Renormalization in the Unitarity Method

Transcription

1 CEA Saclay NBI Summer Institute: Strings, Gauge Theory and the LHC August 23, 2011

2 One-loop amplitudes Events / 10 GeV tth(120) ttjj ttbb (QCD) ttbb (EW) m bb (GeV) Higgs Signal With Background (top quark associated production) NLO (Next-to-Leading-Order) Calculations provide some essential precision reduce scale dependence

3 Prioritized Wish List, Next-to-Leading Order [Les Houches Physics at TeV colliders 2007, NLO multileg working group: Summary report; updated 2009] Done [a] p p t t b b background for t th Done [b] p p t t + 2 jets relevant for t th p p V V b b relevant for benchmark processes Done [c] p p V V + 2 jets VBF H VV Done [d] p p V + 3 jets new physics Done [e] p p b b b b Higgs and new physics p p t t t t new physics p p W b b j new physics [a]: Bredenstein, Denner, Dittmaier, Pozzorini; Bevilacqua, Czakon, Papadopoulos, Pittau, Worek [b]: Bevilacqua, Czakon, Papadopoulos, Worek [c]: Jager, Melia, Melnikov, Nason, Rontsch, Zanderighi [d]: Berger, Bern, Dixon, Febres Cordero, Forde, Gleisberg, Ita, Kosower, Maître; Ellis, Melnikov, Zanderighi [e]: Greiner, Guffanti, Reiter, Reuter

4 2001 Experimenter s Wish List [Knuteson, Campbell] single boson diboson triboson heavy flavor W + 5j WW + 5j WWW + 3j t t + 3j Wb b + 3j WWb b + 3j WWWb b + 3j t tγ + 2j Wc c + 3j WWc c + 3j WWW γγ + 3j t tw + 2j Z + 5j ZZ + 5j Zγγ + 3j t tz + 2j Zb b + 3j ZZb b + 3j WZZ + 3j t th + 2j Zc c + 3j ZZc c + 3j ZZZ + 3j t b + 2j γ + 5j γγ + 5j b b + 3j γb b + 3j γγb b + 3j b bt t γc c + 3j γγc c + 3j WZ + 5j WZb b + 3j WZc c + 3j W γ + 3j Zγ + 3j

5 Numerical Programs for Virtual Corrections BlackHat [ Berger, Bern, Diana, Dixon, Febres Cordero, Forde, Gleisberg, Höche, Ita, Kosower, Maître, Ozeren ] CutTools/HELAC-NLO [Ossola, Papadopoulos, Pittau; Actis, Bevilacqua, Czakon, Draggiotis, Garzelli, van Hameren, Worek,...] GOLEM [ Binoth, Guillet, Heinrich, Pilon, Reiter, Schubert ] MadLoop [Hirschi, Frederix, Frixione, Garzelli, Maltonin, Pittau] NGluon [Badger, Biedermann, Uwer] Rocket [Ellis, Giele, Kunszt, Melnikov, Zanderighi] SAMURAI [Mastrolia, Ossola, Reiter, Tramontano]

6 Analytic one-loop amplitudes Completed recently: pp Higgs + 2 jets. [Badger, Berger, Campbell, Del Duca, Dixon, Ellis, Giele, Glover, Mastrolia, Risager, Sofianatos, Williams, Zanderighi] pp t t [Badger, Sattler, Yudin] 0 dūq Q ll, W -mediated. [Badger, Campbell, Ellis] gg WW νllν including SM Higgs [Campbell, Ellis, Williams] Analytic techniques may have advantages in stability and speed, extend readily to larger numbers of external particles, and have numerical counterparts.

7 Outline 1. Structure of one-loop amplitudes, and the unitarity method 2. When cuts diverge: a prescription for treating the wavefunction renormalization of external massive fermions [work with E. Mirabella, to appear soon]

8 One-Loop Amplitudes In 4-dimensional massless theories, reduction of Feynman integrals brings the one-loop amplitude to the form A = i d i (box) + i c i (triangle) + i b i (bubble) + rational where the master integrals have scalar structure and are known explicitly. [in dim. reg.: Bern, Dixon, Kosower] K2 K3 K1 K4 box triangle bubble 1 loop A c + c + c

9 One-Loop Amplitudes In D = 4 2ɛ dimensions, and allowing for internal masses, the result of reduction is A = i e i (pentagon) + i d i (box) + i c i (triangle) + i b i (bubble) + i a i (tadpole) K 3 K2 K 3 K2 K 2 K 4 K 1 K 5 K1 K4 K 1 K 3 K K 0 m 2

10 Singularities in One-Loop Amplitudes As kinematic functions, one-loop amplitudes have poles, branch cuts, and divergences. These singularities constrain the amplitude enough to bypass a Feynman diagram expansion. This is the principle of the unitarity method and its numerical cousins.

11 Amplitudes from unitarity cuts Tree level input. A 1 loop = c i I i c + c + c Matching 4-dimensional cuts can suffice to determine reduction coefficients! Logarithms with unique arguments. Cut-constructibility [Bern, Dixon, Dunbar, Kosower] But: we get several coefficients together in the same equation. How do we evaluate a unitarity cut?

12 Unitarity Cuts: Loops from Trees A 1 loop = dµ A tree Left Atree Right where dµ = d 4 l 1 d 4 l 2 δ (4) (l 1 + l 2 K) δ(l 2 1) δ(l 2 2) l l 1 By unitarity, this is the discontinuity of the amplitude across a branch cut. [Cutkosky] K

13 Cut integrals The cut integrand is a rational function of the loop momentum. A 1 loop = dµ A Left (l) A Right (l) dµ = d 4 l δ + (l 2 ) δ + ((l K) 2 ) Change to homogeneous (CP 1 ) spinor variables with Integration measure: d 4 l δ + (l 2 ) ( ) = [Cachazo, Svrček, Witten] l aȧ = t λ a λȧ. 0 dt t λ= λ λ dλ [ λ d λ] ( )

14 Spinor variables From Lorentz 4-vector to spinor indices with Pauli matrices: p aȧ = σ µ aȧ p µ a, ȧ = 1, 2 For a null vector (massless particle): 0 = p 2 = det(p aȧ ) = p aȧ = λ a λȧ. Lorentz-invariant spinor products: λ λ ɛ ab λ a λ b [ λ λ ] ɛȧḃ λȧ λ ḃ

15 Spinor integration [Anastasiou, RB, Buchbinder, Cachazo, Feng, Kunszt, Mastrolia] Change variables, l = tλ λ, and use the spinor measure, d 4 l δ(l 2 )δ((l K) 2 ) = dt t λ dλ [ λ d λ]δ((tλ λ K) 2 ) Use 2nd delta function to perform t-integral. Each term of integrand takes the form: (K 2 ) n+1 λ K λ] n+2 Evaluate with residue theorem. n+k i=1 λ R i λ] k j=1 λ Q j λ] Identify cuts of basis integrals and read off coefficients. We have given formulas for the resulting coefficients.

16 Cuts of Master Integrals I 2 = K 2 λ dλ [ λ d λ] λ K λ] 2 I 3 = λ dλ [ λ d λ] 1 λ K λ] λ Q 1 λ] I 4 = λ dλ [ λ d λ] 1 K 2 1 λ Q 1 λ] λ Q 2 λ] Q j K j + K j 2 K 2 K

17 Formulas for coefficients From a general cut integrand T (N) (l). N is the degree. Original integral: i(4π) D 2 d D l (2π) D δ(+) (l 2 ) δ (+) ((K l) 2 )T (N) (l), To define the target master integrals: D i (l) = (l K 2 i )

18 Box coefficients C[K r, K s, K] = 1 2 ( ) T (N) (l)d r (l)d s (l) + {P sr,1 P sr,2 } λ Psr,1, λ Psr,2 ( Qs Q r + sr P sr,1 = Q s + Q 2 r ( Qs Q r sr P sr,2 = Q s + Q 2 r sr = (Q s Q r ) 2 Q 2 s Q 2 r. ) Q r, ) Q r,

19 Triangle coefficients C[K s, K] = 1 2(N + 1)! s N+1 Ps,1 P s,2 N+1 d N+1 dτ N+1 +{P s,1 P s,2 } ( T (N) (l)d s (l) λ K λ] N+1 λ Qs λ,λ P s,1 τp s,2 ) τ 0 ( Qs K + s P s,1 = Q s + K 2 ( Qs K s P s,2 = Q s + K 2 s = (Q s K) 2 Q 2 s K 2. ) K, ) K,

20 Bubble coefficient B (0) N,m (s) C[K] = K 2 d N dτ N N X q=0 ( 1) q q! d q ds q B (0) N,N q (s) + kx NX r=1 a=q B (r;a q;1) N,N a (2η K) m+1 λ K λ] N N![η η K η] N (m + 1)(K 2 ) m+1 λ η N+1 T (N) (l) (s) B (r;a q;2) N,N a (s)!, s=0 λ (K+sη) λ λ (K τη ) η 1 A, τ 0 (r;b;1) n,m (s) (r;b;2) n,m (s) ( 1) b+1 b!(m + 1) r b+1 Pr,1 P r,2 b d b λ η P r,1 ] m+1 λ Q r η λ b λ K λ] N+1 dτ b λ K P r,1 ] m+1 λ ηk λ n+1 ( 1) b+1 b!(m + 1) r b+1 Pr,1 P r,2 b d b λ η P r,2 ] m+1 λ Q r η λ b λ K λ] N+1 dτ b λ K P r,2 ] m+1 λ ηk λ n+1 T (N) (l)d r (l)! λ (K+sη)λ, λ P r,1 τp r,2 τ=0 T (N) (l)d r (l)! λ (K+sη)λ, λ P r,2 τp r,1 τ=0

21 Unitarity in D = 4 2ɛ dimensions Orthogonal decomposition, keeping external momenta in 4 dimensions. [Bern, Chalmers, Mahlon, Morgan] where l 2 2ɛ = K 2 4 u. d 4 2ɛ l 4 2ɛ = (4π)ɛ Γ( ɛ) 1 0 du u 1 ɛ d 4 l 4. The integral over u will remain. The u-dependence is controlled: A = 1 0 du u 1 ɛ d 4 l δ(l 2 ) δ( 1 u K 2 2K l) Recognize and perform the 4-d integral as before. (Cf. methods by Ossola, Papadopoulos, Pittau; Forde; Ellis, Giele, Kunszt; Kilgore; Giele, Kunszt, Melnikov; Badger)

22 Massive particles Cut amplitude: 1 ( ) du u 1 ɛ λ dλ [ λ d λ] [K 2, M1 2, M2 2 ] (K 2 ) n+1 n+k j=1 λ R j λ] 0 K 2 λ K λ] n+2 k i=1 λ Q i λ] The formalism/formulas for integral coefficients have the same form.. Integral coefficients are polynomials in u. New master integrals.

23 The special massive master integrals m 2 m m m 2 m 2 These integrals have cuts only in the m 2 channel. 2 2ɛ Γ(1 + ɛ) I 1 = m ɛ(ɛ 1) I 2 (0; m 2, m 2 2ɛ Γ(1 + ɛ) ) = m ɛ I 2 (m 2 ; 0, m 2 2ɛ Γ(1 + ɛ) ) = m ɛ(1 2ɛ)

24 Tadpole approaches 1. Universal divergent behavior Examples: 4-gluon amplitude with a massive fermion loop [Bern, Morgan]. t tgg, using Mitov-Moch small-mass factorization result [Badger]. 2. Add an auxiliary propagator [RB, Feng] 3. Generalized unitarity: single cuts [RB, Mirabella] Other single cut studies: [Catani, Gleisberg, Krauss, Rodrigo, Winter; Glover, Williams; Caron-Huot]

25 Divergent cuts Try to apply (generalized) unitary cuts to the special massive master integrals Single cut of tadpole, and cut of massless on-shell bubble diverge, due to internal on-shell propagators Single cut can be regularized specially for the tadpole. Double cut of bubble involves adding the counterterms explicitly.

26 Masses, fermions and unitarity [EGKM] [Ellis, Giele, Kunszt, Melnikov (2008)] Isolate and remove the divergent diagrams Implicit use of counterterm Feynman-diagram decomposition is gauge dependent Embedded in a numerical algorithm

27 Our proposal [RB, Mirabella (to appear)] Use an off-shell continuation of the fermion mass. The cut is finite until we take the on-shell limit. Power series expansion in the off-shell parameter In the on-shell limit, divergences are guaranteed to cancel: keep only finite terms Explicit use of counterterms. Gauge dependence enters only in tree level currents. Clean analytic results

28 Off-shell continuation Momentum-conserving shift: k ˆk = k + ξ k, r ˆr = r ξ k. Propagator of interest: k 2 m 2 ξ(2k k) Can choose k = r for some null external momentum r, so it stays on shell: ˆr 2 = r 2 = 0. Cut diverges as 1/ξ.

29 Reduction of the shifted divergent diagram A L = 1 ( ) ˆk 2 m 2 ūˆk l /ε l (m + /ˆk) Jˆ, A 3 = ū k /ε l uˆk l For internal helicity sum, use Feynman gauge as in EGKM: ( εν) λ = gµν λ=± ε λ µ

30 Reduction of the shifted divergent diagram Simple reduction of linear bubble gives A 3 A L 1 ξ(2k k) (4m2 ū k ˆ J + 2ξm ū k /k ˆ J ) B 0 (ˆk 2 ). Also expand off-shell current and scalar bubble to 1st order: Result: 1 4m 2 ū k J B 0 ξ (2k k) ˆ J = J + ξj B 0 (ˆk 2 ) = B 0 (m 2 ) + ξ(2k k)b 0(m 2 ) + 4m2 (2k k) ū k J B 0 + 4m2 ū k J B 0 + 2m ū k /kj B0 (2k k).

31 Reduction: tadpole part Similar, except gluon polarization sum propagator, ig µν /l 2. Result: [( 2 ξ(2k k) 1 ) m 2 ū k J + 1 (2k k)m ūk / k J + ] 2 J (2k k)ūk A 0.

32 Counterterm 1 ξ(2k k)ūk ) (/kδz ψ + ξ/ kδzψ mδz ψ mδz m (/k + ξ/ k + m) Ĵ Renormalization constants in on-shell scheme: δz m = A 0 m 2 + 2B 0, δz ψ = A 0 m 2 4m2 B 0. Verify total cancellation of divergent diagram.

33 Small examples with Feynman Diagrams H b b 3 loop diagrams + 2 counterterm diagrams q q t t 12 loop diagrams + 2 counterterm diagrams 1. Implemented momentum shift 2. Computed bubble and tadpole coefficients from unitarity cut 3. Checked cancellation of divergences against counterterm and agreement of finite result with Passarino-Veltman reduction.

34 The fermion-channel cut in the spinor-helicity formalism The spinor-helicity convention for the polarization vectors requires axial gauge: ε (p) = 2 p [q + q] p, ε + (p) = 2 [qp] p] q + q [p. qp The completeness relation is λ=± ( ) ε λ µ(p) ε λ ν(p) = gµν + p µq ν + q µ p ν p q Cf. Feynman gauge, where the 2nd term is absent. Specific gauge choice = choice of q for each p..

35 Additional counterterm in axial gauge, for spinors The double cut gets an extra O(ξ 0 ) contribution: ( ) ( 1 ū k /l uˆk l ūˆk l /q (m + ˆ/k) ) Jˆ ξ(2k k) dµ 2,k q l ( ) ( ū k /q uˆk l ūˆk l / l (m + ˆ/k) ) Jˆ +. q l Second term vanishes by Ward identity with cut gluon. First term is cancelled by a new (non-divergent) counterterm: M k = 1 [ ] ū k ( /k m) 2k k ˆ/k/q δz k (ˆ/k + m) J ˆ, δz k = B 0 q k

36 Example from t t gg amplitude Full analytic result computed previously by other methods. [Körner, Merabashvili; Badger, Sattler, Yundin] Color decomposition 3-point tree 5-point tree for m 2 on-shell bubbles Checked cancellation of divergence and evaluated on-shell bubble coefficient, for equal-helicity gluons

37 On-shell recursion for off-shell currents Finite parts of the counterterm need off-shell Ĵ explicitly, not the gauge-invariant A L A R. The current Ĵ depends on gauge choices of external gluons these cancel among counterterms. Generate Ĵ by BCFW-type relations, starting purely from 3-point vertices with the polarizations ε (p) = 2 p [q + q] p, ε + (p) = 2 [qp] p] q + q [p. qp BCFW shifts available for any pair of massless quarks/gluons.

38 On-shell recursion for off-shell currents Example: J = ˆ4 [q 4 + q 4 ] ˆ4 [q 4 4] (p 2 p 3 + m) 2p 2 p 3 3 [4 + 4] 3 [ 4ˆ3 ] u 2 = p [q 4 2] + m q 4 ] 43 [42] + (m 2 p ) q 4] 32] 2p 2 p 3 [34] [q 4 4]

39 Summary Unitarity method at one loop constructs amplitudes from branch cuts, using the fact of the expansion in known master integrals. Residue theorem gives analytic formulas for coefficients of most master integrals. Massive particles require an expanded set of master integrals. Challenge for the unitarity method. Tadpole techniques still merit further study; On-shell bubbles via unitarity cuts are available from an off-shell continuation and 1-term Taylor expansion. Must check efficiency.