QCD threshold corrections for gluino pair production at NNLL

Transcription

1 Introduction: Gluino pair production at fixed order QCD threshold corrections for gluino pair production at NNLL in collaboration with Ulrich Langenfeld and Sven-Olaf Moch, based on arxiv: Munich, September 6, 2012

2 Outline 1 Introduction: Gluino pair production at fixed order 2 3

3 1 Introduction: Gluino pair production at fixed order 2 3

4 Gluino production at the LHC The MSSM as one of the most favorite extensions of the SM provides a rich spectrum of heavy new particles. Hadron colliders are especially appropriated to produce color-charged particles, such as the gluino. The search for SUSY particles at the LHC has only produced exclusion limits so far. The limits depend on the model under consideration. As we hope to find evidences with an increased cms energy, a precise prediction of the production cross section near the threshold is required. So far we know the production cross section at NLO (fixed order) [Beenakker, Hopker, Spira, Zerwas, 1997] and the threshold result, with combined soft and Coulomb resummation at NLL, matched onto approximated NLO. [Falgari, Schwinn, Wever, 2012] The production of gluino-bound states has been discussed at NLO. [Kauth, Kuhn, Marquard, Steinhauser, 2011] The production cross section for squark antisquark is known to NNLL accuracy. [Beenakker, Brensing, Kramer, Kulesza, Laenen, 2012]

5 Gluino production at LO gg eg eg, q q eg eg, q = d, u, s, c, b. q g q g (a) (b) (c) g g (d) (e) (f) At NLO, also the gq channel opens. However, it is strongly suppressed near threshold. Our assumption for the NLO calculation: All squark masses are equal.

6 Inclusive hadronic cross section Recall the standard factorization of the hadronic cross section (with the hadronic cms energy s and partonic cms energy ŝ) σ pp/p p eg egx (s, m 2 eg, m2 eq, µ2 f, µ2 r ) = X Zs i,j=q, q,g 4m eg 2 into the parton luminosity function dŝ L (ŝ, s, µ 2 f ) ˆσ eg eg (ŝ, m2 eg, m2 eq, µ2 f, µ2 r ) L (ŝ, s, µ 2 f ) = 1 Zs dz s z f i/p µ 2 f, z «f j/p µ 2 f s, ŝ «z ŝ µ 2 /meg 2 and µ r = µ f = µ and the partonic cross section with L µ = ln " ˆσ = α2 s m eg 2 f (00) + 4πα s f (10) The scaling functions f (ab) ρ = 4m2 eg ŝ + f (11) L µ + (4πα s ) 2 f (20) + f (21) depend on the kinematic variables, β = p 1 ρ, r = m2 eq m eg 2 Note that for gluon fusion the functions f (00) gg and f (11) gg are independent of r! # L µ + f (22) L 2 µ +

7 1 Introduction: Gluino pair production at fixed order 2 3

8 Resummation formula The partonic cross sections contains threshold logarithms ln k (β) which stem from soft-gluon emission. NLO: k 2 NNLO: k 4 For gluino masses of several hundred GeV, the partonic cms energy is expected to be close to the production threshold. The resulting large logarithms can be resummed to all orders in perturbation theory. This can be done in Mellin space [Sterman 1987; Catani, Trentadue, 1989], where one introduces moments N w.r.t. ρ : ˆσ(N, m 2 eg ) = Z 1 dρ ρ N 1 ˆσ(ŝ, m 2 eg ), ˆσ eg eg = X I ˆσ, I 0 The threshold limit β 0 corresponds to N. I labels the admissible SU(3) c representations of the LO scattering reactions. h i ˆσ, I (N, meg 2 ) = ˆσ, B I(N, meg 2 ) g, 0 I(N, meg 2 ) exp G, I (N + 1) + O(N 1 ln n N)

9 Exponent of the resummation formula G, I (N) = Z 1 0 dz z N z ( Z 4m 2 eg (1 z) 2 µ 2 f dq 2 q 2 A i`αs(q 2 ) + A j`αs(q 2 ) + D, I`αs(4m 2 eg (1 z) 2 ) ). where D, I (α s) = 1 2 `Di (α s) + D j (α s) + D egeg, I (α s) Anomalous dimension functions: A i,j : initial-state collinear gluon radiation (process independent) D i,j : initial-state soft radiation (process independent) D egeg, I : final-state soft radiation The process independent functions are known up to O(α 3 s ) from top-quark pair production [Kodaira, Trentadue, 1982; S. Moch, J..A. M. Vermaseren, A. Vogt, 2004] The functions D egeg, I are known to O(α 2 s ) [Beneke, Falgari, Schwinn, 2010]

10 Summation of threshold logarithms Defining λ = a s β 0 ln N, the function G, I (N) can be expanded according to: G, I (N) = ln(n) g 1 (λ) {z } + g, 2 I(λ) {z } + as g, 3 I(λ) {z } +... LL NLL NNLL The expansion coefficients g k are known up to to k = 3. [Moch, Uwer, 2008]

11 Summation of threshold logarithms Defining λ = a s β 0 ln N, the function G, I (N) can be expanded according to: G, I (N) = ln(n) g 1 (λ) {z } + g, 2 I(λ) {z } + as g, 3 I(λ) {z } +... LL NLL NNLL The expansion coefficients g k are known up to to k = 3. [Moch, Uwer, 2008] The explicit results involve constants γ E, which cancel between the g k expansion in α s. The appearance of γ E can be avoided by defining after en = N exp(γ e) e λ = as β 0 ln e N and rearranging the sum according to G, I (N) = ln( e N) g 1 ( e λ) + g 2, I( e λ) + a s g 3, I( e λ) +... Starting from NLL the coefficients depend on the color index I and the logarithmic ratios L fr = ln µ 2 f µ 2 r! and L egr = ln 4m 2 eg µ 2 r!.

12 g 1 ii = A (1) β 1 i ln(1 2λ) e + ln(1 2λ) e λ e 1 g 2 ii, I = A(1) i + A (2) i β 3 0 β 1 2λ e + ln(1 2λ) e ln2 (1 2λ) e β 2 0 2λ e ln(1 2λ) e + A (1) i + β ln(1 2e λ) g 3 ii, I = A(1) β 4 i 0 β λ e 1 1 «1 2λ e ln(1 2λ) e +! + A (1) β 3 i 0 β λ e + ln(1 2λ) e + + A (1) β 2 i 0 β 1 L egr A (2) β 3 3 i 0 β λ e 1 + A (2) β 1 i 0 2λ e L fr + L egr (1 2 e λ) β 1 0 2λ e L fr + ln(1 2λ) e L egr D (1) i + D (1) gg, I 1 2(1 2 e λ) A (1) π 2 i ln(1 2λ) e «A (1) i 1 2λ e ««3 1 2λ e 2 + ln(1 2e λ) ««+ β 2 0 A (3) i 1 2 e λ + (D (1) + D (1) i gg, I )β 2 0 β (D (1) + D (1) i gg, I ) L egr 2(1 2λ) e! (1 2 e λ) 1 + ln(1 2λ) e! + (D (2) + D (2) i gg, I )β ln 2 (1 2λ) e! e λ eλ L 2 fr + L2 egr «1! 2 λ e 1 + 2(1 2λ) e (1 2 e λ)! (1 2λ) e!!

13 Coulomb corrections Besides threshold enhanced logarithms, one has Coulomb corrections proportional to inverse powers of β, which stem from soft gluon exchange between the final state particles (here gluinos): The resummation of Coulomb terms accounts for bound-state effects of the gluino pair. From NRQCD it follows that (pure) Coulomb corrections are summed to all orders by a Sommerfeld factor C : [Fadin, Khoze,Sjostrand, 1990] C = C π αs β D I «, C (x) = x exp(x) 1 Here, D I = C I /2 C A with C A = 3 and C I = {0, 3, 8} for I = {1, 8, 27}. A formal expansion in α s reproduces the fixed order results. [Kulesza, Motyka, 2009] For small β however, the expansion does not converge!

14 Hard matching constant In Mellin space, the fixed order Coulomb terms are formally treated as part of the hard cross section, and therefore enter the hard matching constant. Techniques for a combined soft- and Coulomb resummation in β-space are given in the literature. [Beneke, Falgari, Schwinn, 2010; 2011] h i ˆσ, I (N, meg 2 ) = ˆσ, B I(N, meg 2 ) g, 0 I(N, meg 2 ) exp G, I (N + 1) + O(N 1 ln n N) The matching constant can be separated into a [Kawamura, Lo Presti, Moch, Vogt, 2012] hard coefficient g 0, I(α s): Collects all constants of the fixed order result. Coulomb coefficient g 0, C, I (αs, N): Collects all Coulomb terms and their interference with hard and soft contributions: g, 0 I(α s, N) = g, 0 I(α s) g 0, C, I (αs, N) = 1 + a s g 0 (1) 0, C (1), I + g, I (N) + as 2 g 0 (2) 0, C (2), I + g, I (N) + g 0 (1) 0, C (1), I g, I (N) + O αs 3

15 NLO scaling functions near threshold Near the threshold (β < 1), the (color-summed) NLO functions can be factorized according to: f (10) = f (00) A ln 2 (β) + B ln(β) + C «β + C 1 + O(β) At NLO it is also common to write ln(8β 2 ) instead of ln(β). Of course, there is also the freedom to pull a constant out of the parenthesis. Having specified the convention, the object of interest is the matching coefficient C 1, which has to be determined from a dedicated one-loop calculation, where higher powers in β can be skipped near the threshold. The color-summed scaling functions of the full NLO cross section are known for a long time and have been implemented in the code Prospino. [Beenakker, Höpker, Spira, 1996] The matching constant can be extracted numerically from Prospino by a fit in the threshold region.

16 Color-decomposed NLO scaling functions The color decomposition for gluino pair production has not been given so far. However, there are analytic results for the decomposed NLO cross section of gluino-bound state production. [Kauth, Kuhn, Marquard, Steinhauser, 2011] Omitting corrections of O(β), the hard function factorizes near the threshold into the Born term and the infrared-finite parts of the UV-regularized virtual and real quantum corrections: F T, I = F T Born, I 1 + αs(µr ) V, I δ(1 z) + αs(µr ) R, I (z) π π

17 Color-decomposed NLO scaling functions The color decomposition for gluino pair production has not been given so far. However, there are analytic results for the decomposed NLO cross section of gluino-bound state production. [Kauth, Kuhn, Marquard, Steinhauser, 2011] Omitting corrections of O(β), the hard function factorizes near the threshold into the Born term and the infrared-finite parts of the UV-regularized virtual and real quantum corrections: F T, I = F T Born, I 1 + αs(µr ) V, I δ(1 z) + αs(µr ) R, I (z) π π Note that the difference between the 2 1 and 2 2 kinematics (labeled by the final state T ) only comes from the Born Term! The hard kernel for gluino pair production is obtained by F Born T, I ˆσ B, I

18 LO results and one-loop matching constants gg-channel: 8 8 = 1 s + 8 s + 8 a s (Only symmetric color configurations I = 1 s, 8 s, 27 s contribute to first approx. near threshold.) f (10) gg, I gg MS C 1, I = f (00) gg, I 4π 2 A gg (r) = 9 I 6 ln 2 `8β 2 (24 + C I ) ln `8β 2 π! 2 2β D I + C gg 1, I! ln 2 (2) π 2 + n f (r) I = C I 4 + ln(2) π2 8 b 1 (r) b 4 (r) + 2 b 1 (r) (1 r) Agg 9r (C I + 1) c 5 (r) Scalar n-point integrals b 1 (r), etc. defined in [Kauth, Kuhn, Marquard, Steinhauser, 2011]

19 LO results and one-loop matching constants gg-channel: 8 8 = 1 s + 8 s + 8 a s (Only symmetric color configurations I = 1 s, 8 s, 27 s contribute to first approx. near threshold.) f (10) gg, I gg MS C 1, I = f (00) gg, I 4π 2 A gg (r) = 9 I 6 ln 2 `8β 2 (24 + C I ) ln `8β 2 π! 2 2β D I + C gg 1, I! ln 2 (2) π 2 + n f (r) I = C I 4 + ln(2) π2 8 b 1 (r) b 4 (r) + 2 b 1 (r) (1 r) Agg 9r (C I + 1) c 5 (r) Scalar n-point integrals b 1 (r), etc. defined in [Kauth, Kuhn, Marquard, Steinhauser, 2011] q q-channel: 3 3 = 1 s + 8 s + 8 a (Only the antisymmetric octet I = 8a contributes.) f (10) q q, 8a q q MS C 1, 8a = f (00) q q, 8a 4π 2 8 `8β 2 3 ln ln `8β 2 + 3π 2 4β! + C q q 1, 8a 0 1 = n f ln(2) 5 « ln2 (2) π2 1 3 m2 t m eg 2 A q q (r) = lengthy expression; involves more n-point integrals 8a A n f 6 ln(r) + Aq q 8a (r)

20 Cross check We have calculated analytic expressions for the required scalar one-, two-, and three-point integrals a 1(r), b i (r), b i (r), c i (r). The one-loop matching constants C 1, I depend on the chosen renormalization scheme. Kauth. et. al use DR(full MSSM). Beenakker et. al. use MS(n l = 5), which is also implemented in Prospino. In order to cross check our (color-summed) analytic results with the numerical output of Prospino, we had to decouple the heavy particles and switch the renormalization scheme. Comparison with Prospino: gg-channel: relative numerical agreement at the per mill level! q q-channel: deviations of a few per cent due to a different treatment of the top mass (only logarithms are kept by Kauth et. al.). The function f (11) on the other hand directly follows from the running of α s and the parton evolution and is computed by a Mellin convolution: f (11) = 1 16π 2 2β 0 f (00) f (00) kj, I P (0) ki f (00) ik P (0) kj where P (x) = α s P (0) (x) + α 2 s P(1) (x) +....

21 Approximated NNLO The NNLO cross section in the threshold approximation can be constructed by expanding the r.h.s. of the general resummation formula in α s, and changing to β-space by an inverse Mellin transformation. The general answer for an arbitrary color-representation has already been given in the literature as a function of the one-loop matching constant, including non-relativistic kinetic energy corrections, which depend on the spin configuration. [Beneke, Czakon, Falgari, Mitov, Schwinn, 2010] Now we know the explicit result for gluino-pair production! The scale-dependent part again follows from the parton evolution: [van Neerven, A. Vogt, 2000; Kidonakis, Laenen, Moch, R. Vogt, 2001] f (21) f (22) 1 = (16π 2 ) 2 2β 1 f (00) π 2 1 = (16π 2 ) 2 3β 0 f (10) f (00) kl f (00) kj f (10) kj P (0) ki P (0) lj +3β 2 0 f (00) P (1) ki P (0) ki f (00) ik f (10) ik f (00) in 5 2 β 0f (00) ik P (1) kj P (0) kj P (0) nl P (0) kj P (0) lj f (00) nj 5 «2 β 0f (00) P (0) kj ki P (0) nk P(0) ki

22 f (20) gg, I = f (00) 2 gg, I 4D2 (16π 2 ) 2 4 I π4 3β 2 + D I π 2 ( 192 ln 2 (β) C I 8! n l 192 ln(2) β 3 ln(β) 8C gg 1, I C I + 4! 5 n l 2 ln(2) + 44 ln(2) + 8C I ln(2) 48 ln 2 (2) 16π 2 ) ln 4 (β) ( C I ) n l ln(2) 3 ln 3 (β) + ( 384 C gg 1, I C I + 32C2 I +n l C I 3 3 ( «+ 384 ln(2) ln(2) 3456C I ln(2) ln 2 (2) 2400π 2 ) ln 2 (β) C I + «768 32C I ln(2) C gg 1, I n l C I 1088 ln(2) 32C I ln(2) ln 2 (2) 32π 2! ln(2) C I ln(2) + 96C I 2 ln(2) ln 2 (2) 5184C I ln 2 (2) ln 3 (2) π C I π ln(2)π ζ 3 48C I ζ π 2 «D I 3 ) 3 2D I (1 + v spin ) ln(β) + C gg 5 2, I f (20) q q, 8a = f (00) 2 q q 3π4 (16π 2 ) 2 4 β 2 + π 2 ( β 102 ln(2) + 32 ln 2 (2) + 32π2 ) +n l ln 2 (β) n l ln(2) ln(β) + 12 C q q 1, 8a n l ln 4 (β) «ln(2) ln(2) ln 2 (2) 4480 π 2 ) ln 2 (β) «+ 4 ln(2) 3 ) ( ln(2) + 2n l ln 3 ( (β) ( C q q 1, 8a «+ 512 ln(2) C q q 1, 8a n l 5216 ln(2) ln 2 (2) 128 π 2 « ln(2) ln 2 (2) ln 3 (2) π ln(2)π ζ π 2 «D I 3 ) 3 2D I (1 + v spin ) ln(β) + C q q 5 2, 3 9 8a

23 Exact scaling functions (except for f (20) ) " # ˆσ = α2 s m eg 2 f (00) +4πα s f (10) +f (11) L µ +(4πα s ) 2 f (20) +f (21) L µ+f (22) L 2 µ +, m eg = 750 GeV, r = f gg 10 f gg 11 f gg Η s 4m g f q q 10 f q q 11 f q q Η s 4m g f gg 21 f gg 22 f gg f q q 21 f q q 22 f q q Η s 4m 2 g Η s 4m 2 g 1

24 Hard matching coefficients Finally, the general resummation formula is expanded to NNLL and matched onto the approximated NNLO cross section. Working in MS(n l = 5) we find: j ff g 0 gg, I = 1 + as 4C gg 1, I C I ( 8 4 ln(2)) + 24 ln 2 (2) + 12π 2 j + a 2 s 16C gg C 1, I I ( 1 + ln(2)) + 72 ln(2) 48 ln 2 (2) + 3π C I ( ln(2)) 3 «4000 ln 2 (2) ln 3 (2) + 126π ln(2)π 2 + ( ln(2)) ζ 3 + C 2 I 32 ln(2) 64 ln 2 (2) + 4π n l C I ln(2) + 48 ln 2 (2) 3π ln(2) ln 2 (2) 624 ln 3 (2) 102π ln(2)π 2 336ζ 3 « ( ln(2)) ln2 (2) ln 3 (2) ln 4 (2) π ln(2)π ln 2 (2)π π 4 + ( ln(2)) ζ 3 + L µ 192C gg ( 1 + ln(2)) 1, I + C I ln(2) 576 ln 2 (2) + 48π n l C I ( 1 + ln(2)) + 92 ln(2) 48 ln 2 (2) + 3π ln(2) ln 2 (2) ln 3 (2) π ln(2)π ζ 3 «+ L 2 µ 16 ( 1 + ln(2)) n l ln(2) 1152 ln 2 (2) + 144π 2 «ff

25 g 0 q q, 8a = 1 + as j 4C q q + a 2 s 82 9 ln(2) + 8 ln 2 (2) + 4π 2 + L µ 14 4 «ff 3 n l 1, 8a j 32 3 C q q ln(2) + 32 ln 2 (2) 2π , 8a 81 n l ln(2) ln 2 (2) «2496 ln 3 (2) 489π ln(2)π ζ ln 2 (2) + 8 ( ln(2)) «79424 ln 3 (2) ln 4 (2) π ln(2)π ln 2 (2)π π ( ln(2)) ζ 3 + L µ 3 C q q 16 ( 1 + ln(2)) + 1, 8a 27 n l ln(2) ln 2 (2) 18π ln(2) 8640 ln 2 (2) ln 3 (2) + 502π ln(2)π 2 «+ 1792ζ 3 9 «ff L 2 32 µ 10n l ( 1 + ln(2)) ln 2 (2) 245 ln(2) 8π 2 9 The formula for Coulomb terms (also including non-coulomb kinetic energy corrections) holds for both production channels: s g 0, C, I (N) = 1 as 4D Iπ 2 N π + a2 s j 8 3 D2 I π4 N + D I π 2 s ln(2) n l 5 6 ln(2) L µ 44 8 «ff 3 n l +16π 2 D I 3 2D I (1 + v spin ) 1 ln(2) 1 N) ff 2 ln(e. N π ln( N) e n l

26 1 Introduction: Gluino pair production at fixed order 2 3

27 Fixed order cross sections σ [pb] TeV, PDF set: MSTW2008NNLO NNLO approx NLO LO m g ~ [GeV] σ [pb] TeV, PDF set: MSTW2008NNLO NNLO approx NLO LO m g ~ [GeV] MS(n l = 5) scheme, m eq = 4/5m eg, error bands correspond to a variation m eg /2 µ 2m eg ; We have also included the threshold-suppressed gq production channel at NLO. We observe an increase in the predicted rates due to the approximate NNLO corrections of the order of O(15 20)% More stringent bounds on sparticle masses? As we will see below, the biggest uncertainty comes from the choice of the PDF set and the related value of α s(m Z ).

28 Fixed order scale uncertainty and PDF comparison σ [pb] TeV MSTW 2008 NNLO LO NLO NNLO approx σ [pb] TeV, σ(pp ->gg ~~ MSTW2008NNLO ABM11_5n_NNLO 1 m g ~ = 750 GeV m q ~ = 600 GeV µ/m g ~ m g ~ [GeV] Vertical bars: total scale variation in the range [m eg /2, 2m eg ] Vertical dashed gray line: cross section at the nominal scale µ = m eg. σ(µ = m eg ) = { pb, pb, pb} for LO, NLO, NNLO K NLO = σ NLO/σ LO = 1.46, K NNLO = σ NNLO/σ NLO = 1.13

29 Gluino pair-production cross section with s = 7 TeV m eg σ(lo)[ pb] σ(nlo)[ pb] σ(nnlo)[ pb] [ GeV] x = 1 x = 1 x = 2 x = 2 x = 1 x = 2 x = 2 x = 1 x = 2 2 MSTW 2008 NNLO ABM NNLO x = µ/m eg Differences between the PDF sets ABM11 and MSTW amount to the order of O(30 60)%! This issue is not treated in current experimental analysis!

30 Summary We have derived analytic results for the color-decomposed NLO and approximated NNLO cross section of gluino-pair production near the threshold. The resummation of threshold logarithms has been performed in Mellin space to NNLL accuracy. Alternatively, one could resum in β-space [Becher, Neubert, 2006], where general results for a combined soft- and Coulomb resummation are given in the literature. [Beneke, Falgari, Schwinn, 2010] We have investigated the phenomenological implications by a calculation of the approximated NNLO hadronic production cross section at fixed order in perturbation theory. A further phenomenology paper with results from resummation performed in both approaches is in preparation. At the scale µ = m eg, the approximated NNLO cross section increases the NLO result of about 15 20%. The results are rather stable in the range µ [ 1 2 m eg, 2m eg ]. The dominant source of uncertainty comes from the non-perturbative input. This should be kept in mind in forthcoming investigations.

31 Thank you for listening!