Cross-Section Formulae for Specific Processes (Rev.)

1 50. Cross-Section Formulae for Speciﬁc Processes

50. Cross-Section Formulae for Speciﬁc Processes

Revised August 2019 by H. Baer (Oklahoma U.) and R.N. Cahn (LBNL).

PART I: STANDARD MODEL PROCESSES Setting aside leptoproduction (for which, see Sec. 16 of this Review), the cross sections of primary interest are those with light incident particles, e+e−, γγ, qq, gq , gg, etc., where g and q represent gluons and light quarks. The produced particles include both light particles and heavy ones - t, W , Z, and the Higgs boson H. We provide the production cross sections calculated within the Standard Model for several such processes. 50.1 Resonance Formation Resonant cross sections are generally described by the Breit-Wigner formula (Sec. 18 of this Review). " # 2J + 1 4π Γ 2/4 σ(E) = 2 2 2 BinBout, (50.1) (2S1 + 1)(2S2 + 1) k (E − E0) + Γ /4 where E is the c.m. energy, J is the spin of the resonance, and the number of polarization states of the two incident particles are 2S1 + 1 and 2S2 + 1. The c.m. momentum in the initial state is k, E0 is the c.m. energy at the resonance, and Γ is the full width at half maximum height of the resonance. The branching fraction for the resonance into the initial-state channel is Bin and into the ﬁnal-state channel is Bout. For a narrow resonance, the factor in square brackets may be replaced by πΓ δ(E − E0)/2. 50.2 Production of light particles The production of point-like, spin-1/2 fermions in e+e− annihilation through a virtual photon, e+e− → γ∗ → ff, at c.m. energy squared s is given by

dσ α2 = N β1 + cos2 θ + (1 − β2) sin2 θQ2 , (50.2) dΩ c 4s f where β is v/c for the produced fermions in the c.m., θ is the c.m. scattering angle, and Qf is the charge of the fermion. The factor Nc is 1 for charged leptons and 3 for quarks. In the ultrarelativistic limit, β → 1,

2 2 4πα 2 86.8 nb σ = NcQ = NcQ . (50.3) f 3s f s (GeV2) The cross section for the annihilation of a qq pair into a distinct pair q0q0 through a gluon is com- pletely analogous up to color factors, with the replacement α → αs. Treating all quarks as massless, averaging over the colors of the initial quarks and deﬁning t = −s sin2(θ/2), u = −s cos2(θ/2), one ﬁnds [1]

dσ α2 t2 + u2 (qq → q0q0) = s . (50.4) dΩ 9s s2 Crossing symmetry gives

dσ α2 s2 + u2 (qq0 → qq0) = s . (50.5) dΩ 9s t2 If the quarks q and q0 are identical, we have

P.A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020) 1st June, 2020 8:30am 2 50. Cross-Section Formulae for Speciﬁc Processes

" # dσ α2 t2 + u2 s2 + u2 2u2 (qq → qq) = s + − , (50.6) dΩ 9s s2 t2 3st and by crossing " # dσ α2 t2 + s2 s2 + u2 2s2 (qq → qq) = s + − . (50.7) dΩ 9s u2 t2 3ut Annihilation of e+e− into γγ has the cross section

dσ α2 u2 + t2 (e+e− → γγ) = . (50.8) dΩ 2s tu The related QCD process also has a triple-gluon coupling. The cross section is

! dσ 8α2 1 9 (qq → gg) = s (t2 + u2) − . (50.9) dΩ 27s tu 4s2 The crossed reactions are

dσ α2 1 9 (qg → qg) = s (s2 + u2)(− + ) (50.10) dΩ 9s su 4t2 and

dσ α2 1 9 (gg → qq) = s (t2 + u2)( − ) . (50.11) dΩ 24s tu 4s2 Finally,

dσ 9α2 ut su st (gg → gg) = s (3 − − − ) . (50.12) dΩ 8s s2 t2 u2 Lepton-quark scattering is analogous (neglecting Z exchange)

dσ α2 s2 + u2 (eq → eq) = e2 . (50.13) dΩ 2s q t2 where eq is the charge of the quark. For neutrino scattering with the four-Fermi interaction

dσ G2 s (νd → `−u) = F , (50.14) dΩ 4π2 where the Cabibbo angle suppression is ignored. Similarly

dσ G2 s (1 + cos θ)2 (νu → `−d) = F . (50.15) dΩ 4π2 4 To obtain the formulae for deep inelastic scattering (presented in more detail in Section 18) we consider quarks of type i carrying a fraction x = Q2/(2Mν) of the nucleon’s energy, where ν = E − E0 is the energy lost by the lepton in the nucleon rest frame. With y = ν/E we have the correspondences 1 + cos θ → 2(1 − y) ,

dΩcm → 4πfi(x)dx dy , (50.16) where the latter incorporates the quark distribution, fi(x). In this way we ﬁnd

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dσ 4πα2xs 1 h i (eN → eX) = 1 + (1 − y)2 dx dy Q4 2 h4 1 i × (u(x) + u(x) + ... ) + (d(x) + d(x) + ...) (50.17) 9 9 where now s = 2ME is the cm energy squared for the electron-nucleon collision and we have suppressed contributions from higher mass quarks. Similarly,

dσ G2 xs (νN → `−X) = F [(d(x) + ...) + (1 − y)2(u(x) + ...)] (50.18) dx dy π and dσ G2 xs (νN → `+X) = F [(d(x) + ...) + (1 − y)2(u(x) + ...)] . (50.19) dx dy π − + Quasi-elastic neutrino scattering (νµn → µ p, νµp → µ n) is directly related to the crossed reaction, neutron decay. The formula for the diﬀerential cross section is presented, for example, in N.J. Baker et al., Phys. Rev. D23, 2499 (1981). 50.3 Hadroproduction of heavy quarks For hadroproduction of heavy quarks Q = c, b, t, it is important to include mass eﬀects in the formulae. For qq¯ → QQ¯, one has s dσ α2 4m2 (qq¯ → QQ¯) = s 1 − Q dΩ 9s3 s h 2 2 2 2 2 i (mQ − t) + (mQ − u) + 2mQs , (50.20) while for gg → QQ¯ one has s dσ α2 4m2 6 (gg → QQ¯) = s 1 − Q (m2 − t)(m2 − u)− dΩ 32s s s2 Q Q

2 2 mQ(s − 4mQ) − 2 2 + 3(mQ − t)(mQ − u) 2 2 2 2 4 (mQ − t)(mQ − u) − 2mQ(mQ + t) 2 2 3 (mQ − t) 2 2 2 2 4 (mQ − t)(mQ − u) − 2mQ(mQ + u) + 2 2 3 (mQ − u) 2 2 2 (mQ − t)(mQ − u) + mQ(u − t) −3 2 s(mQ − t)

2 2 2 # (mQ − t)(mQ − u) + mQ(t − u) −3 2 . (50.21) s(mQ − u)

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50.4 Production of Weak Gauge Bosons 50.4.1 W and Z resonant production

Resonant production of a single W or Z is governed by the partial widths

√ 2G m3 Γ (W → ` ν ) = F W (50.22) i i 12π

√ 2G |V |2m3 Γ (W → q q ) = 3 F ij W (50.23) i j 12π

√ 2G m3 Γ (Z → ff) = N F Z c 6π h 2 2 2 2i × (T3 − Qf sin θW ) + (Qf sin θW ) . (50.24)

The weak mixing angle is θW . The CKM matrix elements are indicated by Vij and Nc is 3 for qq final states and 1 for leptonic final states. ¯ ¯ The full differential cross section for fifj → (W, Z) → fi0 fj0 is given by

f 2 dσ Nc 1 s = i · 2 · 2 2 2 dΩ Nc 256π s (s − M ) + sΓ h × (L2 + R2)(L02 + R02)(1 + cos2 θ) i + (L2 − R2)(L02 − R02)2 cosθ (50.25)

√ √ 2 1/2 where M is the mass of the W or Z. The couplings for the W are L = (8GF mW / 2) Vij/ 2; R = 0 0 0 where Vij is the corresponding CKM matrix√ element, with an analogous expression√ for L and R . 2 1/2 2 2 1/2 2 For Z, the couplings are L = (8GF mZ / 2) (T3 − sin θW Q); R = −(8GF mZ / 2) sin θW Q, where T3 is the weak isospin of the initial left-handed fermion and Q is the initial fermion’s electric 0 0 i,f charge. The expressions for L and R are analogous. The color factors Nc are 3 for initial or ﬁnal quarks and 1 for initial or ﬁnal leptons.

50.4.2 Production of pairs of weak gauge bosons

The cross section for ff¯ → W +W − is given in term of the couplings of the left-handed and right- handed fermion f, ` = 2(T3 − QxW ), r = −2QxW , where T3 is the third component of weak isospin

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2 for the left-handed f, Q is its electric charge (in units of the proton charge), and xW = sin θW :

(" !2 dσ 2πα2 ` + r s = 2 Q + 2 dt Ncs 4xW s − mZ !2# ` − r s + 2 A(s, t, u) 4xW s − mZ ! 1 ` s + Q + 2 2xW 2xW s − mZ (Θ(−Q)I(s, t, u) − Θ(Q)I(s, u, t)) ) 1 + 2 (Θ(−Q)E(s, t, u) + Θ(Q)E(s, u, t)) , 8xW (50.26) where Θ(x) is 1 for x > 0 and 0 for x < 0, and where

! 2 4 ! tu 1 mW mW s A(s, t, u) = 4 − 1 − + 3 2 + 2 − 4, mW 4 s s mW ! 2 4 ! 2 tu 1 mW mW s mW I(s, t, u) = 4 − 1 − − + 2 − 2 + 2 , mW 4 2s st mW t ! 4 ! tu 1 mW s E(s, t, u) = 4 − 1 + 2 + 2 , mW 4 t mW (50.27)

2 2 2 and s, t, u are the usual Mandelstam variables with s = (pf +pf¯) , t = (pf −pW − ) , u = (pf −pW + ) . The factor Nc is 3 for quarks and 1 for leptons. ± 0 The analogous cross-section for qiq¯j → W Z is

2 2 ( !2 " dσ πα |Vij| 1 9 − 8xW 2 2 = 2 2 2 ut − mW mZ dt 6s xW s − mW 4 # 2 2 + (8xW − 6) s mW + mZ

" 2 2 2 2 # ut − mW mZ − s(mW + mZ ) `j ì + 2 − s − mW t u 2 2 " 2 2 # 2 2 ) ut − mW mZ `j ì s(mW + mZ ) ì`j + 2 + 2 + , 4(1 − xW ) t u 2(1 − xW ) tu (50.28) where ì and `j are the couplings of the left-handed qi and qj as defined above. The CKM matrix element between qi and qj is Vij. 0 0 The cross section for qiq¯i → Z Z is

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2 4 4 " 2 # dσ πα `i + ri t u 4mZ s 4 1 1 = 2 2 2 2 + + − mZ 2 + 2 . (50.29) dt 96 xW (1 − xW ) s u t tu t u 50.5 Production of Higgs Bosons 50.5.1 Resonant Production The Higgs boson of the Standard Model can be produced resonantly in the collisions of quarks, leptons, W or Z bosons, gluons, or photons. The production cross section is thus controlled by the partial width of the Higgs boson into the entrance channel and its total width. The branching fractions for the Standard Model Higgs boson are shown in Fig. 1 of the “Searches for Higgs bosons” review in the Particle Listings section, as a function of the Higgs boson mass. The partial widths are given by the relations

2 GF m mH Nc 3/2 Γ (H → ff) = f√ 1 − 4m2 /m2 , (50.30) 4π 2 f H

3 + − GF mH βW 2 Γ (H → W W ) = √ 4 − 4aW + 3a , (50.31) 32π 2 W

3 GF mH βZ 2 Γ (H → ZZ) = √ 4 − 4aZ + 3a , (50.32) 64π 2 Z 2 2 2 2 where Nc is 3 for quarks and 1 for leptons and where aW = 1−βW = 4mW /mH and aZ = 1−βZ = 2 2 4mZ /mH . The decay to two gluons proceeds through quark loops, with the t quark dominating [2]. Explicitly,

2 α2G m3 Γ (H → gg) = s F√ H X I(m2/m2 ) , (50.33) 3 q H 36π 2 q where I(z) is complex for z < 1/4. For z < 2 × 10−3, |I(z)| is small so the light quarks contribute negligibly. For mH < 2mt, z > 1/4 and " # 1 2 I(z) = 3 2z + 2z(1 − 4z) sin−1 √ , (50.34) 2 z which has the limit I(z) → 1 as z → ∞. 50.5.2 Higgs Boson Production in W ∗ and Z∗ decay The Standard Model Higgs boson can be produced in the decay of a virtual W or Z (“Hig- gstrahlung”) [3,4]: In particular, if k is the c.m. momentum of the Higgs boson,

πα2|V |2 2k k2 + 3m2 ij √ W σ(qiqj → WH) = 4 2 2 (50.35) 36 sin θW s (s − mW ) 2πα2(`2 + r2) 2k k2 + 3m2 ¯ f f √ Z σ(ff → ZH) = 4 4 2 2 , 48Nc sin θW cos θW s (s − mZ ) (50.36) where ` and r are deﬁned as above.

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50.5.3 W and Z Fusion Just as high-energy electrons can be regarded as sources of virtual photon beams, at very high energies they are sources of virtual W and Z beams. For Higgs boson production, it is the longitudinal components of the W s and Zs that are important [5]. The distribution of longitudinal W s carrying a fraction y of the electron’s energy is [6]

g2 1 − y f(y) = , (50.37) 16π2 y where g = e/ sin θW . In the limit s mH mW , the partial decay rate is Γ (H → WLWL) = 2 3 2 (g /64π)(mH /mW ) and in the equivalent W approximation [7]

3 + − 1 α σ(e e →νeνeH) = 2 2 16mW sin θW " 2 ! 2 # mH s mH × 1 + log 2 − 2 + 2 . (50.38) s mH s There are significant corrections to this relation when m is not large compared to m [8]. For √ H √W m = 150 GeV, the estimate is too high by 51% for s = 1000 GeV, 32% too high at s = 2000 H √ GeV, and 22% too high at s = 4000 GeV. Fusion of ZZ to make a Higgs boson can be treated similarly. Identical formulae apply for Higgs production in the collisions of quarks whose charges permit the emission of a W + and a W −, except that QCD corrections and CKM matrix elements are required. Even in the absence of QCD corrections, the fine-structure constant ought to be evaluated at the scale of the collision, say mW . All quarks contribute to the ZZ fusion process. 50.6 Inclusive hadronic reactions One-particle inclusive cross sections Ed3σ/d3p for the production of a particle of momentum p are conveniently expressed in terms of rapidity y (see above) and the momentum pT transverse to the beam direction (in the c.m.): d3σ d3σ E 3 = . (50.39) d p dφ dy pT dpT In appropriate circumstances, the cross section may be decomposed as a partonic cross section multiplied by the probabilities of finding partons of the prescribed momenta: X Z σhadronic = dx1 dx2 fi(x1) fj(x2) dσbpartonic, (50.40) ij The probability that a parton of type i carries a fraction of the incident particle’s that lies between x1 and x1 + dx1 is fi(x1)dx1 and similarly for partons in the other incident particle. The partonic collision is specified by its c.m. energy squared sˆ = x1x2s and the momentum transfer squared tˆ. The final hadronic state is more conveniently specified by the rapidities y1, y2 of the two jets resulting from the collision and the transverse momentum pT . The connection between the differentials is sˆ dx dx dtˆ= dy dy dp2 , (50.41) 1 2 1 2 s T so that

d3σ sˆ dσˆ dσˆ = f (x )f (x ) (ˆs, t,ˆ uˆ) + f (x )f (x ) (ˆs, u,ˆ tˆ) , (50.42) 2 i 1 j 2 ˆ i 2 j 1 ˆ dy1dy2dpT s dt dt

1st June, 2020 8:30am 8 50. Cross-Section Formulae for Speciﬁc Processes where we have taken into account the possibility that the incident parton types might arise from either incident particle. The second term should be dropped if the types are identical: i = j. 50.7 Two-photon processes In the Weizsäcker-Williams picture, a high-energy electron beam is accompanied by a spectrum of virtual photons of energies ω and invariant-mass squared q2 = −Q2, for which the photon number density is

" # α ω ω2 m2 ω2 dω dQ2 dn = 1 − + − e , (50.43) π E E2 Q2E2 ω Q2 where E is the energy of the electron beam. The cross section for e+e− → e+e−X is then [9]

2 dσe+e−→e+e−X (s) = dn1dn2dσγγ→X (W ), (50.44)

ω2 where W 2 = m2 . Integrating from the lower limit Q2 = m2 i to a maximum Q2 gives X e Ei(Ei−ωi)

α2 Z 1 dz σe+e−→e+e−X (s) = 2 π zth z   2 !2 Qmax 1 3 ×  ln 2 − 1 f(z) + (ln z)  σγγ→X (zs), (50.45) zme 3 where 1 2 1 f(z) = (1 + 2 z) ln(1/z) − 2 (1 − z)(3 + z). (50.46) 2 The appropriate value of Qmax depends on the properties of the produced system X. For 2 2 2 2 production of hadronic systems, Qmax ≈ mρ, while for lepton-pair production, Q ≈ W . For production of a resonance with spin J 6= 1, we have

2 8α ΓR→γγ σe+e−→e+e−R(s) = (2J + 1) 3 mR " 2 !2 !3# 2 mV s 1 s × f(mR/s) ln 2 2 − 1 − ln 2 , memR 3 MR (50.47) where mV is the mass that enters into the form factor for the γγ → R transition, typically mρ.

PART II: PROCESSES BEYOND THE STANDARD MODEL 50.8 Production of supersymmetric particles In supersymmetric (SUSY) theories (see Supersymmetric Particle Searches in this Review), every boson has a fermionic superpartner, and every fermion has a bosonic superpartner. The minimal supersymmetric Standard Model (MSSM) is a direct supersymmetrization of the Standard Model (SM), although a second Higgs doublet is needed to avoid triangle anomalies [10]. Under soft SUSY breaking, superpartner masses are lifted above the SM particle masses. In weak scale SUSY, the superpartners are invoked to stabilize the weak scale under radiative corrections, so the superpartners are expected to have masses of order the TeV scale.

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50.8.1 Gluino and squark production 1 The superpartners of gluons are the color octet, spin− 2 gluinos (g˜), while each helicity component of quark flavor has a spin-0 squark partner, e.g. q˜L and q˜R. Third generation left- and right- squarks are expected to have large mixing, resulting in mass eigenstates q˜1 and q˜2, with mq˜1 < mq˜2 (here, q denotes any of the SM flavors of quarks and q˜i the corresponding flavor and type (i = L, R or 1, 2) of squark). Gluino pair production (g˜g˜) takes place via either glue-glue or quark-antiquark annihilation [11]. The subprocess cross sections are usually presented as differential distributions in the Mandel- stam variables s, t and u. Note that for a 2 → 2 scattering subprocess ab → cd, the Mandelstam 2 2 variable s = (pa + pb) = (pc + pd) , where pa is the 4-momentum of particle a, and so forth. The 2 variable t = (pc − pa) , where c and a are taken conventionally to be the most similar particles in 2 the subprocess. The variable u would then be equal to (pd − pa) . Note that since s, t and u are squares of 4-vectors, they are invariants in any inertial reference frame. Gluino pair production at hadron colliders is described by: ( dσ 9πα2 2(m2 − t)(m2 − u) (gg → g˜g˜) = s g˜ g˜ dt 4s2 s2

2 2 2 2 (mg˜ − t)(mg˜ − u) − 2mg˜(mg˜ + t) + 2 2 (mg˜ − t) 2 2 2 2 2 2 (mg˜ − t)(mg˜ − u) − 2mg˜(mg˜ + u) mg˜(s − 4mg˜) + 2 2 + 2 2 (mg˜ − u) (mg˜ − t)(mg˜ − u) 2 2 2 2 2 2 ) (mg˜ − t)(mg˜ − u) + mg˜(u − t) (mg˜ − t)(mg˜ − u) + mg˜(t − u) − 2 − 2 , (50.48) s(mg˜ − t) s(mg˜ − u) where αs is the strong ﬁne structure constant. Also,

 2 2 !2 2 !2 dσ 8παs 4 mg˜ − t 4 mg˜ − u (qq¯ → g˜g˜) = 2 2 + 2 dt 9s 3 mq˜ − t 3 mq˜ − u 3 h i + (m2 − t)2 + (m2 − u)2 + 2m2s s2 g˜ g˜ g˜ h 2 2 2 i (mg˜ − t) + mg˜s − 3 2 s(mq˜ − t) h i  (m2 − u)2 + m2s 2 g˜ g˜ 1 mg˜s  − 3 2 + 2 2 . (50.49) s(mq˜ − u) 3 (mq˜ − t)(mq˜ − u)

Gluinos can also be produced in association with squarks: g˜q˜i production, where q˜i represents any of the various types (left, right- or mixed) and ﬂavors of squarks. The subprocess cross section is independent of whether the squark is the right-, left- or mixed type: h i 2 16 (s2 + (m2 − u)2) + 4 s(m2 − u) dσ πα 3 q˜i 3 q˜i (gq → g˜q˜ ) = s dt i 24s2 s(m2 − t)(m2 − u)2 g˜ q˜i 2sm2(m2 − m2)! × (m2 − u)2 + (m2 − m2)2 + g˜ q˜i g˜ . g˜ q˜i g˜ 2 (mg˜ − t) (50.50)

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There are many different subprocesses for production of squark pairs. Since left- and right- squarks generally have different masses and different decay patterns, we present the differential cross section for each subprocess of q˜i (i = L, R or 1, 2) separately. (In early literature, the following formulae were often combined into a single equation which didn’t differentiate the various squark types.) The result for gg → q˜iq˜¯i is:

 2 m2 + t!2 m2 + u!2 dσ ¯ παs 1 q˜ 1 q˜ (gg → q˜iq˜i) = 2 2 + 2 dt 4s 3 mq˜ − t 3 mq˜ − u 3 7 + 8s(4m2 − s) + 4(u − t)2 + 32s2 q˜ 12 2 2 1 (4mq˜ − s) − 2 2 48 (mq˜ − t)(mq˜ − u) h 2 2 2 i 3 (t − u)(4mq˜ + 4t − s) − 2(mq˜ − u)(6mq˜ + 2t − s) + 2 32 s(mq˜ − t) h 2 2 2 i 3 (u − t)(4mq˜ + 4u − s) − 2(mq˜ − t)(6mq˜ + 2u − s) + 2 32 s(mq˜ − u) h 2 i h 2 i 7 4mq˜ + 4t − s 7 4mq˜ + 4u − s  + 2 + 2 , (50.51) 96 mq˜ − t 96 mq˜ − u  which has an obvious u ↔ t symmetry. For qq¯ → q˜iq˜¯i with the same initial and ﬁnal state ﬂavors, we have

2 ( ) dσ ¯ 2παs 1 2 2/3 (qq¯ → q˜iq˜i) = 2 2 2 + 2 − 2 dt 9s (t − mg˜) s s(t − mg˜) h 2 2i × −st − (t − mq˜i ) , (50.52)

0 ¯0 while if initial and final state flavors are different (qq¯ → q˜iq˜i) we instead have

2 dσ 0 ¯0 4παs h 2 2i (qq¯ → q˜iq˜i) = −st − (t − mq˜0 ) . (50.53) dt 9s4 i If the two initial state quarks are of diﬀerent ﬂavors, then we have

dσ 2πα2 −st − (t − m2 )2 0 ¯0 s q˜i (qq¯ → q˜iq˜i) = 2 2 2 . (50.54) dt 9s (t − mg˜) If the initial quarks are of different flavor and final state squarks are of different type (i 6= j) then

2 m2s dσ 0 ¯0 2παs g˜ (qq¯ → q˜iq˜j) = 2 2 2 . (50.55) dt 9s (t − mg˜)

For same-ﬂavor initial state quarks, but ﬁnal state unlike-type squarks, we also have

2 m2s dσ ¯ 2παs g˜ (qq¯ → q˜iq˜j) = 2 2 2 . (50.56) dt 9s (t − mg˜)

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There also exist cross sections for quark-quark annihilation to squark pairs. For same flavor quark- quark annihilation to same flavor/same type final state squarks, dσ (qq → q˜ q˜ ) = dt i i 2 ( ) παs 2 1 1 2/3 = 2 mg˜s 2 2 + 2 2 − 2 2 , (50.57) 9s (t − mg˜) (u − mg˜) (t − mg˜)(u − mg˜) while if the final type squarks are different (i 6= j), we have dσ (qq → q˜ q˜ ) = dt i j

2πα2 s × 9s2 ([−st − (t − m2 )(t − m2 )] [−su − (u − m2 )(u − m2 )]) q˜i q˜j q˜i q˜j 2 + 2 . (t − mg˜) (u − mg˜) (50.58)

If initial/final state flavors are different, but final state squark types are the same, then

2 2 dσ 0 0 2παs mg˜s (qq → q˜iq˜i) = 2 2 2 . (50.59) dt 9s (t − mg˜) If initial quark flavors are different and final squark types are different, then 2 −st − (t − m2 )(t − m2 ) dσ 0 0 2παs q˜i q˜j (qq → q˜iq˜j) = 2 2 2 . (50.60) dt 9s (t − mg˜) 50.8.2 Gluino and squark associated production 1 ± In the MSSM, the charged spin- 2 winos and higgsinos mix to make chargino states χ1 and ± 1 χ , with m ± < m ± . The spin− neutral bino, wino and higgsino fields mix to give four neu- 2 χ1 χ2 2 0 tralino mass eigenstates χ1,2,3,4 ordered according to mass. We sometimes denote the charginos and neutralinos collectively as -inos for notational simplicity For gluino and squark production in association with charginos and neutralinos [12], the quark- squark-neutralino couplings1 are defined by the interaction Lagrangian terms f ˜† ¯0 f ˜† ¯0 Lff˜ χ˜0 = iA 0 fLχ˜i PLf + iB 0 fRχ˜i PRf + h.c. i χ˜i χ˜i

f f , where A 0 and B 0 are coupling constants involving gauge couplings, neutralino mixing elements χ˜i χ˜i and in the case of third generation fermions, Yukawa couplings. Their form depends on the con- ventions used for setting up the MSSM Lagrangian, and can be found in various reviews [14] and textbooks [13, 15]. PL and PR are the usual left- and right- spinor projection operators and f denotes any of the SM fermions u, d, e, νe, ··· . The fermion-sfermion- chargino couplings have d † − u ˜† c d the form L = iA − u˜Lχ˜i PLd + iA − dLχ˜i PLu + h.c. for u and d quarks, where the A − and χ˜i χ˜i χ˜i 1The couplings Af and Bf are given explicitly in Ref. [13] in Eq. (8.87). Also, the couplings Ad and Au are χ˜0 χ˜0 χ˜− χ˜− i i i i j j given in Eq. (8.93). The couplings Xi and Yi are given by Eq. (8.103), while the xi and yi couplings are given in Eq. (8.100). Finally, the couplings Wij are given in Eq. (8.101).

1st June, 2020 8:30am 12 50. Cross-Section Formulae for Speciﬁc Processes

u A − couplings are again convention-dependent, and can be found in textbooks. The superscript c χ˜i denotes “charge conjugate spinor”, deﬁned by ψc ≡ Cψ¯T . The subprocess cross sections for chargino-squark associated production occur via squark exchange and are given by

dσ − ˜¯ αs u 2 (¯ug → χ˜ d ) = |A − | ψ(m ˜ , m − , t), (50.61) i L 2 χ˜ dL χ˜ dt 24s i i

dσ − αs d 2 (dg → χ˜i u˜L) = 2 |Aχ˜− | ψ(mu˜L , mχ˜− , t), (50.62) dt 24s i i while neutralino-squark production is given by dσ 0 αs q 2 q 2 (qg → χ˜i q˜) = 2 |Aχ˜0 | + |Bχ˜0 | ψ(mq˜, mχ˜0 , t), (50.63) dt 24s i i i where

2 2 2 s + t − m1 m1(m2 − t) ψ(m1, m2, t) = − 2 2 2s (m1 − t) 2 2 2 2 2 t(m2 − m1) + m2(s − m2 + m1) + 2 . (50.64) s(m1 − t) Here, the variable t is given by the square of “squark-minus-quark” four-momentum. The neutralino- gluino associated production cross section also occurs via squark exchange and is given by

 2 2 (m 0 − t)(mg˜ − t) dσ 0 αs q 2 q 2 χ˜i (qq¯ → χ˜i g˜) = 2 |Aχ˜0 | + |Bχ˜0 |  2 2 dt 18s i i (mq˜ − t) 2 2  (m 0 − u)(mg˜ − u) 2ηiηg˜mg˜m 0 s χ˜i χ˜i + 2 2 − 2 2  , (50.65) (mq˜ − u) (mq˜ − t)(mq˜ − u) where ηi is the sign of the neutralino mass eigenvalue and ηg˜ is the sign of the gluino mass eigenvalue. We also have chargino-gluino associated production:

 2 2 (m − − t)(mg˜ − t) dσ − αs u 2 χ˜i (¯ud → χ˜i g˜) = 2 |Aχ˜− | 2 2 dt 18s i (m ˜ − t) dL 2 2 u d  (m − − u)(mg˜ − u) 2ηg˜Re(A − A − )mg˜mχ˜i s d 2 χ˜i χ˜i χ˜i +|Aχ˜− | 2 2 + 2 2  , (50.66) i (mu˜ − u) (m ˜ − t)(mu˜ − u) L dL L where tˆ = (˜g − d)2 and in the third term one must take the real part of the in general complex coupling constant product. 50.8.3 Slepton and sneutrino production ˜ ¯ The subprocess cross section for `Lν˜`L production (` = e or µ) occurs via s-channel W exchange and is given by 4 2 dσ ˜ ¯ g |DW (s)| 2 2 (du¯ → `Lν˜`L ) = tu − m˜ mν˜ , (50.67) dt 192πs2 `L `L 2 where DW (s) = 1/(s − MW + iMW ΓW ) is the W -boson propagator denominator. The production of τ˜1ν˜¯τ is given as above, but replacing m˜ → mτ˜ , mν˜ → mν˜ and multiplying by an overall `L 1 `L τ

1st June, 2020 8:30am 13 50. Cross-Section Formulae for Speciﬁc Processes

Table 50.1: The constants αf and βf that appear in in the SM neutral current Lagrangian. Here t ≡ tan θW and c ≡ cot θW .

f qf αf βf 1 1 ` -1 4 (3t − c) 4 (t + c) 1 1 ν` 0 4 (t+c) − 4 (t + c) 2 5 1 1 u 3 − 12 t + 4 c − 4 (t + c) 1 1 1 1 d − 3 12 t − 4 c 4 (t + c)

2 factor of cos θτ (where θτ is the tau-slepton mixing angle). Similar substitutions hold for τ˜2ν˜¯τ 2 production, except the overall factor is sin θτ . ˜ ˜¯ The subprocess cross section for `L`L production occurs via s-channel γ and Z exchange, and 0 depends on the neutral current interaction, with fermion couplings to γ and Z given by Lneutral = ¯ µ ¯ µ −eqf fγ fAµ + efγ (αf + βf γ5)fZµ (with values of qf , αf , and βf given in Table-50.1. The subprocess cross section is given by

4 dσ ˜ ˜¯ e 4 (qq¯ → `L`L) = tu − m˜ × dt 24πs2 `L (q2q2 ` q + (α − β )2(α2 + β2)|D (s)|2 s2 ` ` q q Z ) 2q q α (α − β )(s − M 2 ) + ` q q ` ` Z |D (s)|2 , (50.68) s Z

2 where DZ (s) = 1/(s − MZ + iMZ ΓZ ). The cross section for sneutrino production is given by the same formula, but with α , β , q and m˜ replaced by α , β , 0 and m , respectively. The ` ` ` `L ν ν ν˜L cross section for τ˜ τ˜¯ production is obtained by replacing m˜ → m and β → β cos 2θ . The 1 1 `L τ˜1 ` ` τ ¯ cross section for `˜ `˜ production is given by substituting α − β → α + β and m˜ → m˜ in R R ` ` ` ` `L `R ˜ ˜¯ the equation above. The cross section for τ˜2τ˜¯2 production is obtained from the formula for `R`R production by replacing m˜ → m and β → β cos 2θ . `R τ˜2 ` ` τ Finally, the cross section for τ˜1τ˜¯2 production occurs only via Z exchange, and is given by dσ dσ (qq¯ → τ˜ τ˜¯ ) = (qq¯ → τ˜¯ τ˜ ) = dt 1 2 dt 1 2 e4 (α2 + β2)β2 sin2 2θ |D (s)|2(ut − m2 m2 ). (50.69) 24πs2 q q ` τ Z τ˜1 τ˜2 50.8.4 Chargino and neutralino pair production − 0 50.8.4.1 χ˜i χ˜j production − 0 The subprocess cross section for du¯ → χ˜i χ˜j depends on Lagrangian couplings

g +µ LW ud¯ = −√ uγ¯ µPLdW + h.c. 2 , θj − j j 0 −µ L − 0 = −g(−i) χ˜ i[X + Y γ5]γµχ˜j W + h.c. W χ˜i χ˜j i i

1st June, 2020 8:30am 14 50. Cross-Section Formulae for Speciﬁc Processes

, d † − u ˜† c Lqq˜χ˜− = iA − u˜Lχ˜i PLd + iA − dLχ˜i PLu + h.c. i χ˜i χ˜i and q † 0 Lqq˜χ˜0 = iA 0 q˜Lχ˜ jPLq + h.c. j χ˜j ˜ j . Contributing diagrams include W exchange and also dL and u˜L squark exchange. The Xi j and Yi couplings are new, and again convention-dependent: the cross section formulae works if the interaction Lagrangian is written in the above form, so that the couplings can be suitably extracted. The term θj = 0 (1) if m 0 > 0 (< 0); it comes about because the neutralino field must χ˜j be re-defined by a −iγ5 transformation if its mass eigenvalue is negative [13]. The subprocess cross section is given in terms of dot products of four momenta, where particle labels are used to denote their four-momenta; note that all mass terms in the cross section formulae are positive definite, so that the signs of mass eigenstates have been absorbed into the Lagrangian couplings, as for instance in Ref. [13]. We then have dσ 1 (du¯ → χ˜−χ˜0) = dt i j 192πs2 " # T + T ˜ + T + T ˜ + T + T ˜ (50.70) W dL u˜L W dL W u˜L dLu˜L where

4 2 n j2 j2 0 − 0 − TW = 8g |DW (s)| [Xi + Yi ](˜χj · dχ˜i · u¯ +χ ˜j · u¯χ˜i · d) j j 0 − 0 − + 2(Xi Yi )(˜χj · dχ˜i · u¯ − χ˜j · u¯χ˜i · d) j2 j2 o + [X − Y ]m − mχ˜0 d · u¯ , i i χ˜i j (50.71)

u 2 d 2 4|A − | |Aχ˜0 | χ˜i j 0 − T ˜ = d · χ˜ χ˜ · u,¯ (50.72) dL − 2 2 2 j i [(˜χ − u¯) − m ˜ ] i dL 4|Ad |2|Au |2 χ˜− χ˜0 T = i j u¯ · χ˜0χ˜− · d (50.73) u˜L [(˜χ0 − u¯)2 − m2 ]2 j i j u˜L √ 2 d∗ u θj 2 2 − 2g Re[Aχ˜0 A − (−i) ](s − MW )|DW (s)| j χ˜i T ˜ = W dL − 2 2 (˜χ − u¯) − m ˜ i dL n j j 0 − j j o × 8(X + Y )˜χj · du¯ · χ˜ + 4(X − Y )m − mχ˜0 d · u¯ (50.74) i i i i i χ˜i j

√ 2g2Re[Ad∗ Au (−i)θj ](s − M 2 )|D (s)|2 χ˜− χ˜0 W W T = i j W u˜L (˜χ0 − u¯)2 − m2 j u˜L n j j 0 − j j o × 8(X − Y )˜χj · ud¯ · χ˜ + 4(X + Y )m − mχ˜0 d · u¯ (50.75) i i i i i χ˜i j

1st June, 2020 8:30am 15 50. Cross-Section Formulae for Speciﬁc Processes and d u∗ d∗ u 4Re[Aχ˜0 A − A − Aχ˜0 ]mχ˜− mχ˜0 d · u¯ j χ˜i χ˜i j i j T ˜ = − . (50.76) dLu˜L − 2 2 0 2 2 [(˜χ − u¯) − m ˜ ][(˜χ − u¯) − mu˜ ] i dL j L 50.8.4.2 Chargino pair production ¯ − + The subprocess cross section for dd → χ˜i χ˜i (i = 1, 2) depends on Lagrangian couplings L = − − µ − − µ d † − u ˜† −c eχ˜i γµχ˜i A − e cot θW χ˜i γµ(xi − yiγ5)˜χi Z and also L 3 iA − u˜Lχ˜i PLd + iA − dLχ˜i PLu + h.c.. χ˜i χ˜i 0 Contributing diagrams include s-channel γ, Z exchange and t-channel u˜L exchange [16, 17]. The couplings xi and yi are again new and as usual convention-dependent. The subprocess cross section is given by dσ (dd¯→ χ˜−χ˜+) = dt i i 1 [T + T + T + T + T + T ] (50.77) 192πs2 γ Z u˜L γZ γu˜L Zu˜L where 4 2 32e qd + ¯ − − ¯ + 2 ¯ Tγ = 2 d · χ˜i d · χ˜i + d · χ˜i d · χ˜i + mχ˜− d · d (50.78) s i 4 2 2 TZ = 32e cot θW |DZ (s)| ( 2 2 2 2 + ¯ − − ¯ + 2 ¯ (αd + βd)(xi + yi ) d · χ˜i d · χ˜i + d · χ˜i d · χ˜i + m − d · d χ˜i

) h + ¯ − − ¯ +i 2 2 2 2 ¯ ∓ 4αdβdxiyi d · χ˜i d · χ˜i − d · χ˜i d · χ˜i −2yi (αd + βd)m − d · d , (50.79) χ˜i

d 4 4|A − | χ˜i − + Tu˜ = d · χ˜ d¯· χ˜ (50.80) L [(d − χ˜−)2 − m2 ]2 i i i u˜L 64e4 cot θ q (s − M 2 )|D (s)|2 T = W d Z Z × γZ s ( + ¯ − − ¯ + 2 ¯ αdxi d · χ˜i d · χ˜i + d · χ˜i d · χ˜i + m − d · d χ˜i

) − ¯ + + ¯ − ±βdyi d · χ˜i d · χ˜i − d · χ˜i d · χ˜i (50.81)

d 2 2 |A | 8e q χ˜− d i ¯ + − 2 ¯ Tγu˜L = ∓ − 2 2d · χ˜i d · χ˜i + mχ˜− d · d (50.82) s [(d − χ˜ )2 − m ] i i u˜L and

d 2 2 |A − | (s − MZ ) 2 2 χ˜i TZu˜ = ∓8e cot θW |DZ (s)| (α − β ) L [(d − χ˜−)2 − m2 ] d d i u˜L − ¯ + 2 ¯ × 2(xi ∓ yi)d · χ˜i d · χ˜i + m − (xi ± yi)d · d (50.83) χ˜i using the upper of the sign choices.

1st June, 2020 8:30am 16 50. Cross-Section Formulae for Speciﬁc Processes

+ − The cross section for uu¯ → χ˜i χ˜i can be obtained from the above by replacing αd → αu, ˜ d u ¯ βd → βu, qd → qu, u˜L → dL, A − → A − , d → u¯, d → u and adopting the lower of the sign choices χ˜i χ˜i everywhere. − + + − The cross section for qq¯ → χ˜1 χ˜2 , χ˜1 χ˜2 can occur via Z and q˜L exchange. It is usually much − + smaller than χ˜1,2χ˜1,2 production, so the cross section will not be presented here. It can be found in Appendix A of Ref. [13]. 50.8.4.3 Neutralino pair production Neutralino pair production via qq¯ fusion takes place via s-channel Z exchange plus t- and u- channel left- and right- squark exchange (5 diagrams) [17,18]. The Lagrangian couplings (see previ- 0 θi+θj +1 0 µ ous footnote*) needed include terms given above plus terms of the form L = Wijχ˜ iγµ(γ5) χ˜j Z . The couplings Wij depend only on the higgsino components of the neutralinos i and j. The subprocess cross section is given by: dσ 1 (qq¯ → χ˜0χ˜0) = [T + T + T + T + T ] (50.84) dt i j 192πs2 Z q˜L q˜R Zq˜L Zq˜R where 2 2 2 2 2 TZ = 128e |Wij| (αq + βq )|DZ (s)| h 0 0 0 0 i q · χ˜ q¯ · χ˜ + q · χ˜ q¯ · χ˜ − ηiηjm 0 m 0 q · q¯ , (50.85) i j j i χ˜i χ˜j

( 0 0 0 0 q 2 q 2 q · χ˜i q¯ · χ˜j q · χ˜j q¯ · χ˜i Tq˜L = 4|Aχ˜0 | |Aχ˜0 | 0 2 2 2 + 0 2 2 2 i j [(˜χ − q) − m ] [(˜χ − q) − m ] i q˜L j q˜L ) mχ˜0 mχ˜0 q · q¯ − η η i j (50.86) i j [(˜χ0 − q)2 − m2 ][(˜χ0 − q)2 − m2 ] i q˜L j q˜L

( 0 0 0 0 q 2 q 2 q · χ˜i q¯ · χ˜j q · χ˜j q¯ · χ˜i Tq˜R = 4|Bχ˜0 | |Bχ˜0 | 0 2 2 2 + 0 2 2 2 i j [(˜χ − q) − m ] [(˜χ − q) − m ] i q˜R j q˜R ) mχ˜0 mχ˜0 q · q¯ − η η i j (50.87) i j [(˜χ0 − q)2 − m2 ][(˜χ0 − q)2 − m2 ] i q˜R j q˜R 2 2 TZq˜L = 16e(αq − βq)(s − MZ )|DZ (s)| ( q∗ q Re(WijAχ˜0 Aχ˜0 ) i j h 0 0 i 2q · χ˜i q¯ · χ˜j − ηiηjmχ˜0 mχ˜0 q · q¯ [(˜χ0 − q)2 − m2 ] i j i q˜L q q∗ ) Re(WijAχ˜0 Aχ˜0 ) i j h 0 0 i + ηiηj 2q · χ˜j q¯ · χ˜i − ηiηjmχ˜0 mχ˜0 q · q¯ (50.88) [(˜χ0 − q)2 − m2 ] i j j q˜L 2 2 TZq˜R = 16e(αq + βq)(s − MZ )|DZ (s)| ( q∗ q Re(WijBχ˜0 Bχ˜0 ) i j h 0 0 i 2q · χ˜i q¯ · χ˜j − ηiηjmχ˜0 mχ˜0 q · q¯ [(˜χ0 − q)2 − m2 ] i j i q˜R q q∗ ) Re(WijBχ˜0 Bχ˜0 ) i j h 0 0 i − 2q · χ˜j q¯ · χ˜i − ηiηjmχ˜0 mχ˜0 q · q¯ . (50.89) [(˜χ0 − q)2 − m2 ] i j j q˜R

As before, ηi = ±1 corresponding to whether the neutralino mass eigenvalue is positive or negative. When i = j in the above formula, one must remember to integrate over just 2π steradians of solid angle to avoid double counting in the total cross section.

1st June, 2020 8:30am 17 50. Cross-Section Formulae for Speciﬁc Processes

50.9 Universal extra dimensions In the Universal Extra Dimension (UED) model of Ref. [19] (see Ref. [20] for a review of models with extra spacetime dimensions), the Standard Model is embedded in a five dimensional theory, where the fifth dimension is compactified on an S1/Z2 orbifold. Each SM chirality state is then the zero mode of an infinite tower of Kaluza-Klein excitations labelled by n = 0 − ∞. A KK parity is usually assumed to hold, where each state is assigned KK-parity P = (−1)n. If the compactification scale is around a TeV, then the n = 1 (or even higher) KK modes may be accessible to collider searches. Of interest for hadron colliders are the production of massive n ≥ 1 quark or gluon pairs. These production cross sections have been calculated in Ref. [21,22]. We list here results for the n = 1 case only with M1 = 1/R (R is the compactification radius) and s, t and u are the usual Mandelstam variables; more general formulae can be found in Ref. [22]. The superscript ∗ stands for any KK excited state, while • stands for left chirality states and ◦ stands for right chirality states.

dσ 1 = T (50.90) dt 16πs2 where " ! 2g4 4s3 57s 108 T (qq¯ → g∗g∗) = s M 2 − + − 27 1 t02u02 t0u0 s

# 20s2 108t0u0 + − 93 + (50.91) t0u0 s2 and T (gg → g∗g∗) =

" 0 0 0 0 9g4 s2 + t 2 + u 2 s2 + t 2 + u 2 s 3M 4 − 3M 2 + 1 27 1 t02u02 1 st0u0

0 0 # (s2 + t 2 + u 2)3 t0u0 + − (50.92) 4s2t02u02 s2

0 2 0 2 where t = t − M1 and u = u − M1 . Also, 4 " 2 02 02 # 0 0 4g 2M t + u T (qq¯ → q∗ q¯∗ ) = s 1 + , 1 1 9 s s2 " ! g4 4 s 1 T (qq¯ → q∗q¯∗) = 2 2M 2 + − 1 1 9 1 s t02 t0

0 # 23 2s2 8s 6t0 8t 2 + + + + + , 6 t02 3t0 s s2

" g4 t0 u0 s T (qq → q∗q∗) = s M 2 6 + 6 − 1 1 27 1 u02 t02 t0u0

0 0 !# t 2 u 2 s2 +2 3 + 3 + 4 − 5 , u02 t02 t0u0

1st June, 2020 8:30am 18 50. Cross-Section Formulae for Speciﬁc Processes

" ! −4 s2 3 T (gg → q∗q¯∗) = g4 M 4 − 1 1 s 1 t0u0 6t0u0 8 ! # 4 s2 3 s2 17 3t0u0 +M 2 − + − + , 1 s 6t0u0 8 6t0u0 24 4s2

" 0 0 # −g4 5s2 s3 11su0 5u 2 u 3 T (gq → g∗q∗) = s + + + + , 1 3 12t02 t02u0 6t02 12t02 st02

4 " 2 # 0 g s s s T (qq¯0 → q∗q¯∗ ) = s 4M 4 + 5 + 4 + 8 , 1 1 18 1 t02 t02 t0

4 " 2 # 0 2g s 1 s T (qq0 → q∗q∗ ) = s −M 2 + + , 1 1 9 1 t02 4 t02 • ◦ T (qq → q1q1) = " ! # g4 2s3 4s s4 s2 s M 2 − + 2 − 8 + 5 , 9 1 t02u02 t0u0 t02u02 t0u0

4 " 0 0 02 # 0 g 1 u 5 4u 2u T (qq¯0 → q•q¯ ◦) = s 2M 2 + + + + , 1 1 9 1 t0 t02 2 t0 t02 and 4 " 0 02 # 0 g 1 u 1 2u T (qq0 → q•q ◦) = s −2M 2 + + + . 1 1 9 1 t0 t02 2 t02 50.10 Large extra dimensions In the ADD theory [23] with large extra dimensions (LED), the SM particles are conﬁned to a 3-brane, while gravity propagates in the bulk. It is assumed that the n extra dimensions are compactiﬁed on an n-dimensional torus of volume (2πr)n, so that the fundamental 4+n dimensional 2 n+2 n Planck scale M∗ is related to the usual 4-dimensional Planck scale MP l by MP l = M∗ (2πr) . If M∗ ∼ 1 TeV, then the MW − MP l hierarchy problem is just due to gravity propagating in the large extra dimensions. n In these theories, the KK-excited graviton states Gµν for n = 1− ∞ can be produced at collider experiments. The graviton couplings to matter are suppressed by 1/MP l, so that graviton emission 2 cross sections dσ/dt ∼ 1/MP l. However, the mass splittings between the excited graviton states can be tiny, so the graviton eigenstates are usually approximated by a continuum distribution. A 2 summation (integration) over all allowed graviton emissions ends up cancelling the 1/MP l factor, so that observable cross section rates can be attained. Some of the fundamental production formulae for a KK graviton (denoted G) of mass m at hadron colliders include the subprocesses

αQ2 2 dσm ¯ f 1 t m (ff → γG) = 2 F1( , ), (50.93) dt 16Nf sMP l s s where Qf is the charge of fermion f and Nf is the number of QCD colors of f. Also,

2 dσm αs 1 t m (qq¯ → gG) = 2 F1( , ), (50.94) dt 36 sMP l s s

2 dσm αs 1 t m (qg → qG) = 2 F2( , ), (50.95) dt 96 sMP l s s

1st June, 2020 8:30am 19 50. Cross-Section Formulae for Speciﬁc Processes

2 dσm 3αs 1 t m (gg → gG) = 2 F3( , ), (50.96) dt 16 sMP l s s where 1 h F (x, y) = −4x(1 + x)(1 + 2x + 2x2)+ 1 x(y − 1 − x) i y(1 + 6x + 18x2 + 16x3) − 6y2x(1 + 2x) + y3(1 + 4x) (50.97)

x y F (x, y) = −(y − 1 − x)F , (50.98) 2 1 y − 1 − x y − 1 − x and 1 h F (x, y) = 1 + 2x + 3x2 + 2x3 + x4 3 x(y − 1 − x) i −2y(1 + x3) + 3y2(1 + x2) − 2y3(1 + x) + y4 . (50.99)

These formulae must then be multiplied by the graviton density of states formula 2 MP l n−1 dN = Sn−1 n+2 m dm to gain the cross section M∗

2 2 d σ MP l n−1 dσm = Sn−1 n+2 m (50.100) dtdm M∗ dt

(2π)n/2 where Sn = Γ (n/2) is the surface area of an n-dimensional sphere of unit radius. Virtual graviton processes can also be searched for at colliders. For instance, in Ref. [24] the cross section for Drell-Yan production of lepton pairs via gluon fusion was calculated, where it is found that, in the center-of-mass system

2 3 dσ + − λ s 2 2 (gg → ` ` ) = 8 (1 − z )(1 + z ) (50.101) dz 64πM∗ where z = cos θ and λ is a model-dependent coupling constant ∼ 1. Formulae for Drell-Yan production via qq¯ fusion can also be found in Refs. [24, 25]. 50.11 Warped extra dimensions In the Randall-Sundrum model [26] of warped extra dimensions, the arena for physics is a 5- d anti-deSitter (AdS5) spacetime, for which a non-factorizable metric exists with a metric warp −2σ(φ) factor e . It is assumed that two opposite tension 3-branes exist within AdS5 at the two ends of an S1/Z2 orbifold parametrized by co-ordinate φ which runs from 0 − π. The 4-D solution of the Einstein equations yields σ(φ) = krc|φ|, where rc is the compactiﬁcation radius of the extra 3 2 M −2krcπ dimension and k ∼ MP l. The 4-D eﬀective action allows one to identify M P l = k (1 − e ), where M is the 5-D Planck scale. Physical particles on the TeV scale (SM) brane have mass −krcπ m = e m0, where m0 is a fundamental mass of order the Planck scale. Thus, the weak scale- Planck scale hierarchy occurs due to the existence of the exponential warp factor if krc ∼ 12. In the simplest versions of the RS model, the TeV-scale brane contains only SM particles plus −krcπ a tower of KK gravitons. The RS gravitons have mass mn = kxne , where the xi are roots of Bessel functions J1(xn) = 0, with x1 ' 3.83, x2 ' 7.02 etc. While the RS zero-mode graviton couplings suppressed by 1/M P l and are thus inconsequential for collider searches, the n = 1 and −krcπ higher modes have couplings suppressed instead by Λπ = e M P l ∼ T eV . The n = 1 RS 2 2 graviton should have width Γ1 = ρm1x1(k/M P l) , where ρ is a constant depending on how many

1st June, 2020 8:30am 20 50. Cross-Section Formulae for Speciﬁc Processes decay modes are open. The formulae for dilepton production via virtual RS graviton exchange can be gained from the above formulae for the ADD scenario via the replacement [27]

λ i2 ∞ 1 → X . (50.102) M 4 8Λ2 s − m2 + im Γ ∗ π n=1 n n n

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