ColorFull and other tools for color space

Transcription

1 ColorFull and other tools for color space Motivation Trace type bases ColorFull Orthogonal bases ColorMath Conclusions and outlook Manchester October 28, 203 Malin Sjödahl

2 Motivation With the LHC follows an increased demand of accurately calculated processes in QCD This is applicable to NLO calculations and resummation...but my perspective is from a parton shower point of view First SU(3) parton shower in collaboration with Simon Plätzer JHEP 07(202)042, arxiv: color structure treated using my C++ ColorFull code Malin Sjödahl 2

3 Dealing with color space We never observe individual colors we are only interested in color summed/averaged quantities For given external partons, 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 = g (t g ) a b(t g ) c d = a b g c d, then A A = a,b,c,d,g,h (th ) b a(t h ) d c(t g ) a b(t g ) c d Malin Sjödahl 3

4 One way of dealing with color space is to just square the amplitudes as one encounters them Alternatively, we may use any basis (spanning set) Malin Sjödahl 4

5 The standard treatment: Trace bases Every 4g vertex can be replaced by 3g vertices: a, α b, β = + + c,γ d, δ ig 2 s(g αδ g βγ g αγ g βδ ) ig 2 s(g αβ g γδ g αδ g βγ ) ig 2 s(g αβ g γδ g αγ g βδ ) (read counter clockwise) Every 3g vertex can be replaced using: b a if abc = T R ( c ) After this every internal gluon can be removed using: = T R T R Nc Malin Sjödahl 5

6 This can be applied to any QCD amplitude, tree level or beyond In general an amplitude can be written as linear combination of different color structures, like A + B +... For example for 2 (incoming + outgoing) gluons and one qq pair = A + A 2 + A 3 (an incoming quark is the same as an outgoing anti-quark) Malin Sjödahl 6

7 The above type of color structures can be used as a spanning set, a trace basis. (Technically it s in general overcomplete, so it is rather a spanning set.) These bases have some nice properties The effect of gluon emission is easily described: = Convention: + when inserting after, minus when inserting before. 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 Malin Sjödahl 7

8 ColorFull For the purpose of treating a general QCD color structure I have written a C++ color algebra code, ColorFull, based on this type or trace basis. ColorFull is used in the color shower with Simon Plätzer, JHEP 07(202)042, arxiv: ColorFull is interfaced to Herwig++ ( 2.7) via Matchbox, and is now publicly available in a pre-release version (0.90) at colorfull.hepforge.org. Malin Sjödahl 8

9 ColorFull can automatically create a trace basis for any number and kind of partons, and to arbitrary order in α s. For example we may create a basis for qq-pair, 3 gluons and 0 loops (in pure QCD): Trace basis MyBasis(,3,0); MyBasis.write out Col basis to cout(); MyBasis.write out Col basis("colorresults/mybasis"); This results in a basis with basis vectors: 0 [{,3,4,5,2}] [{,3,5,4,2}] 2 [{,4,3,5,2}] 3 [{,4,5,3,2}] 4 [{,5,3,4,2}] 5 [{,5,4,3,2}] g 3 g 4 g 5 Here [{,3,4,5,2}]=(t g3 t g4 t g5 ) q q2= q q 2 Malin Sjödahl 9

10 Having a basis one probably wants to calculate the scalar product matrix MyBasis.scalar product matrix(); The Polynomial and numerical scalar product matrices are then calculated and saved in MyBasis.P smp and MyBasis.d spm The result is Mathematica readable: { { Nc CFˆ(3), *- TR CFˆ(2), *- TR CFˆ(2), TRˆ(2) Ncˆ(-) CF, TRˆ(2) Ncˆ(-) CF,( TRˆ(2) Nc CF + TRˆ(2) Ncˆ(-) CF)}, { *- TR CFˆ(2), Nc CFˆ(3), TRˆ(2) Ncˆ(-) CF,( TRˆ(2) Nc CF + TRˆ(2) Ncˆ(-) CF), *- TR CFˆ(2), TRˆ(2) Ncˆ(-) CF}, { *- TR CFˆ(2), TRˆ(2) Ncˆ(-) CF, Nc CFˆ(3), *- TR CFˆ(2),( TRˆ(2) Nc CF + TRˆ(2) Ncˆ(-) CF), TRˆ(2) Ncˆ(-) CF}, { TRˆ(2) Ncˆ(-) CF,( TRˆ(2) Nc CF + TRˆ(2) Ncˆ(-) CF), *- TR CFˆ(2), Nc CFˆ(3), TRˆ(2) Ncˆ(-) CF, *- TR CFˆ(2)}, { TRˆ(2) Ncˆ(-) CF, *- TR CFˆ(2),( TRˆ(2) Nc CF + TRˆ(2) Ncˆ(-) CF), TRˆ(2) Ncˆ(-) CF, Nc CFˆ(3), *- TR CFˆ(2)}, {( TRˆ(2) Nc CF + TRˆ(2) Ncˆ(-) CF), TRˆ(2) Ncˆ(-) CF, TRˆ(2) Ncˆ(-) CF, *- TR CFˆ(2), *- TR CFˆ(2), Nc CFˆ(3)} } Malin Sjödahl 0

11 ColorFull can also square any color amplitudes and calculate any interference, Col amp Ca("[{,3,2}(4,5)]"); // (t g3 ) q q2tr(t g4 t g5 ) Col amp Ca2("[{,3,4,5,2}]"); // (t g3 t g4 t g5 ) q q2 Col functions Col fun; Col fun.scalar product(ca,ca); Col fun.scalar product(ca,ca2); giving the results TR Nc^2 CF^2 and TR Nc CF^2, respectively. Malin Sjödahl

12 In the context of parton showers we like to be able to describe the effect of gluon emission: Col fun.emit gluon(ca,3,77); (recall Ca=[{,3,2}(4,5)] ) resulting in: [{,3,77,2}(4,5)] + *-[{,77,3,2}(4,5)] Convention +, when inserting before, - when inserting after and gluon exchange: Col fun.exchange gluon(ca,,4); giving: TR[,4,5,3,2] + *- TR[,5,4,3,2] Malin Sjödahl 2

13 For resummation type calculations one may also like to calculate the soft anomalous dimension matrices. To do so, we need the full basis Trace basis MyBasisFull(,3); // Create an all order basis MyBasisFull.Col gamma(,4); // Effect of gluon exchange results in {{ TR Nc, 0, 0, 0, 0, 0, 0, 0, TR, TR, 0}, { 0, 0, 0, 0, 0, *- TR, 0, 0, 0, 0, 0}} {{ 0, 0, 0, 0, 0, 0, 0, 0, *- TR, 0, 0}, { 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, *- TR}, { 0, 0, TR Nc, 0, 0, 0, 0, 0, TR, 0, TR}, { 0, 0, 0, TR Nc, 0, 0, TR, 0, 0, TR, 0}, { 0, 0, 0, 0, 0, 0, 0, 0, 0, *- TR, 0}, { 0, 0, 0, 0, 0, 0, *- TR, 0, 0, 0, 0}, { 0, 0, 0, 0, 0, *- TR, 0, 0, 0, 0, 0}, { 0, TR, 0, 0, TR, 0, 0, TR Nc, 0, 0, 0}, { *- TR, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, { 0, 0, 0, 0, *- TR, 0, 0, 0, 0, 0, 0}, { 0, *- TR, 0, 0, 0, 0, 0, 0, 0, 0, 0}} Malin Sjödahl 3

14 Apart from this ColorFull can also Read in and use orthogonal bases Create tree level gluon bases (using charge conjugation invariance) Be used with arbitrary N c Be used with any T R = Tr(t a t a ) (no sum) Take the strict leading N c limit Take the leading N c limit, but keep C F to it s N c = 3 value (shower like) To learn more download from colorfull.hepforge.org, and see the toy program ColorPlay, or browse the doxygen documentation Malin Sjödahl 4

15 Use case: NLO electroweak Higgs + 3 jet production In collaboration with Francisco Campanario, Terrance Figy and Simon Plätzer, arxiv: , accepted for publication in PRL dσ/dz 3 [fb] 0 3 ystar normed LO NLO dσ/dp,3 [fb/gev] 0 2 Transverse momentum of the third jet 0 0 LO NLO K HJets z 3 K HJets p,3 [GeV] Here, normalized centralized rapidity distribution of third jet w.r.t. two hardest jets (left) and transverse momentum of third jet (right). Malin Sjödahl 5

16 However... This type of basis is non-orthogonal and overcomplete (for more than N c gluons plus qq-pairs)... and the number of spanning vectors grows as a factorial in N g +N qq when squaring amplitudes we run into a factorial square scaling Hard to go beyond 8 gluons plus qq-pairs Would be nice with minimal orthogonal basis Malin Sjödahl 6

17 Minimal orthogonal bases for color spaces In collaboration with Stefan Keppeler Want orthogonal minimal basis for color space Basis vectors can be enumerated using Young tableaux multiplication, for example for gg gg = (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 7

18 One may think that the problem of constructing group theory based multiplet bases should have been solved a long time ago The 2g 2g case was solved in the 60 s (N c = 3) However, until recently only a few cases had been dealt with, those for which (loosely speaking) nothing more complicated than two gluon projection operators is needed About one year ago me and Stefan Keppeler presented a general recipe for constructing gluon projection operators. From these we also know how to construct orthogonal bases for any number and kind of partons, JHEP09(202)24, arxiv: Malin Sjödahl 8

19 For many partons the size of the vector space is much smaller for N c = 3 (exponential), compared to for N c (factorial) Case Vectors N c = 3 Vectors, general case 4 gluons gluons gluons gluons Number of basis vectors for N g N g gluons without imposing vectors to appear in charge conjugation invariant combinations Malin Sjödahl 9

20 Multiplet bases can potentially speed up exact calculations in color space very significantly, as squaring amplitudes is very quick... but before squaring, the amplitudes must be decomposed in the bases (master thesis of Johan Thorén) For resummation and higher order calculations one also wants to quickly compute the effect of gluon exchange (master thesis of Fritiof Persson) It seems that both these issues can be dealt with relatively quickly I m optimistic Malin Sjödahl 20

21 ColorMath Calculations are done using my Mathematica package, ColorMath, Eur. Phys. J. C 73:230 (203), arxiv: ColorMath is an easy to use Mathematica package for color summed calculations in QCD, SU(N c ) Repeated indices are implicitly summed In[2]:= Amplitude If g, g2, g t g, q, q2 Out[2]= t g q q2 f g,g2,g In[3]:= CSimplify Amplitude Conjugate Amplitude. g h Out[3]= 2 Nc Nc 2 TR 2 The package and tutorial can be downloaded from or Malin Sjödahl 2

22 Conclusions and outlook One way of dealing with color space is to use trace bases This method is pursued in my and Simon Plätzer s N c = 3 parton shower (JHEP 07(202)042, arxiv: ) This is also the type of basis ColorFull, colorfull.hepforge.org, is built on Alternatively one may want to use orthogonal bases (JHEP09(202)24, arxiv: ) This may have the potential to very significantly speed up exact calculations in the color space of SU(N c ), but some work still remains... I have also written a Mathematica package ColorMath for color summed calculations of moderate complexity (Eur. Phys. J. C 73:230 (203), arxiv:2.2099) Malin Sjödahl 22

23 Backup: Full program Malin Sjödahl 23

24 Backup: 2 gluon solutions For two gluons, there are two octet projectors, one singlet projector, and 4 new projectors, 0, 0, 27, and for general N c, 0 It turns out that the new projectors can be seen as corresponding to different symmetries w.r.t. quark and anti-quark units, for example the decuplet can be seen as corresponding to P octet(s) (singlet) Similarly the anti-decuplet corresponds to 2 2, the 27-plet corresponds to 2 2 and the 0-plet to 2 2 Malin Sjödahl 24

25 Backup: 2 gluon projectors Problem first solved for two gluons by MacFarlane, Sudbery, and Weisz 968, however only for N c = 3 General N c solution for two gluons by Butera, Cicuta and Enriotti 979 General N c solution for two gluons by Cvitanović, in group theory books, 984 and 2008, using polynomial equations General N c solution for two gluons by Dokshitzer and Marchesini 2006, using symmetries and intelligent guesswork Malin Sjödahl 25

26 Backup: Could this work in general? On the one hand side g g 2... g n (q q ) (q 2 q 2 )... (q n q n ) so there is hope... On the other hand... Why should it? In general there are many instances of a multiplet, how do we know we construct all? Malin Sjödahl 26

27 Backup: Key observation: Starting in a given multiplet, corresponding to some qq symmetries, such as 0, from 2, it turns out that for each 2 way of attaching a quark box to 2 and an anti-quark box to, 2 to there is at most one new multiplet! For example, the projector P 0,35 can be seen as coming from P P after having projected out old multiplets In fact, for large enough N c, there is precisely one new multiplet for each set of qq symmetries Malin Sjödahl 27

28 Backup: 2 gluon projectors P = N 2 c, P 8s = N c 2T R (N 2 c 4), P 8a = 2N c T R, P 0 = 2 P 0 = 2 P 27 = 2 P 0 = 2 + 2T 2 R 2T 2 R + 2T 2 R 2T 2 R 2 P8a 2 P8a N c 2 2N c P 8s N c 2N c P N c +2 2N c P 8s N c + 2N c P Malin Sjödahl 28

29 Backup: Some 3g 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 29

30 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 Multiplets for g 4 g 5 g 6 Malin Sjödahl 30

31 Backup: Construction of 3 gluon projectors We start out by enumerating all projectors in (8 8 2 ) 8 3 Starting in a singlet, the result is trivial = 8 23 If we start in an octet 8 2, is known from before: Nc- Nc- Nc Nc- Nc- Nc-2 Nc- Nc- 2 Nc- Nc- Nc-2 2 = Malin Sjödahl 3

32 The 3g multiplets from (anti-) decuplets Malin Sjödahl 32

33 The 3g multiplets from 27- and 0-plets Malin Sjödahl 33

34 Backup: Projector construction Construct projectors corresponding to old multiplets Construct the tensors which will give rise to new projectors T 0,35 P P From these, project out old multiplets new projectors P 0,35 T 0,35 P m T 0,35 m 0 8 Malin Sjödahl 34

35 Backup: Projecting out old multiplets This would give us a way of constructing all projectors corresponding to new multiplets, if we knew how to project out all old multiplets. In g g 2 g 3, there are many 27-plets. How do we separate the various instance of the same multiplet? By the construction history! P M 2 P M P M ng P M P M We make sure that the n g ν first gluons are in a given multiplet! Then the various instances are orthogonal as, at some point in the construction history, there was a different projector! (More complicated for multiple occurrences...) Malin Sjödahl 35

36 It turns out that the proof of this is really interesting: We find that the irreducible representations 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 column notation elsewhere... have you? Malin Sjödahl 36

37 Backup: Number of projection operators and basis vectors In general, for many partons the size of the vector space is much smaller for N c = 3, compared to for N c Case Projectors N c = 3 Projectors N c = Vectors N c = 3 Vectors N c = 2g 2g g 3g g 4g g 5g 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 Malin Sjödahl 37

38 The size of the vector spaces asymptotically grows as an exponential in the number of gluons/qq-pairs for finite N c For general N c the basis 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! Malin Sjödahl 38

39 Backup: First occurrence n f SU(3) = Young diagrams Examples of SU(3) Young diagrams sorted according to their first occurrence n f. Malin Sjödahl 39

40 Backup: The importance of Hermitian projectors P 6,8 Y = 4 3, P 6,8 = 4 3 P 3,8 Y = 4 3, P 3,8 = 4 3 The standard Young projection operators P 6,8 Y and P3,8 Y compared to their Hermitian versions P 6,8 and P 3,8. Clearly P 6,8 P 3,8 = P 6,8 P 3,8 = 0. However, as can be seen from the symmetries, P 6,8 Y P 3,8 Y 0. Malin Sjödahl 40

41 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 4

42 Backup: Number of emissions First, simply consider the number of emissions for a LEP-like setting 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 42

43 Backup: 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 NOTE: Larger effects expected at LHC τ Malin Sjödahl 43

44 Backup: Angular distribution Cosine of angle between third and fourth jet N dn/dcos α Angle between softest jets full shower strict large-n c x/full DipoleShower + ColorFull NOTE: Larger effects expected at LHC cos α 34 Malin Sjödahl 44

45 Backup: 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 NOTE: Larger effects expected at LHC y Malin Sjödahl 45

46 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 = = T R = T R = T R C F = T R C F N c = T R T R N 2 c N c N c N 2 c = = = T R T R Nc = T R T R Nc C F N c = 0 T R T R N 2 c N c N c Malin Sjödahl 46

47 Backup: /N c -suppressed terms For a parton shower there may also be terms which only are suppressed by one power of N c = = = T R T R Nc Is 0 without emission, with N 2 c did not enter in any form, genuine shower contribution Is N c without emission, with N 2 c included in shower, contribution from hard process The leading N c contribution scales as N 2 c before emission and N 3 c after Malin Sjödahl 47