A NLO Calculation of pqcd: Total Cross Section of P P W + + X. C. P. Yuan. Michigan State University CTEQ Summer School, June 2002

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1 A NLO Calculation of pqcd: Total Cross Section of P P W X C. P. Yuan Michigan State University CTEQ Summer School, June Outline. Parton Model Born Cross Section. Factorization Theorem How to organize a NLO calculation of pqcd 3. Feynman rules and Feynman diagrams Cut diagram notation 4. Immediate Problems Singularities Dimensional Regularization 5. Virtual Corrections 6. Real Emission Contribuation 7. Perturbative Parton Distribution Functions 8. Summary of NLO [Oα s ] Corrections

2 Appendices: A. γ-matrices in n dimensions B. Some integrals and special functions C. Angular integrals in n dimensions D. Two-particle phase space in n dimensions E. Explicit Calculations Typesetting: prepared by Qing-Hong Cao at MSU. A few references can be found in Handbook of pqcd on CTEQ website cteq/

3 Parton Model hp q W σ hh W X = h P X hp = p = x P p = x P q h P σ hh W X = f,f =q, q dx dx { φf/h x ˆσ ff φ f /h x x x } Partonic Born Cross Section of ff W The probablility of finding a parton f with fraction x of the hadron h momentum

4 ˆσ q q = ŝ where [ M = Born Cross Section d 3 q π 3 q π 4 δ 4 p p q M 3 3 }{{} spin color average color and spin Or, im = v p ig w γ µ γ 5 u p Cut-diagram notation p p q ] * P = P = p p ν μ p p ig w p fl ν P L ig w p fl μ P L P L fl 5 = g p W Tr[6 p fl μ P L 6 p fl ν P L ] g μν qμ q ν M TrI 3 3 Doesn't contribute for m q =, due to Ward identity Color

5 Tr[ p γ µ P L p γ µ P L ] = Tr[ p γ µ p γ µ P L ] P L P L = P L = γ 5 = Tr[ p p P L ] γ µ p γ µ = p = 4 p p Tr p p =4p p =ŝ Tr p p γ 5 = Tr[I 3 3 ] =3 ŝ p p = q and p = p = = gw ŝ3 =3g w ŝ d 3 q δ 4 p p q = q d 4 qδ 4 p p q δ q M = δ q M where M is the mass of W -boson. Thus, ˆσ q q = ŝ π δ ŝ M = π g w δ ŝ M = π ŝ g wδ ˆτ 3 g wŝ ˆτ = M /ŝ, ŝ = x x S for S = P P and P = P =

6 Factorization Theorem σ hh = i,j dx dx φ i/h x, Q H ij Q x x S φ j/h x,q Nonperturbative, but universal, hence, measurable IRS, Calculable in pqcd Procedure: Compute σ kl in pqcd with k, l partons not h, h hadron σ kl = i,j dx dx φ i/k x,q Q H ij x x S Compute φ i/k,φ j/l in pqcd 3 Extract H ij in pqcd φ j/l x,q H ij IRS H ij indepent of k, l same H ij with k h, l h 4 Use H ij in the above equation with φ i/h,φ j/h

7 Extracting H ij in pqcd Expansions in α s : σ kl = H ij = n= n= αs π αs π n σ n kl n H n ij φ i/k x = δ ik δ x n= αs π n φ n i/k α s = g 4π φ i/k α s = Parton k stays itself Consequences: H = ff ij ij = Born" suppress "^" from now on H ij = ff ij " ff il ffi l=j ffi ff k=i kj # Computed from Feynman diagrams process dependent Computed from the definition of perturbative parton distribution function process independent, scheme dependent

8 Feynman Rules Quark Propagator p i, α j, β Gluon Propagator i pm βα p m iɛ δ ij i,j=,,3 Take m= in our calculation k ν, a µ, b i g µν k iɛ δ ab Quark-W Vertex a,b=,...,8 W µ i, α j, β i gw γ µ βα γ 5 δ ij g w = e sin θ w, weak coupling Quark-Gluon Vertex G c, µ i, α ig t c ji γ µ βα j, β t c is the SUN N N generator Quark Color Generators Tr [t a,t b ] = if abc t c c c t c = C F I N N C F = N N = 4 3 t c =NC F, N =3

9 Feynman Diagrams Born level α s u _ d qq Born W * NLO: α s virtual corrections qq virt NLO: α s real emission diagrams qq real NLO: α s real emission diagrams qg real NLO: α s real emission diagrams Gq real

10 In Cut-diagram notation qq Born qq virt Re qq real qg real Gq real Same as qg real after replacing q by q.

11 Feynman rules for cut-diagrams quark line p gluon line i; ff j; fi ßffi p m 6 p m fiff ffi ij ffip m p k ν, a µ, b πδ k g µν δ ab W -boson line q ν µ πδ q M g µν q µq ν M Doesn t contribute for m f = because of Ward identity ν µ i g W γ ν γ 5 i g W γ µ γ 5 p p i pm p m iɛ i pm p m iɛ

12 Immediate problems Singularities Ultraviolet singularity UV k d 4 k k k k k k Infrared singularities IR k p as k µ soft divergence or k µ p µ collinear divergence p k m = for m =or p k m p k as k or k µ p µ for m = k for m Similar singularities also exist in virtual diagrams. Solutions Compute H ij in pqcd in n =4 ε dimensions dimensional regularization n 4 UV & IR divergences appear as ε poles in σ ij Feynman diagram calculation H ij is IR safe no ε in H ij H ij is UV safe after renormalization.

13 Dimensional Regularization Revisit the Born Cross Section in n dimensions ˆσ q q = ŝ d n q π n q π n δ n p p q m m = 3 3 In n-dim, the polarization degree of freedom is for a quark, and n- for a gluon. Using the Naive-γ 5 prescription: Tr[ p γ µ P L p γ µ P L ] = Tr[ p γ µ p γ µ P L ] γ µ p γ µ = ε p = ε Tr[ p p P L ] = ε 4 p p = ε ŝ In n dimensions ˆσ q q = π ŝ g w ε δ ˆτ σ δ ˆτ

14 Strong Coupling g in n dimensions In n dimensions d n xl d n x { ψi ψ 4 G µνg µν gt a ψγ µ G µ ψ } The dimension in mass unit µ [ψ] µ n [G] µ n [ ψgψ ] µ n n = µ 3n Since [ g ψgψ ] µ n,so [g] µ n n =4 ε = µ ε g has a dimension in mass when ε Feynman rules should read g gµ ε

15 Calculations Tools needed for a NLO calculation are collected in Appendices A-D The detailed calculation for each subprocess can be found in Appendices E In the following, I shall summarize the result for each subprocess

16 Virtual Corrections qq virt in Feynman Gauge = ε IR and ε UV poles cancel when ε UV = ε IR ε ε UV ε IR cancel Electroweak coupling is not renormalized by QCD interactions at one-loop order Ward identity, a renormalizable theory poles remain σ virt is free of ultraviolent singularity. σ virt = α σ s 4πµ ε Γ ε δ ˆτ π M Γ ε } ε 3ε { 8π C F 3 ε : soft and collinear singularities 3 : soft or collinear singularities ε C F : color factor σ π ŝ g w ε

17 Real Emission Contribution qq real ε Collinear ε Soft and Collinear σ real q q = σ α s π { 4πµ ε δ ˆτ ε M ε Γ ε Γ ε C F ˆτ 4 ˆτ ln ˆτ ˆτ ˆτ ˆτ ˆτ ln ˆτ } Note: [ ] is a distribution, [ ] dz f z = z dz f z f, which is finite. z In the soft limit, ˆτ ˆτ = M, σ real q q σ α s π { 4πµ ε δ ˆτ 4 ε ŝ M ε Γ ε Γ ε C F ˆτ 8 ln ˆτ ˆτ }

18 q q virt q q real at NLO σ q q = σ virt q q σ real q q = σ α s π { ε 4πµ M ε Γ ε Γ ε C F ˆτ ˆτ π 3 8 δ ˆτ ˆτ ˆτ ln ˆτ 4 ˆτ ln ˆτ ˆτ } Where we have used [ ˆτ 3 ] δ ˆτ ε ˆτ = ε ˆτ ˆτ All the soft singularities, ε ε cancel in σ q q KLN theorem Kinoshita-Lee-Navenberg σ q q term finite terms ε Collinear Singularity

19 Factorization Theorem Perturbative PDF φ i/k = δ ikδ x α s π φ i/k can be calculated from the definition of PDF. Process independent,but factorization scheme dependent φ i/k σ kl = H ij k l φ j/l i j H kl = σ kl φ i/k σ kl = H ij k l φ j/l i j k φ i/k H ij l φ j/l i j k φ i/k H ij l φ j/l i j Factorization scheme dependent H kl = σ kl [ φ i/k H il H kj φ ] j/l Finite Divergent

20 Perturbative PDF In MS-scheme modified minimal subtraction φ q/q φ q/g z = φ q/ q z = φ q/g z = ε z = ε 4πe γ E ε P q q z 4πe γ E ε P q g z q where the splitting kernel for φ p q q q zp is P q q z z = C F z z = C F 3 δ z z, and for φ p g q g zp q is P q g z = T R z z, where C F = 4 3 and T R =. Note:The Pole part in the MS scheme is ˆε = ε 4πe γ E ε = ε ln4π γ E In the MS scheme, the pole part is just ε

21 Find H qq in the MS scheme Take off the factor α s π { [ ] σ q q = σ P M q q ˆτ ln µ ε γ E ln 4π C F [ [ H q q ˆτ = σ q q φ where =ˆσ [ C F ˆτ ˆτ ln ˆτ τ ln ˆτ ˆτ { pqcd prediction σ hh = { f=q, q f=q, q f=q, q ] q qσ q q P q q ˆτ ln M µ ˆτ ˆτ ln ˆτ τ ln ˆτ ˆτ ˆτ = M ŝ = M x x S, σ =ˆσ ε, ˆσ = π ŝ g w = πg w. S x x dx dx φ f/h x,µ [ σ δ ˆτ ] φ f/h x,µ dx dx φ f/h x,µ [ α s µ dx dx φ f/h x,µ [ α s µ π π H f f ˆτ ] ]} π 3 4 δ ˆτ ]} π 3 4 δ ˆτ φ f/h x,µ H fg ˆτ ] φ G/h x,µ x x }

22 Find H qg in the MS scheme Take off the factor α s π { σ qg = σ 4 P q g ˆτ [ ε 3ˆτ ˆτ 7 } Γ ε Γ ε lnm ˆτ 4πµ ˆτ ] H qg [ ] ˆτ = σ qg σ q q φ q G { [ = ˆσ P q g ˆτ ln 4 3 ˆτ 7 4ˆτ } M ln µ ˆτ ˆτ ] Similarly, [ ] H G q = σ G q φ q G σ q q = H qg Note: If we choose the renormalization scale µ = M, M then ln = µ

23 W and Z production * CDF and D would like to use their W and Z cross sections for luminosity determination * D cross sections close to center of PDF prediction ellipse; not the case with CDF MSU study PDF uncertainty ellipse T= CDF D CDF rescaled to D lum definition J. Pumplin, D. Stump, R. Brock, D. Casey, J. Huston, J. Kalk, H.L. Lai, W.K. Tung: hep-ph/5

24 Summary φ f/h x, µ depends on scheme MS,DIS,... H ij scheme dependent Evolution equations allow us to perdict q dependent of φx, q Essentially identical procedure for hh jets, inclusive QQ,... But, when the Born level process involves strong interaction eg. qq tt, it is also necessary to renormalize the strong coupling α s, etc, to eliminate ultraviolate singularities

25 Appendix A γ-matrices in n dimensions Dirac algebra {γ µ,γ ν } γ µ γ ν γ ν γ µ =g µν µ, ν =,,,n g µν = diag,,, g µν g µν = n { γ µ,γ 5} = Naive-γ 5 prescription This works in calculating the inclusive rate of W -boson, but fails in the differential distributions of the leptons from the W -boson decay. Matrix identities γ µ aγ µ = ε a γ µ a bγ µ =4a b ε a b γ µ a b cγ µ = c b a ε a b c n =4 ε Traces Tr[ a b] =4a b Tr[ a b c d] =4{a bc d a cb d a db c} Tr[γ 5 a b] =

26 Appendix B Some integrals and special functions The Gamma function Γ z = Γ z = Γ z z Poles in Γz dxx z e x Re z > for all z z Γ n = n! Γ ε = ε γ E ε γ E Γ π 6 - = π - γ E =.577, Euler constant Γ ε = εγ ε =εγ E π ε Γ ε Γ ε =ε π 6 O ε 4 z ε = e ln zε = e ε ln z =ε ln z The Beta function B α, β = = π B α, β = dy y α y β = Γ α Γ β Γ α β dθ sin θ α cos θ β 6 γ E O ε 3 dy y α y α β

27 Feynman trick ab = dx [ax b x] Γ α β = a α bβ Γ α Γ β dx xα x β [ax b x] αβ n-dimension integrals l µ d n l l M α = d n l µ l ν l l M α = l g µν d n n l l M α d n l π n l M α = i α Γ α n n α 4π n/ Γ α M l d n l l M α = d n l l M M l M α Re [ ε ] = ε π O ε 4

28 plus distribution to isolate ε poles Consider z ε [ = z ε δ z = [ z ε because ] z ε dz z ε ] δ z ε cancel dz = for ε z ε ε δ z ε [ ] ln z z because [ ] is a distribution O ε δ z ε = ε z ε z = εln z = [ ] dz f z z dz f z z dz f z δ z dz f z f, which is finite. z dz z

29 Appendix C Angular integrals in n dimensions In n dimensions d n x = r n dω n dω n = dω = dω = π π π π dθ n sin n θ n dθ n sin n θ n dφ Ω =π dθ sin θ dω Ω =4π π π dθ sin θ dφ. dω n = π Ω n = n π n n Γ Γ n = dθ n sin n θ n n π Γ n dω n from repeated use of B α, β because Γ n = n Γ n Γ n Γ

30 Appendix D Two-particle phase space in n dimensions d n k d n q dk dq = PS p π n k π n π n δ n p q k q with k µ = k, k, etc. Use dn q = d n qδ q Q,weget q PS p d n k dk dq = PS p π n δ p k Q k dk k n 3 = π n dω n δ ŝ k ŝ Q p ŝ, k =,k = k ŝ, Use n =4 ε,theninthec.m. frame p µ =, dk dq = Ω n 3 π ε Use new variables: PS p dk k ε 4 ŝ π z = Q ŝ,y = cosθ k = dk dq = 8π ε 4π z ε z ε Q Γ ε dθ sin θ k ε δ ŝ Q ŝ ŝ z, dy [y y] ε

31 Appendix E Explicit Calculations Consider k = = p p-k d n k γ µ p k γ µ π n k iɛ p k iɛ d n k π n d n l π n [ n = = i 6π d n l π n n p k dx [k k xp] l k xp n[ x p l] dx [l iɛ] ] d n l p π n [l iɛ] }{{ } ε IR = Because there is no mass scale Due to cancellation of ε UV and ε IR Trick:A = A B B [ ] [ ] l l Λ }{{ l } Λ }{{ } IR div. UV div. i n 4=εIR, 6π 4 n =ε UV ε UV

32 consider the real emission process g p 3 q g p p q q p p 3 p p k W q W Define the Mandelstam variables ŝ = p p = p p ˆt = p p 3 = p p 3 û = p p 3 = p p 3 After averaging over colors and spins M = ε } {{ } Spin 3 8 Trt a t a } {{ } Color gµ ε g w [ ε ε ŝ ˆt ˆt ŝ ûm ˆtŝ ε ] Note: The d.o.f. of gluon polarization is ε, and that of quark polarization is.

33 In the parton c.m. frame, the constituent cross section ˆσ = ŝ M PS = ŝ { { 4 6 g s µ ε gw ε [ 8π ε 4π M ŝ ˆt ˆt ŝ ûm ˆtŝ ε ε Γ εˆτ ε ˆτ ε ]} dy [y y] ε } where Using and y cosθ ˆt = ŝ M y ŝ û = ŝ M y ŝ dy y α y β = Γ α Γ β, Γ α β p p 3 we get ˆσ qg =ˆσ α s 4π { P q g ˆτ [ ε 3 ˆτ 3 } ˆτ, Γ ε Γ ε lnm ˆτ 4πµ ˆτ ] with P q g ˆτ = [ˆτ ˆτ ] ˆσ π g w ŝ