarisation measurements have not yet been performed with Run2data,butforthefullRun1datasetwehaveP = 0.96±0.05 (stat)±0.10 (syst)[26]. z Taking into account the fact that part of the systematic uncertainty in the polarisation measurementresultsfromthesizeofMonteCarlosamples,wewillassumeanimprovement to a 5% uncertainty in its measurement. Under these conditions, it is clear that P is a far z more sensitive probe of the presence of four-fermion contributions involving right-handed top fields than the total cross section: not only the variation in P is larger but also the z precision in its measurement is better. As it is well known, the effect of four-fermion interactions at high energies is enhanced, because these non-renormalisable terms do not include a propagator as the SM contri- butions do. Setting for instance a lower cut p ≥ 200 GeV on the top quark transverse T 5 1.3 1.3 t σ / σSM σ / σSM 1.2 t P / PSM 1.2 P / PSM t 1.1 1.1 t 1.0 1.0 t 0.9 t 0.9 0.8 0.8 t 0.7 0.7 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 C C qq du 1.3 1.3 σ / σSM σ / σSM 1.2 P / PSM 1.2 P / PSM 1.1 1.1 t t 1.0 1.0 t ≈t t t 0.9 0.9 0.8 0.8 0.7 0.7 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 C , C C , C qu qudR qd qudL Figure 2: Dependence of the cross section and polarisation on effective operator coeffi- cients, normalised to their SM values. 1 1 0.9 0.9 dutb SM RLLR dutb L RLR SM 0.8 dLuLtLbL ddRuuLttLbbR 0.8 Pz L RL R Pz dutb L LL L 0.7 dLuRtRbL 0.7 dutb dutb L RRL RLRL dutb RLRL 0.6 dRuRtRbR 0.6 dRuRtRbR single t single t 0.5 0.5 0.7 0.8 0.9 1 1.1 1.2 1.3 0.7 0.8 0.9 1 1.1 1.2 1.3 σ / σ σ / σ SM SM Figure3: Effectoffour-fermioncontributionsinthesingletopcrosssection(normalisedto the SM value) and polarisation, for top quarks (left) and anti-quarks (right). The black dots and ellipses represent the SM predictions and expected uncertainties. The points corresponding to operator coefficients C/Λ2 = 1 TeV−2 are indicated. momentum, the SM contributions are reduced by an order of magnitude, A = 4.0 pb, 0 B = 2.7 pb for top quarks, and A = 1.8, B = 1.2 for anti-quarks. The interference with 0 0 0 four-fermion operators is similarly suppressed, while the quadratic four fermion contribu- 6 tions are of the same order, see Table 2. Therefore, the sensitivity to the presence of these contributions is greatly enhanced. For example, for C /Λ2 = 0.2 TeV−2, the variation in du the polarisation is as large as 20% of its value. Of particular interest are these deviations in the longitudinal polarisation: in contrast with the total cross section, they are expected to be rather insensitive to the uncertainties on the PDFs at the tails of the distributions, although the measurement for very boosted tops may involve other uncertainties. An estimate of the precision in high-p polarisation measurements is difficult, and we look T forward to actual measurements by the ATLAS and CMS Collaborations. t t¯ B B A A B B A A int 2 int 2 int 2 int 2 C −0.6 7.1 −0.7 8.2 ∼ 0 2.2 ∼ 0 2.6 qq C 0 −7.2 0 8.2 0 −2.4 0 2.6 du C , C 0 −4.6 0 4.8 0 −1.5 0 1.8 qu qudR C , C 0 4.4 0 4.8 0 1.6 0 1.8 qd qudL Table 2: Numerical values of the coefficients in Eqs. (7) for the different four-fermion operators, with a lower cut p ≥ 200 GeV for the top quark. A , B are given in units T int int of pb TeV2, and A , B in units of pb TeV4. 2 2 To summarise, in this Letter we have addressed the influence of four-fermion contri- butions in the cross section and polarisation of top quarks in the t-channel process at the LHC. The possible effect of operators involving t in the longitudinal polarisation R is significant, especially at high energies. Our results highlight the importance of polar- isation measurements, and especially motivate further measurements at high transverse momenta. Suchmeasurementswillbefeasiblewiththehighstatisticsthatwillbeachieved at the LHC Run 2. Acknowledgements S.K. thanks the CERN Theory Department and the University of Granada for their hospitality during the realisation of this work. This research has been supported by MINECO Projects FPA 2016-78220-C3-1-P, FPA 2013-47836-C3-2-P (including ERDF), Junta de Andaluc´ıa Project FQM-101, Iran National Science Foundation (INSF) and European Commission through the contract PITN-GA-2012-316704 (HIGGSTOOLS). 7 References [1] T. M. P. Tait and C.-P. Yuan, Phys. Rev. D 63 (2000) 014018 [hep-ph/0007298]. [2] E. Boos, V. Bunichev, L. Dudko and M. Perfilov, Phys. Lett. B 655 (2007) 245 [hep-ph/0610080]. [3] A.Arhrib, K.Cheung, C.W. Chiang andT.C.Yuan, Phys.Rev.D73(2006)075015 [hep-ph/0602175]. [4] E. Drueke, J. Nutter, R. Schwienhorst, N. Vignaroli, D. G. E. Walker and J. H. Yu, Phys. Rev. D 91 (2015) no.5, 054020 [arXiv:1409.7607 [hep-ph]]. [5] E. Boos, L. Dudko and T. Ohl, Eur. Phys. J. C 11 (1999) 473 [hep-ph/9903215]. [6] C. R. Chen, F. Larios and C.-P. Yuan, Phys. Lett. B 631 (2005) 126 [hep- ph/0503040]. [7] J. A. Aguilar-Saavedra, Nucl. Phys. B 804 (2008) 160 [arXiv:0803.3810 [hep-ph]]. [8] C. Zhang and S. Willenbrock, Phys. Rev. D 83 (2011) 034006 [arXiv:1008.3869 [hep- ph]]. [9] M. Hosch, K. Whisnant and B. L. Young, Phys. Rev. D 56 (1997) 5725 [hep- ph/9703450]. [10] T. Han, M. Hosch, K. Whisnant, B. L. Young and X. Zhang, Phys. Rev. D 58 (1998) 073008 [hep-ph/9806486]. [11] N. Kidonakis and A. Belyaev, JHEP 0312 (2003) 004 [hep-ph/0310299]. [12] P. M. Ferreira, O. Oliveira and R. Santos, Phys. Rev. D 73 (2006) 034011 [hep- ph/0510087]. [13] P. M. Ferreira and R. Santos, Phys. Rev. D 73 (2006) 054025 [hep-ph/0601078]. [14] J. A. Aguilar-Saavedra, Nucl. Phys. B 837 (2010) 122 [arXiv:1003.3173 [hep-ph]]. [15] S. M. Etesami and M. Mohammadi Najafabadi, Phys. Rev. D 81 (2010) 117502 [arXiv:1006.1717 [hep-ph]]. [16] J. A. Aguilar-Saavedra, Nucl. Phys. B 843 (2011) 638 Erratum: [Nucl. Phys. B 851 (2011) 443] [arXiv:1008.3562 [hep-ph]]. [17] G. Mahlon and S. J. Parke, Phys. Lett. B 476 (2000) 323 [hep-ph/9912458]. 8 [18] J. A. Aguilar-Saavedra and S. Amor Dos Santos, Phys. Rev. D 89 (2014) no.11, 114009 [arXiv:1404.1585 [hep-ph]]. [19] D. Espriu and J. Manzano, Phys. Rev. D 66 (2002) 114009 [hep-ph/0209030]. [20] S. Gopalakrishna, T. Han, I. Lewis, Z. g. Si and Y. F. Zhou, Phys. Rev. D 82 (2010) 115020 [arXiv:1008.3508 [hep-ph]]. [21] A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks, Comput. Phys. Commun. 185 (2014) 2250 [arXiv:1310.1921 [hep-ph]]. [22] J. Alwall et al., JHEP 1407 (2014) 079 [arXiv:1405.0301 [hep-ph]]. [23] P. M. Nadolsky, H. L. Lai, Q. H. Cao, J. Huston, J. Pumplin, D. Stump, W. K. Tung and C.-P. Yuan, Phys. Rev. D 78 (2008) 013004 [arXiv:0802.0007 [hep-ph]]. [24] M. Aaboud et al. [ATLAS Collaboration], arXiv:1609.03920 [hep-ex]. [25] A. M. Sirunyan et al. [CMS Collaboration], [arXiv:1610.00678 [hep-ex]]. [26] ATLAS collaboration, ATLAS-CONF-2016-097. 9