pcut T , and the regular piece below pT . The regular piece is the difference between the real correction and the asymptotic piece. The contribution to the total cross-section from the regular piece is small if pcut T is taken to be small. Throughout this work, pcut T will be chosen to be small such that the regular piece provides a negligible contribution and will therefore be ignored. For the leading order calculation, obtaining the fixed-order calculation from the resummed result is trivial, and therefore the first order discussed will be NLO. Throughout the following calculation, without loss of generality, the choices for the resummation constants (C1 , C2 , and C3 ) will be chosen to be the canonical values for simplicity (C1 = C3 = b0 , C2 = 1). As will be discussed in the following section (Sec. 3.6), all C1 , C2 , and C3 dependence should cancel in the expansion of the resummation formalism to some fixed-order in Îąs . Therefore, the asymptotic, singular, and real piece do not depend on the choice of resummation constants. The asymptotic piece consists of terms that are at least as singular as qTâ2 . The singular piece consists of terms that are at least as singular as qTâ2 plus it also includes the Î´(qT ) terms. 91 The expansion of the A, B, C, and H coefficients to O (Îąsn ) can be explicitly found up to O Îąs3 in Section 3.4.1, Section 3.4.2, and Appendix F for the CSS and CFG formalisms. The expansion of both the CSS and CFG formalism result in the same singular and asymptotic piece, so it is sufficient to only consider the CSS formalism. The lepton variables and angle between ~b and ~qT are integrated out to simplify the discussion, but do not modify the results. After these simplifications, the resummation formalism becomes: Z â dĎ 1 âS Îˇ/qT ,Q lim â dÎˇÎˇJ (Îˇ) e 0 qT â0 dQ2 dydq 2 2ĎqT2 0 T 2 2 2 2 Ă C â fj x1 , qT /Îˇ C â fkĚ x2 , qT /Îˇ + j â kĚ, (3.79) where terms that are not of importance in the derivation have been dropped, and terms that are less singular than 12 or Î´(qT ) have also been dropped. The asymptotic piece is obtained qT by integrating over Îˇ = bqT . This can be performed by using the following integration by parts identity: Z â 0 Z â dÎˇÎˇJ0 (Îˇ) F (Îˇ) = â 0 dÎˇÎˇJ1 (Îˇ) dF (Îˇ) , dÎˇ which is true given that the boundary term vanishes, ÎˇJ1 (Îˇ) F (Îˇ)â Îˇ=0 (3.80) = 0 . Additionally, the following integral results will be important in obtaining both the asymptotic and singular 92 piece up to O Îąs3 : Z â 0 dÎˇJ1 (Îˇ) lnm ďŁą ďŁ´ ďŁ´ ďŁ´ ďŁ´ 1, if m = 0 ďŁ´ ďŁ´ ďŁ´ ďŁ´ ďŁ´ ďŁ´ 2 ďŁ´ ďŁ´ ďŁ´ ln Q2 , if m = 1 ďŁ´ ďŁ´ qT ďŁ´ ďŁ´ ďŁ´ ďŁ´ ďŁ´ 2 ďŁ´ ďŁ´ ďŁ´ ln2 Q2 , if m = 2 ďŁ´ ďŁ´ qT ďŁ´ ! ďŁ´ ďŁ´ ďŁ˛ 2 Îˇ 2 Q2 = ln3 Q2 â 4Îś(3), if m = 3 2 2 ďŁ´ qT b0 q T ďŁ´ ďŁ´ ďŁ´ ďŁ´ ďŁ´ 2 2 ďŁ´ ďŁ´ ďŁ´ ln4 Q2 â 16Îś(3) ln Q2 , if m = 4 ďŁ´ ďŁ´ qT qT ďŁ´ ďŁ´ ďŁ´ ďŁ´ ďŁ´ 2 2 ďŁ´ ďŁ´ ln5 Q2 â 40Îś(3) ln2 Q2 â 48Îś(5), if m = 5, ďŁ´ ďŁ´ ďŁ´ qT qT ďŁ´ ďŁ´ ďŁ´ ďŁ´ ďŁ´ ďŁ´ Q2 Q2 Q2 ďŁ´ ďŁłln6 2 â 80Îś(3) ln3 2 â 288Îś(5) ln 2 + 160Îś(3)2 , if m = 6, qT qT qT (3.81) where b0 = eâÎłE , and ÎłE is the Euler constant. Up through m = 2 is needed for the O (Îąs ) calculations, through m = 4 is needed for the O Îąs2 calculations, and all of the above will be needed for the O Îąs3 calculations. Secondly, the singular piece is obtained by taking the integral of qT for the O (Îąsn ) corrections from 0 to pcut T , and is calculated in a manner similar to the asymptotic piece, with one key modification. Instead of using the integration by parts identity, the order of integration between qT and b is interchanged, giving us the following relationship: Z pcut 2 T 1 (2Ď)2 0 dqT2 Z ~ d2 bei~qT Âˇb F Z â 1 cut F (b) . (b) = dbpcut J bp 1 T T 2Ď 0 (3.82) Finally, the calculation of the real corrections is needed. The additional jets then need to be integrated out, and the singularities need to be canceled in order to obtain a finite prediction. For NLO this is straightforward, and there is an analytic form. However, at 93 NNLO this is not possible, and an external code with the parameters tuned to match that of the ResBos2 code will be used to obtain this contribution. Additionally, to ensure the external code is tuned correctly, it will be validated against the total cross-section at NLO. Once the functional form for C (3) is calculated, the ResBos2 code along with the Z plus jet to NNLO calculation will be able to predict the N3 LO total cross-section. 3.5.1 O (Îąs ) Singular, Asymptotic Piece, and NLO Result In this subsection, the computational details of obtaining the singular and asymptotic piece needed for the NLO calculation are shown. Firstly, to obtain the asymptotic piece the Sudakov factor is expanded to O (Îąs ), S(b, Q) = S (1) (b, Q) + O Îąs2 , with S (1) given as: S (1) (b, Q) Îąs Q2 = Ď " # 1 (1) 2 Q2 b2 Q 2 b2 (1) A ln + B ln 2 . 2 b20 b0 (3.83) Also, the convolution of the PDF with the C function needs to be expanded to O (Îąs ). At O (Îąs ), the C function does not appear in the asymptotic piece at this order, but will appear at higher orders. For the asymptotic piece, at this order it is sufficient to use the PDF evolution equation given as: dfj (x, Âľ2 ) Îąs (Âľ2 ) (1) 2 + O Îą2 . = P â f x, Âľ a s jâa 2Ď d ln Âľ2 (3.84) Using the integration by parts identity in Eq. 3.80, requires the derivatives with respect to Îˇ of the Sudakov factor and the PDF, given as: " # 2Îˇ2 d âS(Îˇ/q ,Q) â2 Îąs (Q2 ) Q T e = A(1) ln 2 2 + B (1) + O Îąs2 , dÎˇ Îˇ Ď b0 q T 94 (3.85) and d 2 2 2 â2 Îąs (Q2 ) (1) fj x, b0 qT /Îˇ = Pjâa â fa x, Âľ2 + O Îąs2 . dÎˇ Îˇ 2Ď (3.86) Combining these results with those from Eq. 3.81, the final results for the asymptotic piece is given as: dĎ Ď0 1 Îąs (Q2 ) nh 2 2 lim = f j x1 , Q PkĚâb â fb x2 , Q S 2ĎqT2 Ď qT â0 dQ2 dydq 2 T i + fkĚ x2 , Q2 Pjâa â fa x1 , Q2 (3.87) " # ) Q2 + 2 A(1) ln 2 + B (1) fj x1 , Q2 fkĚ x2 , Q2 + j â kĚ + O Îąs2 . qT For simplicity, it is useful to introduce the following definition, n â 2nâ1 dĎ Ď0 1 X X X Îąs (Âľ2 ) (i,j) m = n Cm ln 2 2 2 S Ď dQ dydqT 2ĎqT i,j n=1 m=0 Q2 qT2 ! , (3.88) which becomes very useful for organization beyond O (Îąs ). The definition above differs from s that found in Ref. [175] by expanding in factors of ÎąĎs instead of Îą 2Ď , and the overall factor 1 instead of 1 . Using these definitions the above results are given as: for the 12 term is 2Ď Ď q T (i,j) 1 C1 = 2A(1) fi fj , (i,j) 1 C0 = 2B (1) fi fj + fj (Piâb â fb ) fi Pjâa â fa . Secondly, calculating the singular piece begins by expanding the Sudakov factor in the same manner. However, instead of using the evolution of the PDF as a shortcut to obtain 95 the derivative with respect to Îˇ, the PDF is directly expanded to O (Îąs ) as: 2 Îą Q Âľ2 s 2 + O Îą2 , fj x, Âľ2 = fj x, Q2 + ln P x, Q jâaâfa s 2Ď Q2 (3.89) where Âľ2 is the factorization scale3 . It will be convenient to again introduce the following definition: Z pcut 2 T 0 n â 2n dĎ Ď0 X X X Îąs (Âľ2 ) (i,j) m 2 dqT = ln n Vm 2 2 S Ď dQ dydqT i,j n=0 m=0 Q2 qT2 ! (3.90) Combining this with Eq. 3.82 we obtain the following results using the above definition of (i,j) n Vm : (i,j) 0 V0 = fi fj , (i,j) 1 V2 1 = â A(1) fi fj , 2 1 = âB (1) fi fj â fj (Piâb â fb ) + fi Pjâa â fa , 2 (i,j) (1) (1) = fj Ciâb â fb + fi Cjâa â fa . 1 V0 (i,j) 1 V1 Finally, the regular terms and real corrections can be found in Ref. [176] in Section 2.2.5. By combining these pieces, the NLO prediction can be calculated, and is implemented into the ResBos2 code. As mentioned above, there is a pcut T that is artificially introduced to separate the singular region from the regular region. The numerical result at NLO is for the total cross section for Drell-Yan in the invariant mass region of 66 GeV < Mll < 116 GeV â at a collider energy of S = 8 TeV. Furthermore, comparisons to other publicly available codes tuned to have the same Electroweak parameters as defined in Ref. [68], and mentioned 3 The higher order expansion for the PDF is detailed in App. F 96 NLO Calculation ResBos2 (qTcut = 0.1GeV) ResBos2 (qTcut = 3GeV) MCFM FEWZ Cross-Section 1113.3 pb 1115.1 pb 1112.6 pb 1113.2 pb Table 3.3: Total inclusive NLO Drell-Yan Cross-sections for 66 GeV < Mll < 116 GeV. MCFM calculation using version 8.0 [3]. FEWZ calculation using version 3.1 rc [4] in Sec. 1.1.4, and using the CT14nnlo PDF [10], can be seen in Table 3.3. 3.5.2 O Îąs2 Singular, Asymptotic, and the NNLO Total CrossSection At NNLO, the procedure above is extended to the next order in Îąs . The most important difference from NLO is that the expansion of Îąs (Âľ2 ) needs to be considered. This expansion is given by the following to O Îąs3 : Îąs (Âľ2 ) Ď Îąs (Âľ2R ) Îąs (Âľ2R ) = â Ď Ď !2 Îąs (Âľ2R ) Î˛0 ln 2 + Ď ÂľR Âľ2 !3 Î˛02 log2 Âľ2 Âľ2R Âľ2 ! â Î˛1 log 2 +O ÂľR Îąs4 (3.91) where Î˛0 and Î˛1 are defined in App. E, and ÂľR is the renormalization scale4 . The Sudakov factor then can be expressed as S = S (1) + S (2) + O Îąs3 with: S (2) (b, Q) Îąs2 Q2 = Ď2 " # 1 (2) 2 Q2 b2 Q2 b2 (2) A ln + B ln 2 , 2 b20 b0 4 The complete derivation of the result can be found in Appendix F 97 (3.92) , and after expanding Îąs , the Sudakov factor is ! 2 b2 1 (1) 2 Q2 b2 Q A ln + B (1) ln 2 2 b20 b0 Îą 2 1 Q2 b2 Âľ2 1 Q 2 b2 1 Q2 b2 s + A(1) Î˛0 ln2 2 ln R2 + A(1) Î˛0 ln3 2 + A(2) ln2 2 Ď 4 6 2 Q b0 b0 b0 ! 1 Q2 b2 Âľ2 1 Q2 b2 Q2 b2 + B (1) Î˛0 ln 2 ln R2 + B (1) Î˛0 ln2 2 + B (2) ln 2 + O Îąs3 . 2 4 Q b0 b0 b0 Îąs S= Ď (3.93) A similar expansion as above is carried out for the convolution of the C function with the PDF. The results can then be expressed as: !! Âľ2F Q2 b2 ln 2 + ln 2 Q b0 ! Âľ2 1 Îąs 2 Q 2 b2 + 4Î˛0 C (1) â f ln 2 + ln F2 8 Ď Q b0 ! ! ! 2 2 2 2 2 b2 2 b2 2 b2 Âľ Âľ Âľ Q Q Q â 2Î˛0 P (1) â f ln 2 + ln F2 ln 2 + ln R2 + Î˛0 P (1) â f ln 2 + ln F2 Q Q Q b0 b b0 ! 0 Âľ2 Q2 b2 + 4C (1) â P (1) â f ln 2 + ln F2 + 8C (2) â f Q b0 !2 !ďŁś 2 2 2 2 2 2 Âľ Âľ Q b Q b + P (1) â P (1) â f ln 2 + ln F2 + 2P (2) â f ln 2 + ln F2 ďŁ¸ + O Îąs3 , Q Q b0 b0 Îąs C âf =f + Ď 1 C (1) â f â P (1) â f 2 (3.94) where ÂľF is the factorization scale. 98 Following the procedure used at NLO, the n Cm âs for the asymptotic piece, are given as: 2 = â2 A(1) fi fj , (i,j) (1) (1) (1) = â6A B + 2A Î˛0 fi fj â 3A(1) fj (Piâb â fb ) fi Pjâa â fa , 2 C2 ! 2 2 Âľ (i,j) = A(1) Î˛0 ln R2 + 2A(2) â 2 B (1) + B (1) Î˛0 fi fj + 4A(1) C (1) â fi fj 2 C1 Q Âľ2F (1) (1) (1) (1) â 2A P â fi fj ln 2 â 4B P â fi fj Q + Î˛0 P (1) â fi fj â P (1) â P (1) â fi fj â P (1) â fi P (1) â fj + i â j, ! 2 2 Âľ (i,j) = 4 A(1) Îś(3) + B (1) Î˛0 ln R2 + 2B (2) fi fj 2 C0 Q ! 2 Âľ + B (1) 4 C (1) â fi fj â 2 P (1) â fi fj ln F2 Q ! 2 Âľ â Î˛0 4 C (1) â fi fj â 2 P (1) â fi fj ln R2 Q Âľ2 + 2 C (1) â P (1) â fi fj + 2 C (1) â fi P (1 ) â fj â P (1) â P (1) â fi fj ln F2 Q Âľ2 + P (1) â fi P (1) â fj ln F2 + P (2) â fi fj + i â j, Q (i,j) 2 C3 where the i â j corresponds to the same terms with i and j interchanged. These results above are consistent with the results of [177]. Also, using the expansions above, the singular 99 piece is given by 1 (1) 2 A fi fj , 8 1 (1) (1) 1 (1) 1 (1) (1) (i,j) (1) = A B â A Î˛0 f i f j + A P â fi fj + P â fj fi , 2 V3 2 6 4 ďŁŤ ďŁś ! (1) 2 2 B ÂľR Î˛0 B (1) A(2) ďŁˇ (i,j) ďŁŹ 1 = ďŁâ Î˛0 A(1) ln + â â ďŁ¸ fi fj 2 V2 8 4 8 4 Q2 (i,j) 2 V4 = ! ! Âľ2F 1 (1) B (1) Î˛0 (1) â f + 1 P (1) â P (1) â f f + A ln + â f P j i i j 4 2 8 8 Q2 1 1 (1) â A(1) C (1) â fi fj + P â fi P (1) â fj + i â j, 2 8 ! ! 2 2 1 (2) Âľ B (i,j) (1) (1) ln (1) C (1) â f f R â V = âÎś(3) A â Î˛ B f f â B 2 1 i j i j 4 0 2 Q2 ! Âľ2F (1) 1 (1) 1 (1) 1 (1) (1) â f f + B ln P â f f + Î˛ C â f f â C â P i j i j i j 2 2 0 2 Q2 ! ! 2 1 Âľ2F (1) Âľ 1 (1) â f (1) â P (1) â f f F + ln P â f P + ln P i j i j 4 4 Q2 Q2 ! Âľ2R (1) 1 1 (2) â Î˛0 ln P â f f â P â f i j i fj + i â j, 4 4 Q2 1 (i,j) (1) (1) (1) (1) (1) = Î˛ Îś(3)A â Îś(3)A B fi fj â Îś(3)A P â fi fj 2 V0 3 0 ! ! 1 Âľ2F (1) Âľ2F (1) 1 (1) (1) â ln C â fi P â fj â ln C â P â fi jj 2 2 Q2 Q2 ! Âľ2F (1) 1 1 (1) (1) + Î˛0 ln C â fi fj + C â fi C â fj 2 2 Q2 ! Âľ2F (1) 1 2 + Î˛0 ln P â fi fj 8 Q2 ! ! 1 Âľ2F (1) 1 2 Âľ2F (1) (1) 2 (1) + ln P â fi P â fj + ln P â P â fi fj 8 8 Q2 Q2 ! ! 2 2 Âľ Âľ 1 (1) (2) F R â Î˛0 ln ln P â fi fj + C â fi fj 4 Q2 Q2 ! 2 Âľ 1 (2) F â ln P â fi fj + i â j. 4 Q2 100 NNLO Calculation ResBos2 (qTcut = 1.5GeV) ResBos2 (qTcut = 4.5GeV) MCFM FEWZ Cross-Section 1111.0 pb 1115.6 pb 1116.6 pb 1111.0 pb Table 3.4: Total inclusive NNLO Drell-Yan Cross-sections for 66 GeV < Mll < 116 GeV. MCFM calculation using version 8.0 [3]. FEWZ calculation using version 3.1 rc [4] Similar to the NLO calculation, the results can be compared to that from other publicly available tools. Everything in the comparison is the same as it was in the NLO calculation, with the exception that the real correction is obtained from SHERPA [178]. The results are shown in Tab. 3.4. The results for the O Îąs3 calculation can be found in the App. F. 3.6 Scale Dependence In the resummation formalism, there exists three constants that are a result of solving the renormalization group equations. These constants are arbitrary, and should therefore not appear in the expansion of the resummation formalism to a fixed-order in Îąs . This implies that the resummation coefficients, A, B, C, and H should depend on these parameters. Here the calculation is done in the CSS scheme, but the CFG scale dependence can be obtained using Eq. 3.73. To obtain the coefficients, the resummation formalism is expanded for arbitrary scales and is compared to the canonical choice, (C1 = C3 = b0 , and C2 = 1). In other words: W (b, Q, C1 , C2 , C3 ) |O(Îąn ) = W (b, Q, C1 = b0 , C2 = 1, C3 = b0 ) |O(Îąn ) , s s 101 (3.95) where the definitions of C1 , C2 , and C3 can be found in the scale dependent resummation formalism given as: ! ! Z C 2 Q2 2 C22 Q2 dÂľ 2 W = exp â A(Âľ; C1 ) log + B(Âľ; C1 , C2 ) 2 Âľ2 C12 /b2 Âľ C1 C3 C1 C 3 C â f a x1 , , C â f b x2 , , . C2 b C2 b (3.96) performing the series expansion of the previous equation, and using Eq. 3.95, to O Îąs3 , the scale dependence is given by: A(1) = A(1,c) (3.97) ! 2 b 0 A(2) = A(2,c) â Î˛0 A(1,c) log C12 b0 b0 (3) (3,c) 2 (1,c) 2 A =A + 4Î˛0 A log â 2 log Î˛1 A(1,c) + 2Î˛0 A(2,c) C1 C1 ! b2 C 2 B (1) = B (1,c) â A(1,c) log 0 22 C1 ! 2C 2 b B (2) = B (2,c) â A(2,c) log 0 22 C1 b0 (1,c) 2 (1,c) 2 (1,c) + Î˛0 2A log â 2A log (C2 ) + 2B log (C2 ) C1 ! 2C 2 b B (3) = B (3,c) â A(3,c) log 0 22 C1 b0 (1,c) 2 (1,c) (1,c) + 2Î˛1 A log + log (C2 ) B âA log (C2 ) C1 4 2 b0 (1,c) 3 2 (1,c) (1,c) â Î˛0 2A log + log (C2 ) 2A log (C2 ) â 3B 3 C1 b0 (2,c) 2 (2,c) (2,c) + 4Î˛0 A log + log (C2 ) B âA log (C2 ) C1 102 (3.98) (3.99) (3.100) (3.101) (3.102) C (1) = (1,c) Cja (Îž) + Î´ja Î´(1 â Îž) 1 â A(1,c) log2 4 b20 C22 C12 ! 1 + B (1,c) log 2 b20 C22 C12 C2 1 (1) â Pja log 23 2 b0 !! (3.103) ! ! 2 Âľ2 2C 2 b b 1 0 2 log F C (2) = â Î˛0 A(1,c) log2 4 C12 b20 ! ! ! 2 Âľ2 2 2C 2 2C 2 b b b 1 1 F 0 2 + Î˛0 B (1,c) log 0 22 log + A(1,c) log4 2 32 C1 b20 C12 ! ! 2C 2 2C 2 b b 1 1 0 2 â A(1,c) B (1,c) log3 0 2 + 1 B (2,c) log â Î˛0 A(1,c) log3 2 12 8 2 C1 C12 ! ! 1 (2,c) 2 b20 C22 1 (1,c) 2 2 b20 C22 1 â A log + B log + Î˛0 B (1,c) log2 2 2 4 8 4 C1 C1 ! ! b20 C22 1 (1,c) (1,c) 1 (1,c) (1,c) 2 b20 C22 + B Cja log â A Cja log 2 4 C12 C12 C2 1 (1,c) 1 (2) (1,c) (1) + Î˛0 Cja â Cjb â Pba â Pja log 23 2 4 b0 ! 2 2C 2 C b C2 1 (1,c) (1) 1 (1) (1) + A Pjb â Pba log2 23 â B (1,c) Pja log 0 22 log 23 8 4 b C1 b0 !0 b20 C22 C32 Î˛0 (1) 2 C32 1 (1) + A(1,c) Pja log2 log â Pja log 2 , 8 4 C12 b20 b0 (2,c) Cja (Îž) + Î´ja Î´(1 â Îž) ! b20 C22 C12 !! b20 C22 C12 (3.104) Comparing these results to that from Ref. [179], it is important to note the differences in the definition of Î˛0 and Î˛1 . In Ref. [179], the Î˛ functions are Î˛0 = (11CA â 2nf )/6 and 2 â 5C n â 3C n )/6, while here Î˛ = (11C â 2n )/12 and Î˛ = (17C 2 â Î˛1 = (17CA 0 1 A f F f A f A 5CA nf â 3CF nf )/24. Note that this result is consistent with Ref. [179], except for the scale dependence in C (2) . Additionally, the calculation is extended to include A(3) and B (3) . The maximum uncertainty for the Sudakov factor arises for the choice C1 = b0 /2 and C2 = 2 and C1 = 2b0 and C2 = 1/2, which can be understood from the fact that this has the largest impact on the value of the Sudakov integral. The dependence of C3 for the uncertainty is more complicated, because it deals with the complex energy and x-dependence of the PDFs. 103 With the these scale dependence calculations, and the calculations of the previous sections for the asymptotic piece, the comparison of the ResBos2 code to the LHC data for W and Z physics is possible to a higher precision then previously. The comparison to the LHC data can be found in Chap. 4 and 5, for the Z and W results respectively. 104 Chapter 4 Z Boson Resummed Predictions At the LHC, one of the most important precision Standard Model processes to study is DrellYan, specifically, with a focus on the Z boson mass peak. The precision of the data at the LHC is at the sub-percent precision for the normalized distributions, and a few percent for unnormalized distributions due to the luminosity uncertainty at the Z-peak. The transverse momentum distribution is important for the study of soft gluon resummation. However, a new observable was recently proposed at the Tevatron which has been shown to be more accurate, known as ĎâÎˇ [180, 25] and is discussed in Sec. 4.2. 4.1 Z pT Distribution As mentioned in Chapter 3, the transverse momentum of the Z boson requires resummation to describe the small pT region. With the precision of the LHC data, the current ResBos code is insufficient, and the order of resummation needs to be increased, from NNLL to N3 LL. A comparison of the calculation using NNLL resummation to the data at ATLAS at â s = 8 TeV can be seen in Fig. 4.1. Clearly, there is a disagreement between the theory and the data. There are two regions of disagreement between the theory prediction and the data. These two regions can be categorized as: the intermediate transverse momentum region, between about 10 GeV and 100 GeV, and the high transverse momentum region, 105 T 1/Ď dĎ/dp ll RESBOS / Data 1.4 1.3 1.2 ATLAS s = 8 TeV, 20.3 fb-1 Data - statistical uncertainty Data - total uncertainty RESBOS uncertainty 66 GeV â¤ mll < 116 GeV, |y | < 2.4 ll 1.1 1 0.9 1 10 102 pll [GeV] T Figure 4.1: Comparison of pll T data from ATLAS compared to ResBos predictions, reproduced from Ref. [23] above 100 GeV. The resolution to each of these disagreements arises for different physics reasons. Firstly, in the high transverse momentum region, the ResBos code used the invariant mass of the lepton pair for the factorization and renormalization scale. However, when the transverse momentum is above the mass of the Z boson, the major contribution comes from the process of a Z boson with one hard jet. In this process, the hard jet will have an energy close to that of the Z boson, and the scale that appears in the calculation is the transverse mass of the Z boson, defined as: Âľ2 = Mll2 + p2T , (4.1) where Mll is the invariant mass of the lepton pair, and pT is the transverse momentum of the lepton pair. Changing the scale to the transverse mass results in better agreement between the theory prediction and the data as seen in Fig. 4.2. However, in the intermediate transverse momentum region simply changing the scale 106 1dĎ Ďdq (pb/GeV) t 10â5 10â6 10â7 Theory CSS321 10â8 1.2 1.15 200 300 400 500 600 700 800 200 300 400 500 600 700 800 900 qt (GeV) 1.1 1.05 1 0.95 0.9 0.85 0.8 900 qt (GeV) Figure 4.2: Comparison of the ResBos2 calculation with a scale choice of Âľ2 = Mll2 + p2T to the ATLAS data given in Ref. [23], focusing on the high transverse momentum region to the transverse mass does not resolve the problem. Therefore, the precision of the theory calculation needs to be improved. The order of the resummation calculation needs to be done to N3 LL, which means that the A coefficient is done to O Îąs3 , the B coefficient is done to O Îąs3 , and the C coefficient is done to O Îąs2 . The order of B needs to be one order higher than that of the C coefficient in order to have good agreement between the CSS and CFG prediction as seen in Fig. 4.3. In the previously mentioned figure, only the W -piece is included to focus solely on the difference between the two schemes (the Y -piece is the same in both schemes). The fixed order calculation for Z+jet at NLO is obtained from the Sherpa [178] program to correctly predict the angular distributions which is discussed in Section 4.3. Since the fixed order calculation contains a jet, the transverse momentum of the Z boson is required to be greater than 2 GeV to remove the singularity at pT = 0. Additionally, it has been shown that at this order the fixed order and asymptotic calculations cancel to a sub-percent accuracy for pT â 2 GeV, as shown in Fig. 4.4. Finally, the matching of the resummed prediction to the fixed order prediction needs to be performed at high transverse momentum. In the ResBos2 code, matching occurs when the asymptotic expansion cancels with the resummed calculation, i.e. W â A = 0. As seen in Fig. 4.5, there are multiple 107 T dĎ dp 100 CSS Prediction CFG Prediction 80 60 40 Ratio to CSS 20 1.008 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 p (GeV) 1.006 1.004 1.002 1 0.998 0.996 0.994 0.992 0.99 T dĎ (pb/GeV) dp Figure 4.3: Comparison of the CSS and CFG formalisms, focusing on only the small transverse momentum, and not including the Y -piece. T 102 Fixed Order Asymptotic Piece 10 1 10 20 30 40 50 60 70 p (GeV) T Figure 4.4: Comparison of the fixed order piece up to O Îąs2 to the asymptotic piece up to O Îąs2 . The cutoff on the transverse momentum of both calculations is set to 2 GeV. crossing points, and the first crossing past the peak is used as the matching condition. Above the matching point, the prediction is set to be only the fixed order prediction. In this matching scheme, the crossing occurs approximately around 20 GeV for the invariant mass of the lepton pair near the Z peak, and in the central rapidity region, for a resummed order of N3 LL and an asymptotic order of O Îąs3 . With the resummation calculation to N3 LL, asymptotic piece to O Îąs2 , the fixed order to NNLO, and the matching procedure as described above, the ResBos2 code can be compared to the data. The comparison of the ResBos2 code for both the CSS and CFG schemes to the ATLAS 108 T dĎ (pb/GeV) dp Resummed 10 Asymptotic Piece 1 0 10 20 30 40 50 60 70 p (GeV) T T 10-1 1 dĎ Ď dp 1 dĎ Ď dp T Figure 4.5: Comparison of the Asymptotic piece to O Îąs3 to the N3 LL resummed piece. 10-2 -3 -3 10 10 10-4 10-4 -5 10 -6 -5 10 -7 10-7 -8 10 1.3 10 -6 10 10 -8 10 1.3 Ratio to Theory Ratio to Theory 10-1 10-2 1.2 1.1 1 0.9 0.8 1 10 102 1.2 1.1 1 0.9 0.8 * ĎÎˇ 1 10 102 * ĎÎˇ Figure 4.6: Comparison of both the CSS(left) and CFG(right) prediction to data. The PDF and scale uncertainty is given by the error bands. The lighter error band is the PDF uncertainty, and the darker error band is the combination of the scale and PDF uncertainty. 109 data [23] can be found in Fig. 4.6. In the figure, both the scale and PDF uncertainties are shown. As mentioned previously, the high transverse momentum region is resolved by switching the scale of the calculation, and this is supported by the results shown in the previously mentioned figure. Additionally, by increasing the order to N3 LL accuracy, the intermediate transverse momentum is improved. There exists a calculation for Z+jet up to NNLO [24], however, their results are not public enough yet to perform a detailed comparison to the asymptotic piece up to O Îąs3 . However, in their calculation, the kfactor is approximately flat above the matching region, in which the fixed order is the only calculation. To approximate these corrections at high transverse momentum, this k-factor is included. One step for the future of the ResBos2 calculation, is to perform a matching to the NNLO Z+jet calculation, when the results are public enough to check that the asymptotic piece cancels exactly in the limit of the transverse momentum going to zero. The results from the NNLO Z+jet calculation for the same data set as compared to above can be found in Fig. 4.7, and shows improvement in the intermediate transverse momentum region over the NLO result. Therefore, the matching to this prediction would further improve the prediction of the ResBos2 code. 4.2 Z ĎâÎˇ Distribution The ĎâÎˇ observable is a new observable proposed at the Tevatron [180, 25] that only depends on the angular distribution of the final state leptons, but directly correlates with the transverse momentum of the Z boson. The definition of this observable is given as: ĎâÎˇ = tan Ď â âĎ 2 110 sin Î¸Îˇâ , (4.2) ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ ďż˝ ďż˝ďż˝ ďż˝ďż˝ ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ Figure 4.7: Comparison of the fixed order calculation of Z+jet to NNLO compared to the ATLAS data. Reproduced from [24]. Figure 4.8: The frame definition for ĎâÎˇ , reproduced from [25]. where âĎ is the azimuthal separation of the two leptons and Î¸Îˇâ is the measurement of the scattering angle with respect to the proton beam direction in the rest frame of the Z boson. â + Îˇ âÎˇ Î¸Îˇâ is given in terms of lab observables as cos Î¸Îˇâ = tanh , where Îˇ â and Îˇ + are 2 the pseudorapidities of the negatively and positively charged lepton respectively, see Fig. 4.8 for a depiction of the frame. Taking the limit of the transverse momentum to zero, the ĎâÎˇ prediction can be approximated by: ĎâÎˇ â qT sin ĎCS , Mll 111 (4.3) 1/Ď dĎ/dĎ* Îˇ RESBOS / Data 1.4 1.3 1.2 ATLAS s = 8 TeV, 20.3 fb-1 Data - statistical uncertainty Data - total uncertainty RESBOS uncertainty 66 GeV â¤ mll < 116 GeV, |y | < 2.4 ll 1.1 1 0.9 10-2 10-1 1 Ď* Îˇ Figure 4.9: Comparison of ĎâÎˇ data from ATLAS compared to ResBos predictions, reproduced from Ref. [23] where ĎCS is the Ď angle in the Collins-Soper Frame, qT is the transverse momentum of the Z boson, and Mll is the invariant mass of the lepton pair. Using the equation above, the correspondence between the ĎâÎˇ distribution and the transverse momentum distribution may be approximated, e.g. for Mll â MZ , the range of 10â 3 â¤ ĎâÎˇ â¤ 0.1 radians corresponds to a transverse momentum from 0.1 GeV to 10 GeV. From Eq. 4.3, the features can be mapped to Fig. 4.9, and therefore can be explained in the same manner as before. Looking first at the high ĎâÎˇ region, the disagreement is again resolved by a change of scale, from Mll to MT . For the intermediate ĎâÎˇ region (0.1 â¤ ĎâÎˇ â¤ 1), the calculation needs to be improved as detailed above. Again Sherpa is used to obtain the perturbative prediction for Z+jet at NLO for the transverse momentum above 2 GeV. The matching is done in the same way as mentioned above, again matching in the transverse momentum distribution. Since the results are fully differentiable, the ĎâÎˇ distribution can be calculated from the lepton and anti-lepton momentums. Similar to the discussion above for the transverse momentum distribution of the Z boson, 112 1 dĎ Ď * dĎÎˇ 1 dĎ Ď * dĎÎˇ 10 1 1 10-1 10-1 10-2 10-2 -3 -3 10 Ratio to Theory 10 Ratio to Theory 10 1.3 1.2 1.1 1 0.9 0.8 10-4 -3 10 10-2 10-1 1.2 1.1 1 0.9 0.8 10-4 * 1 1.3 Ď10 Îˇ -3 10 10-2 10-1 * 1 Ď10 Îˇ Figure 4.10: Comparison of both the CSS(left) and CFG(right) prediction to data. The PDF and scale uncertainty is given by the error bands. The lighter error band is the PDF uncertainty, and the darker error band is the combination of the scale and PDF uncertainty. the improvement from the ResBos2 code is again noticeable. The PDF and scale uncertainties for the ResBos2 prediciton of ĎâÎˇ can be found in Fig. 4.10. The improvement in the large ĎâÎˇ region is again due to using the correct scale for the calculation. Also, the improvement in the intermediate region is due to the increased accuracy of the ResBos2 code over the ResBos code. Finally, the Z+jet to NNLO fixed order calculation [24] compared to the data can be found in Fig. 4.11. The matching cannot yet be completed to this order, until the results of the group become more public. However, once the matching can be performed, the ResBos2 prediction should be even further improved. 4.3 Angular Functions In Drell-Yan, the angular distributions of the charged lepton pairs allow for an additional handle on precision QCD studies. The fully differential cross-section describing the kinematics of the leptons can be decomposed into nine harmonic polynomials in the Collins-Soper Frame [181, 182, 183, 184]. It is convenient to factor out the unpolarized cross-section, and write the differential cross-section as a function of the harmonic polynomials and dimension- 113 ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ ďż˝ ďż˝ ďż˝ďż˝ ďż˝ďż˝ Figure 4.11: Comparison of the fixed order calculation of Z+jet to NNLO compared to the ATLAS data. Reproduced from [24]. 114 less angular coefficients, A0â7 as: dĎ dĎ = 2 dpT dydQ d cos Î¸dĎ dpT dydQ2 1 2 2 1 + cos Î¸ + A0 1 â 3 cos Î¸ + A1 sin 2Î¸ cos Ď 2 (4.4) 1 + A2 sin2 Î¸ cos 2Ď + A3 sin Î¸ cos Ď + A4 cos Î¸ 2 2 A5 sin Î¸ sin 2Ď + A6 sin 2Î¸ sin Ď + A7 sin Î¸ sin Ď . It is important to note that only A4 is non-zero in the limit of pT goes to zero. There is a well established relationship that states that A0 â A2 = 0, known as the Lam-Tung relation [142, 143, 144], and is expected to hold up to O (Îąs ), but is known to break down at O Îąs2 as will be shown later. The coefficients A5,6,7 are zero at NLO, and small at NNLO, and thus will not be included in the comparisons to data. The coefficients A3 and A4 depend on the relationship between the vector and axial couplings and are therefore sensitive to the weak mixing angle (sin2 Î¸W ). To obtain the theory predictions for each of the angular coefficients, the moments of each coefficient are calculated by: R hPi (cos Î¸, Ď)i = Pi (cos Î¸, Ď) dĎd cos Î¸dĎ R , dĎd cos Î¸dĎ (4.5) where the Pi âs are the angular functions associated with each Ai respectively. There is a 115 Figure 4.12: The predictions for A2 â A0 for the DYNNLO code [26], at NLO and NNLO. Reproduced from [27]. direct relationship between each moment and the angular coefficient given by: 1 3 2 hP0 (cos Î¸, Ď)i = h 1 â 3 cos Î¸ i = 2 20 1 hP1 (cos Î¸, Ď)i = hsin 2Î¸ cos Ďi = A1 , 5 1 hP2 (cos Î¸, Ď)i = hsin2 Î¸ cos 2Ďi = A2 , 10 1 hP3 (cos Î¸, Ď)i = hsin Î¸ cos Ďi = A3 , 4 1 hP4 (cos Î¸, Ď)i = hcos Î¸i = A4 , 4 1 hP5 (cos Î¸, Ď)i = hsin2 Î¸ sin 2Ďi = A5 , 5 1 hP6 (cos Î¸, Ď)i = hsin 2Î¸ sin Ďi = A6 , 5 1 hP7 (cos Î¸, Ď)i = hsin Î¸ sin Ďi = A7 . 4 2 A0 â , 3 (4.6) The precision of the LHC shows the breaking of the Lam-Tung relation at large transverse momentum [27], as predicted by the NNLO calculation by DYNNLO [26] of the angular 116 A2 A0 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 1 10 102 1 p (GeV) 10 102 p (GeV) T 0.4 A0-A2 A4 T 0.35 0.4 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 -0.05 -0.05 -0.1 1 10 102 -0.1 1 p (GeV) T 10 102 p (GeV) T Figure 4.13: Theoretical prediction from the ResBos2 program for the angular coefficients of A0 , A2 , A4 , and the breaking of the Lam-Tung Relationship for the CSS Scheme. The lighter error band is the PDF uncertainty, and the darker error band is the combination of the scale and PDF uncertainty. functions in Drell-Yan, see Fig. 4.12. The original ResBos code is unable to appropriately predict the angular coefficients to the precision required by the LHC, due to the fact that the k-factor obtained using the calculation by Arnold and Kauffman [177] can only be applied to the symmetric and anti-symmetric leading order angular functions (L0 and A4 ), but not to the other angular distributions. To include these corrections in the ResBos2 code, the fixed order prediction is obtained by using the Z+jet prediction at NLO from the Sherpa code [178], which includes the full angular dependence. From the prediction in Figs. 4.13 and 4.14, the ResBos2 prediction shows the LamTung relationship breaking down, as expected at this order. Additionally, A4 is non-zero as expected. There is some disagrement with the data, but it should improve with the inclusion of higher order corrections. 117 A2 A0 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 1 10 102 1 p (GeV) 10 102 p (GeV) T 0.4 A0-A2 A4 T 0.35 0.4 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 -0.05 -0.05 -0.1 1 10 102 -0.1 1 p (GeV) T 10 102 p (GeV) T Figure 4.14: Theoretical prediction from the ResBos2 program for the angular coefficients of A0 , A2 , A4 , and the breaking of the Lam-Tung Relationship for the CFG Scheme. The lighter error band is the PDF uncertainty, and the darker error band is the combination of the scale and PDF uncertainty. 118 Chapter 5 W Mass One of the most interesting places to look for new physics signals is in the W mass. The precision of the Standard Model prediction of the mass is at the 0.01% level, and the experimental measurement is of similar precision. In order to improve the experimental measurement of the W boson, it is important to understand the transverse momentum of the W boson, and its relationship with the Z boson transverse momentum. Currently, the dominate uncertainty in the direct measurement of the W mass at the LHC arises from the theoretical predictions of the transverse mass of the W boson, or the transverse momentum of the lepton. In this chapter, the Standard Model prediction from Electroweak precision tests will be discussed, followed by a discussion of the experimental measurements at the Tevatron and the LHC. Finally, the improvements to the ResBos2 calculation for the W mass measurement are introduced. 5.1 SM EW Precision Fit Using the Electroweak input scheme defined by on MZ , GÂľ , and Îą, the W boson mass is a predicted parameter. The Standard Model calculation is detailed in Ref. [185], and is given by: 2 MW 1â 2 MW MZ2 ! ĎÎą =â (1 + âr) , 2GÂľ 119 (5.1) with the loop corrections contained in âr. At the one-loop order, the calculation of âr is given as: c2 âr = âÎą â W âĎ + ârrem (MH ) , s2W (5.2) M2 where c2W = W , s2W = 1 â c2W , and sW is the Weinberg angle defined in Chap. 1 as the M2 Z rotation angle needed to diagonalize the neutral boson sector of the Standard Model. The correction to Îą, given by âÎą is due to light fermions and is proportional to the logarithm of their masses. Additionally, there is a modification to the Ď parameter as defined in Sec. 1.1.3. Finally, the last term contains all the dependence of the Higgs boson mass. The calculation of âr has been done to two-loops in the Electroweak coupling, contains the fermionic contributions [186, 187, 188] and the bosonic contributions [189, 190, 191, 192]. As for the QCD corrections to âr, they are known to O ÎąÎąs2 [193, 194, 195, 196]. Furthermore, âĎ has been calculated to O ÎąÎąs3 [197, 198, 199]. The fit to all of the Electroweak observables was done at NNLO in Ref. [5], and a summary of the results can be found reproduced in Table 5.1. The precision obtained by the global fit on the mass of the W boson sets the goal for the LHC. It is possible to break down the uncertainty of the fit to the W mass measurement into the contributions from the top mass, the theory uncertainty of the top mass, the Z mass, âÎąhad , Îąs , the Higgs mass, and the theory uncertainty of the W mass. These contributions are broken down in Table 5.2. The largest uncertainties on the indirect determination of the W mass come from both the experimental and theoretical determination of the top quark mass, followed by the theory uncertainty on the W mass, and then the experimental measurement of the mass of the Z boson. The LHC should provide an improvement on the experimental measurement of the top quark mass, and should improve significantly at a future electron-positron collider. There is not going to be any improvement 120 Parameter Input Value MH [GeV] a 125.14 Âą 0.24 Fit Result w/o exp. input w/o exp. input or theory unc. 125.14 Âą 0.24 93+25 â21 93+24 â20 MW [GeV] ÎW [GeV] MZ [GeV] ÎZ [GeV] 80.385 Âą 0.015 2.085 Âą 0.042 91.1875 Âą 0.0021 2.4952 Âą 0.0023 80.364 Âą 0.007 2.091 Âą 0.001 91.1880 Âą 0.0021 2.4950 Âą 0.0014 80.358 Âą 0.008 2.091 Âą 0.001 91.200 Âą 0.011 2.4946 Âą 0.0016 80.358 Âą 0.006 2.091 Âą 0.001 91.200 Âą 0.010 2.4945 Âą 0.0016 mt [GeV] 173.34 Âą 0.76 173.81 Âą 0.85 b 170.0+2.3 â2.4 177.0 Âą 2.3 b a Average of the ATLAS [200] and CMS [201] measurements, ignoring any correlation between systematic uncertainties b The theoretical top-mass uncertainty is excluded. Table 5.1: Electroweak fit. The fourth column gives the fit results without using any experimental or phenomenological estimate for the parameter when performing the fit. The fifth column is the same as the fourth, but ignores all theory uncertainties. The table is reproduced from Ref. [5]. Source mt Î´theory mt MZ âÎąhad Îąs MH Î´theory MW Value 0.0046 0.0030 0.0026 0.0018 0.0020 0.0001 0.0040 Table 5.2: The values are reported in GeV, and taken from Ref. [5] on the experimental measurement of the Z boson at the LHC or at a future electron-positron collider, due to the fact that the LEP measurements are very precise [202]. Finally, the experimental measurement of the W mass ideally should be close to the current uncertainty of the indirect measurement by the end of the LHC running [203, 204]. The details of the experimental goals and the limits that are expected are discussed in the following section. 121 5.2 Experimental Measurement The experimental measurement of the W mass is one of the most important measurements at a hadron collider. The most precise channel to study the W mass at the LHC comes from the process, pp â eÎ˝ + X. However, the fact that there is a neutrino in the final state leads to the inability to directly reconstruct the invariant mass of the lepton-neutrino system. Therefore, two different observables are proposed to study the mass of the W boson. The first observable is the transverse mass of the W boson defined as: / (1 â cos Î¸) , MT2 = 2pT E l T (5.3) / T is the missing transverse where pT is the transverse momentum of the final state lepton, E l momentum (attributed to the neutrino), and Î¸ is the angle between the lepton and missing transverse momentum. The other observable is the transverse momentum of the lepton. Both of these observables are not accurately predicted by a fixed order calculation due to the dependence on the transverse momentum of the W boson, which requires resummation. In measuring the W mass, an appropriate prediction of the Jacobian peak region for the lepton is important to obtain the correct mass. One of the most important features in determining the W mass is the tail of the distribution, and the Jacobian peak. Having an accurate theoretical prediction is required to match the shape of the data. The traditional method of obtaining the W mass is through the use of a template fit technique. To estimate the uncertainty that arises from the PDF, the following 4 step method is used [205]1 : 1 For real data, a similar procedure is used to estimate the central value, but replacing the pseudodata with the actual data. 122 Figure 5.1: Results of the EW Fit, reproduced from Ref. [5] 123 1. Generate the observable to be used for the fitting for fixed W mass for each PDF error set of interest. Treat the generated observable as pseudodata. 2. Generate the mass templates for the distributions used to fit the pseudodata as a function of the W boson mass using the central prediction for the PDF. 3. Given the PDF set i, and the mass template j, calculate the Ď2 for each combination of i and j, given by: Ď2i,j = 1 Nbins Nbins X k=1 2 j Ok â Oki 2 j 2 , i Ďk + Ďk (5.4) where Ok and Ďk are the value of the observable and the standard deviation of it in a given bin (k), respectively. 4. The minimum of the Ď2i,j distribution for each PDF set i gives the mass j that the PDF best fits. The difference from the central value is the shift in the mass induced by PDF i. Combining all of the PDF sets, using the method to obtain the uncertainty as defined for the given PDF set, the allowed mass range is obtained. This procedure will be demonstrated below for the CT14nnlo PDF set [10] for the lepton transverse momentum. Similar results are obtained if the transverse mass of the W boson is used. The CTEQ PDFs are based off of the Hessian method. Therefore, each eigenvector does not give the full uncertainty, and the uncertainty needs to be calculated by using the 124 master equation as given in [206] and below: v u n uX 2 + âX = t max Xi+ â X0 , Xiâ â X0 , 0 , (5.5) i v u n uX 2 â âX = t max X0 â Xi+ , X0 â Xiâ , 0 , (5.6) i where i goes over each eigenvector, and the Âą refers to whether the shift is along the positive or negative direction respectively. 5.3 ResBos2 Results One important prediction to make is the ratio of the transverse momentum of the Z boson to the transverse momentum of the W boson. With an accurate prediction of this ratio, the experimentalists can use data driven methods to obtain the transverse momentum predictions of the W boson through: ďŁŤ dĎ ďŁś ) ďŁŹ dp(W ďŁˇ dĎ T th ďŁˇ dĎ ďŁŹ = ďŁ dĎ , ďŁ¸ (Z) (W ) dpT dp (Z) T data dpT where dĎ (W ) dpT (5.7) th is the data driven prediction for the W boson transverse momentum, the theory prediction for the W boson transverse momentum, dĎ (Z) for the Z boson, and dĎ (Z) dpT data dpT dĎ (W ) dpT th is is the theory prediction th is the Z boson transverse momentum data. An accurate prediction of this ratio allows to attempt to apply a transverse momentum cut on the W boson. This cut reduces the theoretical uncertainty arising from the PDFs greatly [205]. The normalized ratio (RW/Z ) is shown in Fig. 5.2 for the ResBos2 prediction. 125 T RW/Z(p ) 1.06 1.04 1.02 1 0.98 0.96 0.94 0.92 0 5 10 15 20 25 30 p (GeV) T Figure 5.2: The ratio of the normalized transverse momentum distributions of the W boson to the Z boson. The ResBos code can be used to calculated the theoretical uncertainty of the PDFs, as described in the section above. An example of the Ď2 distribution for one error set can be found in Fig. 5.3. After calculating the shift for all of the eigenvectors, and using the Hessian master equation as given in Eq. 5.5, the CT14nnlo PDF uncertainty can be calculated. For â a collider center of mass energy of 13 TeV, the W mass uncertainty is given as Î´ + = 37 MeV and Î´ â = 34 MeV. 126 Ď2 500 450 400 350 300 250 200 150 80260 80280 80300 80320 80340 80360 80380 80400 80420 80440 W Mass (MeV) Figure 5.3: An example of the Ď2 distribution for an error set, others are similar to this. The central input mass was MW = 80.358 GeV. The minimum here occurs around 80.372, a shift of 14 MeV. 127 Chapter 6 Color Singlet Boson Plus Jet Resummation at Hadron Colliders Soft gluon resummation for the class of color singlet boson plus jet processes has its own interest in perturbative QCD. To deal with the divergence in low transverse momentum hard processes, the transverse momentum resummation formalism is employed [138]. However, the qT resummation formalism has been mainly applied to color-neutral particle production, such as inclusive vector boson W/Z and Higgs boson productions. Extensions to jet production in the final state has been limited, not only because of the technical issues associated with the jets in the final state, but also because jets carry color. Therefore, the soft gluon interactions are more complicated than those for color neutral particle production. Nevertheless, there has been progress made in the last few years on the qT resummation for dijet production in hadronic collisions [207, 208, 209]. Here the details of expanding the calculation to single jets plus a color singlet boson are examined. The resummation formula for V + j resummation can be summarized as X d5 Ď = Ď0 dyV dyj dPT2 d2 qâĽ ab "Z d2~bâĽ âi~q Âˇ~b e âĽ âĽ Wabâcd (x1 , x2 , bâĽ ) + Yabâcd (2Ď)2 # , (6.1) where yV and yj are rapidities for the boson and the jet, respectively, PT is the jet transverse 128 momentum, and ~qâĽ = P~V âĽ + P~J is the total transverse momentum of the boson and the jet. The first term W contains the all order resummation, the second term Y comes from the difference between the fixed order corrections and the asymptotic piece, and Ď0 represents the normalization of the differential cross section. Higgs plus jet qT resummation will be used as an example of the procedure for calculating the resummation results for this class of processes. 6.1 Higgs Plus Jet Resummation The effective Lagrangian in the heavy top quark mass limit is used to describe the coupling between the Higgs boson and gluon given by: Lef f = â Îąs a aÂľÎ˝ F F H, 12Ďv ÂľÎ˝ (6.2) where v is the vacuum expectation value, H the Higgs field, F ÂľÎ˝ the gluon field strength tensor, and a the color index. To begin, the W term can be written as: 2 WggâHg (x1 , x2 , b) = HggâHg (Q)x1 fg (x1 , Âľ)x2 fg (x2 , Âľ)eâS(Q ,bâĽ ) , (6.3) 2 WgqâHq (x1 , x2 , b) = HgqâHq (Q)x1 fg (x1 , Âľ)x2 fq (x2 , Âľ)eâS(Q ,bâĽ ) + x1 â x2 , (6.4) 2 WqqĚâHg (x1 , x2 , b) = HqqĚâHg (Q)x1 fq (x1 , Âľ)x2 fq (x2 , Âľ)eâS(Q ,bâĽ ) + x1 â x2 , (6.5) at the next-to-leading logarithmic (NLL) accuracy. Here Q2 = s = x1 x2 S and represents the hard momentum scale, b0 = 2eâÎłE , with ÎłE being the Euler constant, fa,b (x, Âľ) are the parton distributions for the incoming partons a and b, and x1,2 are momentum fractions of 129 the incoming hadrons carried by the partons. Beyond the NLL, a C function associated with the parton distribution functions will also need to be included. The Sudakov form factor can be written as SSud (Q2 , b Z C 2 Q2 2 2 dÂľ Q 1 2 ln A + B + D ln 2 , âĽ) = 2 Âľ2 R C 2 /b2 Âľ 1 (6.6) âĽ where R represents the cone size of the jet. Here the parameters A, B, D can be expanded perturbatively in Îąs , for example in the gg â Hg channel, A = CA ÎąĎs , B = s â2CA Î˛0 ÎąĎs , and D = CA Îą 2Ď . The hard coefficient H can be calculated order by or- der, from the leading Born diagrams. For example, the gluon-gluon channel is given by: H (0) = s4 + t4 + u4 + m8h /(stu) [210, 211], where s = Q2 , t and u are the Mandelstam variables for the partonic 2 â 2 process. 6.1.1 Virtual Corrections Firstly, W (b) needs to be calculated at the one-loop order and shown that it can be factorized into the parton distribution, jet, soft, and hard factors. The virtual corrections have been calculated in the literature, see for example Refs. [210, 212]. For convenience, the results are summarized below to demonstrate the resummation of the Higgs plus jet system. Since the virtual diagrams are proportional to Î´(qT ), the b-space result can be expressed as (1)v Wab (bT ) = x1 f (x1 , Âľ) x2 f (x2 , Âľ) Îvab , (6.7) where f1 and f2 are the PDFs, x1 and x2 are the momentum fractions of the partons from each hadron, and the expressions for Îvab can be extracted from the calculations in the 130 references mentioned above. For the gluon-gluon channel, Îvgg = (0) Îąs CA Hgg " 3 1 â 2+ P2 Q2 2 ln 2 + ln J2 Âľ Âľ ! 2Ď !2 PJ2 1 Q2 PJ2 Q2 Q2 11 2 + ln 2 â 2 ln 2 ln 2 â 2 ln ln + Ď 2 ât âu 6 Âľ Âľ Âľ u M2 t + 2Li2 1 â 2 + 2Li2 + 2Li 2 Q M2 M2 2 2 # tĚ tĚ uĚ 2 uĚ 2 + ln 2 â ln + ln 2 â ln ât âu M M (1) + Î´Hgg , (6.8) (1) (0) where uĚ, tĚ = M 2 â u, t respectively, and Î´Hgg is the finite term not proportional to Hgg and is given by: (1) Î´Hgg i 1 M2 h 2 2 = (N â nf ) N (N â 1) stu + M (st + su + tu) . 3 stu (6.9) For the quark-gluon channel, Îvqg = 2 âu 10 â2 2 âu ât s 4CA Î˛0 + ln 2 â + CA 2 â ln 2 â ln 2 â ln 2 2Ď 3 Âľ Âľ Âľ Âľ (0) Îąs Hqg s u âu ât s uĚ uĚ +2 ln 2 ln + ln2 2 â ln2 2 â ln2 2 â 2 ln2 + 2 ln2 2 t âu Âľ Âľ Âľ Âľ M u 2 Ď 14 +4Li2 + 18Îś (2) â + 2 3 3 M â4 2 âu âu s âu +CF â â2 ln 2 + 3 â 2 ln2 2 + 2 ln2 + 6 ln 2 2 t Âľ Âľ Âľ 2 M t tĚ tĚ 2Ď 2 2 2 +4Li2 1 â + 4Li2 â 2 ln + 2 ln + 4Îś (2) â â 16 s ât 3 M2 M2 (1) + Î´Hqg , (6.10) 131 (1) (1) again with Î´Hqg analogous to Î´Hgg above, given by: (1) Î´Hqg = (CA â CF ) CA CF (s + t) . (6.11) Finally, for the quark-anti-quark channel, ÎvqqĚ 2 s 10 â2 2 sÂľ2 = 4CA Î˛0 â 2 ln 2 + + CA 2 â ln 3 ut Âľ 2 ut s ât âu M ln2 2 + 2 ln2 2 â 2 ln2 2 â 2 ln2 2 + 4Li2 1 â s sÂľ Âľ Âľ Âľ Ď2 â4 2 s â14Îś (2) â â 8 +CF â â2 ln 2 + 3 2 3 Âľ u ât âu s ut s ut t 2 â18 ln 2 ln 2 + 6 ln 2 + 2 ln 2 + 4 ln 2 ln 2 + 4Li2 + 4Li2 Âľ Âľ Âľ Âľ s Âľ sÂľ M2 M2 tĚ uĚ tĚ uĚ 2 â2 ln2 â 2 ln2 + 2 ln2 2 + 2 ln2 2 + 36Îś (2) â Ď 2 â 16 ât âu 3 M M (0) Îąs HqqĚ 2Ď (1) + Î´HqqĚ , (6.12) (1) with Î´HqqĚ given as: (1) Î´HqqĚ = (CA â CF ) CA CF (ât â u) . (6.13) It is important to note the 12 and 1 divergences that appear in the above equations. These will be canceled by the divergences in the remaining terms. At NLO, the strong coupling needs to be renormalized, yielding the following contribution: Îąs CA 2Ď 2 â Q 3 â 2Î˛0 . 2 Âľ Here, the renormalization scale is set to Q2 to simplify the final expression. 132 (6.14) 6.1.2 Jet Corrections Next, since this process contains a final state jet, the jet function needs to be applied in order to cancel out the collinear divergences from the final state. In order to perform this calculation, the Narrow Jet Approximation (NJA) is applied as derived in the calculation of inclusive jet production [213]. This approximation allows an analytic result for the jet production cross section to be obtained. For completeness, the quark and gluon jet contributions are listed below: Jg Jq " Îąs CA 1 1 = + 2 2Ď " Îąs CF 1 1 = + 2 2Ď ! # PJ2 R2 2Î˛0 â ln + Ig , Âľ2 ! # PJ2 R2 3 â ln + Iq , 2 Âľ2 (6.15) where I g,q depends on the jet algorithm used. For the kt -family of jet algorithms the values of I q and I g are, 1 Ig = 2 P 2 R2 ln J 2 Âľ !2 1 = 2 PJ2 R2 ln Âľ2 !2 Iq P 2 R2 67 3 2 23 â 2Î˛0 ln J 2 + â Ď â Nf , 9 4 54 Âľ (6.16) 3 PJ2 R2 13 3 2 â ln + â Ď . 2 2 4 Âľ2 (6.17) Their contributions to W (bâĽ ) can be included in the following manner, (1)J Wgg (bâĽ ) = x1 fg (x1 , Âľ)x2 fg (x2 , Âľ)Jg , (1)J Wqg (bâĽ ) = x1 fq (x1 , Âľ)x2 fg (x2 , Âľ)Jq , (1)J WqqĚ (bâĽ ) = x1 fq (x1 , Âľ)x2 fqĚ (x2 , Âľ)Jg . 133 (6.18) (6.19) (6.20) Again, there are divergences proportional to 1 as expected for a collinear jet function, which will be canceled with similar contributions from the soft corrections, see Sec. 6.1.4. 6.1.3 Collinear Corrections Thirdly, the collinear radiation associated with the incoming partons needs to be calculated. These contributions can be written in terms of parton splitting functions: 2 dx0 0 0 , Âľ) Îž(1 + Îž ) , x f (x q x0 (1 â Îž)+ Z Îąs 1 dx0 0 0 2 Pgq = 2 2 CF x fq (x , Âľ) 1 + (1 â Îž) , x0 2Ď qâĽ Z Îąs 1 dx0 0 2Îž 2(1 â Îž) 0 Pgg = 2 2 CA x fg (x , Âľ)Îž + + 2Îž(1 â Îž) , x0 (1 â Îž)+ Îž 2Ď qâĽ Z Îąs 1 1 dx0 0 0 , Âľ) Îž 2 + (1 â Îž)2 . Pqg = 2 2 x f (x g x0 2Ď qâĽ 2 Îąs 1 Pqq = 2 2 CF 2Ď qâĽ Z (6.21) (6.22) (6.23) (6.24) where Îž = x/x0 . Here the Îž 6= 1 contributions are taken into consideration, whereas the Îž = 1 part will be evaluated together with the soft gluon radiation contribution. These terms introduce divergences of the form 1 Pab into Wab (b). 6.1.4 Soft Corrections For soft gluon radiations, the leading power expansion can be applied to derive the dominant contribution by the Eikonal approximation, as discussed in Sec. F. This analysis has been applied in Ref. [214] to obtain the leading double logarithmic contributions to dijet production. For outgoing quark, antiquark, and gluon lines, the Eikonal approximation gives the 134 Figure 6.1: Feynman diagrams for the gluon-gluon scattering channel to a Higgs Boson. following Feynman rules: Âľ 2ki g, 2ki Âˇ kg + i Âľ â Âľ 2ki 2ki g, g, 2ki Âˇ kg + i 2ki Âˇ kg + i (6.25) respectively, where ki represents the momentum of the outgoing particles. Similarly, for incoming quark, antiquark, and gluon lines, the rules are: Âľ 2p1 g, 2p1 Âˇ kg + i Âľ â Âľ 2p1 2p1 g, g 2p1 Âˇ kg + i 2p1 Âˇ kg â i (6.26) respectively, where p1 represents the momentum for the incoming particle, g is the strong coupling constant, and kg is the momentum of the soft gluon. In the gluon-gluon scattering channel, the relevant Feynman diagrams can be found in 135 Fig. 6.1, from which the following contributions for soft gluon radiation can be calculated, g2 Z d3 kg C Î´ (2) (qâĽ â kgâĽ ) A Sg (p1 , p2 ) + Sg (k1 , p1 ) + Sg (k1 , p2 ) 3 2 (2Ď) 2Ekg (6.27) where Sg (p, q) is a short-handed notation for Sg (p, q) = 2p Âˇ q . p Âˇ kg q Âˇ kg (6.28) In order to evaluate the contributions from soft gluon radiation, the phase space of the gluon with restriction that gluon transverse momentum leads to the imbalance between the Higgs-jet system needs to be integrated out, i.e., g2 Z d3 kg Î´ (2) (qâĽ â kgâĽ )Sg (p1 , p2 ) . (2Ď)3 2Ekg (6.29) The derivation of the above term is straightforward, by noticing that the lower limit in the longitudinal momentum fraction integral, is given by: Z 1 dx 1 , 2 xmin x kgâĽ (6.30) where x is the momentum fraction of p1 carried by the soft gluon. Because of momentum k2 conservation, the lower limit for the x-integral is given by: xmin = gâĽ . Therefore, the Q2 above integral leads to the following leading contribution, 1 Q2 ln . 2 2 qâĽ qâĽ 136 (6.31) Substituting the above equation into Eq. (6.29) results in, g2 Z d3 kg Îąs 1 Î´ (2) (qâĽ â kgâĽ )Sg (p1 , p2 ) = 2 2 3 (2Ď) 2Ekg 2Ď qâĽ Q2 2 ln 2 qâĽ ! . (6.32) The other terms are not as straightforward to calculate, because the simple phase space integral will contain jet contributions which have already been taken into account by the jet functions above (Sec. 6.1.2). Therefore, the jet contributions need to be subtracted in order to avoid double counting. That means the phase space integral has to exclude the jet region, noting that this exclusion does not depend on the jet algorithm at this order. This is because, this contribution arises from the soft gluon radiation, whereas the jet algorithm focuses on the collinear gluons associated with the jet. As a general discussion, take the example of one term, given by: Z d3 kg (2) Î´ (qâĽ â kgâĽ )Sg (k1 , p1 ) = 2Ekg Z d2 k gâĽ Î´ (2) (q Z âĽ â kgâĽ ) dÎž1 2 , (6.33) Îž1 (kgâĽ â Îž1 k1âĽ )2 where k1âĽ represents the transverse momentum for the final state jet, and Îž1 = kg Âˇp1 /k1 Âˇp1 . Clearly, there is a collinear divergence associated with the jet, if the gluon radiation is within the jet cone a collinear divergence will be generated. In order to regulate this collinear jet divergence, the phase space integral can be limited to require that the gluon radiation be outside of the jet cone. Under this restriction, there will be no divergence associated with the jet, instead, the jet size R will be introduced to regulate the collinear divergence from the jet. There are different ways to regulate the above integral. The central point is to identify the jet cone requirement. In the above example, a divergence exists when kg is parallel to 137 k1 , which means the invariant mass of k1 + kg is small. The out of cone radiation requires that the invariant mass has a minimum, (k1 + kg )2 > Î2 . (6.34) Clearly, Î depends on the jet size, if (k1 + kg )2 is smaller than Î2 , its contribution needs to be excluded, because it belongs to the jet contribution calculated in the previous section. Following a similar analysis as for the jet contribution, the size of Î can be worked out. For example, if the kinematics of k1 and kg are substituted into the above equation, the results are: 2 , (k1 + kg )2 â k1âĽ kgâĽ ey1 âyg + eyg ây1 â 2k1âĽ kgâĽ cos(Ď1 â Ďg ) â k1âĽ kgâĽ R1g (6.35) where y1 and yg are rapidities for k1 and kg , Ď1 and Ďg are the azimuthal angles, for the jet and additional gluon respectively, and R1g represents the seperation of k1 and kg q (R1g = (y1 â yg )2 + (Ď1 â Ďg )2 ). In other words, if R1g is smaller than R, the gluon radiation will be considered inside the jet cone and vice versa. Therefore, in the phase space integral of Eq. (6.33), the following kinematic restriction is imposed: Î(2k1 Âˇ kg â Î2 ) with Î2 = k1âĽ kgâĽ R2 . Equivalently, a slight off-shell-ness for the jet momentum k1 can 2 R2 . By doing so there is no need be adopted to regulate the divergence: k12 = m21 = k1âĽ to impose any kinematic constraints, and the phase space integral can be carried out in a straightforward manner. The choice of m21 is made to ensure (k1 + kg )2 is always larger than 138 Î2 , shown by, (k1 + kg )2 q y1 âyg + eyg ây1 â 2k k 2 + m2 k = k1âĽ (6.36) 1âĽ gâĽ cos(Ď1 â Ďg ) 1 gâĽ e q q 2 2 2 2 2 + m2 â k â k1âĽ + m1 kgâĽ (ây) + k1âĽ kgâĽ (âĎ) + 2kgâĽ k1âĽ . 1âĽ 1 2 R2 , (k + k )2 is guaranteed to be larger than Î2 for any values of By choosing m21 = k1âĽ g 1 ây and âĎ. In addition,the narrow jet approximation, i.e., R â 0 limit, is taken. In this limit, the phase space cut-off technique results in the same leading contributions in terms of ln(1/R), as the introduction of a mass term. After adding an off-shell-ness to the jet momentum, the above integral can be written as Z dÎž1 2 . 2 (1 + R2 ) + k 2 â 2Îž k Îž1 Îž12 k1âĽ Âˇ k 1 1âĽ gâĽ gâĽ (6.37) To further integrate, the azimuthal angle of the jet is averaged over but the azimuthal angle is fixed for kgâĽ , giving: Z dÎž1 Ď dĎ 2 , 2 2 2 2 Îž1 0 Ď Îž1 k1âĽ (1 + R ) + kgâĽ â 2Îž1 k1âĽ kgâĽ cos(Ď) Z dÎž1 1 r = . 2 Îž1 2 2 2 2 2 2 2 Îž1 k1âĽ (1 + R ) + kgâĽ â 4Îž1 k1âĽ kgâĽ Z (6.38) Again, the integration over Îž1 has a lower limit, and taking the limit of qâĽ Q and R â 0, the leading power contribution is: " # 1 1 Q2 t 1 ln 2 + ln + ln 2 . 2 2 u R qâĽ qâĽ 139 (6.39) Therefore, the final result for the Sg (k1 , p1 ) term can be written as g2 Z " # 2 d3 kg Îą 1 Q 1 t s Î´ (2) (qâĽ â kgâĽ )Sg (k1 , p1 ) = 2 2 ln 2 + ln 2 + ln . (6.40) 3 u (2Ď) 2Ekg 2Ď qâĽ qâĽ R1 Evaluation of the other terms follow the same procedure, and the results are summarized below: Îąs 1 Sg (p1 , p2 ) â 2 2Ď 2 qâĽ Q2 2 ln 2 qâĽ ! , " !# Îąs 1 Q2 1 t 1 2 1 Ď2 Sg (k1 , p1 ) â ln 2 + ln 2 + ln + ln 2 + , 2 u 2 6 2Ď 2 qâĽ qâĽ R1 R1 " !# u Îąs 1 Q2 1 1 2 1 Ď2 Sg (k1 , p2 ) â ln 2 + ln 2 + ln + ln 2 + , 2 t 2 6 2Ď 2 qâĽ qâĽ R1 R1 (6.41) (6.42) (6.43) where the terms are kept since they contribute to the finite piece after Fourier transforming into bâĽ -space1 . 6.1.5 Total Real Correction Combining the soft and collinear gluon radiation together, the asymptotic behavior at small qT can be obtained. The gluon channel is given by: (0) Îąs Hgg 2Ď 2 qT2 Z dz1 dz2 z1 fg (z1 ) z2 fg (z2 ) Î´ (Îž2 â 1) Îž1 P (1) (Îž1 ) + Îž1 â Îž2 z1 z2 !! Q2 1 1 2 1 Ď2 + Î´ (Îž1 â 1) Î´ (Îž2 â 1) CA 2 ln 2 â 4Î˛0 + ln 2 + ln 2 + , 2 6 R R qT 1 Details on Fourier transforming into b-space can be found in App. F 140 (6.44) (0) where Îži = xi /zi , xi is the momentum fraction of the gluon from the proton, and Hgg is the tree level contribution. Similar equations exist for the gluon-quark and the quark-anti-quark channel. Combining the three channels gives: Îąs 1 2Ď 2 qT2 Z n dz1 dz2 z1 fa (z1 ) z2 fb (z2 ) Î´ (Îž2 â 1) Îž1 P (1) (Îž1 ) + Îž1 â Îž2 z1 z2 " ! Q2 1 (0) + Î´ (Îž1 â 1) Î´ (Îž2 â 1) Hgg CA 2 ln 2 â 4Î˛0 + ln 2 R qT Q2 3 u 1 (CA + CF ) ln 2 â 2Î˛0 â CF + (CA â CF ) ln + CF ln 2 2 t R qT ! Q2 3 u 1 (CA + CF ) ln 2 â 2Î˛0 â CF + (CA â CF ) ln + CF ln 2 2 t R qT !#) Q2 1 (0) +HqqĚ 2CF ln 2 â 3CF + CA ln 2 . R qT ! (0) +Hqg (0) +Hgq (6.45) This will be compared to the fixed order calculation in the limit qT goes to zero in Sec. 6.1.7. Now that the complete asymptotic piece is calculated, the Fourier Transform into b-space is performed. After the Fourier Transform, the poles from all of the different pieces cancel. The gluon-gluon channel is shown below, but similar results exist for the other channels, and can be found in App. G. The poles for the virtual correction are given by: Îąs CA H (0) 2Ď 3 1 â 2+ P2 Q2 2 ln 2 + ln J2 Âľ Âľ !! , (6.46) the coupling renormalization is given in Eq. 6.14, and needs to be multiplied by the tree level matrix element (H (0) ). The poles for the jet, collinear, PDF renormalization, and soft 141 corrections are given by: Îąs CA H (0) 2Ď 1 1 + 2 P 2 R2 2Î˛0 â ln J 2 Âľ !! , Îąs CA 1 Î´ (Îž2 â 1) Îž1 Pgg (Îž1 ) + Îž1 â Îž2 , 2Ď Îą C s A1 H (0) Î´ (Îž2 â 1) Îž1 Pgg (Îž1 ) + 2Î˛0 Î´ (1 â Îž1 ) + Îž1 â Îž2 , 2Ď Îąs CA 2 2 Âľ2 1 1 (0) H + ln â ln 2 , 2Ď R 2 Q2 â H (0) (6.47) (6.48) (6.49) (6.50) respectively. All the terms above are proportional to Î´(Îž1 â 1)Î´(Îž2 â 1) unless otherwise noted. Combining Eqs. 6.46, 6.14, and 6.50, it is clear that all of the poles cancel, and the remaining result is finite as required. 6.1.6 Resummation Calculation After Fourier Transforming into b-space, the finite contribution at the one-loop order for the gluon-gluon channel is given as: (1) WggâHg (b) ( i b2 h (1) ln 2 0 2 Î´(Îž2 â 1)Îž1 Pgg (Îž1 ) + (Îž1 â Îž2 ) + Î´(Îž1 â 1)Î´(Îž2 â 1) 2Ď b ÂľĚ ďŁŽ ďŁšďŁź !2 2 2 2 2 Q bâĽ Q bâĽ ďŁ˝ 1 (1) ďŁ° Ă â ln 2 + 4Î˛0 â ln 2 ln 2 ďŁť + Hgg Î´(Îž1 â 1)Î´(Îž2 â 1), ďŁž R b0 b0 = (0) Îąs CA Hgg (6.51) where the integral over the parton distributions is implicit, and similar results exist for the gluon-quark and quark-anti-quark channels. The hard coefficient, H (1) for the gluon-gluon 142 channel is: (1) Hgg ! 2 Q Q2 1 Q2 ât âu = ln2 + 2Î˛ ln + ln ln â 2 ln ln 0 2 2 2 2 2 2Ď s s R PT PT R PT ! ! ! 2 m tĚ tĚ uĚ uĚ + ln2 â ln2 + ln2 â ln2 + 2Li2 1 â 2h ât âu Q m2h m2h ! ! # t u 67 Ď 2 23 (1) +2Li2 + 2Li + + â Nf + Î´Hgg , (6.52) 2 2 2 9 2 54 mh mh (0) Îąs CA Hgg " (1) where tĚ = m2H â t, uĚ = m2H â u, and Î´Hgg can be found in Sec. 6.1.1. performing the calculation for the gluon-quark channel results in: (1) Hgq = (0) Îąs Hgq 2Ď ( " 1 2 CA ln 2 ÂľĚ2 2 PJâĽ ! + ln 2 PJâĽ ÂľĚ2 ! ln u t + ln 2 PJâĽ ÂľĚ2 ! ln s ÂľĚ2 ! # 2 ât âu âu ÂľĚ2 u uĚ uĚ 7 4Ď â2 ln 2 ln 2 â 4Î˛0 ln 2 â 6Î˛0 ln 2 + 2Li2 â ln2 + ln2 2 + + 2 âu 3 ÂľĚ ÂľĚ ÂľĚ ÂľĚ mh mh 3 " ! 2 2 PJâĽ PJâĽ 1 2 ÂľĚ2 3 ÂľĚ2 1 ÂľĚ2 u s + CF ln + ln + ln ln â ln ln â ln ln 2 2 R2 2 2 2 PJâĽ t R2 PJâĽ ÂľĚ2 ÂľĚ2 ÂľĚ2 PJâĽ ! ! ! m2h âu t tĚ tĚ 3 2 2 +3 ln 2 + 2Li2 1 â + 2Li2 â ln + ln â s ât 2 ÂľĚ m2h m2h 5Ď 2 (1) â + 20Î˛0 + Î´Hgq , (6.53) 6 143 (1) where Î´Hqg can again be found in Sec. 6.1.1. Finally, the quark-anti-quark channel is given by: (1) (0) Îąs HqqĚ = HqqĚ 2Ď ( " 1 2 CA ln 2 ÂľĚ2 2 PJâĽ ! + ln 2 PJâĽ ÂľĚ2 ! ln u t + ln 2 PJâĽ ÂľĚ2 ! ln s ÂľĚ2 ! ât âu âu ÂľĚ2 u â2 ln 2 ln 2 â 4Î˛0 ln 2 â 6Î˛0 ln 2 + 2Li2 ÂľĚ ÂľĚ ÂľĚ ÂľĚ m2h # uĚ uĚ 7 4Ď 2 â ln2 + ln2 2 + + âu 3 mh 3 " ! 2 2 PJâĽ PJâĽ 1 2 ÂľĚ2 3 ÂľĚ2 1 ÂľĚ2 u s +CF ln + ln + ln ln â ln ln â ln ln 2 2 R2 2 2 2 PJâĽ t R2 PJâĽ ÂľĚ2 ÂľĚ2 ÂľĚ2 PJâĽ ! ! ! # 2 m2h âu t tĚ tĚ 3 5Ď +3 ln 2 + 2Li2 1 â + 2Li2 â ln2 + ln2 â â 2 s ât 2 6 ÂľĚ mh m2h (1) +20Î˛0 } + Î´Hgq , (6.54) (1) where Î´HqqĚ is given in Sec. 6.1.1. With the above results, it is possible to calculate the resummation result in the CFG formalism by solving the evolution equations as discussed in Sec. 3.4. This results in a similar Sudakov factor to that for color singlet final states. however, the cone size needs to be introduced. This gives the following form: Z ÂľË2 dÂľ2 S(b) = 2 b2 /b2 Âľ 0 s 1 A ln 2 + B + D ln 2 Âľ R , (6.55) where R is the cone size for the jet. The coefficients for A, B, and D depend on the process, and can be expanded order by order in perturbation theory. For the gluon-gluon channel, A(1) = CA , B (1) = â2Î˛0 , and D = CA . For the gluon-quark channel, A(1) = 12 (CA + CF ), B (1) = âÎ˛0 â 34 CF â 12 (CA â CF ) ln ut , and D = CF . Finally, for the quark-anti-quark channel, A(1) = CF , B (1) = â 32 CF , and D = CA . 144 The dependence of ln 12 in the Sudakov factor can be understood by looking at the R soft radiation, which is what is contained in the Sudakov factor. As mentioned during the calculation of the soft factors (Sec. 6.1.4), the phase space needed to be separated to ensure that the soft gluons inside the jet were not double counted. This can also be understood by considering the contribution of the soft gluons inside the jet cone to the transverse momentum of the Higgs plus jet system. If the gluon falls within the jet cone, then the momentum of the gluon contributes to the jet, and does not contribute to the imbalance of transverse momentum between the jet and the Higgs boson. Instead, if it falls outside the cone it does contribute to the imbalance in transverse momentum. It is interesting to note that many of the logarithms in H (1) can be eliminated if the resummation scale ÂľĚ is chosen to be PJâĽ . To illustrate this point, the ratio of H (1) /H (0) is plotted as functions of the Higgâs rapidity (yH ) as shown in Fig. 6.2 with the jet rapidity fixed at yj = 0. This result shows that H (1) is much larger than H (0) in the large yH region if ÂľĚ2 = s. In contrast, the ratio of H (1) /H (0) becomes less sensitive to yH with 2 . This is because when the difference of y and y becomes large, the invariant ÂľĚ2 = PJâĽ H J mass Q2 of the Higgs boson and the leading jet can become much larger than the transverse momentum of the jet. Hence, the scale of the results will be taken to be ÂľĚ = PJâĽ in order to resum the large logarithms in the perturbative contributions. In the following, this scale choice is adopted in the theory predictions for comparison to fixed order and parton showers. However, results with ÂľĚ2 = s will be shown for the sake of comparison, and to support the 2 . choice of ÂľĚ2 = PJâĽ 145 (0) H /H 3 g g -> H g at Âľ =Q g g -> H g at Âľ =PT g q -> H q at Âľ =Q g q -> H q at Âľ =PT (1) S=8 TeV RJet=0.5 2.5 y Jet T =0 P > 30 (GeV) 2 p + p -> H + Jet 1.5 1 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 yH Figure 6.2: Rapidity dependence of the hard factor for Higgs+Jet for different scale choices 6.1.7 Results The above resummation formula to compute the differential and total cross sections of the Higgs boson plus jet will be used to compare to both fixed order and parton shower results. While data exists for this process, it is currently not at an accuracy that will be able to discriminate against different predictions. In the numerical calculations shown, only the gluon-gluon and the gluon-quark channel are used. This is a valid approximation because the contribution for the quark-anti-quark channel is less than 1%, while the gluon-gluon and gluon-quark channels make up 71% and 29% respectively. The anti-kT algorithm is used in defining the jet with a cone size of R = 0.5. Firstly, before the results are given, a cross check of the total cross-section is performed. This cross check is done in a manner similar to that described in Sec. 3.5. In this cross check, 146 it is evident that using the narrow jet approximation results in a 2% discrepancy between the fixed order result from MCFM, and the result from the calculation described here. This discrepancy varies as a function of R supporting the fact that it is due to the narrow jet approximation. Therefore, the correction is modeled by an additional R-dependent function inside of H (1) . The correction is obtained by fitting the difference in cross section to a quadratic in R, s C R â 1.1R + 23.3R2 and H (0) Îąs C â 0.8 â R + 22.3R2 giving a correction of H (0) Îą A F 2Ď 2Ď for final state gluon and quark jets respectively. Additionally, for the resummation to be valid, the qT needs to be smaller than PJâĽ , since the qT is defined as the vector sum of the transverse momentum of the Higgs boson and the leading jet. If qT is greater than PJâĽ it is not clear what the leading jet is, and the fixed order calculation needs to be used in this region. Finally, since A(2) is global and only depends on the color factor in the initial state, this is also included into the calculation. 6.1.7.1 Comparison to Fixed Order In Fig. 6.3, the comparison of the Higgs plus jet system at the LHC is shown for the total transverse momentum and the transverse momentum of the Higgs. The fixed order calculation is given by MCFM, while the resummation calculation is shown for the two scales discussed above. For the total transverse momentum distribution, the leading order (LO) MCFM prediction is from the first non-zero prediction of the transverse momentum of the Higgs plus jet system, and the NLO result is the one-loop correction to that. In other words, the LO prediction is given by Higgs plus two jets, while the NLO prediction is given by Higgs plus two jets at NLO. The MCFM NLO prediction is cut-off at 20 GeV due to the numerical difficulties of obtaining an accurate prediction below this value. Comparing the total transverse 147 dĎ/dPH dĎ/dq 103 Full LO from MCFM Full NLO from MCFM resum at Âľ=PJ resum at Âľ=Q Full NLO from MCFM resum at Âľ=PJ resum at Âľ=Q 120 100 102 p + p -> H + Jet PJ > 30 (GeV) -4.4 < y < 4.4 Jet RJet=0.5 S=8 TeV 10 p + p -> H + Jet 80 PJ > 30 (GeV) -4.4 < y 60 Jet < 4.4 RJet=0.5 40 S=8 TeV 1 20 10-1 0 20 40 60 80 100 120 140 160 180 0 200 20 40 60 80 100 120 140 160 180 200 PH (GeV) q (GeV) Figure 6.3: The differential cross sections of Higgs boson plus one jet production at the LHC as functions of the total transverse momentum qâĽ (left) and the Higgs boson transverse momentum PHâĽ (right). The resummation predictions (resum) with resummation scale set to be PJâĽ (black line) and Q (green line) respectively are compared to the LO result from MCFM (pink line) with non-zero qâĽ , and the NLO result from MCFM (red line) which is the production rate of Higgs boson plus two separate jets up to one-loop in QCD. 148 momentum distribution of the resummed and fixed order calculation, it is seen that the fixed order fails to describe the small qT region. Additionally, it is clear that the scale choice of ÂľĚ = Q results in a calculation that is too hard (too large of a cross-section for large qT ), and can never be matched to fixed order. On the other hand, the scale choice of ÂľĚ = PJâĽ gives a much more reasonable prediction. For the transverse momentum of the Higgs boson, only the NLO prediction from MCFM is shown. This prediction is given by the Îąs corrections to the Higgs plus jet result. Again, the resummation calculation is given for the two different scale choices. The choice for ÂľĚ = Q is too hard in the high transverse momentum region as also seen in the total transverse momentum distribution. Here, there is a discontinuity at the value of pHâĽ = pJcut , known as the Sudakov shoulder singularity. This is due to a integrable singularity that arises when the leading jet is close to back to back to the Higgs boson, or in other words, when qT approaches zero. This is resolved in the resummation calculation, by appropriately resumming the logarithms to all orders that result in the divergence to give a finite result. 6.1.7.2 Comparison to Parton Showers Similar to the fixed order calculation, the comparison of the resummation result can be compared to that from parton shower predictions. In this study, the Higgs boson was set to 125 GeV and left undecayed, the MMHT2014nlo68clas118 PDF set was used [215]. Finally, the theoretical uncertainties were estimated by varying the scales by factors of 12 and 2 around the central scale, but left the renormalization scale (ÂľĚ) fixed to PJâĽ as discussed above. The setup and details of the other predictions used in this comparison can be found in [28]. The jets for this calculation are taken to have a cone size of R = 0.4, with pJâĽ > 30 GeV and |yJ | < 4.4. The codes that are used in this comparison are: SHERPA [178] plus 149 GOSAM [216, 217], MINLO [218], BFGLP [219], HEJ [220, 221, 222, 223, 224], Herwig 7.1 [225], Madgraph5 aMC@NLO [226], POWHEG-BOX [227], STWZ [228], and SHERPA NNLOPS [229, 230]. Exact details of the setups for each of the individual codes can be found in [28]. Firstly, the total rates for the jet multiplicities are compared across all of the codes, shown in Fig. 6.4. The ResBos2 code is able to predict both the total inclusive rate for Higgs along with the inclusive rate for Higgs plus 1 jet. Comparing across all of the codes, it is clear that the ResBos2 prediction is consistent with the others as expected from the fixed order comparison above. The differential distributions that are compared are the transverse momentum of the Higgs, the leading jet, the Higgs plus leading jet system, and the rapidity of the leading jet. Here the transverse momentum of the Higgs requires the presence of at least one hard jet. The comparison of the transverse momentum of the Higgs boson in the presence of an additional jet can be seen in Fig. 6.5. The fixed order calculations again show the Sudakov shoulder singularity, but the all orders calculations do not have this singularity. In the ratio plot, at the high transverse momentum, there is a deviation of the resummation calculation from that of POWHEG due to a different choice of scales. In the RESBOS2 calculation, the central scale is Âľ0 = 12 mh as mentioned above, but the scale choice for POWHEG is Âľ0 = q 1 m2 + p2 . This scale choice softens the high transverse momentum tail as discussed in 2 T h detail in Sec. 4.1. Next, the comparison of the leading jet transverse momentum and rapidity distributions can be found in Fig. 6.6 and Fig. 6.7, respectively. Both of these distributions do not contain any large Sudakov effects, and therefore all the calculations should be close to that of the fixed order calculation. This is clearly seen in the aforementioned figures, with all the calculations falling within each others uncertainty bands. Additionally, the size of the uncertainty bands 150 1 STWZ ResBos 2 Sherpa h Nnlo BFGLP hj Nnlo GoSam+Sherpa NLO hj NLO hjj hjjj NLO 1 MiNLO MiNLO MiNLO Powheg NnloPs Sherpa NnloPs MG5_aMC FxFx Sherpa MePs@Nlo Herwig 7.1 Hej STWZ ResBos 2 Sherpa h Nnlo BFGLP hj Nnlo GoSam+Sherpa NLO hj NLO hjj hjjj NLO nNLO nNLO MiNLO MiNLO MiNLO nNLO nNLO Ratio to Powheg 10â1 1.4 1.2 1 0.8 1.2 1 0.8 0.6 1.4 1.4 Ratio to Powheg 0.6 1.2 1 0.8 1.2 1 0.8 0.6 0.6 1.4 1.4 Ratio to Powheg Ratio to Powheg Ratio to Powheg Inclusive jet multiplicity 10 1 10â1 1.4 Ratio to Powheg 10 2 LH15 pp â h + jets comparison Powheg NnloPs Sherpa NnloPs MG5_aMC FxFx Sherpa MePs@Nlo Herwig 7.1 Hej 10 1 dĎ/dn j [pb] Inclusive jet multiplicity LH15 pp â h + jets comparison dĎ/dn j [pb] 10 2 1.2 1 0.8 0.6 1.2 1 0.8 0.6 âĽ0 âĽ1 âĽ2 âĽ3 âĽ0 nj âĽ1 âĽ2 âĽ3 nj Figure 6.4: The central predictions(left panel) and with theoretical uncertainties (right panel) for the inclusive jet multiplicities as predicted by fixed-order calculations, resummed calculations, NNLO and NLO Monte Carlos. The bottom panel is divided up into three subplots all showing the ratios with respect to the POWHEG NNLOPS prediction. The upper of these plots contains the HEJ and SHERPA NNLOPS ratios, while the middle one includes all NLO merged predictions (Madgraph5 aMC@NLO, Herwig 7.1 and SHERPA) and the lower one shows all those listed in the bottom left legend of the main panel. Reproduced from [28]. are consistent with each other and are of the order of 20%. Again the difference in the high transverse momentum tail between the POWHEG and RESBOS2 calculations arise from the difference in scale choices. For the rapidity, the calculation is consistent throughout the rapidity range, and the offset between the POWHEG and RESBOS2 calculations is due to 151 10â1 ResBos 2 BFGLP hj Nnlo 10â2 1 Higgs boson transverse momentum (n j âĽ 1) Powheg NnloPs Sherpa NnloPs MG5_aMC FxFx Sherpa MePs@Nlo Herwig 7.1 10â1 ResBos 2 BFGLP hj Nnlo 10â2 GoSam+Sherpa hj NLO MiNLO nNLO GoSam+Sherpa hj NLO MiNLO nNLO Ratio to Powheg 10â3 1.4 1.2 1 0.8 1.2 1 0.8 0.6 1.4 1.4 Ratio to Powheg 0.6 1.2 1 0.8 1.2 1 0.8 0.6 1.4 1.4 Ratio to Powheg 0.6 1.2 1 0.8 1.2 1 0.8 0.6 1.4 1.4 Ratio to Powheg 0.6 1.2 1 0.8 1.2 1 0.8 0.6 1.4 1.4 Ratio to hj Nlo 0.6 1.2 1 0.8 1.2 1 0.8 0.6 0.6 1.4 1.4 Ratio to hj Nlo Ratio to hj Nlo Ratio to Powheg Ratio to Powheg Ratio to Powheg Ratio to Powheg 10â3 1.4 Ratio to hj Nlo LH15 pp â h + jets comparison Powheg NnloPs Sherpa NnloPs MG5_aMC FxFx Sherpa MePs@Nlo Herwig 7.1 dĎ/dp âĽ [pb/GeV] Higgs boson transverse momentum (n j âĽ 1) LH15 pp â h + jets comparison dĎ/dp âĽ [pb/GeV] 1 1.2 1 0.8 0.6 1.2 1 0.8 0.6 0 50 100 150 200 pâĽ (h) [GeV] 0 50 100 150 200 pâĽ (h) [GeV] Figure 6.5: The Higgs boson transverse momentum in the presence of at least one jet central predictions(left) with uncertainty bands(right). The ratio plot panel is divided into six parts where the upper four exhibit the ratios to the POWHEG NNLOPS result while the lower two show them to the NLO calculation for h + 1 jet as provided by GOSAM+SHERPA. Reproduced from [28]. 152 10â1 STWZ ResBos 2 BFGLP hj Nnlo 10â2 1 Leading jet transverse momentum (n j âĽ 1) Powheg NnloPs Sherpa NnloPs MG5_aMC FxFx Sherpa MePs@Nlo Herwig 7.1 10â1 STWZ ResBos 2 BFGLP hj Nnlo 10â2 GoSam+Sherpa hj NLO MiNLO nNLO GoSam+Sherpa hj NLO MiNLO nNLO Ratio to hj Nnlo 10â3 1.4 1.2 1 0.8 1.2 1 0.8 0.6 1.4 1.4 Ratio to hj Nnlo 0.6 1.2 1 0.8 1.2 1 0.8 0.6 1.4 1.4 Ratio to hj Nnlo 0.6 1.2 1 0.8 1.2 1 0.8 0.6 0.6 1.4 1.4 Ratio to hj Nnlo Ratio to hj Nnlo Ratio to hj Nnlo Ratio to hj Nnlo 10â3 1.4 Ratio to hj Nnlo LH15 pp â h + jets comparison Powheg NnloPs Sherpa NnloPs MG5_aMC FxFx Sherpa MePs@Nlo Herwig 7.1 dĎ/dp âĽ [pb/GeV] Leading jet transverse momentum (n j âĽ 1) LH15 pp â h + jets comparison dĎ/dp âĽ [pb/GeV] 1 1.2 1 0.8 0.6 1.2 1 0.8 0.6 40 60 80 100 120 140 160 180 200 pâĽ ( j1 ) [GeV] 40 60 80 100 120 140 160 180 200 pâĽ ( j1 ) [GeV] Figure 6.6: The leading jet transverse momentum distribution for h + âĽ 1-jet production, to the right (left) shown with (without) the uncertainty bands provided by the various calculations. The part below the main plot contains four ratio plots taken wrt. the NNLO result of the BFGLP group following the same strategy for grouping the predictions as before (NNLOPS versus NLO ME+PS versus fixed-order and resummation results). Reproduced from [28]. the fact that the total cross-sections differ by about 5%, which is within the theoretical uncertainty at this order, as seen when the uncertainty bands are overlaid. 153 3 2.5 2 1.5 4 Powheg NnloPs Sherpa NnloPs MG5_aMC FxFx Sherpa MePs@Nlo Herwig 7.1 3.5 3 2.5 2 1.5 ResBos 2 1 ResBos 2 1 GoSam+Sherpa hj NLO MiNLO nNLO 0.5 Ratio to Powheg 0 1.4 1.2 1 0.8 1.2 1 0.8 0.6 1.4 1.4 Ratio to Powheg 0.6 1.2 1 0.8 1.2 1 0.8 0.6 0.6 1.4 1.4 Ratio to Powheg Ratio to Powheg Ratio to Powheg GoSam+Sherpa hj NLO MiNLO nNLO 0.5 0 1.4 Ratio to Powheg LH15 pp â h + jets comparison Powheg NnloPs Sherpa NnloPs MG5_aMC FxFx Sherpa MePs@Nlo Herwig 7.1 3.5 dĎ/dy [pb] Leading jet rapidity LH15 pp â h + jets comparison dĎ/dy [pb] Leading jet rapidity 4 1.2 1 0.8 0.6 1.2 1 0.8 0.6 0 0.5 1 1.5 2 2.5 3 3.5 4 0 y( j1 ) 0.5 1 1.5 2 2.5 3 3.5 4 y( j1 ) Figure 6.7: The rapidity distribution for the leading jet in h + âĽ 1-jet production, shown without (left) and with (right) theoretical uncertainties. Ratio plots are displayed in the lower part of the plot using the POWHEG NNLOPS result for Higgs boson production as their reference. Predictions are grouped, from top to bottom, according to the categories NNLOPS, ME+PS at NLO and NLO fixed order as well as resummation. Reproduced from [28]. Finally, the transverse momentum of the Higgs plus jet system is examined. The comparison between the different results can be found in Fig. 6.8. Here comparing the RESBOS2 calculation to POWHEG has some interesting features that need to be discussed. In the high transverse momentum region, there is a discrepancy between the two calculations which arise from the fact that in the tail of the distribution, the RESBOS2 calculation only is matched to 154 the LO prediction for this observable as described in Sec. 6.1.7, and the agreement should be improved if the matching is done to a higher order prediction. Additionally, in the RESBOS2 calculation the logarithms dealing with the jet cone size are resummed as mentioned in Sec. 6.1.6. This results in a broader Sudakov peak than the other parton shower predictions, and an upward shift in the Sudakov peak value. When the LHC data becomes precise enough, this difference will be a strong test of the resummation formalism versus the parton shower model. Overall, the parton shower predictions and the resummed predictions are consistent with each other. This was expected, since the formal accuracy of the two predictions for all observables with the exception of the transverse momentum of the Higgs plus jet system are of the same order. The major difference arises in the prediction of the Higgs plus jet system transverse momentum, due to the dependency of the jet cone size in the RESBOS2 prediction. Once the LHC data becomes precise enough, these codes and their differences can be tested against data. 6.1.8 Future of Higgs Plus Jet Resummation In this section, some brief future steps are discussed. Firstly, there has been some work to begin to extend this calculation from Higgs plus one jet to Higgs plus two jets [29]. Additionally, at higher order calculations for the Higgs plus one jet system non-global logarithms are introduced into the calculation. The calculation of the Higgs plus two jets system is an important calculation to separate the vector boson fusion (VBF) production of the Higgs boson from the gluon fusion production of the Higgs boson. As shown in [29], the peak of the transverse momentum of the system peaks at drastically different values. This can be used to apply an additional cut 155 10â1 10â2 ResBos 2 GoSam+Sherpa hj NLO MiNLO nNLO 1 0.5 1 0.5 1 0.5 0 20 40 60 80 100 120 140 pâĽ (hj1 ) [GeV] Ratio to Powheg 10â3 1.5 Ratio to Powheg Ratio to Powheg 1.5 Powheg NnloPs Sherpa NnloPs MG5_aMC FxFx Sherpa MePs@Nlo Herwig 7.1 10â1 1.5 Ratio to Powheg Ratio to Powheg Ratio to Powheg 1.5 Transverse momentum of Higgs and leading jet system (n j âĽ 1) 10â2 ResBos 2 GoSam+Sherpa hj NLO MiNLO nNLO 10â3 1.5 1 LH15 pp â h + jets comparison Powheg NnloPs Sherpa NnloPs MG5_aMC FxFx Sherpa MePs@Nlo Herwig 7.1 dĎ/dp âĽ [pb/GeV] Transverse momentum of Higgs and leading jet system (n j âĽ 1) LH15 pp â h + jets comparison dĎ/dp âĽ [pb/GeV] 1 1.5 1 0.5 1 0.5 1 0.5 0 20 40 60 80 100 120 140 pâĽ (hj1 ) [GeV] Figure 6.8: The transverse momentum of the Higgs-boson-leading-jet system in the presence of at least one jet. For better visibility, results are shown without (left) and with (right) theoretical uncertainties. The plot layout exactly corresponds to that of Figure 6.7, except for the extended yĚ-axis range in the ratio plots. Reproduced from [28]. on the Higgs production events to increase the purity of the VBF signal. The normalized distributions for this observable can be seen in Fig. 6.9. Finally, the goal is to extend the Higgs plus one jet calculation to NNLL accuracy. However, going beyond NLL accuracy introduces what are known as non-global logarithms. Non-global logarithms are logarithms that may not appear at each order in the calculation, and do not have a specific form to predict the higher order coefficients of them. Therefore, 156 Figure 6.9: Noramlized distributions of the vector boson fusion and gluon-fusion contributions toâthe Higgs boson plus two jets production in the typical kinematics at the LHC with S = 13T eV , where the jet transverse momenta k1âĽ = k2âĽ = 30GeV , yj1 = âyj2 = 2 and yh = 0: as functions of the total transverse momentum qâĽ (left); the total rate as function of the upper limit of qâĽ (right). Reproduced from [29]. it is not possible to resum these logarithms to all orders [231, 232, 11, 233]. This results in additional complications that need to be considered. 157 Chapter 7 Conclusion The improvement of the ResBos code from the precision used for the Tevatron, to that of the new ResBos2 code, will allow for precision resummation calculations at the LHC. These improvements come from many different improved predictions, as mentioned above. Below a review of the improvements to the predictions, along with the experimental comparisons are summarized for the reader. First, the improvements to color singlet resummation will be reviewed. In this section, the difference between schemes will again be pointed out. Also, the predictions for both the Z and W boson at the LHC will be reviewed. Finally, the future steps for this project will be outlined. Afterwards, the addition of non-color singlet resummation calculation will be reviewed. In this section, the calculation for the Higgs plus jet system will be reviewed, along with some mentions of other works using these ideas. Finally, the future prospects for this new type of qT resummation will be emphasized. 158 7.1 7.1.1 Color Singlet Resummation Resummation Schemes Two main schemes in qT resummation are again the Collins-Soper-Sterman and the Catanide-Florian-Grazzini schemes. The differences between these schemes have both an order by order, and an all orders relationship, to map from one scheme to the other. It was shown that as long as the calculation is preformed such that the orders of the B, C, and H coefficient are such that the order of B is one higher than C and H, then the two calculations are in very good agreement. This is supported by the conversion between the two schemes as seen in Eq. 3.73, and the results of Fig. 4.3. 7.1.2 Z Boson Predictions Previously at the LHC, the prediction of the ResBos code was unable to reproduce the data. The resolution to this issue took many different steps as outlined above. These steps, included the improvement of the precision of the calculation, and the change of the scale of the fixed order calculation. The improvement in the precision of the calculation, was from NNLL to N3 LL, and the fixed order was improved from approximate NNLO to NNLO, with a k-factor included in the high transverse momentum region to match the NNLO Z plus jet calculation. One improvement that will be made to the ResBos2 code is the matching fully with the NNLO Z plus jet calculation. This will be included once the results of the NNLO Z plus jet calculation become sufficiently available. The other improvement was the change of the fixed order scale from the invariant mass of the lepton pair to the transverse mass of the lepton pair, defined in the text above. This 159 choice of scale makes more sense for the high transverse momentum region due to the fact that at high transverse momentum, the energy of the jet is comparable to the invariant mass of the lepton pair. Upon using these improvements to the ResBos prediction, the agreement between the theory prediction and the data is greatly improved. Further improvements will arise with the matching to the higher order fixed order calculation. However, this improvement will only help in the intermediate region. One more improvement that can be made to the ResBos code, is to further improve the non-perturbative piece. Currently, the non-perturbative piece is fixed as a function of the resummation scales. This has recently been believed to give too conservative an estimate of the uncertainty of the prediction at small transverse momentum. Therefore, another future improvement is to include the resummation scale variations into the non-perturbative fit, allowing the non-perturbative function to depend on these scales, as makes physical sense. 7.1.3 W Boson Predictions The ResBos2 predictions for the W -transverse momentum, and the ability to use the code to help predict the W mass at the LHC will be helpful in understanding the EW sector of the SM. As mentioned in the text above, the indirect measurement is much smaller than the direct measurement of the W mass. It has been proposed to use resummation tools to improve the direct measurement, by predicting the ratio of the W transverse momentum to the Z transverse momentum. With these, the experimentalists will be able to reduce the uncertainties in the W mass measurement. These reductions will allow the measurement to be competitive with the indirect measurement. Additionally, the ResBos2 prediction allows for a template fit of the experimental mea160 surements of the transverse momentum of the leptons to the theoretical prediction. This will allow for a precise determination of the W mass. Some future improvements that could be included into the ResBos2 prediction are again similar to that of the Z boson, from the matching to the higher order fixed order calculation, and improving the non-perturbative piece. While these may not have a large effect on the determination of the W boson mass, there are not any improvements that can be made in the theoretical prediction in the near future. The major improvements that would help, included calculations to much higher orders, which are not feasible at the writing of this thesis. 7.2 Non-Color Singlet Resummation Recently, there has been work done to allow for non-color singlet qT resummation in QCD. The first of these being the dijet calculation, and another related to the work of this thesis being the Higgs plus jet resummation as mentioned previously. With these calculations completed, a new set of processes are now able to be calculated that previously were thought not possible. As mentioned above, the Higgs plus jet calculation was an important step in understanding QCD, and an important contribution towards obtaining the Higgs plus two jet resummation calculation. The Higgs plus two jet calculation is important in the study of the Higgs boson, in allowing one to separate the gluon-gluon fusion production mechanism of the Higgs boson from the EW production of the Higgs boson. Additionally, it is shown that the current prediction for the qT resummed prediction for the Higgs plus jet system is consistent with the parton shower results. However, there is the one major difference in that 161 the analytic resummation knows about the cone size of the jet, resulting in a shape that differs from the parton shower codes. When the data becomes more precise, a great test of this method would be to see if the data can distinguish whether the paron shower method, or the analytic resummed method is correct. 7.3 Final Remarks In conclusion, the improvement of the qT resummation calculations in QCD are included in the ResBos2 code. With these improvements, the calculations are now at a level that is sufficient for the precision required at the LHC. In the future, the ResBos2 will continue to improve building off of the work of this thesis, and the ideas that have formed as a result of this work. 162 APPENDICES 163 APPENDIX A The Dirac Equation and Îł Matrices The Dirac equation is the equation of motion to describe fermions, and is given by: ĎĚ iâ Âľ ÎłÂľ + m Ď = 0, (A.1) where Ď is the wavefunction of a fermion, m is the mass of that fermion, and ÎłÂľ 1 are the Dirac matrices defined by their anti-commutation relation: {Îł Âľ , Îł Î˝ } = 2g ÂľÎ˝ , (A.2) where g ÂľÎ˝ is the metric tensor defined using the mostly negative choice, g ÂľÎ˝ = diag(1, â1, â1, â1), such that squares of time-like four momenta result in positive results. The Îł or Dirac matrices are used for infinitesimal transformations of spinors under spatial rotations and Lorentz boosts. There are many representations for these matrices. The Weyl representation is given by: ďŁŤ ďŁś ďŁŹ 0 1ďŁˇ Îł0 = ďŁ ďŁ¸, 1 0 ďŁŤ ďŁŹ 0 Îłi = ďŁ âĎ i Ďi 0 ďŁś ďŁˇ ďŁ¸, (A.3) 1 In Eq. A.1, it is common to rewrite â Âľ Îł as â. / This slashed notation will be used through the rest of Âľ this work. Also, additional details on the Îł matrices, along with one representation of them can be found in App. A 164 where Ď i are the Pauli matrices, and 1 is the 2 Ă 2 Identity matrix. Additionally, a fifth Îł matrix can be defined as: ďŁŤ ďŁś ďŁŹâ1 0ďŁˇ Îł 5 = iÎł 0 Îł 1 Îł 2 Îł 3 = ďŁ ďŁ¸. 0 1 (A.4) Important relationships of Îł matrices are: 1. Îł Âľ ÎłÂľ = 414 , 2. Îł Âľ Îł Î˝ ÎłÂľ = â2Îł Î˝ , 3. Îł Âľ Îł Î˝ Îł Ď ÎłÂľ = 4g Î˝Ď 14 , 4. Îł Âľ Îł Î˝ Îł Ď Îł Ď ÎłÂľ = â2Îł Ď Îł Ď Îł Î˝ , 5. trace of any odd product of Îł matrices excluding Îł 5 is zero, 6. trace of any odd product of Îł matrices times Îł 5 is also zero, 7. Tr (Îł Âľ Îł Î˝ ) = 4g ÂľÎ˝ , 8. Tr (Îł Âľ Îł Î˝ Îł Ď Îł Ď ) = 4 (g ÂľÎ˝ g ĎĎ â g ÂľĎ g Î˝Ď + g ÂľĎ g Î˝Ď ), 9. Tr Îł 5 = Tr Îł Âľ Îł Î˝ Îł 5 = 0, 10. Tr (Îł Âľ Îł Î˝ Îł Ď Îł Ď ) = â4iÂľÎ˝ĎĎ , 11. Tr (Îł Âľ1 Âˇ Âˇ Âˇ Îł Âľn ) = Tr (Îł Âľn Âˇ Âˇ Âˇ Îł Âľ1 ), where 14 is the 4 Ă 4 unit matrix and ÂľÎ˝ĎĎ is the completely anti-symmetric Levi-Civita Tensor. When moving beyond 4 dimensions, the contraction identities are modified as follows: 165 1. Îł Âľ ÎłÂľ = d1d , 2. Îł Âľ Îł Î˝ ÎłÂľ = â(d â 2)Îł Î˝ , 3. Îł Âľ Îł Î˝ Îł Ď ÎłÂľ = 4g Î˝Ď â (4 â d)Îł Î˝ Îł Ď , 4. Îł Âľ Îł Î˝ Îł Ď Îł Ď ÎłÂľ = â2Îł Ď Îł Ď Îł Î˝ + (4 â d)Îł Î˝ Îł Ď Îł Ď . The trace identities not involving Îł 5 are not modified since they are independent of dimensionality. There are many complications of dealing with Îł 5 in dimensions not equal to 4. There have been many discussions on the appropriate way to handle this situation. The reader is referred to [234, 235] for detailed discussions on this topic. 166 APPENDIX B Spin, Helicity, and Chirality Spin ~ ~ For a fermion, S ~ = sigma Spin is the eigenvalue of S. 2 . For a single particle, the spin and the angular momentum operators are the same. The scalar spin s is the eigenvalue in ~ 2 = s(s + 1). When saying a particle is spin-1/2 refers to the value of s. S Helicity Helicity is the projection of the spin on the direction of momentum. Helicity eigenstates are ~ p given by the operator H = SÂˇ~ , and exist for any spin particle. |~ p| Chirality Chirality is a concept that only exists for fermionic particles. A particle is chiral if it is not symmetric under a mirror symmetry. This leads to the definition of left-/right-handed particles. The projection of the fermionic wavefunction to the left-/right-handed components is given by: PL = 1 + Îł5 , 2 PR = 167 1 â Îł5 . 2 (B.1) ďŁŤ ďŁś ďŁŹ ĎL ďŁˇ In the Weyl representation, the spinors can be represented by ďŁ ďŁ¸. In the massless limit ĎR chirality and helicity are the same. 168 APPENDIX C Standard Model Higgs Boson and Higgs Mechanism As discussed in Chapter 1, the Higgs Mechanism is used to introduce masses for vector bosons, in a manner that preserves the gauge theory. There, the calculation was demonstrated for a Higgs boson added to QED. The full details for adding a Higgs boson into SU (2)L Ă U (1)Y is detailed below. First, the addition of a scalar doublet is required to be added to the Standard Model, given by: ďŁŤ ďŁś 1 ďŁŹĎ1 + iĎ2 ďŁˇ Ď= â ďŁ ďŁ¸, 2 Ď3 + iĎ4 (C.1) which is a multiplet of SU (2)L timesU (1)Y . The hypercharge of the above doublet is chosen to be Y = 1. The potential for the scalar doublet that is responsible for the spontaneous breaking of the symmetry is the generalized form from that in Chapter 1, and is given as: 2 V (Ď) = Âľ2 Ďâ Ď + Îť Ďâ Ď , 169 (C.2) where Âľ2 < 0. With a Lagrangian given as: L = (DÂľ Ď)â DÂľ Ď â V (Ď) , (C.3) where DÂľ is the covariant derivative given as: 1 ~ Âľ + ig 0 1 Y BÂľ . DÂľ = âÂľ + ig ~Ď Âˇ W 2 2 (C.4) The vacuum state (Ď0 ) is given by: ďŁŤ ďŁś 1 ďŁŹ0ďŁˇ Ď0 = â ďŁ ďŁ¸ , 2 v (C.5) where v is known as the vacuum expectation value. With this choice of the vacuum, SU (2)L Ă U (1)Y is broken, but since the vacuum has charge Q = I3 + 12 Y = 0, there remains a U (1)EM symmetry as required. The Lagrangian above can be rewritten into terms of the physical gauge bosons (W + , W â , Z, Îł). The W Âą boson can be related to W 1 and W 2 , by the following relationship: 1 W Âą = â W 1 â iW2 . 2 (C.6) Additionally, the Z and Îł can be rewritten in terms of the W 3 and B bosons as given by: 1 A= p g 0 W 3 + gB , g 2 + g 02 1 Z=p gW 3 â g 0 B . g 2 + g 02 170 (C.7) (C.8) Combining these relationships, the Lagrangian for the scalar can be rewritten as: 1 2 2 (DÂľ Ď)â DÂľ Ď = v 2 g 2 W + + g 2 W â + g 2 + g 02 Z 2 + 0 Âˇ A2 , 8 (C.9) plus terms that involve the Higgs boson. The rest of the results follow as discussed in Chapter 1. 171 APPENDIX D Structure of SU (3) A given representation of the Gell-Mann matrices is given by the following 8 matrices: ďŁŤ Îť1 Îť4 Îť7 ďŁŹ0 ďŁŹ ďŁŹ = ďŁŹ1 ďŁŹ ďŁ 0 ďŁŤ ďŁŹ0 ďŁŹ ďŁŹ = ďŁŹ0 ďŁŹ ďŁ 1 ďŁŤ ďŁŹ0 ďŁŹ ďŁŹ = ďŁŹ0 ďŁŹ ďŁ 0 ďŁś 1 0 0 0 0 0 0 0 i ďŁŤ ďŁś ďŁŤ ďŁś 0ďŁˇ ďŁŹ0 âi 0ďŁˇ ďŁŹ1 0 0 ďŁˇ ďŁˇ ďŁŹ ďŁˇ ďŁŹ ďŁˇ ďŁˇ ďŁŹ ďŁˇ ďŁŹ ďŁˇ , Îť2 = ďŁŹ i 0 0 ďŁˇ , Îť3 = ďŁŹ0 â1 0ďŁˇ , 0ďŁˇ ďŁˇ ďŁŹ ďŁˇ ďŁŹ ďŁˇ ďŁ¸ ďŁ ďŁ¸ ďŁ ďŁ¸ 0 0 0 0 0 0 0 ďŁś ďŁŤ ďŁś ďŁŤ ďŁś 1ďŁˇ ďŁŹ0 0 âiďŁˇ ďŁŹ0 0 0 ďŁˇ ďŁˇ ďŁŹ ďŁˇ ďŁŹ ďŁˇ ďŁˇ ďŁŹ ďŁˇ ďŁŹ ďŁˇ , Îť = , Îť = ďŁˇ ďŁŹ ďŁˇ ďŁŹ ďŁˇ, 5 6 0ďŁˇ ďŁŹ0 0 0 ďŁˇ ďŁŹ0 0 1 ďŁˇ ďŁ¸ ďŁ ďŁ¸ ďŁ ďŁ¸ 0 i 0 0 0 1 0 ďŁś ďŁŤ ďŁś 0ďŁˇ ďŁŹ1 0 0 ďŁˇ ďŁˇ ďŁˇ 1 ďŁŹ ďŁˇ ďŁŹ ďŁˇ , Îť8 = â ďŁŹ0 1 0 ďŁˇ . âiďŁˇ ďŁˇ ďŁˇ 3ďŁŹ ďŁ¸ ďŁ ďŁ¸ 0 0 0 â2 (D.1) The structure functions are completely anti-symmetric in the choices of a, b, and c, and are a generalization of the Levi-Cevita Tensor from SU (2) to SU (3). They are defined as: â 1 3 f123 = 1, f147 = f165 = f246 = f257 = f345 = f376 = , f458 = f678 = , 2 2 (D.2) and in general they are zero, unless they contain an odd number from the set of {2, 5, 7}. 172 APPENDIX E QCD Feynman Rules Fermion Lagrangian To find the interaction terms for the fermions, we start from the free Lagrangian. For all of this, we will assume that the fermions are massless. It is easy for one to extend the following to include fermion masses. Lf = ĎĚiâ/Ď (E.1) We will now require that the Lagrangian remain invariant under a SU(3) transformation. To begin, a local SU(3) transformation can be written as: a a X = eâiT Ď (x) (E.2) Where Ďa (x) are all real. We can then act this transformation on the above Lagrangian: Ď â XĎ âÂľ Ď â XâÂľ Ď + âÂľ X Ď (E.3) To bring back the desired SU(3) invariance, we introduce the covariant derivative which is defined such that: DÂľ Ď â XDÂľ Ď 173 (E.4) Working through the equations above to find the definition of the covariant derivative, one finds that the form the covariant derivative must take is given by: DÂľ = âÂľ + igs T a AaÂľ (E.5) Where in the above equation, gs is the strong coupling constant, and AaÂľ are the vector boson fields for the strong force, known as gluons. Therefore, the SU(3) invariant Lagrangian for fermions is defined as: a / Lf = ĎĚ iâ/ â gs T a A Ď (E.6) Looking at this equation, one can see where the interaction between the quarks and the gluons arises. Details on the interaction and the Feynman rules for QCD will be calculated in later sections. Vector Boson Lagrangian The vector bosons, or gluons, are in the adjoint representation of SU(3). Therefore, under a SU(3) transformation the fields transform as: a a b c A0a Âľ = AÂľ â âÂľ Ď (x) + gs fabc Ď (x)AÂľ (E.7) We can also define the field strength tensor for QCD in an anologous way to that of QED, 174 but we have to include additional terms since unlike QED, QCD is non-Abelian. a = â Aa â â Aa + g f abc A A FÂľÎ˝ Âľ Î˝ Î˝ Âľ s Âľ,b Î˝,c (E.8) Also, similar to QED, the kinetic part of the Lagrangian is formed from a Lorentz invariant using two field strength tensors. 1 a Lkin = â F ÂľÎ˝,a FÂľÎ˝ 4 (E.9) To simplify the expressions, we can define: AÂľ âĄ âiT a AaÂľ and a . Using these definitions, we can rewrite the kinetic part of the Lagrangian FÂľÎ˝ âĄ âiT a FÂľÎ˝ by using the following relations: FÂľÎ˝ = âÂľ AÎ˝ â âÎ˝ AÂľ + gs [AÂľ , AÎ˝ ] = 1 [DÂľ , DÎ˝ ] gs (E.10) Where DÂľ is the covariant derivative from Eq. E.5. Using the trace property of the T a âs, one can rewrite the Lagrangian as: 1 Lkin = â Tr(FÂľÎ˝ FÂľÎ˝ ) 2 (E.11) Putting the fermion Lagrangian E.6 together with the kinetic term for the boson Lagrangian E.9, one obtains the complete Lagrangian for QCD. 1 a + ĎĚ iD / âm Ď LQCD = â F ÂľÎ˝,a FÂľÎ˝ 4 175 (E.12) QCD Feynman Rules In order to find the Feynman Rules for QCD, we first need to expand the kinetic term of the QCD Lagrangian in order to find the quartic, cubic, and quadratic terms for the vector bosons in the Lagrangian. a = â Âľ AÎ˝,a â â Î˝ AÂľ,a + g f abc AÂľ AÎ˝ a â â Aa + g f ab0 c0 A F ÂľÎ˝,a FÂľÎ˝ â A A s Âľ Î˝ Î˝ Âľ s Âľ,b0 Î˝,c0 b c 0 0 = â Âľ AÎ˝,a âÂľ AaÎ˝ â â Âľ AÎ˝,a âÎ˝ AaÂľ + gs f ab c â Âľ AÎ˝,a AÂľ,b0 AÎ˝,c0 â â Î˝ AÂľ,a âÂľ AaÎ˝ + â Î˝ AÂľ,a âÎ˝ AaÂľ 0 0 Âľ Âľ â gs f ab c â Î˝ AÂľ,a AÂľ,b0 AÎ˝,c0 + gs f abc Ab AÎ˝c âÂľ AaÎ˝ â gs f abc Ab AÎ˝b âÎ˝ AaÂľ 0 0 Âľ + gs2 f abc f ab c Ab AÎ˝c AÂľ,b0 AÎ˝,c0 (E.13) The equation above can be further simplified by noting: â Âľ AÎ˝,a âÂľ AaÎ˝ = â Î˝ AÂľ,a âÎ˝ AaÂľ and â Âľ AÎ˝,a âÎ˝ AaÂľ = â Î˝ AÂľ,a âÂľ AaÎ˝ We can also interchange the A-fields since we represented them in terms of group components instead of matrices. Reorganizing the above equation and simplifying, one obtains: a = 2 â Âľ AÎ˝,a â Aa â â Âľ AÎ˝,a â Aa F ÂľÎ˝,a FÂľÎ˝ Âľ Î˝ Î˝ Âľ h i + 2gs fabc (â Âľ AÎ˝,a ) AbÂľ AcÎ˝ â fabc (â Î˝ AÂľ,a ) AbÂľ AcÎ˝ 0 0 + gs2 fabc fab0 c0 AÂľ,b AÎ˝,c AbÂľ AcÎ˝ 176 (E.14) Putting this back into Eq. E.12, the Lagrangian can be written as: 1 Âľ Î˝,a â A âÂľ AaÎ˝ â â Âľ AÎ˝,a âÎ˝ AaÂľ 2 i 1 h â gs fabc (â Âľ AÎ˝,a ) AbÂľ AcÎ˝ â fabc (â Î˝ AÂľ,a ) AbÂľ AcÎ˝ 2 0 0 1 â gs2 fabc fab0 c0 AÂľ,b AÎ˝,c AbÂľ AcÎ˝ 4 + ĎĚ iâ/ â m Ď â gs T a AaÂľ ĎĚÎł Âľ Ď LQCD = â (E.15) Now that we have the Lagrangian in this form, we can begin calculating the Feynman Rules for QCD. We will calculate the Feynamn Rules for: 1. Gluon propagator 2. Triple gluon coupling 3. Quartic gluon coupling 4. Fermion propagator 5. Fermion-gluon vertex Gluon Propagator To find the gluon propagator Feynman Rule, one must first find all the quadratic terms in AaÂľ . These terms are as follows: 1 Âľ Î˝,a â A âÂľ AaÎ˝ â â Âľ AÎ˝,a âÎ˝ AaÂľ 2 i 1 Âľ,a h 2 =â A â gÂľÎ˝ Î´ab + âÂľ âÎ˝ Î´ab AÎ˝,b 2 â (E.16) However, to find the propagator one needs to find the inverse of this operator, but due to gauge invariance this operator does not have an inverse. Therefore, in order to obtain the 177 gluon propagator we must first break the gauge invariance of the Lagrangian. To do this, we will insert a new term by hand, known as the gague fixing term. Here we will use the following gauge breaking term: Lgf = â 2 1 1 âÂľ AÂľ,a = â AaÂľ (ââ Âľ â Î˝ Î´ab ) AbÎ˝ 2Îą 2Îą (E.17) Combining these two equations, we are able to find an operator that is invertible. The new term that we need to invert is given by: 1 Âľ,a 2 1 â A â gÂľÎ˝ Î´ab + (1 â )âÂľ âÎ˝ Î´ab AÎ˝,b 2 Îą (E.18) The propagator is found by taking two derivatives with respect to the Fourier transform of the Lagarangian. Pâ1 â 2 F(L) 1 2 = âi = âi âk gÎąÎ˛ Î´cd + (1 â )kÎą kÎ˛ Î´cd Îą âAcÎą âAdÎ˛ (E.19) By observation, it is easy to see that the form that the inverse of this equation must take the form of: kÂľ kÎ˝ P = i AgÂľÎ˝ + B 2 Î´ab k One can solve the equations for A and B to obtain the result for the propagator. Âľ PPâ1 = 1 = gĎ Î´ab 178 (E.20) kÂľkÎ˝ = +B 2 Î´ab k 1 1 Âľ Âľ 2 Âľ Âľ â â Ak gĎ â Bk kĎ + A 1 â k kĎ + B 1 â k Âľ kĎ = gĎ Îą Îą 1 1 2 â â Ak = 1 âB+A 1â +B 1â =0 Îą Îą k 2 gÎ˝Ď 1 â (1 â )kÎ˝ kĎ Îą 1 âA = â 2 k Ag ÂľÎ˝ (E.21) 1 B = (1 â Îą) 2 k Plugging the solutions to A and B back into Eq. E.20, we obtain the Feynman Rule for the gluon propagator. (1 â Îą) kÂľ kÎ˝ Î´ab = i 2 âgÂľÎ˝ + k k2 (E.22) Triple Gluon Coupling To find the triple gluon coupling, first we need to find all the terms that are cubic in the gauge fields. We will also be using the all incoming momentum convention when calculating the Feynman Rules. These terms are as follows: 1 L3 = â gs fabc [â Âľ AÎ˝,a â â Î˝ AÂľ,a ] AbÂľ AcÎ˝ 2 (E.23) We then follow a similar procedure to that used for the propagator. We need to find: Î(3) = âiâ 3 F(L3 ) n âAlÎą âAm Î˛ âAÎł (E.24) The array of (l, m, n) has 3! = 6 permutations, but due to fabc being antisymmetric, we can reduce the number of terms that need to be consider 3! 2 = 3 cases. In deriving the Feynman rule, we will go through each of the three cases to determine it. Also, the gluons 179 will have the momenta p, k, and q, as seen in the diagram of the vertex below. 1. Consider (l, m, n) â (a, b, c): The Fourier transform brings down the momentum associated with the gluon that has the derivative acting on it. So in this case we get factors of (ip). âi âi gs fabc Î´mb g Î˛Âľ Î´nc g ÎłÎ˝ Î´al gÎąÎ˝ ipÂľ â gÂľÎą Î´al (ipÎ˝ ) = gs flmn ipÎ˛ gÎąÎł â ipÎł gÎąÎ˛ 2 2 2. Consider (l, m, n) â (b, c, a): Similar to above, but here the momentum factor is (ik). âi âi gs fabc Î´lb g ÎąÂľ Î´mc g Î˛Î˝ Î´an gÎłÎ˝ ikÂľ â gÂľÎł Î´an (ikÎ˝ ) = gs fnlm (ikÎą ) gÎ˛Îł â ikÎ˛ gÎąÎł 2 2 Note: fnlm = âflnm = flmn 3. Consider (l, m, n) â (c, a, b): Here the momentum factor is (iq) âi âi gs fabc Î´nb g ÎłÂľ Î´lc g ÎąÎ˝ Î´an gÎ˛Î˝ iqÂľ â gÂľÎ˛ Î´an (iqÎ˝ ) = gs fmnl iqÎł gÎąÎ˛ â (ipÎą ) gÎłÎ˛ 2 2 Putting the three terms together along with manipulating the structure constants to match, one obtains the Feynman rule for the 3 point function as: h i = gs flmn (q â k)Îą Î´Î˛Îł + (k â p)Î˛ Î´ÎłÎą + (p â q)Îł Î´ÎąÎ˛ 180 (E.25) Quartic Gluon Coupling The last piece that includes on the gauge field are terms that contain quartic terms in the fields. There is one such term shown below: 1 L4 = â gs2 fabc fade AÂľ,b AÎ˝,c AdÂľ AeÎ˝ 4 (E.26) Therefore, following the same approach in the sections above, we will need to calculate the following term in order to get the Feynman rule: Î(4) = iâ 4 F(L4 ) n âAkÎą âAlÎ˛ âAm Îł âAĎ (E.27) Here we can see that there will be 4! = 24 different permutations that can occur for the indices (k, l, m, n). Again, due to the properties of the structure function we are able to relate (k, l, m, n) to (k, l, n, m), (k, m, l, n), and (k, n, m, l). However, to illustrate the relationship, we will still show the work for all 24 permutations. 1. â˘ Consider (k, l, m, n) â (b, c, d, e) i i â gs2 fabc fade Î´bk g ÂľÎą Î´cl g Î˝Î˛ Î´dm gÂľÎł Î´en gÎ˝Ď = â gs2 fakl famn gÎąÎł gÎ˛Ď 4 4 â˘ Consider (k, l, m, n) â (b, c, d, e) i i â gs2 fabc fade Î´bk g ÂľÎą Î´cl g Î˝Î˛ Î´dn gÂľĎ Î´em gÎ˝Îł = â gs2 fakl fanm gÎąĎ gÎ˛Îł 4 4 181 â˘ Consider (k, m, l, n) â (b, c, d, e) i i â gs2 fabc fade Î´bk g ÂľÎą Î´cm g Î˝Îł Î´dl gÂľÎ˛ Î´en gÎ˝Ď = â gs2 fakm faln gÎąÎ˛ gÎłĎ 4 4 â˘ Consider (k, m, n, l) â (b, c, d, e) i i â gs2 fabc fade Î´bk g ÂľÎą Î´cm g Î˝Îł Î´dn gÂľĎ Î´el gÎ˝Î˛ = â gs2 fakm faln gÎąĎ gÎ˛Îł 4 4 â˘ Consider (k, n, m, l) â (b, c, d, e) i i â gs2 fabc fade Î´bk g ÂľÎą Î´cn g Î˝Ď Î´dm gÂľÎł Î´el gÎ˝Î˛ = â gs2 fakn faml gÎąÎł gÎ˛Ď 4 4 â˘ Consider (k, n, l, m) â (b, c, d, e) i i â gs2 fabc fade Î´bk g ÂľÎą Î´cn g Î˝Ď Î´dl gÂľÎ˛ Î´em gÎ˝Îł = â gs2 fakn falm gÎąÎ˛ gÎłĎ 4 4 2. From here out, we will just write the result for each term. This can be done by recognizing the pattern of the terms above. â˘ Consider (l, m, n, k) â (b, c, d, e) i â gs2 falm fank gÎ˛Ď gÎłÎą 4 â˘ Consider (l, m, k, n) â (b, c, d, e) i â gs2 falm fakn gÎąÎ˛ gÎłĎ 4 182 â˘ Consider (l, n, m, k) â (b, c, d, e) i â gs2 faln famk gÎ˛Îł gÎąĎ 4 â˘ Consider (l, k, n, m) â (b, c, d, e) i â gs2 falk fanm gÎ˛Ď gÎąÎł 4 â˘ Consider (l, n, k, m) â (b, c, d, e) i â gs2 faln fakm gÎąÎ˛ gÎłĎ 4 â˘ Consider (l, k, m, n) â (b, c, d, e) i â gs2 falk famn gÎ˛Îł gÎąĎ 4 3. â˘ Consider (m, n, k, l) â (b, c, d, e) i â gs2 famn fakl gÎąÎł gÎ˛Ď 4 â˘ Consider (m, n, l, k) â (b, c, d, e) i â gs2 famn falk gÎ˛Îł gÎąĎ 4 183 â˘ Consider (m, k, n, l) â (b, c, d, e) i â gs2 famk fanl gÎłĎ gÎąÎ˛ 4 â˘ Consider (m, l, k, n) â (b, c, d, e) i â gs2 faml fakn gÎąÎł gÎ˛Ď 4 â˘ Consider (m, k, l, n) â (b, c, d, e) i â gs2 famk faln gÎ˛Îł gÎąĎ 4 â˘ Consider (m, l, n, k) â (b, c, d, e) i â gs2 faml fank gÎłĎ gÎąÎ˛ 4 4. â˘ Consider (n, k, l, m) â (b, c, d, e) i â gs2 fank falm gÎ˛Ď gÎąÎł 4 â˘ Consider (n, k, m, l) â (b, c, d, e) i â gs2 fank faml gÎłĎ gÎąÎ˛ 4 184 â˘ Consider (n, l, k, m) â (b, c, d, e) i â gs2 fanl fakm gÎąĎ gÎ˛Îł 4 â˘ Consider (n, m, l, k) â (b, c, d, e) i â gs2 fanm falk gÎąÎł gÎ˛Ď 4 â˘ Consider (n, l, m, k) â (b, c, d, e) i â gs2 fanl famk gÎłĎ gÎąÎ˛ 4 â˘ Consider (n, m, k, l) â (b, c, d, e) i â gs2 fanm fakl gÎąĎ gÎ˛Îł 4 Combining the above 24 terms together, and simplifing the expressions, we get the following for the Feynman Rule: = âigs2 [fakl famn gÎąÎł gÎ˛Ď â gÎąĎ gÎ˛Îł +fakm faln gÎąÎ˛ gĎÎł â gÎąĎ gÎ˛Îł +fakn falm gÎąÎ˛ gĎÎł â gÎąĎ gÎ˛Îł ] 185 (E.28) Fermion Propagator Now we will consider terms in the Lagrangian containing fermions. The first term that we will calculate is the fermion propagator. Unlike the gluon propagator, the fermion propagator does not have the issue of being zero due to gague invariance. It is therefore, a straightforward calculation. The term needed in the Lagrangian is: Lf = ĎĚ iâ/ â m Ď (E.29) The propagator is found by the same method as above. In other words, we need to calculate: P= â 2 F Lf âi â ĎĚâĎ !â1 (E.30) Preforming this calculation, one obtains: â1 P = i p/ â m = i(p/ + m) i = 2 p/ â m p â m2 (E.31) Where to get to the last equation, we multiplied the top and the bottom by p/ + m to bring it to the typical form. Thus, the Feynman rule for the Fermion propagator is given by: i(p/ + m) = 2 p â m2 186 (E.32) Fermion-Gluon Vertex The remaining term that has not been accounted for in the Lagrangian thus far is the interaction term between gluons and fermions. This term is given by: Lint = gs T a AaÂľ ĎĚÎł Âľ Ď = gs AaÂľ ĎĚi Îł Âľ (T a )ij Ďj (E.33) Where in the second equation we wrote in the indices corresponding to the color of the fermions. The index a goes from 1 to 8, where the indicies i, j can be 1,2, or 3. Again, we preform the same procedure as before: Îint = iâ 3 F(Lint ) â ĎĚâĎâAaÂľ (E.34) This gives us the Feynman rule, which is shown below: = igs ÎłÂľ (T a )ij (E.35) Note: The order of (ij) is important because T a is not a symmetric matrix. Fadeev-Popov Ghost Fields Arising from an artifact of insiting on Lorentz invariance and unitarity for massless spin-1 particles are unphysical particles known as Fadeev-Popov ghosts. These also arise in QED, but since the theory is Abelian, they do not couple to the photon. Here in the non-Abelian theory they do couple to the gluon, but only appear at higher orders, since they are non- 187 physical. There are ways to derive the Feynman rules to not have any ghost fields knwon as Axial gauges and a discussion on these can be found in [236]. The gauge fixing term can be expressed as: 1 a 2 (c ) 2Îą (E.36) LF P = ĎĚa Mab Ďb (E.37) Lgf = â The ghost Lagrangian is defined as: a where Mab = Î´c b . We can see that there will be one ghost and one anti-ghost for each of Î´Î the gluons. Therefore, there will be 8 ghost fields and 8 anti-ghost fields. Let us examine how the ca fields change under a local gauge transformation. ca = â Âľ AaÂľ â â Âľ (AaÂľ + Î´AaÂľ ) = ca + Î´ca (E.38) To find the equation for Mab we will need to calculate the Î´C a term in the above equation. The derivation is shown in the equations below. Î´C a =â Âľ (Î´AaÂľ ) = â Âľ (ââÂľ Îa + gs fabc Îb AcÂľ ) = â â 2 Îa + gs fabc â Âľ (Îb AcÂľ ) h i = ââ 2 Î´ ab + gs fabc AcÂľ â Âľ + â Âľ AcÂľ Îb âMab = Î´ca 2Î´ + g f c â Âľ + â Âľ Ac = ââ A s ab abc Âľ Âľ Î´Îb 188 (E.39) (E.40) Plugging the above result into Eq. E.37, we get the resulting ghost Lagrangain. h i LF P =ĎĚa ââ 2 Î´ab + gs fabc AcÂľ â Âľ + â Âľ AcÂľ Ďb = âÂľ ĎĚa (â Âľ Ďa ) â gs fabc (â Âľ ĎĚa ) AcÂľ Ďb (E.41) Now we will begin to calculate the Feynman rules for the ghost fields. We begin with the calculation of the propagator. P= âiâ 2 F(âĎĚa â 2 Î´ab Ďb ) â ĎĚa âĎb !â1 1 = âi 2 Î´ab k (E.42) (E.43) The only remaining Feynman rule for QCD to calculate is the ghost-gluon interaction term. This term is given by the same procedure as above, giving: = gs fabc pÂľ (E.44) It is important to note that the anti-ghost fields are not the anti-particles of the ghost fields. 189 QCD Î˛ Functions The Î˛ functions in QCD arise from the calculation of the renormalization of the strong coupling constant. They have been calculated up to four-loops [237, 238]. The results up to that order are listed below: 11 4 CA â TF nF , (E.45) 3 3 34 2 20 = CA â CA TF nF â 4CF TF nF , (E.46) 3 3 2857 3 1415 2 158 44 = CA â CA TF nF + CA TF2 n2F + CF TF2 n2F 54 27 27 9 205 â C C T n + 2CF2 TF nF , (E.47) 9 A FF F 17152 448 4204 352 424 2 2 2 =CA CF TF nF + Îś3 + CA CF TF nF â + Îś3 + C T 3 n3 243 9 27 9 243 A F F 7073 656 7930 224 1232 2 2 2 2 + CA CF TF nF â Îś3 + CA TF nF + Îś3 + CF TF3 n3F 243 9 81 9 243 39143 136 150653 44 1352 704 3 4 2 2 2 CA TF nF â + Îś + CA â Îś3 + CF TF nF â Îś 81 3 3 486 9 27 9 3 dabcd dabcd dabcd dabcd 512 1664 704 512 3 2 F A F F + 46CF TF nF + nF â Îś + nF â + Îś NA 9 3 3 NA 9 3 3 dabcd dabcd 80 704 A A + â + Îś , (E.48) NA 9 3 3 Î˛0 = Î˛1 Î˛2 Î˛3 where the coefficients in the Î˛ equations can be expressed in the specific values for an SU (N ) group are: 1 TF = , 2 CF = N2 â 1 , 2N abcdA dabcd N (N 2 + 6) F d = , NA 48 CA = N, abcdF dabcd N 4 â 6N 2 + 18 F d = , NA 96N 2 abcdA dabcd N 2 (N 2 + 36) A d = , NA 24 190 NA = N 2 â 1. (E.49) APPENDIX F Calculation Details Plus Functions The Plus function is a distribution to handle singularities of functions that occur for x = 1. Given the function F (x) which is singular for x = 1, the plus function is defined as: ! Z 1âÎ˛ F (x)+ âĄ lim Î˛â0 F (x)Î(1 â x â Î˛) â Î´(1 â x â Î˛) dyF (y) . (F.1) 0 When a plus function is convoluted with a test function, the final result is well defined. For example, given a test function G(x), the convolution is given by: Z 1 Z 1 dxF (x) (G(x) â G(1)) . dxF (x)+ G(x) = 0 (F.2) 0 Additionally, a property of the plus function is: Z 1 dxF (x)+ = 0, 0 191 (F.3) which can be easily derived with a choice of G(x) = 1. Finally, it is possible to obtain the result for a lower bound that is non-zero by: Z 1 Z 1 dxF (x) (G(x) â G(1)) + G(1) dxF (x)+ G(x) = a Z a a F (x). (F.4) 0 NLO Real Corrections to Drell-Yan Similar to the quark-anti-quark channel, the singular terms for the (anti-)quark-gluon channel are given by: |M |2gq 2 Q 2 + g2 2 + f 2 K L â g2 â g2 2 â f 2 K (2A ) , = 64 gL f f 3 0 4 4 R L R L R L R 2 sing (F.5) with 1 2 K3 = 2 zA + (1 â zA )2 Î´ (1 â zB ) qT 1 â 2 Î´ (1 â zB ) zA (1 â ZA ) + zA â zB , qT 1 2 K4 = 2 zA + (1 â zA )2 Î´ (1 â zB ) + zA â zB . qT (F.6) (F.7) Loop Integrals Here a brief discuss of loop integrals is given, along with the calculation of the loop integral involved in calculating the NLO correction to Drell-Yan. For additional discussions of loop integrals, and the more modern approach that has been developed to tackle the calculations of higher number of loops can be found in [239, 240, 241]. 192 p1 q p1 + k q Îł k p2 â k q p2 Figure F.1: The hadronic virtual correction Feynman Diagram to Drell-Yan at NLO. For Drell-Yan at NLO accuracy, there is only one virtual diagram to calculate if dimensional regularization is used. The hadronic component of the calculation is given the following Feynman diagram in Fig. F.1. Since this diagram is O (Îąs ), the matrix element squared will be of O Îąs2 . Since this is a higher order correction, only the cross term of the virtual diagram and the tree level diagram are used up to NLO. The result is obtained from â 2Re Mborn Mvirt . Using the Feynman rules, the virtual correction is given as: 2Re â Mborn Mvirt dd k Z = CF (2Ď)d igÎąÎ˛ k2 Tr p/2 ÎłÎą i p/2 â k/ i 2 (p2 â k) (p1 + k)2 ÎłÂľ p/1 + k/ ÎłÎ˛ p gÂľÎ˝ (2Ď) gs2 Q2 e2 , /1 ÎłÎ˝ (F.8) where CF is the color factor, gs is the strong coupling constant, Q is the charge of the quark, and e is the electric charge. Consider the integral: Z Tr p/2 Îł Îą p/2 â k/ Îł Âľ p/1 + k/ Îł Î˛ p/1 Îł Î˝ . k 2 (p2 â k)2 (p1 + k)2 (2Ď)d dd k 193 (F.9) Using the Feynman parameter trick1 , gives: Z I= Tr p/2 Îł Îą p/2 â k/ Îł Âľ p/1 + k/ Îł Î˛ p/1 Îł Î˝ dxdy . (2Ď)d k 2 (1 â x â y) + (p2 â k)2 x + (p1 + k)2 y dd k (F.10) The denominator of the above equation can be simplified, using the substitution: lÂľ = Âľ Âľ k Âľ â p2 x + p1 y, and the definition q 2 = 2p1 Âˇ p2 , giving: D = l2 + q 2 xy. (F.11) Now consider the numerator of Eq. F.10, the numerator can be simplified, using the substitution for l given above, to: h i N = â4q 4 (1 â y â x) (1 â ) + xy â 22 xy +8 1 â 2 2 2 q l â 8 (1 â ) q 2 l2 . 2 (2 â ) (F.12) Combining the numerator and the denominator and putting the integrals back, gives three unique integrals to calculate. The first integral is: Z 1 dd l 1 â y â x (2Ď)d l2 + q 2 xy 3 0 0 âi 4Ď 1 1 1 2 = + + 4 + O () , 2 Î (1 â ) 2 16Ď 2 q 2 âq 2 I1 = Z 1âx dx Z dy 1 Details on the Feynman parameter trick can be found in Appendix A.4 of [42] 194 (F.13) the second is: Z 1 dd l l2 (2Ď)d l2 + q 2 xy 3 0 0 âi 2 â 4Ď 1 1 3 7 = + + + O 2 , 2 2 2 Î (1 â ) 2 2 2 16Ď âq I2 = Z 1âx dx Z dy (F.14) and finally, the third is: Z 1 dd l xy d (2Ď) l2 + q 2 xy 3 0 0 âi 1 4Ď 1 3 = + + O 2 . 2 2 2 2 2 16Ď q 2 âq I3 = Z 1âx dx Z dy (F.15) Plugging all of the results back into Eq. F.10, gives: iq 2 I= 16Ď 2 4Ď 1 â2 3 2 â â 8 + Ď + O () . Î (1 â ) 2 q2 (F.16) Finally, substituting the result of the integral back into Eq. F.8, gives the desired result found in Eq. 3.18. Phase Space In D-Dimensions When working outside of d = 4â2 dimensions, the integrals over the phase space need to be modified. The major calculation that is needed, is the area of a unit sphere in d dimensions. 195 This can be calculated by using the following trick: â d Ď = Z dxeâx 2 ďŁŤ d Z dd x exp ďŁâ = d X ďŁś x2i ďŁ¸ i=1 Z â Z = Solving for R dÎŁd 2 dxxdâ1 eâx Z = 0 dÎŁd 1 Î 2 d . 2 dÎŁd gives: Z dÎŁd = 2Ď d/2 . Î (d/2) (F.17) Using Eq. F.17 reproduces the expected results for the familiar integer dimensions as seen in Table F.1. d 1 2 3 4 Î (d/2) â Ď â1 Ď/2 1 R dÎŁd 2 2Ď 4Ď 2Ď 2 Table F.1: Table of the results of calculating the area of a d-Sphere Resummation Coefficients to N3LL The coefficients up to NNLL can be found in Section 3.4.1 and 3.4.2 for the CSS and CFG formalisms respectively. For the additional terms that appear at N3 LL, they can be found in [167, 168], and are reproduced here for ease. The B anomalous dimension in CSS at O Îąs3 is given as: B3DY = Îł2DY â Îł2r + Î˛1 cDY 1 1 DY 2 DY + 2Î˛0 c2 â c , 2 1 196 (F.18) substituting the numbers into the equation above, the numerical result is given as: B3DY = 114.98 â 11.27nf + 0.32n2f , (F.19) where nf is the number of active flavors. Note the above equation differs from that in [167], Îąs s since the expansion in [167] is for Îą 4Ď , while this work uses Ď . The hard-collinear coefficient at the NNLO for vector boson production is given by five different initial states: q qĚ, q qĚ 0 , qq, qq 0 , and qg. These coefficients are given as: (2) 2CqqĚ (z) + Î´ (1 â z) DY (2) Hq 3 DY (1) 2 CF 2 DY (1) â Hq + Ď â 8 Hq 4 4 1 DY (1) + CF Hq (1 â z) 2 " # 2 CF2 Ď2 â 8 DY (2) = HqqĚâqqĚ (z) â Î´ (1 â z) + Ď 2 â 10 (1 â z) â (1 + z) ln z , 4 4 (F.20) 1 DY (1) (2) DY (2) Cqg (z) + Hq z (1 â z) = HqqĚâqg (z) 4 Ď2 CF 1 2 â z ln z + 1âz + â 4 z (1 â z) , 4 2 4 (2) DY (2) (2) DY (2) Cqq (z) = HqqĚâqq (z) , (F.21) (F.22) C 0 (z) = H (z) , qq q qĚâqq 0 (F.23) (2) C 0 (z) = HqqĚâqqĚ0 , q qĚ (F.24) 197 where the H coefficients are given by: DY (2) HqqĚâqqĚ (z) 7Îś3 101 1 59Îś3 1535 215Ď 2 Ď4 = CA CF â + â + â Î´(1 â z) 2 27 1âz + 18 192 216 240 1 + z2 Li (1 â z) Li (z) log(z) 1 + â 3 + Li3 (z) â 2 â Li2 (z) log(1 â z) 1âz 2 2 2 1 1 1 2 Ď2 3 2 â log (z) â log (1 â z) log(z) + Ď log(1 â z) â 24 2 12 8 1 1 1 + â 11 â 3z 2 Îś3 â âz 2 + 12z + 11 log2 (z) 1âz 4 48 1 2 Ď2z â 83z â 36z + 29 log(z) + 36 4 Li2 (z) 1 z + 100 1 + (1 â z) + log(1 â z) log(z) + + z log(1 â z) 2 2 27 4 14 1 1 + C F nF + 192Îś3 + 1143 â 152Ď 2 Î´(1 â z) 27 1 â z + 864 1 + z2 1 + log(z)(3 log(z) + 10) + (â19z â 37) 72(1 â z) 108 1 511 67Ď 2 17Ď 4 2 + CF â15Îś3 + â + Î´(1 â z) 4 16 12 45 1 + z 2 Li3 (1 â z) 5Li3 (z) 1 3Li2 (z) log(z) + â + Li2 (z) log(1 â z) + 1âz 2 2 2 2 3 1 1 2 5Îś3 2 2 + log(z) log (1 â z) + log (z) log(1 â z) â Ď log(1 â z) + 4 4 12 2 3 2Ď 2 29 1 + (1 â z) âLi2 (z) â log(1 â z) log(z) + â + (1 + z) log3 (z) 2 3 4 24 1 1 1 2 2 2 + â2z + 2z + 3 log (z) + 17z â 13z + 4 log(z) 1âz 8 4 z â log(1 â z) 4 Li (z) 1 1 Ď2 2 2 + CF (1 â z) 2z â z + 2 + log(1 â z) log(z) â z 6 6 36 1 1 + (1 â z) 136z 2 â 143z + 172 â 8z 2 + 3z + 3 log2 (z) 216z 48 1 2 1 3 + 32z â 30z + 21 log(z) + (1 + z) log (z) , (F.25) 36 24 198 DY (2) H (z) q qĚâq qĚ 0 DY (2) HqqĚâqq (z) 1 Ď2 2 = CF (1 â z) 2z â z + 2 Li2 (z) + log(1 â z) log(z) â 12z 6 1 1 + (1 â z) 136z 2 â 143z + 172 + (1 + z) log3 (z) 432z 48 1 1 2 2 2 â 8z + 3z + 3 log (z) + 32z â 30z + 21 log(z) , 96 72 = CF 1 CF â CA 2 (F.26) 1 + z 2 3Li3 (âz) 1 Li (âz) log(z) + Li3 (z) + Li3 â 2 1+z 2 1+z 2 Li2 (z) log(z) 1 1 1 â log3 (z) â log3 (1 + z) + log(1 + z) log2 (z) 2 24 6 4 2 Ď 3Îś3 Li2 (z) 1 15 + log(1 + z) â + (1 â z) + log(1 â z) log(z) + 12 4 2 2 8 Ď2 1 1 â (1 + z) Li2 (âz) + log(z) log(1 + z) + (z â 3) + (11z + 3) log(z) 2 24 8 1 Ď2 2 + CF (1 â z) 2z â z + 2 Li2 (z) + log(1 â z) log(z) â 12z 6 1 1 2 + (1 â z) 136z 2 â 143z + 172 â 8z + 3z + 3 log2 (z) 432z 96 1 1 2 3 + 32z â 30z + 21 log(z) + (1 + z) log (z) , (F.27) 72 48 â DY (2) (z) q qĚâqq 0 H DY (2) (z) q qĚâq qĚ 0 =H 199 , (F.28) DY (2) HqqĚâqg (z) 1 (1 â z) 11z 2 â z + 2 Li2 (1 â z) 12z Li (1 â z) 1 1 3 2 3 + 2z â 2z + 1 â Li2 (1 â z) log(1 â z) + log (1 â z) 8 8 48 1 3Li (âz) Li3 1+z Li (âz) log(z) 1 3 + 2z 2 + 2z + 1 + â 2 â log3 (1 + z) 8 4 8 24 1 1 2 1 2 + log (z) log(1 + z) + Ď log(1 + z) + z(1 + z)Li2 (âz) + zLi3 (z) 16 48 4 1 3 149z 2 2 â zLi2 (1 â z) log(z) â zLi2 (z) log(z) â 2z + 1 Îś3 â 2 8 216 1 2 1 â 44z â 12z + 3 log2 (z) + 68z 2 + 6Ď 2 z â 30z + 21 log(z) 96 72 2 Ď z 43z 43 1 1 + + + + (2z + 1) log3 (z) â z log(1 â z) log2 (z) 24 48 108z 48 2 1 1 â (1 â z)z log2 (1 â z) + z(1 + z) log(1 + z) log(z) 8 4 1 35 + (3 â 4z)z log(1 â z) â 16 48 Li (1 â z) Li3 (z) 1 + CF 2z 2 â 2z + 1 Îś3 â 3 â + Li2 (1 â z) log(1 â z) 8 8 8 = CA + + â + â â Li2 (z) log(z) 1 1 â log3 (1 â z) + log(z) log2 (1 â z) 8 48 16 1 log2 (z) log(1 â z) 16 3z 2 1 2 1 3 2 â 4z â 2z + 1 log (z) + â8z + 12z + 1 log2 (z) 8 96 64 1 5 11z 1 â8z 2 + 23z + 8 log(z) + Ď 2 (1 â z)z + + (1 â z)z log2 (1 â z) 32 24 32 8 1 1 9 (1 â z)z log(1 â z) log(z) â (3 â 4z)z log(1 â z) â , (F.29) 4 16 32 DY (2) HqqĚâgg (z) z =â 2 1 1 â z + (1 + z) log(z) 2 200 , (F.30) where Lik (z) (k = 2, 3) are the polylogarithm functions, Z z dt Li2 (z) = â ln(1 â t) , 0 t Z 1 dt Li3 (z) = ln(t) ln(1 â zt) , 0 t (F.31) and the H factors are the scheme dependent resummation factors. For CSS, H is 1 to all orders, while for CFG, H has Îąs dependence. Fit to SIDIS Data The universality of the parton distribution is a powerful prediction from QCD factorization. According to the TMD factorization, the universality of the TMD parton distributions should exist between SIDIS and Drell-Yan processes as well. Therefore, the non-perturbative functions determined for the TMD parton distributions from the Drell-Yan type of processes shall apply to that in the SIDIS. Of course, the transverse momentum distribution of hadron production in DIS processes also depends on the final state TMD fragmentation functions, which need to be determined by fitting to existing experimental data. Following the universality arguments, the following parameterizations for the non-perturbative form factors for SIDIS process can be assumed, in contrast to Eq. 3.77 for Drell-Yan process, (DIS) SN P = g1 2 b + g2 ln (b/bâ ) ln(Q/Q0 ) + g3 b2 (x0 /xB )Îť + 2 gh 2 b . zh2 (F.32) In the above parameterization, g1 and g2 have been determined from the experimental data of Drell-Yan lepton pair production. The factor of 1/2 in front of the g1 term is due the fact that there is only one incoming hadron in the SIDIS process, while there are two incoming hadrons in the Drell-Yan process. Although there has been evidence from recent studies [242, 243] that 201 gh could be different for the so-called favored and dis-favored fragmentation functions, them will be taken to be the same in this study for simplicity. When more precise data become available, a global analysis with two separate gh parameters may need to be preformed. In principle, g1 , g2 , and gh may be fitted to both Drell-Yan and SIDIS data simultaneously. However, the SIDIS data from HERMES and COMPASS mainly focus in the relative low Q2 range. Because of that, the theoretical uncertainty of the CSS prediction is not well under control, particularly, from the Y -term contribution. There have been several successful phenomenological studies to describe the experimental data from HERMES and COMPASS experiments, using the leading order TMD formalism [242, 244]. The goal of this paper is to check if the non-perturbative form factors determined in the Drell-Yan process can be applied to the SIDIS processes. As shown in Ref. [171], fitting both is not possible with the original BLNY or KN fit, where it was found that the extrapolation of these fits to the kinematic region of HERMES and COMPASS is in conflict with the experimental data. However, it will be shown that the SIYY form will be able to extend to SIDIS experiments from HERMES and COMPASS Collaborations. Therefore, in the following, the parameters (g1,2 ) are set to the fitted values from the Drell-Yan data and compared to the SIDIS data for consistency. In Fig. F.2, the comparisons between the theory predictions with gh = 0.042 and the SIDIS data from HERMES is given, with a total Ď2 around 180. This parameter is consistent with previous analysis when the leading order TMD formalism is considered [242, 244]. It is also consistent with the TMD formalism with truncated evolution effects in Ref. [171]. The differential cross section for the SIDIS process depends on the hadron fragmentation functions, for which the parameterization from the new DSS fit [245, 246] is used here. A normalization factor about 2.0 in the calculation of the multiplicity distributions shown in Fig. F.2 is used, which 202 Multiplicity Multiplicity 4.5 9 8 4 7 3.5 6 3 5 4 0.2 < zh < 0.3 2.5 Q2=3.14GeV2 2 3 2 1.5 HERMES (e+p -> Ď++X) HERMES (e+p -> Ď++X) t Multiplicity t Multiplicity Q2=3.14GeV2 1 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 p (GeV) 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 p (GeV) 1.6 0.3 < zh < 0.4 7 6 1.4 5 1.2 4 1 0.4 < zh < 0.6 0.8 Q2=3.14GeV2 0.6 HERMES (e+p -> Ď++X) 3 Q2=3.14GeV2 2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 p (GeV) HERMES (e+p -> Ď-+X) 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 p (GeV) t Multiplicity t Multiplicity 0.2 < zh < 0.3 1.2 3 1 2.5 0.8 2 1.5 0.6 0.3 < zh < 0.4 0.4 < zh < 0.6 Q2=3.14GeV2 1 0.4 HERMES (e+p -> Ď-+X) Q2=3.14GeV2 HERMES (e+p -> Ď-+X) 0.2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 p (GeV) 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 p (GeV) t t Figure F.2: Multiplicity distribution as function of transverse momentum in semi-inclusive hadron production in deep inelastic scattering compared to the experimental data from HERMES Collaboration at Q2 = 3.14GeV2 . 203 accounts for theoretical uncertainties from higher order corrections for both differential and inclusive cross sections 2 . Here, the Y -term contribution is not included, for the reasons mentioned above. The figures in Sec. 3.4.4 and Fig.F.2 clearly illustrate that the SIYY non-perturbative function is a universal non-perturbative TMD function which can be used to describe both Drell-Yan lepton pair production and semi-inclusive hadron production in DIS processes in the CSS resummation framework. Also, note that that the new functional form for the non-perturbative function is crucial to achieve this conclusion as given in Eqs. (3.77) and (F.32). Îąs Expansion The Taylor series expansion of Îąs around some fixed scale Âľ0 is given by: Îą (Âľ) Îą (Âľ) â sĎ Îąs (Âľ) Îąs (Âľ0 ) Âľ2 1 â 2 sĎ = + log 2 + log2 2 2 Ď Ď â log Âľ Âľ=Âľ Âľ0 2 â log Âľ2 Âľ=Âľ0 0 Âľ2 + O Îąs3 . (F.33) 2 Âľ0 To obtain the final result, we need to calculate the derivatives. However, the first derivative is exactly related to the Î˛ function for the running of the coupling, Îą (Âľ) â sĎ = Î˛ (Îąs (Âľ)) , â log Âľ2 (F.34) 2 Compared to the leading order TMD fit of Ref. [244] where there is no normalization factor, the C (1) coefficient is large and negative in the CSS resummation application to the SIDIS. Phenomenologically, that is the reason to include a factor of 2 in the comparison to the SIDIS data. This could be improved if the differential cross section (instead of multiplicity distributions) are measured in the future. 204 with Î˛ (Îąs (Âľ)) = â â X Îąs (Âľ) k+1 k=1 Ď Î˛kâ1 . (F.35) For the second derivative, Îą (Âľ) â X â 2 sĎ âÎ˛ (Îąs (Âľ)) Îąs (Âľ) k Î˛ (Îąs (Âľ)) = âÎ˛ (Îąs (Âľ)) (k + 1) Î˛kâ1 . 2 = Îą (Âľ) Ď â log Âľ2 â sĎ k=1 (F.36) Inputing these results into Eq. F.33, and keeping terms only up to O Îąs3 , the expansion of Îąs is: Îąs (Âľ) Îąs (Âľ0 ) = Ď Ď Îąs (Âľ0 ) Âľ2 1â Î˛0 log 2 + Ď Âľ0 Îąs (Âľ0 ) 2 Ď Âľ2 Âľ2 Î˛02 log2 2 â Î˛1 log 2 Âľ0 Âľ0 ! + O Îąs3 ! (F.37) PDF Expansion The Taylor series expansion of the parton distribution function around some fixed scale ÂľF is given by: âfi (x, Âľ) Âľ2 fi (x, Âľ) = fi (x, ÂľF ) + log â log Âľ2 Âľ=Âľ Âľ20 F 2 1 â 2 fi (x, Âľ) Âľ2 1 â 3 fi (x, Âľ) 2 3 Âľ + O Îą4 . + log + log s 2 â log Âľ2 2 Âľ=Âľ 6 â log Âľ2 3 Âľ=Âľ Âľ2F Âľ2F F F 205 (F.38) . To obtain the final result, the derivatives need to be calculated. However, the first derivative is exactly related to the DGLAP evolution equations, âfi (x, Âľ) Îąs (Âľ) (1) = P â fi (x, Âľ) 2 2Ď â log Âľ Îąs (Âľ) 2 (2) Îąs (Âľ) 3 (3) + P â fi (x, Âľ) + P â fi (x, Âľ) + O Îąs4 , 2Ď 2Ď (F.39) When calculating the second derivative, it is also necessary to expand Îąs (Âľ) around ÂľR as shown in the previous section. Only up to O Îąs3 will be keep to keep the equations short. The second derivative is given as, Îą 2 1 â 2 fi (x, Âľ) 1 s (1) (1) (1) P â P â f (Âľ) â Î˛0 P â f (Âľ) 2 = Ď 4 2 â log Âľ2 Îą 3 1 1 (2) Âľ2 (1) s (1) (2) (1) 2 + P â P â f (Âľ) + P â P â f (Âľ) + Î˛0 ln 2 P â f (Âľ) Ď 8 8 ÂľR ! 1 Âľ2 (1) 1 1 â Î˛0 ln 2 P â P (1) â f (Âľ) â Î˛1 P (1) â f (Âľ) â Î˛0 P (2) â f (Âľ) + O Îąs4 , 2 2 2 ÂľR (F.40) and the third derivative is given as, Îą 3 â 3 fi (x, Âľ) 3 s Î˛02 P (1) â f (Âľ) â Î˛0 P (1) â P (1) â f (Âľ) 3 = Ď 4 â log Âľ2 1 (1) (1) (1) + P â P â P â f (Âľ) + O Îąs4 . 8 (F.41) Inputing these results into Eq. F.38, and keeping terms only up to O Îąs3 , the expansion of 206 the PDF is: 1 Îąs (ÂľR ) Âľ2 (1) ln 2 P â f (ÂľF ) 2 Ď ÂľF 2 Âľ2F (1) Îąs (ÂľR ) 1 Âľ2 1 Âľ2 + â Î˛0 ln 2 ln 2 P â f (ÂľF ) â Î˛0 ln2 2 P (1) â f (ÂľF ) Ď 2 4 ÂľF ÂľR Âľ ! F 1 Âľ2 1 Âľ2 + ln2 2 P (1) â P (1) â f (ÂľF ) + ln 2 P (2) â f (ÂľF ) 8 4 ÂľF ÂľF 3 Îąs (ÂľR ) 1 2 Âľ2 (1) 1 2 Âľ2 (2) + ln 2 P â P (2) â f (ÂľF ) + ln 2 P â P (1) â f (ÂľF ) Ď 16 16 ÂľF ÂľF f (Âľ) = f (ÂľF ) + 1 2 Âľ2 2 Âľ2F (1) Âľ2F (1) Âľ2 ln P â f Âľ P â f (ÂľF ) f + Î˛0 ln 2 ln 2 Âľ2F Âľ2R ÂľF Âľ2R Âľ2F (1) Âľ2F (1) 1 Âľ2 1 Âľ2 2 (1) â Î˛0 ln 2 ln 2 P â P â f (ÂľF ) â Î˛1 ln 2 ln 2 P â f (ÂľF ) 4 2 ÂľF ÂľR ÂľF ÂľR 1 + Î˛02 ln2 2 1 + Î˛02 ln3 6 Âľ2 (1) 1 P â f (ÂľF ) â Î˛0 ln3 2 8 ÂľF Âľ2 (1) 1 P â P (1) â f (ÂľF ) â Î˛1 ln2 2 4 ÂľF Âľ2 (1) P â f (ÂľF ) Âľ2F 2 2 Âľ2 (1) (1) â P (1) â f (Âľ ) â 1 Î˛ ln Âľ ln ÂľF P (2) â f (Âľ ) P â P F F 2 0 Âľ2F Âľ2F Âľ2R ! 2 2 1 Âľ 1 Âľ â Î˛0 ln2 2 P (2) â f (ÂľF ) + ln 2 P (3) â f (ÂľF ) + O Îąs4 . 4 8 ÂľF ÂľF + 1 3 ln 48 207 (F.42) O Îąs3 Asymptotic Piece Continuing on to O Îąs3 , the calculations become extremely complex. The asymptotic piece can be given by: 1 (1) 3 = A fi fj , 4 2 5 5 5 5 2(1) (i,j) (1) (1) (1) (1) = A B f i f j â Î˛0 f i f j + f i P â f j + A fj P â fi , 3 C4 4 3 8 8 2 7 7 (i,j) (1) (1) (1) 2 =A B â Î˛0 B â A(2) + Î˛0 fi fj + 2B (1) fj P (1) â fi â Î˛0 fj P (1) â fi 3 C3 3 3 1 1 + P (1) â fi P (1) â fj + fj P (1) â P (1) â fi 2 2 ! !! 2 1 Q2 Q2 (1) (1) (1) â A fj C â fi â fj P â fi log â Î˛0 fi fj log + i â j, 2 Âľ2F Âľ2R ! 3 (2) 3 (2) Q2 (i,j) (2) (1) 2 (1) 2 = 2Î˛0 A + B Î˛0 â A +A â B + Î˛1 â 2Î˛0 log 3 C2 2 2 Âľ2R !! Q2 (1) +3Î˛0 B log Âľ2R 3 1 (1) 3 3 (1) 2 3 (1) â5 A Îś3 + B â Î˛0 B fi fj â A(2) fj P (1) â fi 2 2 2 !! 2 3 Q + A(1) 5Î˛0 fj C (1) â fi + B(1) â3fj C (1) â fi â fj P (1) â fi log 2 Âľ2F ! 2 3 (1) 3 Q â C â fi P (1) â fj â fj C (1) â P (1) â fi + Î˛0 fj P (1) â fi log 2 2 Âľ2F ! ! 2 2 3 (1) Q 3 Q â P â fi P (1) â fj log â P (1) â P (1) â fi fj log 4 4 Âľ2F Âľ2F ! ! 2 Q 3 +3Î˛0 fj P (1) â fi log â fj P (2) â fi 4 Âľ2R (i,j) 3 C5 3 3 + Î˛02 fj P (1) â fi â Î˛0 P (1) â fi P (1) â fj â Î˛0 fj P (1) â P (1) â fi 4 4 3 (1) 2 + B fj P (1) â fi 2 3 (1) 3 (1) (1) (1) (1) (1) +B â3Î˛0 fj P â fi + P â fi P â fj + fj P â P â fi 4 4 208 3 1 + P (1) â P (1) â fi P (1) â fj + fj P (1) â P (1) â P (1) â fi 8 8 + i â j, (i,j) 3 C1 â2Î˛0 A(2) log = +B (1) + A(1) Q2 Âľ2R Î˛1 â 2Î˛02 log 2 40 ! + A(3) â 2B (1) B (2) + 2B (2) Î˛0 + 2 B (1) Q2 Âľ2R !! + A(1) Î˛02 log Q2 Âľ2R ! â Î˛1 log 2 Q2 Âľ2R Q2 Âľ2R Î˛0 log ! !! Î˛ â 10B (1) Îś3 fi fj + A(2) 2fj C (1) â fi + fj P (1) â fi log 3 0 ! 2 Q 1 â 4Î˛02 fj C (1) â fi â 2Î˛02 fj P (1) â fi log â P (2) â fi P (1) â fj 2 2 ÂľR Q2 Âľ2F !! 1 â C (1) â P (1) â fi P (1) â fj â C (1) â fi P (1) â P (1) â fj 2 ! 2 3 (1) Q 1 â P â P (1) â fi P (1) â fj log â 2B (2) fj P (1) â fi â fj P (1) â P (2) â fi 2 4 4 ÂľF 1 1 â fj P (2) â P (1) â fi â fj C (1) â P (1) â P (1) â fi 4 2 ! 1 Q2 (1) (1) (1) â fj P â P â P â fi log 4 Âľ2F !! 2 Q2 (1) (1) (1) + B â2fj C â fi â fj P â fi log Âľ2F + 3Î˛0 C (1) â fi P (1) 1 â fj + Î˛0 P (1) â fi P (1) â fj 2 log Q2 Âľ2F ! + 2 log 1 + Î˛0 fj P (2) â fi + 3fj C (1) â P (1) â fi + Î˛0 fj P (1) â P (1) â fi log 2 ! Q2 (1) (1) + Î˛0 fj P â P â fi log Âľ2R 209 Q2 Âľ2R ! Q2 Âľ2F !! + B (1) +Î˛0 fj P (1) +4Î˛0 fj âfj â2C (1) â fi log P (1) P (1) â fi P (1) Q2 Âľ2F â P (1) â fi log â fi P (1) â fj log Q2 Âľ2F ! ! Q2 Âľ2R â fi log â fj â P (1) ! â fj P (2) â fi â 2fj C (1) â P (1) â fi Q2 Âľ2F ! +6Î˛0 fj C (1) â fi + Î˛0 fj P (1) â fi log Q2 Q2 Âľ2 ! F ! + 4 log Q2 !!! Âľ2R Q2 1 (1) 1 P â fi P (1) â fj log2 + fj P (1) â P (1) â fi log2 2 4 4 ÂľF Âľ2F ! ! 1 Q2 Q2 (1) 2 (1) (1) + Î˛0 fj P â fi log + C â fi P â fj log 2 Âľ2F Âľ2F ! 1 Q2 + fj P (2) â fi log 2 Âľ2F ! ! ! 2 2 2 Q Q Q +fj C (1) â P (1) â fi log â 2Î˛0 fj P (1) â fi log log 2 ÂľF Âľ2F Âľ2 !! R Q2 +C (1) â fi C (1) â fj + 2fj C (2) â fi â 4Î˛0 fj C (1) â fi log Âľ2R 2 + Î˛1 fj P (1) â fi â 10 A(1) Îś3 fj P (1) â fi + A(1) + i â j, (i,j) 3 C0 = 2C (1) â fj fi B (2) + P (1) â fj log ! fi B (2) + 2C (1) â fi fj B (2) ! 2 3 Q + P (1) â fi log fj B (2) + P (1) â fj P (1) â P (1) â fi log2 8 Âľ2F ! ! 2 2 3 (1) Q Q + P â fi P (1) â P (1) â fj log2 + P (1) â fj log2 fi Î˛02 8 Âľ2F Âľ2R ! ! Q2 Q2 (1) 2 (1) 2 + 4C â fj log fi Î˛0 + P â fi log fj Î˛02 2 2 ÂľR ÂľR Q2 Âľ2F ! Q2 Âľ2F 210 ! + 4C (1) â fi log Q2 Âľ2R ! fj Î˛02 + C (2) â fj P (1) â fi + C (2) â fi P (1) â fj 1 1 + C (1) â fj P (2) â fi + C (1) â fi P (2) â fj 2 2 + C (1) â fj C (1) â P (1) â fi + C (1) â fi C (1) â P (1) â fj ! ! 2 2 1 (1) Q 1 Q + P â fj P (2) â fi log + P (1) â fi P (2) â fj log 2 2 Âľ2F Âľ2F ! ! 2 2 Q Q + P (1) â fj C (1) â P (1) â fi log + P (1) â fi C (1) â P (1) â fj log Âľ2F Âľ2F ! ! 2 2 1 (1) Q 1 Q + C â fj P (1) â P (1) â fi log + C (1) â fi P (1) â P (1) â fj log 2 2 Âľ2F Âľ2F ! 2 1 (1) Q + P â P (1) â P (1) â fj log2 fi â 2Î˛1 C (1) â fj fi 8 Âľ2F 1 1 + P (3) â fj fi + C (1) â P (2) â fj fi + C (2) â P (1) â fj fi 4 2 ! 2 1 (1) Q 1 + P â P (2) â fj log fi + P (2) â P (1) â fj log 2 4 4 ÂľF ! 2 1 (1) Q + C â P (1) â P (1) â fj log fi â Î˛1 P (1) â fj log 2 Âľ2F ! 2 1 (1) Q + P â P (1) â P (1) â fi log2 fj â 2Î˛1 C (1) â fi fj 8 Âľ2F Q2 Âľ2F ! Q2 Âľ2R fi ! fi 1 1 + P (3) â fi fj + C (1) â P (2) â fi fj 4 2 ! 1 (1) Q2 1 (2) (1) (2) + C â P â fi fj + P â P â fi log fj + P (2) â P (1) â fi log 2 4 4 ÂľF ! ! 2 2 1 Q Q + C (1) â P (1) â P (1) â fi log fj â Î˛1 P (1) â fi log fj 2 2 ÂľF Âľ2R ! 2 1 Q + P (1) â fi P (1) â fj log2 Î˛0 â 4C (1) â fi C (1) â fj Î˛0 2 2 ÂľF ! ! 2 2 Q Q â C (1) â fj P (1) â fi log Î˛0 â C (1) â fi P (1) â fj log Î˛0 2 ÂľF Âľ2F ! ! 2 2 Q Q â 2C (1) â fj P (1) â fi log Î˛0 â 2C (1) â fi P (1) â fj log Î˛0 2 ÂľR Âľ2R 211 Q2 Âľ2F ! fj ! ! Q2 Q2 â fi â fj log log Î˛0 Âľ2F Âľ2R ! ! 1 (1) Q2 Q2 (1) 2 (2) (1) (1) + P â P â fj log fi Î˛0 â 4C â fj fi Î˛0 â C â P â fj log fi Î˛0 4 Âľ2F Âľ2F ! ! Q2 Q2 (2) (1) (1) â P â fj log fi Î˛0 â 2C â P â fj log fi Î˛0 Âľ2R Âľ2R ! ! ! Q2 Q2 1 (1) Q2 (1) (1) (1) 2 â P â P â fj log log fi Î˛0 + P â P â fi log f j Î˛0 4 Âľ2F Âľ2R Âľ2F ! ! 2 2 Q Q â 4C (2) â fi fj Î˛0 â C (1) â P (1) â fi log fj Î˛0 â P (2) â fi log f j Î˛0 2 ÂľF Âľ2R ! ! ! 2 2 2 Q Q Q â 2C (1) â P (1) â fi log fj Î˛0 â P (1) â P (1) â fi log log f j Î˛0 2 2 ÂľR ÂľF Âľ2R ! ! 1 2 2 Q 1 Q + B (1) P (1) â fi P (1) â fj log2 + P (1) â P (1) â fj fi log2 2 2 4 ÂľF Âľ2F ! ! 2 1 Q2 1 (1) 2 Q + P (1) â P (1) â fi fj log2 + P â f f Î˛ log j i 0 4 2 Âľ2F Âľ2F ! ! 1 Q2 1 (2) Q2 + P (1) â fi fj Î˛0 log2 + P â f f log j i 2 2 Âľ2F Âľ2F ! ! 2 2 Q Q +C (1) â fj P (1) â fi log + C (1) â fi P (1) â fj log 2 ÂľF Âľ2F ! ! 2 2 Q Q +C (1) â P (1) â fj fi log + C (1) â P (1) â fi fj log 2 Âľ Âľ2F ! F 1 Q2 + P (2) â fi fj log + 2C (1) â fi C (1) â fj 2 2 ÂľF ! ! ! ! 2 2 2 2 Q Q Q Q â2P (1) â fj log fi Î˛0 log â 2P (1) â fi log fj Î˛0 log 2 2 ÂľR ÂľF Âľ2R Âľ2F ! ! ! 2 2 Q Q +2C (2) â fj fi + 2C (2) â fi fj â 4C (1) â fj log fi Î˛0 â 4C (1) â fi log fj Î˛0 2 ÂľR Âľ2R ! 2 2 Q + A(1) 4C (1) â fj Îś(3)fi + 2P (1) â fj log Îś(3)fi + 4C (1) â fi fj Îś(3) 2 ÂľF ! ! Q2 +2P (1) â fi log fj Îś(3) Âľ2F â 2P (1) P (1) 212 28 f Î˛ Îś(3)P (1) â fj 3 i 0 + B (1) â8P (1) â fj Îś(3)fi â 8P (1) â fi fj Îś(3) â 2P (1) â P (1) â fj fi Îś(3) 28 (1) (1) (1) â2P â P â fi fj Îś(3) + P â fi fj Î˛0 Îś(3) 3 ! Q2 + fi fj â4 log Î˛0 B (2) + 2B (3) 2 ÂľR ! !! 2 2 2 Q Q (1) 2 2 (1) (1) + B 2 log Î˛0 â 2Î˛1 log + A â8Îś(3) B Âľ2R Âľ2R 3 56 (1) 2 + Î˛0 Îś(3) B â 8Î˛0 Îś(3) â 12 A(1) Îś(5) + 8A(1)+(2) Îś(3) 3 ! ! 2 2 Q â8 A(1) log Î˛0 Îś(3) , Âľ2R + A(1) â4P (1) â fi Îś(3)P (1) â fj + Similarly, the singular piece to O Îąs3 can be calculated in a similar method. However, C (3) has not yet been calculated, and involves a complex three-loop calculation. Therefore, it is not possible to obtain the Îąs3 cross-section at this point, and there is no need for the singular piece to be calculated. Eikonal Approximation Eikonalization in the Collinear Limit Given a collinear gluon of momentum l connecting two fermions of momentum p1 and p2 , as shown in Fig. F.3, the matrix element can be given as: p/1 â /l p/2 â /l M = vĚ (p1 ) ÎłÎą ÂˇÂˇÂˇ Îł Îą u (p2 ) , (p1 â l)2 (p2 â l)2 213 (F.43) p1 (p1 â l) (p2 â l) p2 l Figure F.3: Connection of a soft/collinear gluon between two colored particles. where the dots represent any allowed processes between the two fermions. If l is collinear to p2 , then p/2 â /l Îł Îą u (p2 ) = (p2 â l)+ Îł â Îł Îą u (p2 ) , (F.44) where the coordinate system being used is the light-cone coordinates, and the +/â refers to the plus or minus component of the momentum defined as: pÂą = E Âą pz . Consider the Îł matrices in the above equation. If Îą = â, then Îł â u (p2 ) = 0 from the on-shell condition. If Îą = T , then Îł â Îł T = âÎł T Îł â from the anti-commutation relations, and therefore also gives zero. Hence, the only non-vanishing term is for Îą = +, giving: Îł â p/1 â /l (p2 â l)+ Îł â Îł + M = vĚ (p1 ) Âˇ Âˇ Âˇ u (p2 ) . (p1 â l)2 (p2 â l)2 (F.45) Using the on-shell condition and commutation relation, the part before the dots can be simplified to: vĚ (p1 ) 2pâ 1 (p1 â l)2 , (F.46) + Îą â T T and furthermore, if p2 = p+ , 0, 0 , and pÎą 1 = p1 nĚ + p1 nÎą + p1 n , where nĚÂľ = (1, 0, 0), nÂľ = (0, 1, 0), and nT = (0, 0, 1), then the equation can be shown to factorize, giving: vĚ (p1 ) nÎą . (ân Âˇ l) + i 214 (F.47) Eikonalization in the Soft Limit Similar to the collinear limit, the process can be factorized if the gluon connecting the particles is soft. Again, starting from Eq. F.43, taking the limit l â 0, gives: p/2 â /l Îą pÎą 2 Îł u (p2 ) â . (âp2 Âˇ l) + i (p2 â l)2 (F.48) Which can be calculated by using the on-shell condition, and properties of the Îł matrices. It is important to note, that in calculations the sign of the term is important to keep straight, and must always be positive when using this approach. Again, the final result factorizes the left-hand side of Fig. F.3 from the right-hand side. Fourier Transform Details The Fourier Transform in d â 2 dimensions from transverse momentum space to impact parameter space is given by: Z ~ eikT Âˇb 2â2 d kT = (kT2 )Îą 2 +Îąâ1 b Î (1 â â Îą) ĎÎą . 4Ď Î (Îą) (F.49) To obtain the Fourier Transform for functions of the form 12 logn kT2 , derivatives with respect kT to Îą are taken of the previous equation, and then Îą is taken to one. The derivation of the above equation can be found below, and requires the use of the following identity: (kT2 )âÎą Z â 2 1 âxkT = xÎąâ1 e dx, Î (Îą) 0 215 (F.50) using this identity, the first equation is derived as follows: Z ~ ~ eikT Âˇb 2â2 1 d kT = 2 Îą Î (Îą) (kT ) Z ~ eikT Âˇb d2â2 k Z â Z â T 2 âxkT dx xÎąâ1 e 0 Z â 2 1 ik b cos(Î¸)âxkT Îąâ1 = x dx dâŚ kT2â2â1 e T dkT Î (Îą) 0 0 Z â Z â Z 2 1 ik b cos(Î¸)âxkT 2â2â1 Îąâ1 = x dx kT dkT dâŚ2â2 sinâ2 Î¸e T Î (Îą) 0 0 Z â Z â 2 1 2Ď 1â 1 2 2 âxkT 2â2â1 Îąâ1 = x dx kT dkT 0 F1 1 â ; â b kT e Î (Îą) 0 Î (1 â ) 4 0 2 +Îąâ1 b Î (1 â â Îą) = ĎÎą , 4Ď Î (Îą) Z where 0 F1 1 â ; â 14 b2 kT2 is the confluent hypergeometric function. Using this relationship, we can easily derive the Fourier Transform of all the terms needed to check the asymptotic contribution up through O Îąs3 . Here are a list of the results for completeness: 2 b =Ď Î (â) (F.51) 4Ď 2 b b2 (1) I = âĎ Î (â) ÎłE + log â Ď0 (â) (F.52) 4Ď 4 ! 2 2 2 2 b b Ď I (2) = Ď Î (â) ÎłE + log â Ď0 (â) â + Ď1 (â) (F.53) 4Ď 4 6 2 3 b b2 Ď2 b2 (3) I = âĎ Î (â) ÎłE + log â Ď0 (â) â ÎłE + log â Ď0 (â) 4Ď 4 2 4 b2 +3Ď1 (â) ÎłE + log â Ď0 (â) â 2 (Ď2 (1) + Ď2 (â)) (F.54) 4 I (0) 216 2 4 2 b b2 b2 2 =Ď Î (â) ÎłE + log â Ď0 (â) â Ď ÎłE + log â Ď0 (â) 4Ď 4 4 2 b2 Ď4 2 âĎ Ď1 (â) + 6Ď1 (â) ÎłE + log â Ď0 (â) + + Ď3 (â) 4 60 b2 â4 (Ď2 (1) + Ď2 (â)) ÎłE + log â Ď0 (â) (F.55) 4 2 5 3 b b2 5Ď 2 b2 (5) I =Ď Î (â) ÎłE + log â Ď0 (â) â ÎłE + log â Ď0 (â) 4Ď 4 3 4 3 b2 b2 2 +10Ď1 (â) ÎłE + log â Ď0 (â) â 10Ď ÎłE + log â Ď0 (â) Ď1 (â) 4 4 ! 2 b2 Ď2 â10 (Ď2 (1) + Ď2 (â)) ÎłE + log â Ď0 (â) + Ď1 (â) â 4 6 2 2 Ď b2 +15 + Ď1 (â) ÎłE + log â Ď0 (â) 6 4 b2 b2 â30Îś (4) ÎłE + log â Ď0 (â) + 5 ÎłE + log â Ď0 (â) Ď3 (â) 4 4 â Ď4 (1) + Ď4 (â) (F.56) I (4) R d2â2 qt ~ where I (n) = logn qt2 eâiq~t Âˇb , and Ďn is the nth polygamma function defined as: 2 q t Ďn (z) = dn+1 log (Î (z)) dz n+1 217 (F.57) APPENDIX G Higgs Plus Jet Resummation Calculation Details The poles for the quark-gluon channel are given for the virtual, renormalization, jet, collinear, PDF renormalization, and soft corrections as: 24 â2 2 âu ât s Î˛ + CA â ln 2 â ln 2 â ln 2 2Ď 0 2 Âľ Âľ Âľ 4 2 âu +CF â 2 â â2 ln 2 + 3 , Âľ Îąs 18 â H (0) 2Î˛ , 2Ď 0 !! 2 R2 P Îą C 2 2 3 s F H (0) + â ln J 2 , 2Ď 2 2 Âľ Îąs H (0) Îąs â2 CF Î´ (Îž2 â 1) Îž1 Pqq (Îž1 ) + CA Î´ (Îž1 â 1) Îž2 Pgg (Îž2 ) , 2Ď Îąs 2CF 3 2CA (0) H Pqq (Îž1 ) + Î´ (xi1 â 1) + Pgg (Îž2 ) + 2Î˛0 Î´ (xi2 â 1) , 2Ď 2 Îąs 1 1 Âľ2 2 t 1 (0) H 2 (CA + CF ) 2 + ln â (CA â CF ) ln + CF ln 2 , 2Ď s u R H (0) 218 (G.1) (G.2) (G.3) (G.4) (G.5) (G.6) ârespectively. Similarly, for the quark-anti-quark channel, the poles are in the same order as above, and are given as: 1 â2 2 sÂľ2 â4 2 s 24Î˛0 + CA â ln + CF â â2 ln 2 + 3 , 2Ď ut 2 2 Âľ Îąs â6CA (0) H 2Î˛0 , 2Ď !! 2 P Îąs 1 1 1 H (0) 2CA 2 + 2Î˛0 â ln T2 + ln 2 , 2Ď Âľ R Îąs H (0) Îąs â2 CF Î´ (Îž2 â 1) Îž1 Pqq (x) + Îž1 â Îž2 , 2Ď Îąs 2CF 3 (0) H Pqq (Îž1 ) Îž1 + Î´ (Îž1 â 1) + Îž1 â Îž2 , 2Ď 2 2 Îąs 2 2 Âľ 1 1 (0) H 2 CF + ln â CA ln 2 . 2Ď s 2 R H (0) (G.7) (G.8) (G.9) (G.10) (G.11) (G.12) In both of the above set of equations, all the terms are proportional to Î´(Îž1 â 1)Î´(Îž2 â 1) unless otherwise noted. Looking at the above equations, all of the poles cancel, leaving a finite result, and validating the results of the calculations. 219 BIBLIOGRAPHY 220 BIBLIOGRAPHY [1] M. Baak, M. Goebel, J. Haller, A. Hoecker, D. Kennedy, R. Kogler, K. MoĚnig, M. Schott, and J. Stelzer. The electroweak fit of the standard model after the discovery of a new boson at the lhc. 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