A color matrix element corrected parton shower and multiplet color bases

Transcription

1 A color matrix element corrected parton shower and multiplet color bases First results from an SU(3), rather than SU( ) parton shower In collaboration with Simon Plätzer (DESY), arxiv: ColorFull our treatment of the color structure Minimal orthogonal multiplet based color bases for treating the SU(3) color space In collaboration with Stefan Keppeler (Tübingen), soon on the arxiv DESY June, 202 Malin Sjödahl

2 Why SU(3) parton showers? Wisdom from LEP is that parton showers seem to do well with the leading N c approximation At LHC much more energy is available many more colored partons many more squared color suppressed terms Often two quark-lines importance of terms suppressed by /N c rather than /N 2 c should grow Also useful for exact NLO matching Malin Sjödahl 2

3 Basics of our shower Built on the Catani-Seymour dipole factorization (S.Plätzer & S. Gieseke, JHEP 0, 024 (20) & ) Parton ĩj splitting to partons i and j, and parton k absorbs the longitudinal recoil such that all partons remain on shell M n+ (..., p i,..., p j,..., p k,...) 2 M n (..., pĩj,..., 2p i p p k,...) V ij,k (p i, p j, p k ) M n (..., pĩj,..., p k,...) j k i,j In a standard parton shower parton ĩj and k would have to be color connected, V ij,k = 8πα s V ij,k Tĩj T k T 2 ĩj 8πα s V ij,k + δĩj δ(ĩj, k color connected) we keep all pairs (δĩj = for gluon, 0 else, T 2 ĩj is a convention) Malin Sjödahl 3

4 For the emission probability this means that: dp ij,k (p 2, z) = V ij,k (p 2, z) dφ n+(p 2, z) dφ n rather than T 2 ĩj M n Tĩj T k M n M n 2 dp ij,k (p 2, z) = V ij,k (p 2, z) dφ n+(p 2, z) dφ n δ(ĩj, k color connected) + δĩj The splitting kernels read: ( ) 2( z) V qg,k (p i, p j, p k ) = C F ( z) 2 + p 2 /s ( + z) ijk ( z z V gg,k (p i, p j, p k ) = 2C A ( z) 2 + p 2 /s + ijk z 2 + p 2 /s ijk ) 2 + z( z) Malin Sjödahl 4

5 Challenges Three major new challenges Evolution with amplitude information (next) Negative contributions to radiation probability, negative splitting kernels, treated using interleaved veto/competition algorithm (S. Plätzer & M. Sjodahl, EPJ Plus 27 (202) 26) Keeping track of the color structure for an arbitrary number of partons (more towards the end) Malin Sjödahl 5

6 Evolution with amplitude information Assume we have a basis for the color space M n = d n α= c n,α α n M n = (c n,,..., c n,dn ) T (this basis need not be orthogonal) M n is known for the hard process How do we get M n+ after emission? Malin Sjödahl 6

7 Observe that M n 2 = M ns n M n = Tr ( S n M n M ) n (where S n is the color scalar product matrix) and ( ) M n Tĩj T k M n = Tr S n+ T k,n M n M nt ĩj,n Use an amplitude matrix M n = M n M n as basic object M n+ = i =j k i,j 4πα s p i p j V ij,k (p i, p j, p k ) T 2 ĩj T k,n M n T ĩj,n where M hard = M hard M hard Malin Sjödahl 7

8 A proof of concept: Our current implementation e + e jets, a LEP-like setting Fixed α s = 0.2 Up to 6 gluons, only gluon emission, g qq is suppressed anyway, and there is no non-trivial color structure No hadronization, we don t want to spoil our N c = 3 parton shower by attaching an N c hadronization model. Also, comparing showers in a fair way, would require retuning the hadronized N c = 3 shower No virtual corrections, i.e. no color rearrangement without radiation, no Coulomb gluons Malin Sjödahl 8

9 Three different treatments of color space Full, exact SU(3) treatment, all color correlations Shower, resembles standard showers, C F for gluon emission off quarks is exact but non-trivial color suppressed terms are dropped Strict large-n c, all N c suppressed terms dropped, C F = 4/3 3/2 (T R = /2) Malin Sjödahl 9

10 Results: Number of emissions First, simply consider the number of emissions event fraction 0. number of emissions full shower strict large-n c 0.0 x/full DipoleShower + ColorFull n emissions... this is not an observable, but it is a genuine uncertainty on the number of emissions in the perturbative part of a parton shower Malin Sjödahl 0

11 Results: Thrust For standard observables small effects, here thrust T = max n N dn/dτ Thrust, τ = T full shower strict large-n c i p i n i p i x/full DipoleShower + ColorFull τ Malin Sjödahl

12 Results: Angular distribution Cosine of angle between third and fourth jet N dn/d cos α Angle between softest jets full shower strict large-n c x/full DipoleShower + ColorFull cos α 34 Malin Sjödahl 2

13 Results: Some tailored observables For tailored observables we find larger differences average transverse momentum w.r.t. n 3 average rapidity w.r.t. n 3 GeV N dn/d p full shower strict large-n c N dn/d y 0. full shower strict large-n c 0.00 x/full DipoleShower + ColorFull x/full DipoleShower + ColorFull 0 p /GeV Average transverse momentum and rapidity of softer particles with respect to the thrust axis defined by the three hardest partons y Malin Sjödahl 3

14 Parton shower conclusions For standard observables we find small deviations for LEP, of order a few percent Leading N c was probably a very good approximation for standard observables at LEP For tailored observables we find larger differences 20% Keeping C F to its N c = 3 value (4/3) (as is done in standard showers), rather than 3/2, tends to improve the approximation (T R = /2) At the LHC we have many more colored particles, so (many more) 2 possible color suppressed interference terms For full evolution we should include color rearranging virtual corrections, they do have the same IR singularity structure Malin Sjödahl 4

15 The color space For given external particles, the color space is a finite dimensional vector space equipped with a scalar product A, B = A a,b,c,... (B a,b,c,... ) Example: If a,b,c,... A = (t g ) a b(t g ) c d = a b c d, then A A = a,b,c,d,g,f (tg ) a b(t g ) c d(t h ) b a(t h ) d c We may use any basis Malin Sjödahl 5

16 A basis for the color space In general an amplitude can be written as linear combination of different color structures, like A + B +... This is the kind of trace bases used in our current parton shower, and most NLO calculations Malin Sjödahl 6

17 It has some nice properties The effect of gluon emission is easily described: g g g g g g g g g g g g g g g > = (Z. Nagy & D. Soper, JHEP 0807 (2008) 025) So is the effect of gluon exchange: g g 2 g 3 g 4 g g 2 g 3 g 4 g g 2 g 3 g 4 g 2 g 3 g g 4 = T R ( + ) Convention: + when inserting after, - when inserting before (M. Sjödahl, JHEP 0909 (2009) 087 JHEP) Malin Sjödahl 7

18 ColorFull For the purpose of treating a general QCD color structure I have written a C++ color algebra code, ColorFull, which: automatically creates a trace basis for any number and kind of partons, and to any order in α s describes the effect of gluon emission... and gluon exchange squares color amplitudes can be used with boost for optimized calculations is planned to be published separately Malin Sjödahl 8

19 But... this type of basis is non-orthogonal and overcomplete (for more than a few partons)... and the number of basis vectors grows as a factorial in N g when squaring amplitudes we run into a factorial square scaling Hard to go beyond qq + 7 gluons Malin Sjödahl 9

20 But... this type of basis is non-orthogonal and overcomplete (for more than a few partons)... and the number of basis vectors grows as a factorial in N g when squaring amplitudes we run into a factorial square scaling Hard to go beyond qq + 7 gluons Would be nice with minimal orthogonal basis Malin Sjödahl 9

21 Minimal orthogonal bases for the color spaces In collaboration with Stefan Keppeler Want orthogonal minimal basis for color space Basis vectors can be enumerated using Young tableaux multiplication = (0) and constructed if projection operators are known The problem is the construction of the corresponding projection operators; the Young-tableaux operate with quark-units but the physical particles include anti-quarks and gluons Malin Sjödahl 20

22 New idea: Iteratively build up gluon projection operators using quark and anti-quark projection operators and project out already known projection operators We have found a general strategy for constructing gluon projection operators for n g n g gluons! This can be used for constructing orthogonal bases for up to 2n g + gluons or qq-pairs! We have explicitly constructed the 5 3g 3g projection operators for any N c... and the 6 gluon orthonormal bases... and orthonormal bases for all other 6 parton cases Bases can easily be made minimal by crossing out states that are disallowed for N c = 3 Malin Sjödahl 2

23 Number of projection operators and basis vectors Number of projection operators and basis vectors for N g N g gluons without imposing projection operators and vectors to appear in charge conjugation invariant combinations Case Projectors N c = 3 Projectors N c = Vectors N c = 3 Vectors N c = 2g 2g g 3g g 4g g 5g Malin Sjödahl 22

24 Conclusions for the color space We have outlined a general recipe for construction of minimal orthogonal multiplet based bases for any QCD process On the way we found an N c -independent labeling of the multiplets in g n g, and a one to one, or one to zero, correspondence between these for various N c... and an N c independent way of obtaining SU(N c ) Clebsch-Gordan matrices Number of basis vectors grows only exponentially for N c = 3 This has the potential to very significantly speed up exact calculations in the color space of SU(N c ) Malin Sjödahl 23

25 ...and outlook However, in order to use this in an optimized way, we need to understand how to sort QCD amplitudes in this basis in an efficient way...also, a lot of implementational work remains Malin Sjödahl 24

26 ...and outlook However, in order to use this in an optimized way, we need to understand how to sort QCD amplitudes in this basis in an efficient way...also, a lot of implementational work remains Thank you for your attention Malin Sjödahl 24

27 Backup: The Sudakov decomposition In each splitting a parton ĩj splits into i and j whereas a spectator k takes up the longitudinal recoil p i = zpĩj + p2 zs ijk p k + k () p 2 p j = ( z)pĩj + p k k (2) ( z)s ijk ( p 2 ) p k = p k, (3) z( z)s ijk with p 2 ĩj = p2 k = 0, a space like transverse momentum k with k 2 = p2 and k pĩj = k p k = 0. With this parametrization we also have s ijk = (p i + p j + p k ) 2 = (pĩj + p k) 2. Malin Sjödahl 25

28 N dn/dτ Backup: Thrust For standard observables small effects, here thrust T = max n Thrust, τ = T full shower strict large-n c N dn/dτ i p i n p i i Thrust, τ = T full correlations, n = 6 n = 5 n = DipoleShower + ColorFull DipoleShower + ColorFull x/full x/n = τ τ Malin Sjödahl 26

29 Backup: Importance of g qq splitting average transverse momentum w.r.t. n 3 δcollinear (%) average rapidity w.r.t. n y δcollinear (%) p /GeV Influence on average transverse momentum and rapidity w.r.t. the thrust axis defined by the three hardest patrons Malin Sjödahl 27

30 Backup: Jet separation differential Durham two-jet rate differential Durham five-jet rate N dn/dy full shower strict large-n c N dn/dy full shower strict large-n c 00 DipoleShower + ColorFull DipoleShower + ColorFull x/full x/full y 23 Jet separation between 2nd and 3rd jet, and 5th and 6th jet y = 2 min(e 2 i, E2 j )( cos θ ij)/s y 56 Malin Sjödahl 28

31 Backup: N c suppressed terms That non-leading color terms are suppressed by /N 2 c, is guaranteed only for same order α s diagrams with only gluons ( t Hooft 973) 2 = = 2 = 2 = C F = 2 2 C F N = 2 2 N N 2N ~ N 2 * = = = 2 2N = 2N C F N = 2 2 N 2N ~ N Malin Sjödahl 29

32 Backup: N c suppressed terms For a parton shower there may also be terms which only are suppressed by one power of N c * = = = Was 0 before emission, now ~N c 2 Was ~N c before emission, now ~N c 2 did not enter shower in any form, genuine "shower" contribution "Included" in showers, contribution from hard process The leading N c contribution scales as N 2 c before emission and N 3 c after Malin Sjödahl 30

33 Backup: A dipole shower in the trace basis A dipole shower can easily be thought of in the language of the N c limit of the trace basis i Coherent from i,j j k Coherent from j,k No coherent emision from i,k Also, it is easy to see that in this limit only color neighbors radiate, i.e. only neighboring partons on the quark-lines in the basis radiate trace basis well suited for comparing to parton showers Malin Sjödahl 3

34 Backup: Gluon exchange A gluon exchange in this basis directly i.e. without using scalar products gives back a linear combination of (at most 4) basis tensors = 2 2 Fierz = + canceling N c suppressed terms Fierz = _ 2 _ 2 + canceling N c suppressed terms = N_ c 2 0 N c -enhancement possible only for near by partons only color neighbors radiate in the N c limit Malin Sjödahl 32

35 Backup: The size of the vector space and the trace basis For general N c the trace type bases size grows as a factorial N vec [n q, N g ] = N vec [n q, N g ](N g + n q ) + N vec [n q, N g 2](N g ) where N vec [n q, 0] = n q! N vec [n q, ] = n q n q! The size of the vector spaces for finite N c asymptotically grows as an exponential in the number of gluons/qq-pairs. Malin Sjödahl 33

36 Backup: Some example projectors P 8a,8a g g 2 g 3 g 4 g 5 g 6 = T 2 R P 8s,27 g g 2 g 3 g 4 g 5 g 6 = T R 4N 2 c if g g 2 i if i g 3 i 2 if g4 g 5 i 3 if i3 g 6 i 2 N c 2(N 2 c 4) d g g 2 i P 27 i g 3 i 2 g 6 d i2 g 4 g 5 P 27,8 g g 2 g 3 g 4 g 5 g 6 = 4(N c + ) N 2 c (N c + 3) P27 g g 2 i g 3 P 27 i g 6 g 4 g 5 P 27,64=cc g g 2 g 3 g 4 g 5 g 6 = TR 3 Tg 27,64 g 2 g 3 g 4 g 5 g 6 N2 c N c 2 8N c (N c + 2) P27,27s g g 2 g 3 g 4 g 5 g 6 N 2 c 62(N c + )(N c + 2) P27,8 g g 2 g 3 g 4 g 5 g 6 Malin Sjödahl 34

37 Backup: Three gluon multiplets SU(3) dim Multiplet c0c0 cc cc2 c2c cc c2c2 ((45) 8s 6) 2 ((45) 8s 6) 8s or a ((45) 8s 6) 0 ((45) 8s 6) 0 ((45) 8s 6) 27 ((45) 8s 6) 0 ((45) 8a 6) 2 ((45) 8a 6) 8s or a ((45) 8a 6) 0 ((45) 8a 6) 0 ((45) 8a 6) 27 ((45) 8a 6) 0 ((45) 0 6) 8 ((45) 0 6) 0 ((45) 0 6) 0 ((45) 0 6) 27 ((45) 0 6) 0 ((45) 0 6) 8 ((45) 0 6) 0 ((45) 0 6) 0 ((45) 0 6) 27 ((45) 0 6) 0 ((45) 27 6) 8 ((45) 27 6) 0 ((45) 27 6) 0 ((45) 27 6) 27 ((45) 0 6) 0 ((45) 0 6) 8 ((45) 0 6) 0 ((45) 0 6) 0 ((45) 27 6) 27 ((45) 0 6) 0 SU(3) dim Multiplet cc cc2 c2c c2c2 ((45) 27 6) 64 ((45) 0 6) 35 ((45) 0 6) 35 ((45) 0 6) c2c2 ((45) 27 6) 35 ((45) 27 6) 35 ((45) 0 6) c2c2 ((45) 27 6) c2c2 ((45) 0 6) c2c2 SU(3) dim Multiplet cc3 c3c c2c3 c3c2 c3c3 ((45) 0 6) cc3 ((45) 0 6) c3c ((45) 0 6) c2c3 ((45) 0 6) c3c2 ((45) 0 6) c3c3 ((45) 0 6) c2c3 ((45) 0 6) c3c2 Malin Sjödahl 35

38 Backup: SU(N c ) multiplets We find that the irreducible spaces in g n g for varying N c stand in a one to one, or one to zero correspondence to each other! (For each SU(3) multiplet there is an SU(5) version, but not vice versa) Every multiplet in g n g can be labeled in an N c -independent way using the lengths of the columns. For example Nc- Nc- Nc Nc- Nc- Nc-2 Nc- Nc- 2 Nc- Nc- Nc-2 2 = I have not seen this anywhere else.. have you? Malin Sjödahl 36