Heavy Quarks in MRST/MSTW Global Fits

Transcription

1 Heavy Quarks in MRST/MSTW Global Fits Robert Thorne October 6th, 8 University College London Ringberg8-Heavy Flavour

2 Will discuss Charm 1.5GeV, bottom 4.3GeV, and at end strange.3gev as heavy flavours. Two distinct regimes: Near threshold Q m H massive quarks not partons. Created in final state. Described using Fixed Flavour Number Scheme (FFNS). F (x, Q ) = C F F,n f k (Q /m H) f n f k (Q ) Note that n f is effective number of light quarks. Can be 3, 4 or 5. Does not sum αs n lnn Q /m H terms in perturbative expansion. Usually achieved by definition of heavy flavour parton distributions and solution of evolution equations. Additional problem FFNS known up to NLO (Laenen et al), but are not defined at NNLO αs 3 F,3 CF,Hg unknown. Ringberg8-Heavy Flavour 1

3 Variable Flavour High scales Q m H massless partons. Behave like up, down (strange always in this regime. Sum ln(q /m H ) terms via evolution. Zero Mass Variable Flavour Number Scheme (ZM-VFNS). Ignores O(m H /Q ) corrections. F (x, Q ) = C ZM,n f j f n f j (Q ). Partons in different number regions related to each other perturbatively. f n f +1 j (Q ) = A jk (Q /m H) f n f k (Q ), Perturbative matrix elements A jk (Q /m H ) (Buza et al) containing ln(q /m H ) terms relate f n f i (Q ) and f n f +1 (Q ) correct evolution for both. i Want a General-Mass Variable Flavour Number Scheme (VFNS) taking one from the two well-defined limits of Q m H and Q m H. Ringberg8-Heavy Flavour

4 At NLO the partons remain continuous if transition point is taken as Q = m H. ZM-VFNS possible, if inaccurate. At NNLO lead to discontinuities in partons. Heavy flavour no longer turns on from zero at µ = m c (c + c)(x, m c) = A Hg (m c) g(m c) In practice turns on from negative value, (for general gluon). -.5 xc(x,q =.45GeV ) x1-1 Ringberg8-Heavy Flavour 3

5 Evolution of NNLO F c (x,q ) Evolution of NNLO F (x,q ).5.4 F c (x,q ) x=.3 x=.3. x= F (x,q ) x=.1 x=.1 1 x=.1 x=.1.5 x= Q Q Leads to huge discontinuity in F c (x, Q ). Still significant in F T ot (x, Q ). ZM-VFNS not really feasible at NNLO. Want Need. Ringberg8-Heavy Flavour 4

6 The GM-VFNS can be defined by demanding equivalence of the n f light flavour and n f + 1 light flavour descriptions at all orders above transition point n f n f + 1 F (x, Q ) = C F F,n f k (Q /m H ) f n f k (Q ) = C V F,n f +1 j (Q /m H ) f n f +1 j (Q ) Hence, the VFNS coefficient functions satisfy C F F,n f which at O(α S ) gives C V F,n f +1 j (Q /m H ) A jk(q /m H ) f n f k (Q ). k (Q /m H) = C V F,n f +1 j (Q /m H) A jk (Q /m H), C F F,n f,(1),hg ( Q m H ) = C V F,n f +1,(),HH ( Q m H ) Pqg ln(q /m H) + C V F,n f +1,(1),Hg ( Q m ), H The VFNS coefficient functions tend to the massless limits as Q /m H. However, C V F j (Q /m H ) only uniquely defined in this limit. Can swap O(m H /Q ) terms between C V F,,HH (Q /m F,1 H ) and CV,g (Q /m H ). Ringberg8-Heavy Flavour 5

7 Original ACOT prescription violated threshold W > 4m H quark in final state rather than quark-antiquark pair. since only needed one (TR-VFNS) recognised ambiguity in definition of C V F,,HH (Q /m H ) for first time and removed it by making (d F /d ln Q ) continuous at transition (in gluon sector). Smoothness guaranteed at Q = m H but complicated. Various other alternatives. Most recently Tung, Kretzer, Schmidt have come up with the ACOT(χ) prescription which I interpret as C V F,,HH (Q /m H, z) = δ(z Q /(Q + 4m H)). F H, (x, Q ) = (h + h)(x/x max, Q ), x max = Q /(Q + 4m H) C ZM,,HH (z) = δ(1 z) for Q /m H. Also W = Q (1 x)/x 4m H. For VFNS to remain simple (and physical) at all orders is necessary to choose Have adopted this. C V F,n,HH (Q /m H, z) = C ZM,n,HH (z/x max). Ringberg8-Heavy Flavour 6

8 One more problem in defining VFNS. Ordering for F H (x, Q ) different above and below transition point. LO ( NLO α S 4π ) Below Above α S 4π CF F,n f,(1),hg g n f C V F,n f +1,(),HH (C F F,n f,(),hg g n f+c F F,n f,(),hq Σ n f) α S 4π (CV F,n f +1,(1) (h + h),hh h + +C V F,n f +1,(1),Hg g n f+1) NNLO ( α S 4π ) 3 i CF F,n f,(3),hi f n f i ( α S 4π ) j CV F,n f +1,(),Hj f n f +1 j. Switching direct from fixed order to same order when going from n f to n f + 1 flavours discontinuity. Must make some decision how to deal with this. Ringberg8-Heavy Flavour 7

9 ACOT type schemes have used e.g. NLO α S 4π CF F,n f,(1),hg i.e., same order of α S above and below. g n f α S 4π (CV F,n f +1,(1),HH (h + h) + C V F,n f +1,(1),Hg g n f +1 ), But LO evolution below and NLO evolution above. Slope discontinuous. TR have used e.g. LO α S (Q ) 4π C F F,n f,(1),hg (Q /m H ) gn f(q ) α S(M ) 4π C F F,n f,(1),hg (1) g n f(m ) +C V F,n f +1,(),HH (Q /m H ) (h+ h)(q ), i.e. freeze higher order α S term when going upwards through Q = m H. This difference in choice can be phenomenologically important. In order to define our VFNS at NNLO, need O(αS 3 ) heavy flavour coefficient functions for Q m H and to be frozen for Q > m H. However, not calculated. Model using known leading threshold logarithms (Laenen and Moch) and leading ln(1/x) term from k T -dependent impact factors Catani, Ciafaloni and Hautmann. Ringberg8-Heavy Flavour 8

10 Have to also consider ambiguities in definition of scheme for F L (x, Q ). If charm explicitly in proton, zeroth order contribution to C c L (x, Q ) = 4m c Q δ(1 z/(x(1+ m c Q )), which disappears at high Q. Cancels between orders in properly defined GM -VFNS. However, large near m c while other terms suppressed by v 3 (v is the velocity of the heavy quark in the centre-of-mass frame). If implemented can lead to peculiar behaviour for Q slightly above m c particularly at NNLO where charm distributions start off negative. Chosen to be absent in our scheme. Example of competing definitions of schemes for F c L (x, Q ) (Chuvakin, Smith, van Neerven)..5 Fig 18 One choice gives positive bump at NLO and negative bump at NNLO. Other is smooth. All TR-VFNS schemes and SACOT(χ) schemes have no zeroth order contribution. F L,c (x,q ) BMSN NLO BMSN NNLO CSN NLO CSN NNLO Ringberg8-Heavy Flavour 9 Q,GeV

11 Ordering is the main difference in the NLO predictions from MRST and CTEQ in the comparison to (published) H1 data on F b (x, Q ). (Preliminary data lower.) O(αS ) part is dominant at for Q m c. Frozen part remains significant. Clearly improves match to data. _ F bb 8 i 1 1 x=. i=5 x=.5 i=4 x=. i=3 Now raised m b = 4.3GeV to m b = 4.75GeV slight relative suppression at low scales compared to plots. 1 x=.5 i= x=.13 i= H1 Data H1 Data (High Q ) MRST4 MRST NNLO CTEQ6HQ x=.3 i= Q /GeV Ringberg8-Heavy Flavour 1

12 NNLO consequences. NNLO F c (x, Q ) starts from higher value at low Q. 1.5 F c (x,q ) x=.1 At high Q dominated by (c + c)(x, Q ). This has started evolving from negative value at Q = m c. Remains lower than at NLO for similar evolution. 1.5 General trend F c (x, Q ) flatter in Q at NNLO than at NLO. Important effect on gluon distribution going from one to other..4 F c (x,q ) x= Q (GeV ) Ringberg8-Heavy Flavour 11

13 Remember caveat at NNLO. At NNLO also get contribution due to heavy flavours away from photon vertex. γ γ q q h h + h q q h Strictly, left-hand type diagram and soft parts of right-hand type diagram should be light flavour structure function, and hard part of right-hand type diagram contributes to F H (x, Q ) (Chuvakin, Smith, van Neerven). Soft part of right cancels ln 3 (Q /m H ) divergences in virtual corrections (left). Can be implemented (depends on separation parameter), but each contribution tiny. At moment all in light flavours. Ringberg8-Heavy Flavour 1

14 Importance of treating heavy flavour correctly illustrated at NNLO with MRST6 partons. Previous approximate NNLO sets used (declared) approximate VFNS at flavour thresholds. Full VFNS flatter evolution of charm bigger gluon and more evolution of light sea and bigger α S. 6% increase in σ W and σ Z at the LHC. This is a correction not uncertainty. Very important changes nonetheless. Ratio of NNLO to MRST4NNLO Ratio of NNLO to MRST4NNLO x 1 Q = 1 4 GeV Q = 1 4 GeV Gluon Up quark x Ringberg8-Heavy Flavour 13

15 With hindsight check effect of change in flavour prescription for NLO. Compare MRST4 (with 1 uncertainties) to unofficial MRST6 NLO. Fit to same data as 6 NNLO set. Same trend for partons and α S as at NNLO, but a lot smaller. Max of % changes. % increase in σ W and σ Z at the LHC. Can be same size as quoted uncertainties. This is a genuine theory uncertainty due to competing but equally valid choices. Ambiguity decreases at higher orders. 6/4 at NLO for g(x,q ) 6/4 at NLO for u(x,q ) Ringberg8-Heavy Flavour 14

16 Dependence on m c at NLO in 8 fits (very prelim). Vary m c in steps of.1gev. m c (GeV) χ global χ F c 699 pts 83 pts α s (M Z ) Clear correlation between m c and α S (M Z ). For low m c overshoot low Q medium x data badly. Preference for m c = 1.4GeV. Towards lower end of pole mass determinations. Uncertainty from fit.1.15gev. Ringberg8-Heavy Flavour 15

17 Dependence on m c at NNLO in 8 fits (prelim). m c (GeV) χ global χ F c 615 pts 83 pts α s (M Z ) Again clear correlation between m c and α S (MZ ), but less variation in latter. Preference for m c = 1.3GeV. Very much lower end of pole mass determinations. Uncertainty from fit.1.15gev. More consistency between best global fit and fit to charm data. Ringberg8-Heavy Flavour 16

18 NNLO and NLO fits to charm data (prelim) F c (x,q ) Q =1.75 GeV NNLO NLO ZEUS H Q =3.75 GeV.3. Q =6.75 GeV F c (x,q ) Q =11.5 GeV.6.4 Q =7.5 GeV.6.4 Q =6 GeV x x 1-1 Clearly NNLO improves match to lowest Q data, where NLO always too low. NNLO generally better shape, but undershoots some Q 1 GeV data. Ringberg8-Heavy Flavour 17

19 NNLO shape seems to best match high statistics preliminary H1 charm data as well. _ F cc 4 i H1+ZEUS F cc _ (x,q ) x=. i=5 1 x=.5 i=4 1 x=. i=3 x=.5 i= 1 x=.13 i=1 1-1 H1 HERA I+II 6 e p (prel.) H1 D ZEUS D MRST4 MRST NNLO CTEQ6HQ x=.3 i= Q /GeV Ringberg8-Heavy Flavour 18

20 Extension to charged currents. General procedure works on same principles - can now produce single charm quark from strange (and some down) so threshold at x max = 1/(1 + m c/q ). Fairly straightforward to NLO. However, massive FFNS coefficient functions not known at O(αS ) (only asymptotic limits Buza et al). Needed in our GM-VFNS at low Q at NLO, and at all Q at NNLO - though in later case subtracted so tend to known massless limit for large Q /m c. Initial proposal assume mass-dependence the same as for neutral current functions but with threshold in 4m c replaced by threshold in m c. Various complications. In particular important for processes such as dimuon production at CCFR, NuTeV. Cross-section given by dσ (1 y + y M ( Nxy )F (x) y E ν F L(x) ± y 1 y ) xf 3, where y.3.8, so all terms are important. Ringberg8-Heavy Flavour 19

21 O(α S ) dominated by gluon. Simple prescription gives large corrections to F (x, Q ). Smaller threshold longer convolution length than neutral current. But consider comparison at O(α S ). γ c W + c c s g g For NC coefficient is finite and positive. For CC co-linear divergence from light quark. After subtraction approx factor (1 + log(1 z)) at low Q. Finite parts of NC and CC (after log(q /m c) subtraction) converge to each other high at Q. NC positive at O(αS ) with threshold log enhancements. Obtain CC by change in kinematics and (1 + m c/q log(1 z)) factor. correct large Q /m c limit. Ringberg8-Heavy Flavour

22 O(αS ) contribution to CCC 3g (x, m c, Q ) non-zero for finite Q /m c. However, vanishes at both Q /m c and W /m c. Must be implemented else potentially important at low Q and x. Complication in ordering for longitudinal CC charm production. In massless limit lowest order contribution F CC L,c (x, Q ) = α S (C L,g (x) g(x, Q ) + C L,q (x) s(x, Q )). At low Q zeroth order contribution (ξ = x(1 + m c/q )) FL,c CC(x, Q ) = m c s(ξ, Q ). m c +Q Difference in orders below and above (opposite to NC F (x, Q )). Choose to obtain correct limits in both regimes with continuity, i.e. above LO for Q < m c and for Q > m c ( FL,c CC(x, Q )= m c s(ξ, Q )+ 1 m m c )α c +Q Q S (C L,g (x) g(x, Q )+C L,q (x) s(x, Q )), where first term dies away leading to normal massless limit. Generalise to higher orders. Ringberg8-Heavy Flavour 1

23 Extraction of strange sensitive, to varying degrees, to these details. Find slightly reduced ratio of strange to non-strange sea compared to previous default single factor κ =.5. ) = 1 GeV κ(x, Q MSTW NLO PDF fit (preliminary, 7/11/7) κ(x,q s(x,q ) u(x,q )+s(x,q )+d(x,q ) ) at Q = 1 GeV dx [xs + xs] / dx [xu + xd] = x Ringberg8-Heavy Flavour

24 With chosen prescription for charged current heavy flavour, results rather stable from LO NLO NNLO. NNLO similar overall to NLO. Not completely obvious a priori that this would be so. ) = 1 GeV κ(x, Q MSTW NNLO PDF fit (preliminary, 3/11/7) κ(x,q s(x,q ) u(x,q )+s(x,q )+d(x,q ) ) at Q = 1 GeV dx [xs + xs] / dx [xu + xd] = x Ringberg8-Heavy Flavour 3

25 Strange itself has some non-insignificant mass, and this should qualitatively lead to suppression compared to light sea quarks up and down. When c and b turn on they evolve like massless quarks, but always lag behind. some suppression at all x for finite Q (s(x) + s (x))/(u (x) + d (x)) at Q =1GeV (c(x) + c (x))/(u (x) + d (x)) at Q =15GeV x c + c evolved through 7 8 times input scale similar to s + s at Q = 1GeV. Do not expect exact correspondence, but very good except c + c more suppressed at x.1. (Implication for s + s from recent HERMES K ± data). Ringberg8-Heavy Flavour 4

26 Note c + c is harder as x 1 than ū + d. This is more the case at NNLO compared to NLO. Largely driven by input contribution where as z 1 (c + c)(x, m c) = A Hg (z, Q = m c) g(x/z, Q ), A Hg (z, Q = m c) 1 αs 9 (4π) ln3 (1 z). Very small absolute effect, but sign of high-x intrinsic charm appearing in perturbative series? (Renormalons - higher twist related to perturbative series) question of intrinsic charm. Contribution of O(Λ QCD /m c). In VFNS coefficient functions ambiguous at O(m c/q ). perturbative contributions between orders. Cancels exactly in For intrinsic charm ambiguity of order Λ QCD /m c m c/q = O(Λ QCD /Q ), i.e. genuine higher twist. CTEQ examine possibilities. What about MSTW? Ringberg8-Heavy Flavour 5

27 Intrinsic Charm? First compare to EMC data. F c (x,q ).5.5 x=.75 EMC.4. x=.133 NNLO threshold corrections cause increase At fixed NLO and NNLO with m c = 1.4GeV some room at x >., but more tendency to overshoot.. x=.4. x=.4 Comparison not great overall x=.75 x= x=.37 x= Q Q 1 Ringberg8-Heavy Flavour 6

28 Always have problem with heavy flavours that use W = 4m c as threshold, whereas in reality need to produce mesons, i.e. W a bit greater. In practice higher twist effect, but EMC data not so far from threshold. Implement m c m c(1+λ /m c) in threshold dependent parts of coefficient functions, where Λ is a binding energy, e.g. Λ =.GeV. No change in off-threshold parts of coefficient functions, and certainly not in PDF definitions. Model nonperturbative corrections for EMC data much smaller proportional effect for HERA data (but not completely negligible). Ringberg8-Heavy Flavour 7

29 Green curve has nonperturbative threshold corrections. Clearly matches lower Q EMC data better. Slightly smaller at higher x. F c (x,q ).5.5 x=.75 EMC.4. x=.133 Pink curve now includes.3% contribution of Brodsky, et al model for intrinsic charm (with correct threshold). 1/ of CTEQ upper limit. 1. x= x=.4 Seems about maximum that can be included even with nonperturbative suppression of other contribution x= x=.133 Note Hoffmann and Moore claimed.3% as best fit in 1983 with very old PDFs and only LO perturbative contribution x= x= Q Q Ringberg8-Heavy Flavour 8

30 Conclusions Use a GM-VFNS as default for heavy flavours, as done since Coefficient functions for heavy quarks now based on those in ACOT(χ) leads to physically sensible and simple VFNS. However, ordering different in two competing approaches for a variety of quantities. Higher order effect. Sometimes large. Full NNLO VFNS, has small amount of necessary modelling. Improves fit to lowest x and Q data. Important impact on gluon. (Caveat on uniqueness of F c (x, Q ) beyond NLO- Chuvakin et al). Recent changes in T-Roberts prescription lead to differences in partons of at most % at NLO. Inherent theoretical uncertainty. Decreases with perturbative order. Implemented detailed prescription for charged currents. Needs more modelling at low Q but contains physics constraints and tends to correct asymptotic limit. Order-by-order stability in strange extraction from dimuon data. Strange behaves roughly like mass-suppressed quark with evolution starting from scale.1gev. Look for evidence of intrinsic charm. EMC data require nonperturbative corrections for good comparison. If data at all sensible imply some high-x intrinsic charm, but cannot have much more than.3% contribution. Ringberg8-Heavy Flavour 9