Vector boson production in association with KK modes of the ADD model to NLO in QCD at LHC
aa r X i v : . [ h e p - ph ] A p r Vector boson production in association with KKmodes of the ADD model to NLO in QCD at LHC
M. C. Kumar a , Prakash Mathews b , V. Ravindran a , Satyajit Seth b , a Regional Centre for Accelerator-based Particle PhysicsHarish-Chandra Research Institute, Chhatnag Road, Jhunsi,Allahabad 211 019, India b Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata 700 064, India
Abstract
Next-to-leading order QCD corrections to the associated production of vectorboson ( Z / W ± ) with the the Kaluza-Klein modes of the graviton in large ex-tra dimensional model at the LHC, are presented. We have obtained variouskinematic distributions using a Monte Carlo code which is based on the twocut off phase space slicing method that handles soft and collinear singularitiesappearing at NLO level. We estimate the impact of the QCD corrections onvarious observables and find that they are significant. We also show the re-duction in factorization scale uncertainty when QCD corrections are included. Key words: Large Extra Dimensions, NLO QCD [email protected] [email protected] [email protected] [email protected] Introduction
With the on set of the Large Hadron Collider (LHC) era, unique opportunity toprobe the realm of new physics in the TeV scale has begun. Models with extraspatial dimensions and TeV scale gravity, are proposed to address the large hierarchybetween the electroweak and Planck scale and are expected to provide a plethora ofnew and interesting signals.The extra dimension model proposed by Arkani-Hamed, Dimopoulos and Dvali(ADD) [1], was the first extra dimension model in which the compactified dimen-sions could be of macroscopic size and consistent with present experiments. A viablemechanism to hide the extra spatial dimension, is to introduce a 3-brane with negli-gible tension and localise the Standard Model (SM) particles on it. Only gravity isallowed to propagate in the full 4 + δ dimensional space time. As a consequence ofthese assumptions, it follows from Gauss Law that the effective Planck scale M P in4-dimensions is related to the 4 + δ dimensional fundamental scale M S through thevolume of the compactified extra dimensions [1]. The extra dimensions are compact-ified on a torus of common circumference R S . The compactified extra dimensionsare flat, of equal size and could be large. The large volume of the compactified extraspatial dimensions would account for the dilution of gravity in 4-dimensions andhence the hierarchy. Current experimental limits on deviation from inverse squarelaw [2], constraint the number of possible extra spatial dimensions δ ≥
2. The spacetime is factorisable and the 4-dimensional spectrum consists of the SM confined to4-dimensions and a tower of Kaluza-Klein (KK) modes, of the graviton propagatingin the full 4 + δ dimensional space time.The interaction of the KK modes G ( ~n ) µν with the SM fields localised on the 3-braneis described by an effective theory given by [3, 4] L int = − M P ∞ X ~n =0 T µν ( x ) G ( ~n ) µν ( x ) , (1)where T µν is the energy-momentum tensor of the SM fields on the 3-brane and M P = M P / √ π is the reduced Planck mass in 4-dimensions. The relation betweenthe 4-dimensional coupling, the volume of the extra dimensions and the fundamentalscale M D in 4 + δ -dimensions M P = R δD M δ +2 D . (2)The size of the extra dimension R S is related to the radius R D , R S = 2 πR D . Thefundamental scales in 4 + δ dimensions, as defined in [3] M D is related to M S [4]as: 8 πM δ +2 D = M δ +2 S S δ − , where S δ − = 2 π δ/ / Γ( δ/
2) is the surface area of a unitsphere in δ dimensions.The zero mode corresponds to the usual 4-dimensional massless graviton andhigher massive KK modes are labeled by ~n = ( n , n , · · · , n δ ). The masses of theindividual KK modes are m ~n = ~n /R D and the mass gap between adjacent KK1odes is ∆ m = R − D . For not too large δ the discrete mass spectrum could bereplaced by a continuum, with the density of KK states ρ ( m ~n ) = 12 S δ − R δD m δ − ~n . (3)For an inclusive cross section at the collider, we have to sum over all accessible KKmodes and hence cross section for the production of an individual KK mode of mass m ~n has to be convoluted with the density of states ρ ( m ~n ). The discrete sum P ~n can be replaced by R dm ρ ( m ~n ), and hence the inclusive cross section for the towerof KK modes is dσ = S δ − M P M δD Z dm m δ − dσ ( m ) D , (4)where dσ ( m ) D is the cross section to produce a single KK mode. The cross sectionfor an individual KK mode is suppressed by the coupling factor (2 M P ) − , the highmultiplicity of accessible KK modes at the collider would compensate, leading to theexciting possibility of observing low scale quantum gravity effects at the LHC. Theadditional 1 / P ~n is kinematically constrained to those KK modes which satisfy m ~n = | ~n | /R D < √ s , where √ s is the partonic center of mass energy or as the casemay be the available energy to produce the KK modes.Viable signatures of the ADD scenario at the LHC are possible by the exchange ofvirtual KK modes between the SM particles, leading to an enhanced cross section orby the emission of real KK modes from the SM particles, leading to a missing energysignal. Various such processes have been extensively studied in this model, most ofwhich have been considered only up to leading order (LO) in QCD [3, 4, 5, 6, 7].These LO approximations at the hadron colliders suffer from large factorisation andrenormalisation scale dependence which for some processes could be as large as afactor of two. These issues go beyond normalisation of a cross section by a K -factoras the shapes of distributions may not be modeled correctly and in addition theLO cross sections are strongly dependent on the factorisation scale. It is henceessential to evaluate the next-to-leading order (NLO) corrections to the processof interest to provide quantitatively reliable predictions. NLO QCD correctionsto extra dimension models have been studied for dilepton [8], boson pair [9, 10]productions, and real graviton production processes such as graviton plus jet [11]and graviton plus photon [12]. Searches at the Tevatron using the single photon or jetwith missing transverse energy have been used to put bounds on extra dimensionalscale M D for different number of extra dimensions [13, 14]. The same signal hasbeen simulated for the LHC at the ATLAS detector [15], discovery limit and themethods to determination of the parameters of the extra dimensional models arediscussed.In this paper we consider the graviton production in association with a vectorboson at the hadron colliders at NLO in QCD. Z-boson process to LO had been2onsidered at LEP [16] and simulation studies for the Z + G KK modes productionat the LHC was studied to LO [17] as a complement to the more conventionalchannels. The associated production of Z-boson with the KK modes of the ADD model leadsto missing energy signals at the hadron colliders. We begin by discussing the neutralweak gauge boson ( Z ) production in association with the KK modes of the ADDlarge extra dimensional model to NLO in QCD and would consider the charged weakgauge bosons ( W ± ) towards the end.At the hadron collider, the associated production P P → Z G KK X at LOproceeds via the quark, anti-quark annihilation process q ¯ q → Z G KK . There arefour diagrams that would contribute to this process, which corresponds to the KKmodes of the graviton being emitted of a fermion leg, Z-boson or the q ¯ q Z vertex.The Feynman rules to evaluate the matrix elements are given in [3, 4] and for thevector boson, unitary gauge ( ξ → ∞ ) is used. Summation of the polarisation tensorof the KK modes is given in [3]. It can be seen that the terms proportional to theinverse powers of KK mode mass m vanish on expressing the matrix element squarein terms of independent variable and is an useful check.The NLO calculation presented here uses both analytical and Monte Carlo in-tegration methods and hence is flexible to incorporate the experimental cuts andcan generate the various distributions unlike a fully analytical computation. Ourcode is based on the method of two cutoff phase space slicing [18] to deal withvarious singularities appearing in the NLO computation of the real diagrams andto implement the numerical integrations over phase space. The analytical resultsare evaluated using the algebraic manipulation program FORM [19]. The real andvirtual corrections have been evaluated in the massless quark limit. We use dimen-sional regularisation with space time dimensions d = 4 + ǫ . To deal with γ in d -dimensions, we use the completely anti-commuting γ prescription [20].The order O ( α s ) corrections to the associated production of the Z-boson andKK modes of the graviton come from the following 2 → q ¯ q → Z G KK g , (b) q (¯ q ) g → q (¯ q ) Z G KK . There are 14 diagrams that contribute to thequark antiquark annihilation process and can be classified into diagrams where theKK modes couples to the (i) fermion legs, (ii) Z-boson leg, (iii) gluon leg, (iv) q ¯ q Z vertex and (v) q ¯ q g vertex. The unitary gauge is used for the Z boson propagatorand Feynman gauge for the gluon propagator. Note that the gauge parametersinfluence the coupling of KK modes to the SM fields [4]. For the sum over thepolarisation vectors we have retained only the physical degrees of freedom.The real diagrams involving gluons and massless quarks would be singular insoft and collinear regions of the 3-body phase space integration. Two small slic-ing parameters δ s and δ c are introduced to isolate regions of phase space that are3ensitive to soft and collinear singularities. Rest of the region is finite and can beevaluated in 4-dimensions. Phase space integrations in the mutually exclusive softand collinear regions are performed not on the full matrix element but in the leadingpole approximation of soft and collinear regions in 4 + ǫ dimensions. The soft andcollinear poles now appear as poles in ǫ and in addition the soft part would dependlogarithmically on the soft cut-off δ s while the collinear part would depend on theboth δ s and δ c . All positive powers of the small cut-off parameters are set to be zero.The phase space degrees of freedom that remain, correspond to a 2-body processand can now be combined with the virtual diagram.The virtual corrections to the annihilation process to order O ( α s ) can be obtainedby considering the gluonic virtual corrections to the vertex and wave function renor-malisation for the process q ¯ q → Z and then attaching the KK modes to all possiblelegs and vertex as allowed by the Feynman rules [3, 4]. This would generate 27, oneloop 2 → q ¯ q → Z G KK at LO wouldgive all the virtual contributions for the annihilation process. Performing the loopintegrals in 4 + ǫ dimensions would lead to poles in ǫ in the soft and collinear regionsand combining this with the real diagrams would lead to the cancellation of the softsingularities. The remaining collinear singular terms which appear as poles in ǫ aresystematically removed by collinear counter terms in the M S factorisation scheme,at an arbitrary factorisation scale µ F . The resultant expression is now finite butdepends on the cut-off parameters δ s and δ c logarithmically. Combining this 2 → → δ s and δ c .The q (¯ q ) g → q (¯ q ) Z G KK process begins at O ( α s ) so does not get any virtualcorrections to this order and can be obtained from the 2 → →
2, finite part ofNLO virtual corrections and full 2 → In this section we present the numerical results for the associated production of the Z -boson with the KK modes at the LHC ( √ S = 14 TeV) to NLO in QCD. Themass of the Z -boson and the weak mixing angle are taken to be m Z = 91 . θ w = 0 . α = 1 / n f = 5 light quark flavors. The two loop running strong coupling constant inthe M S scheme has been used with the corresponding Λ
QCD = 0 .
226 GeV. Unlessmentioned other wise, both the renormalization and the factorization scales are setto µ R = µ F = p ZT , the transverse momentum of the Z -boson. Further, the following4uts have been implemented in our numerical code: p ZT , p missT > p minT and | y Z | < . y Z is the rapidity of the Z -boson, p minT = 400 GeV and the missing transversemomentum p missT is given by p missT = p ZT ( p GT ) for p jetT <
20 ( >
20) GeV (6)At LO, the missing transverse momentum p missT is the same as p ZT . The additionaljet at NLO can be soft or hard; For the soft jet the p missT is the same as p ZT while forhard jet p missT is p GT of the KK modes (Eq. (6)). In addition to the above, we haveput a cut on the pseudo rapidity of the jet | η jet | < . p jetT >
20 GeV. -0.05-0.04-0.03-0.02-0.0100.010.020.030.040.05 10 -5 -4 -3 -2Variation with δ s d σ / dp T (Z) (fb/GeV) LHCADDp T = 500 GeV δ c = δ s / 100 δ s δ s -2 -5 -4 -3 -2 Figure 1: Dependence of the order α s contribution to the transverse momentumdistribution of the Z -boson at the LHC, on the slicing parameter δ s (top) with δ c = δ s /
100 and for M D = 3 TeV and δ = 4. Below the variation the sum of 2-bodyand 3-body contributions is contrasted against the value at δ s = 10 − .We check the stability of our results against the variation of the slicing parameters δ s and δ c introduced in the slicing method. In Fig. 1, we present the dependency5f both the 2-body and 3-body pieces of the O ( α s ) contribution on δ s , keeping theratio δ s /δ c = 100 fixed. These results are obtained for a specific choice of the ADDmodel parameters M D = 3 TeV and δ = 4. It can be seen from the figure thatthe sum of these two pieces is positive and is fairly stable for a wide range of δ s .The positive sum implies that the QCD corrections do enhance the leading orderpredictions. In the rest of our work, we choose δ s = 10 − and δ c = 10 − . Transverse momentum distributiond σ / dp TZ ( fb / GeV) LHCADDCTEQ6 L / MM D = 3 TeV δ = 2 (LO) δ = 4 (LO) δ = 6 (LO) δ = 2 (NLO) δ = 4 (NLO) δ = 6 (NLO) p TZ (GeV) -4 -3 -2
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Transverse momentum distributionK-factor
LHCADDCTEQ6 L / MM D = 3 TeV δ = 2 δ = 4 δ = 6 p TZ (GeV) Figure 2: Transverse momentum distribution of the Z -boson (left) and the corre-sponding K-factors (right) for the associated production of the Z -boson and the KKmode at the LHC, for M D = 3 TeV and δ = 2 , , Z -boson atthe LHC to NLO in QCD and its dependence on the number of extra dimensions δ for M D = 3 TeV. In each of the distributions corresponding to δ = 2 , , Z -boson, the K -factor, defined as theratio of NLO cross sections to the LO ones, is found to increase with p ZT and varyfrom 1.1 to 1.46 depending on the number of extra dimensions δ . In a similar way,the missing transverse momentum distribution is shown in the left panel of Fig. 3.The ADD model which is an extension of the SM to address the hierarchy prob-lem is an effective theory and the UV completion of the TeV scale gravity has tobe quantified. For the real KK mode production process, the kinematical constraintdiscussed earlier, provides a natural UV cutoff on the integration over the n-sphere,but the hard scattering scales involved at the LHC energies can be close to the fun-damental scale M D . To study the sensitivity of our results close to the UV region [3],6 issing transverse momentum distributiond σ / dp Tmiss ( fb / GeV)
LHCADDCTEQ6 L / MM D = 3 TeV δ = 2 (LO) δ = 2 (NLO) δ = 4 (LO) δ = 4 (NLO) p Tmiss (GeV) -4 -3 -2
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Transverse momentum distributiond σ / dp TZ ( fb / GeV) LHCADDCTEQ6 L/MM D = 5 TeV δ = 4Truncated (LO)Truncated (NLO)Untruncated (LO)Untruncated (NLO) p TZ (GeV) -6 -5 -4
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Figure 3: Missing transverse momentum distribution (left) at the LHC for M D =3 and for δ = 2 and 4. The truncated as well as the un-truncated transversemomentum distributions of the Z -boson (right) at the LHC for M D = 5 and for δ = 4 for both LO and NLO.we compare at LO and NLO the p T distribution of Z-boson wherein the invariantmass of the KK mode and Z-boson Q ZG is (a) computed only when Q ZG < M D ( truncation ) and (b) computed for all possible values of Q ZG ( un-truncation ). Inthe right panel of Fig. 3, we compare the results for the ADD model parameters, M D = 5 TeV and δ = 4 at LO and NLO for the p T distribution. As compared tothe un-truncated distribution, the percentage difference is tabulated below for LOand NLO for a few values of p ZT . p ZT (GeV) LO NLO500 11 7.81000 23.5 20.21500 44.9 41.8The difference between the un-truncated and truncated results become larger with(a) increase in p ZT , (b) increase in the number of extra dimensions δ and (c) decreasein the fundamental scale M D . Further it is seen that the NLO QCD corrections dodecrease the difference between the un-truncated and truncated results.Finally, we study the dependence of both LO and NLO cross sections on thefactorization scale µ F by varying it from 0 . p ZT to 2 . p ZT . One of the motivationsfor the computation of the QCD corrections is to minimise the scale uncertainties7 actorization scale uncertaintyd σ / dp TZ ( fb / GeV) LHCADDCTEQ6 L / MM D = 3 TeV δ = 4 µ R = p TZ µ F = 0.2 p TZ (LO) µ F = 2.0 p TZ (LO) µ F = 0.2 p TZ (NLO) µ F = 2.0 p TZ (NLO) p TZ Transverse momentum distributiond σ / dp TZ ( fb / GeV) LHCADDCTEQ6 L / MM D = 3 TeV δ = 4 √ S = 7 TeVLONLO p TZ (GeV) -5 -4
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Figure 4: Scale uncertainty in the transverse momentum distribution of the Z -boson at the LHC (left), for a variation of the factorization scale µ F in the range[0 . , . p ZT and for the choice of M D = 3 TeV and δ = 4. The transverse momentumdistribution of the Z-boson for √ S = 7 TeV at the LHC (right).by computing the cross sections to higher orders in the perturbation theory. Asexpected, the scale uncertainties in the leading order predictions are considerablydecreased after incorporating the one-loop QCD corrections to the associated pro-duction of the Z -boson and the KK modes at the LHC. The results are shown in theFig. 4 (left) for the case of transverse momentum distribution of the Z -boson, for M D = 3 TeV and δ = 4. In Fig. 4 (right) we have also plotted the p T distributionfor the current LHC energies of √ S = 7 TeV. The K-factor ranges from 1.05 to 1.13for 500 < p ZT < q (¯ q ) g subprocess for the √ S = 7TeV is much smaller than the contribution at √ S = 14 TeV and that accounts forthe much lower K-factor.It is interesting to note here that a similar analysis goes through for the associatedproduction of W ± and the KK modes in the large extra dimensional model at theLHC. The difference lies both in the couplings of the quarks to the weak bosons andin the respective parton fluxes to be convoluted with the partonic cross sections. InFig. 5 we have plotted the W − distribution for √ S = 14 TeV at LO and NLO. Forthe p WT distribution the K-factor is in the range 1.25 - 1.37 for 500 < p WT < W ± in association withKK modes of the ADD model will be presented in the longer version [21].8 ransverse momentum distributiond σ / dp TW ( fb / GeV) LHCADDCTEQ6 L / MM D = 3 TeV δ = 4 TeVLONLO p TW (GeV) -4 -3 -2
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Figure 5: The p T distribution of W − in association with KK modes at the LHCwith √ S = 14 TeV. To conclude, we have computed the NLO QCD corrections to the associated produc-tion of the vector boson at the LHC, using semi-analytical two cutoff phase spaceslicing method. We have presented results for √ S = 14 and 7 TeV. Our results arechecked for the stability against the variation of the slicing parameters δ s and δ c .We have studied the truncated as well as the un-truncated transverse momentumdistributions of the Z -boson, together with the missing transverse momentum distri-bution and their dependence on the number of extra dimensions δ . The NLO QCDcorrections are found to have not only enhanced the LO cross sections considerably,with the K-factors ranging from 1.1 to 1.46 depending on the δ = 2 , , M D = 3TeV, but also decreased their factorization scale uncertainties significantly. Acknowledgments
The work of V.R. and M.C.K. has been partially supported by funds made availableto the Regional Centre for Accelerator based Particle Physics (RECAPP) by theDepartment of Atomic Energy, Govt. of India. We would like to thank the clustercomputing facility at Harish-Chandra Research Institute where part of computa-tional work for this study was carried out. S.S. would like to thank UGC, NewDelhi for financial support. S.S. would also like to thank RECAPP center for his9isit, where part of the work was done.