Lattice QCD Impact on Determination of CKM Matrix: Status and Prospects
LLattice QCD Impact on Determination of CKMMatrix: Status and Prospects
Steven Gottlieb ∗ Indiana University, Bloomington, IN 47405, USAE-mail: [email protected]
Lattice QCD is an important tool for theoretical input for flavor physics. There have been four re-views by the Flavour Lattice Averaging Group (FLAG). This talk will review the current status ofthe magnitude of eight of the nine CKM matrix elements, borrowing heavily from the most recentFLAG review (co-authored by the speaker). Future prospects for improving the determination ofthe CKM matrix will be discussed. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] F e b QCD Impact on Determination of CKM Matrix
Steven Gottlieb
1. Introduction
I was asked to “review very recent FLAG results on standard model parameters and renormal-ization.” This is a very broad charge and one that is ill-suited to the amount of time available, so Iwill restrict my attention to results from lattice QCD calculations that have an impact on the deter-mination of the CKM mixing matrix. Even with this restriction, it will be necessary to summarizeresults at a rather high level. Many details of the calculations can be found in the latest and recentFLAG reviews [1, 2], and, of course, the original papers cited therein.The Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix is fundamental to the field of flavorphysics within the Standard Model (SM) of Elementary Particle and Nuclear Physics. Kobayashiand Maskawa were awarded the Nobel Prize for their realization that with three generations ofquarks the matrix may contain a complex phase that results in CP violation. Although it seems thatthis is not sufficient to explain the baryon asymmetry of the universe, there are many processes inwhich to test the CKM scenario, and any such tests in which the SM fails to explain the observationscould give a window into new physics beyond the standard model (BSM).A few words about my background may be in order. I am a founding member of the MILCCollaboration and a member of the Fermilab Lattice/MILC effort that is some 16 years old. I amalso a member of the Flavour Lattice Averaging Group (FLAG) where I have been working in the B and D semileptonic working group. However, this is not a FLAG approved talk, so I am solelyresponsible for its content. The two most recent editions of the FLAG report are in Refs. [1] and[2]. I will use many plots from the most recent FLAG report and cover results from several of theworking groups. I will also include several graphs from the Fermilab Lattice/MILC Collaborations.I am grateful for the work of all my FLAG, Fermilab Lattice, and MILC collaborators.
2. CKM Matrix
The CKM matrix describes how quarks mix under the weak interaction, that is, the misalign-ment of mass eigenstates and weak eigenstates. The CKM matrix is shown in expression 2.1. Thematrix elements are shown in bold type, and beneath the elements in the first two rows you will findone or two weak decays that can be used to determine that matrix element, if we can accurately cal-culate the QCD contribution to the decay. Under the last row of elements are two hadronic matrixelements that remind us that B ( s ) mixing allows us to determine V td and V ts . V ud V us V ub π → l ν K → l ν B → l ν K → π l ν B → π l ν V cd V cs V cb D → π l ν D → Kl ν B → D ( ∗ ) l ν D → l ν D s → l ν Λ b → Λ c l ν V td V ts V tb (cid:104) B d | B d (cid:105) (cid:104) B s | B s (cid:105) (2.1)The CKM matrix is unitary so each row and each matrix is a complex unit vector. Each row(column) is orthogonal to the other two rows (columns). Violations of unitarity are evidence of1 QCD Impact on Determination of CKM Matrix
Steven Gottlieb
BSM physics. It is important to use multiple processes to determine each matrix element. If twodifferent processes infer different values for the same CKM matrix element, that would also beevidence for non-standard model physics. Of course, we would very much like to have solid evi-dence for BSM physics, but that requires precise determination of the standard model contribution.Lattice QCD is one of the best tools for calculating those contributions.As a first example, let’s consider the leptonic branching fraction for the D ( s ) meson: B ( D ( s ) → (cid:96) ν (cid:96) ) = G F | V cq | τ D ( s ) π f D ( s ) m (cid:96) m D ( s ) (cid:32) − m (cid:96) m D ( s ) (cid:33) . (2.2)On the right hand side, the Fermi constant, lepton mass, meson mass, and meson lifetime appear.These are all well determined. Also appearing are the CKM matrix element V cq that we would liketo determine, and the so-called hadronic decay constant f D ( s ) that we calculate using lattice QCD.A second example is a semileptonic D meson decay for which the differential decay rate may bewritten: d Γ ( D → P (cid:96) ν ) dq = G F | V cx | π ( q − m (cid:96) ) (cid:113) E P − m P q m D (cid:20) (cid:18) + m (cid:96) q (cid:19) m D ( E P − m P ) | f + ( q ) | + m (cid:96) q ( m D − m P ) | f ( q ) | (cid:21) , (2.3)where x = d , s is the daughter light quark, P = π , K is the daughter light-pseudoscalar meson, q = ( p D − p P ) is the momentum of the outgoing lepton pair, and E P is the light-pseudoscalarmeson energy in the rest frame of the decaying D . A similar formula holds for other heavy-lightmesons such a D s , B or B s . The hadronic physics that we require is expressed in terms of the twoform factors f + ( q ) and f ( q ) . They are defined in terms of the hadronic matrix element of theflavor-changing vector current V µ = x γ µ c , (cid:104) P | V µ | D (cid:105) = f + ( q ) (cid:18) p D µ + p P µ − m D − m P q q µ (cid:19) + f ( q ) m D − m P q q µ . (2.4)The experimental observable depends upon known quantities, a CKM matrix element, and thehadronic information that is encoded the form factors which depend upon q .
3. First Row
Because | V ub | is so small, we can test unitarity of the first row of the CKM matrix quite well(with current precision) looking only at pion and kaon leptonic decays and kaon semileptonicdecay. Figure 1 summarizes results for f π and f K , the pion and kaon decay constants, respectively.This is the first of a number of plots from FLAG [2], so we should explain the color coding. Solidgreen symbols correspond to calculations for which there should be sufficient control of systematicerrors. There are criteria for lattice volume, quark masses (to control the chiral limit) and numberof lattice spacings (to control the continuum limit). Some quantities have additional quality criteriadetailed in the FLAG report [2]. Points in green with open plotting symbols have been superseded,usually because a group has added additional ensembles or statistics. Points plotted in red are2 QCD Impact on Determination of CKM Matrix
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120 125 130 =++=+=
ETM 09 ETM 14D FLAG average for = MILC 04 HPQCD/UKQCD 07 RBC/UKQCD 08 Aubin 08 MILC 09 MILC 09 MILC 09A MILC 09A RBC/UKQCD 10A JLQCD/TWQCD 10 MILC 10 MILC 10 Laiho 11 RBC/UKQCD 12A RBC/UKQCD 14B JLQCD 15C FLAG average for = +
ETM 10EMILC 13AHPQCD 13AFNAL/MILC 14AETM 14E FLAG average for = + + ±
150 155 160 =++=+=
MeV
ETM 09 ETM 14D FLAG average for = MILC 04 HPQCD/UKQCD 07 RBC/UKQCD 08 Aubin 08 MILC 09 MILC 09 MILC 09A MILC 09A RBC/UKQCD 10A JLQCD/TWQCD 10 MILC 10 MILC 10 Laiho 11 RBC/UKQCD 12A RBC/UKQCD 14B JLQCD 15C FLAG average for = +
ETM 10E MILC 13A HPQCD 13A FNAL/MILC 14A ETM 14E FLAG average for = + + ± Figure 1:
FLAG 2019 compilation of results for f π and f K . Meaning of the colors is explained the the text. deemed not to have adequate control of systematic errors. The black points and gray bands arethe FLAG average values. Only solid green points are included in the FLAG average values. Eachgraph is generally divided into sections depending upon the number of dynamical quarks in the sea.The values N f =
2, 2+1, and 2+1+1 are used for two-light flavors, two light plus strange, and twolight plus strange plus charm, respectively. In most cases, there is a FLAG average for each valueof N f . We will restrict our attention to N f = + f K , but none for f π . This is usuallybecause f π has been used to set the scale. Marciano [3] pointed out that the ratio f K / f π can beused to determine the ratio | V us / V ud | as leptonic decays of both pion and kaon are well measured inexperiment. As summarized by the Particle Data Group [4] and Moulson at CKM 2017 [5], (cid:12)(cid:12)(cid:12)(cid:12) V us V ud (cid:12)(cid:12)(cid:12)(cid:12) f K ± f π ± = . ( ) . (3.1)Thus, the experimental error is 0.15%. Figures 2 and 3 display summaries of results from multiplegroups for the decay constant ratio. Figure 2 is from FLAG [2], while Fig. 3 is from a recentFermilab/MILC publication [6]. The former is more comprehensive, but the latter uses a finerscale, and one can actually see the error bars on the most precise calculations, as they are no longerobscured by the plotting symbols. The FLAG 2019 result for N f = + + . ( + − ) . The theoryerror has been reduced to 0.16%, quite comparable to the experimental error.Let us now turn from leptonic decays to the semileptonic decay of the K meson. Since semilep-tonic decays have three-body final states, the kinematics are slightly more complicated, and thereis one kinematic variable usually called q upon which the form factor depends. From energy-momentum conservation, p K = p π + q (cid:96) + q ν where p K is the energy-momentum 4-vector of the3 QCD Impact on Determination of CKM Matrix
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QCDSF/UKQCD 07 ETM 09 ETM 10D (stat. err. only) ALPHA 13A ETM 14D (stat. err. only) FLAG average for = MILC 04 HPQCD/UKQCD 07 RBC/UKQCD 08 Aubin 08 MILC 09 MILC 09A BMW 10 RBC/UKQCD 10A JLQCD/TWQCD 10 MILC 10 Laiho 11 RBC/UKQCD 12 RBC/UKQCD 14B Durr 16 QCDSF/UKQCD 16 FLAG average for = +
ETM 10E (stat. err. only) MILC 11 (stat. err. only) MILC 13A HPQCD 13A ETM 13F FNAL/MILC 14A ETM 14E FNAL/MILC 17 FLAG average for = + + ± / ± Figure 2:
FLAG 2019 compilation of ratio f K / f π . Comparison of calculations of decay constant ratio f K / f π with N f = + +
1, 2 +
1, and 2 sea quark flavors. From Ref. [2]. .
16 1 .
18 1 . f K + /f π + u, d, s, c sea u, d, s sea Figure 3:
Comparison of calculations of decay constant ratio f K / f π with N f = + + + initial state kaon, p π refers to the final state pion, and q (cid:96) and q ν , refers to the final state lepton andneutrino. The momentum transferred to the leptons q = q (cid:96) + q ν , and we have already introduced q . To determine | V us | from experiment, we could determine the vector form factor f + ( q ) and use4 QCD Impact on Determination of CKM Matrix
Steven Gottlieb N o n - l a tt i c e N f = + N f = + + This work FLAG N f =2+1+1 FLAG N f =2+1 ETM 2016 FNAL/MILC 2014 RBC/UKQCD 2015 FNAL/MILC 2012 Bijnens & Ecker 2014 Kastner & Neufeld 2008 Cirigliano et al 2005 Jamin et al 2004 Bijnens & Talavera 2003 Leutwyler & Roos 1984 f + K π - (0) Figure 4:
Comparison of calculations of the vector form factor f + ( q = ) with N f = + + + the differential decay rate of the kaon; however, it is convenient to just calculate f + ( q = ) and useexperimental input that determines | V us | f + ( ) . We would like to start our story in 2014, when theexperimental value was | V us | f + ( ) = . ( ) . At that time, FNAL/MILC had an N f = + + f + ( ) = . ( )( ) where the first error was statistical and the second systematic. So,in 2014, the experimental error was 0.18%, but the theory error was 0.34%. Moving ahead to late2018, the experimental result had been slightly updated [5] to: | V us | f + ( ) = . ( ) . AlsoFNAL/MILC [7] updated their calculation resulting in f + ( ) = . ( ) stat ( ) sys = . ( ) , (3.2)so the theory error is 0.20%, quite comparable to the experimental error. The FLAG 2019 averagefor N f = + + f + ( ) = . ( ) . The FLAG result has a larger error because Ref. [7] hadnot been published before the FLAG deadline for inclusion in the latest edition. Figure 4, takenfrom Ref. [7], depicts the most relevant lattice calculations for N f = + + +
1. It alsocontains results from a number of other theoretical approaches whose errors are larger than thosefrom lattice QCD.We now turn to the V ud - V us plane to consider the implications of what we have just found.The ratio of decay constant determines the ratio | V us / V ud | and thus an angled band in the planeof the two CKM matrix elements. The semileptonic kaon decay determines a horizontal band for | V us | . Nuclear β -decay provides a fairly precise value of | V ud | , i.e., a vertical band. In addition,because | V ub | is so small, unitarity determines a very narrow arc of a circle in the plane. Figure 5describes the situation according to FLAG [2]. We see that the two white ellipses that come fromthe intersection of leptonic and semileptonic decay bands show some tension with both unitarityand V ub from β -decay. This graph has results for both N f = + + + QCD Impact on Determination of CKM Matrix
Steven Gottlieb V ud V u s lattice results for f + (0) , N f =2+1+1 lattice results for f K ± /f π ± , N f =2+1+1 lattice results for f + (0) , N f =2+1 lattice results for f K ± /f π ± , N f =2+1 lattice results for N f =2+1+1 combinedlattice results for N f =2+1 combinednuclear β decay Figure 5:
First row unitarity plot from FLAG [2]. calculations. It should be noted that the decay constant ratio is in reasonable agreement with β -decay and unitarity. In Fig. 6, we show the results from Ref. [7]. In this case, we have N f = + + K (cid:96) , acommon notation for kaon semileptonic decay, while the angled band is labeled K (cid:96) a notationfor kaon leptonic decay. In addition, there is a wide horizontal band which is determined fromthe known value of | V cd | and the assumption of CKM unitarity. (Recall that in the Wolfensteinrepresentation, V cd = − V us .) |V cd | + unitarityUnitarity 0 + → + K l2 K l3 | V u s | |V ud | Figure 6:
First row unitarity plot from FNAL/MILC [7]. QCD Impact on Determination of CKM Matrix
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Hardy 16 nuclear decayHudspith 17 decay and + Hudspith 17 decay and + HFAG 16 decayJLQCD 05 RBC 06 QCDSF 07 (stat. err. only)QCDSF/UKQCD 07 ETM 09 ETM 09A ETM 10D (stat. err. only)ETM 10D (stat. err. only)ALPHA 13A ETM 14D (stat. err. only)FLAG average for = MILC 04 HPQCD/UKQCD 07 RBC/UKQCD 07 RBC/UKQCD 08 Aubin 08 MILC 09 MILC 09A BMW 10 RBC/UKQCD 10 RBC/UKQCD 10A JLQCD/TWQCD 10 MILC 10 JLQCD 11 Laiho 11 JLQCD 12 FNAL/MILC 12I RBC/UKQCD 12 RBC/UKQCD 13 RBC/UKQCD 14BRBC/UKQCD 15ADurr 16 QCDSF/UKQCD 16 JLQCD 17 FLAG average for = +
ETM 10E (stat. err. only) MILC 11 (stat. err. only) MILC 13A HPQCD 13A ETM 13F FNAL/MILC 13C FNAL/MILC 13EFNAL/MILC 14AETM 14E ETM 16 FNAL/MILC 17FLAG average for = + + | | =++=+=
Hardy 16 nuclear decayHudspith 17 decay and + Hudspith 17 decay and + HFAG 16 decayJLQCD 05 RBC 06 QCDSF 07 QCDSF/UKQCD 07 ETM 09 ETM 09A ETM 10D ETM 10D ALPHA 13A ETM 14D FLAG average for = MILC 04 HPQCD/UKQCD 07 RBC/UKQCD 07 RBC/UKQCD 08 Aubin 08 MILC 09 MILC 09A BMW 10 RBC/UKQCD 10 RBC/UKQCD 10A JLQCD/TWQCD 10 MILC 10 JLQCD 11 Laiho 11 JLQCD 12 FNAL/MILC 12I RBC/UKQCD 12 RBC/UKQCD 13 RBC/UKQCD 14B RBC/UKQCD 15A Durr 16 QCDSF/UKQCD 16 JLQCD 17 FLAG average for = +
ETM 10E MILC 11 MILC 13A HPQCD 13A ETM 13F FNAL/MILC 13C FNAL/MILC 13EFNAL/MILC 14AETM 14E ETM 16 FNAL/MILC 17FLAg average for = + + | |
Figure 7:
Determination of | V us | and | V ud | from leptonic and semileptonic decays from FLAG [2]. Squaresdenote results from leptonic decays and triangles are used for semileptonic decays. We have a summary of determination of | V us | and | V ud | from FLAG in Fig. 7. This plot assumesunitarity to go from either band to a value. Squares denote values from leptonic decays, andtriangles are used for semileptonic decays. The tension we saw in the previous two plots is seenas a difference between triangles and squares, which is most noticeable for N f = + +
1. FLAGresults for N f = + + | V us | = . ( ) (3.3) | V ud | = . ( ) (3.4)The blue values near the bottom of the plot show results based on standard model analysis of τ decay and nuclear β -decay. Please see the FLAG report [2] for references.
4. Second Row
Returning to expression 2.1, we find several semileptonic decays on the row below V cd , andleptonic decays of D and D s on the following row. We will consider the leptonic decays first.It has been almost 15 years since the initial test of D meson decay constants at CLEO-c [8].The unquenched calculations were based on N f = + N f = + + . N f = +
1. Figure 8 from Ref. [6] depicts several recent calculations of the D ( s ) QCD Impact on Determination of CKM Matrix
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205 215 225 235 245 255 265 275Fermilab/MILC 17ETM 14Fermilab/MILC 14 χ QCD 14HPQCD 12Fermilab/MILC 11 (
Clover c )HPQCD 10 f D s (MeV) f D + (MeV) u, d, s, c sea u, d, s sea Figure 8:
Comparison of recent calculations of f D + and f D s with N + f = + + + meson decay constants. The interested reader can find a more complete set of values in Ref. [2].The results in Ref. [6] are: f D + = . ( . ) MeV , f D s = . ( . ) MeV . (4.1)The Particle Data Group [4] has compiled the results of several experiments to provide thebest values for the product of decay constant and CKM matrix elements. The experimental valuesare: f D | V cd | = . ( . ) MeV and f D s | V cs | = . ( . ) MeV. These values have errors of 1.6–2.3%, so the experimental error is going to be dominant in determination of the two CKM matrixelements. From Ref. [6], we have | V cd | SM , f D = . ( ) f D ( ) expt ( ) EM , (4.2) | V cs | SM , f Ds = . ( ) f Ds ( ) expt ( ) EM , (4.3)where the errors are from lattice decay constant, experiment, and a structure dependent electromag-netic correction. These values differ slightly from those in FLAG [2]. Earlier this year, some newresults for D s decay were published by BESIII [10]. It was found that f D s | V cs | = . ( . ) MeV.This will bring the world average down and help improve the second row unitary sum. In fact,after Lattice 2019, the Heavy Flavor Averaging Group (HFLAV)) [11] updated its results for manyquantities of interest including the leptonic decays of D and D s mesons. Their current worldaverages are f D | V cd | = . ( . ) MeV and f D s | V cs | = . ( . ) MeV. These errors are 2.4 and1.3%, respectively, which is an improvement for D s . HFLAV uses the FLAG results for the de-cay constants, which have slightly more generous errors than those in Ref. [6]. The FLAG valueof f D is also slightly smaller. With that input, HFLAV finds | V cd | = . ( ) expt ( ) LQCD and | V cs | = . ( ) expt ( ) LQCD .For charm semileptonic decays, a single form factor describes the major contribution to thedecay rate. The analysis is often restricted to q =
0. In that case, we can use the world average8
QCD Impact on Determination of CKM Matrix
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ETM 11BFNAL/MILC 04HPQCD 11 / 10BJLQCD 17BFLAG average for = +
ETM 17DFLAG average for = + + + ( ) + ( ) Figure 9:
FLAG [2] summary of results for D semileptonic decays to π or K . values provided by the Heavy Flavor Averaging Group (HFLAV) [11]: f D π + ( ) | V cd | = . ( ) and f DK + ( ) | V cs | = . ( ) . These values have been updated since Lattice 2019 and do not agreewith the values quoted in my slides on the conference indico site. In particular, the second valuehas decreased. The experimental errors are 1.3% and 0.5% for the two decays. Figure 9 shows theFLAG averages for both form factors at q =
0. For both N f = + + + there is only asingle result. Consider the form factors for D → K (cid:96) ν , as that has the smaller experimental error, the N f = + + . ( ) , i.e., a error of 4.2%. The N f = + q . Also, FNAL/MILC are expecting to have a result with anerror of about 2.1%. FLAG had summarized the results for | V cd | and | V cs | which are displayed inFig. 10. As before, squares are used for results from leptonic decays, and triangles for semileptonicdecays. These FLAG results use an older value of f DK + ( ) | V cs | = . ( ) . The result labeledMeinel 16 is from baryon decay, and the result labeled ETM 17D/Riggio 17 uses non-zero valuesof q to extract | V cs | from the differential decay rate. Experimental errors dominate for the leptonicdecays, while the theory error is dominant for semileptonic decays. Here are some key FLAGresults. leptonic decays , N f = + + | V cd | = . ( )( ) , | V cs | = . ( )( ) , (4.4)leptonic decays , N f = + | V cd | = . ( )( ) , | V cs | = . ( )( ) , (4.5)semileptonic decays , N f = + + | V cd | = . ( ) , | V cs | = . ( ) , (4.6)semileptonic decays , N f = + | V cd | = . ( )( ) , | V cs | = . ( )( ) , (4.7)semileptonic Λ c decay , N f = + | V cs | = . ( )( ) , (4.8)9 QCD Impact on Determination of CKM Matrix
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CKM unitarityneutrino scatteringETM 13BBlossier 18our estimate for = FNAL/MILC 11HPQCD 12A/10AHPQCD 11/10BQCD 14Meinel 16RBC/UKQCD 17our estimate for = +
ETM 14EETM 17D ( = )ETM 17D/Riggio 17FNAL/MILC 17 our estimate for = + + | | | | Figure 10:
FLAG [2] summary of results for | V cd | and | V cs | . Squares are used for results from leptonicdecays, and triangles for semileptonic decays. FLAG2019 , N f = + + | V cd | = . ( ) , | V cs | = . ( ) , (4.9)FLAG2019 , N f = + | V cd | = . ( ) , | V cs | = . ( ) (4.10)Let’s consider second row unitarity. Using their most recent leptonic decay results [6] Fermi-lab/MILC have: | V cd | + | V cs | + | V cb | = . ( ) | V cd | ( ) | V cs | ( ) | V cb | , which is just over 1.5 σ from1. The FLAG 2019 value for N f = + | V cs | than leptonic decays do, so the improved agree-ment with unitarity should not be a surprise. If we consider the latest HFLAV update, we have: | V cd | + | V cs | + | V cb | = . ( ) | V cd | ( ) | V cs | ( ) | V cb | , which is only about 1.1 σ away from 1. Thus,we have reasonable agreement with unitarity for both leptonic and semileptonic charm meson de-cays, with a few percent accuracy. It will, or course, be interesting to increase the precision of thistest in the future. B Hadron Decays
Leptonic and semileptonic decays of hadrons containing a b quark have been studied usinglattice QCD. Mesonic decays have been extensively studied. Recently, Meinel and his collaboratorshave been looking at various baryon semileptonic decay form factors [16]. In addition to tree leveldecays such as B → π (cid:96) ν , which can be used to determine | V ub | , there have been studies of raredecays that involve flavor changing neutral currents [17]. Such decays vanish at tree level in the10 QCD Impact on Determination of CKM Matrix
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175 185 195 205 215 225 235 245 255 Fermilab/MILC 17HPQCD 17 ( pseudoscalar current )ETM 16HPQCD 13 (
NRQCD b )RBC/UKQCD 14HPQCD 12 (
NRQCD b )HPQCD 11 (
HISQ b )Fermilab/MILC 11 (
Clover b ) f B s (MeV) f B + (MeV) u, d, s, c sea u, d, s sea Figure 11:
Comparison of recent results for f B + and f B s from Ref. [6]. Standard Model, so they are a good place to search for new physics. They provide a complementarymethod to B mixing studies for determining | V td | and | V ts | . There have been some interestingtensions between recent Standard Model predictions and LHCb measurements, including hintsof lepton universality violation. (See Ref. [17] for a summary.) Although these issues are veryinteresting, we will need to stay focused on determination of CKM matrix elements.In Fig. 11, we show a compilation of results from Ref. [6]. The latest results from FNAL/MILChave errors under 1.3 MeV, i.e., < . B decay do not agree very well and have large errors, so the determination of | V ub | is very muchlimited by the required experimental input. In the next few years, Belle II should provide muchhigher precision results. Additional details and historical calculations may be found in Refs. [1]and [2].As mentioned above, B semileptonic and rare decays have been studied both experimentallyand theoretically. Several decays such a B → π (cid:96) ν , B s → K ( ∗ ) (cid:96) ν , and Λ b → p (cid:96) ν all depend on | V ub | .These decays can be used to determine | V cb | : B → D ( ∗ ) (cid:96) ν , B s → D ( ∗ ) s (cid:96) ν , and Λ b → Λ c (cid:96) ν . Recallthat when a particular CKM matrix element can be determined from multiple decays, if the standardmodel predictions of the decay rates do not imply identical values of the particular CKM matrixelement, that would be evidence for new physics not in the standard model. In addition, since (cid:96) can be an electron, muon or τ , there is an opportunity to test lepton universality. Recently, sometensions with lepton universality have been seen, and this is a fertile area of study. Unfortunately,we do not have time to discuss this in detail. Interesting rare decays include B → µ + µ − , B s → µ = µ − , and B → K (cid:96) = (cid:96) − .There has been a long standing difference between the values of | V ub | and | V cb | as determinedfrom exclusive and inclusive decay results. The history of the comparison from 2009 to 2018is shown in Fig. 12. When the precision of each determination was low in 2009, there was not atension between the two values; however, by 2014 the difference had grown. In 2015, the difference11 QCD Impact on Determination of CKM Matrix
Steven Gottlieb . . . . . . . . . . . | V u b | × inclusiveexclusiveCKM unitarity Figure 12:
History of | V ub | as determined from both inclusive and exclusive measurements. was reduced, but there has not been much change since then. The plot also has a horizontal bandcorresponding to CKM unitarity.The change in the value of | V ub | as determined from exclusive decays resulted from a newform factor calculation from the Fermilab Lattice and MILC collaborations [18]. Experimentalresults from both BaBar and Belle were used to determine | V ub | = . ( ) × − . Figure 13 fromRef. [18] shows both their calculation (denoted “This work”) and other results. Their result wasin reasonable agreement with other exclusive decay calculations, but with smaller error. However,the black diamond labeled BLNP, which comes from inclusive decays, is larger. There have beenno more recent published results for the B → π (cid:96) ν form factors. The FLAG report summarizesform factor results and their comparison with experiment (which determines | V ub | ). We will notreproduce those graphs here, but Fig. 14 shows the FLAG summary plot for | V ub | , which considersboth leptonic and semileptonic decays, and we quote the average value 3 . ( ) × − , whichis a 3.8% error. The BaBar and Belle leptonic decay results do not agree very well, so FLAGreports results based on each experiment and their average. The experimental error is dominant forall values of N f . Only for N f = + z -expansion with | V ub | as a free parameter. One sees that currently the error from semileptonicdetermination of | V ub | is smaller than the determination from leptonic decays.An interesting story has emerged over the last several years regarding V cb . In this case, bothsemileptonic decays B → D (cid:96) ν and B → D ∗ (cid:96) ν have been measured. The inclusive measurementsalso have been used to determine | V cb | . In 2014, Ref. [19] updated the form factor for B → D ∗ atzero-recoil, and used the experimental average from HFLAV [20] (then using a different acronym)to obtain | V cb | = ( . ± . expt ± . QCD ± . QED ) × − . The inclusive value from Gam-bino and Schwanda [21], was ( . ± . ) × − , which differs by 3 . σ from the exclusivevalue. The next year, results for the B → D semileptonic decay became available [22], which gave12 QCD Impact on Determination of CKM Matrix
Steven Gottlieb | V ub | × UTFit 2014, CKM unitarityBLNP 2004 + HFAG 2014, B → X u l ν Detmold et al . 2015 + LHCb 2015, Λ b → pl ν HPQCD 2006 + HFAG 2014, B → π l ν Imsong et al . 2014 + BaBar12 + Belle13, B → π l ν RBC/UKQCD 2015 + BaBar + Belle, B → π l ν Fermilab/MILC 2008 + HFAG 2014, B → π l ν This work + BaBar + Belle, B → π l ν Figure 13:
Comparison of several determinations of | V ub | from Ref. [18]. =++=+= . HFLAV inclusive (GGOU) (average) (Belle) (BaBar) (average) (Belle) (BaBar) FLAG estimate for = + (average) (Belle) (BaBar) | |
Figure 14:
FLAG summary plot for | V ub | from Ref. [2]. a slightly larger value of | V cb | , but with larger errors. Updated results from Belle became availablefor B → D which Bigi and Gambino [23] analyzed using the BGL form factor parametrization [24].They found | V cb | = ( . ± . ) × − . The next year, two groups looked at new Belle data [25]for B → D ∗ . Both Bigi, Gambino, Schacht [26]; and Grinstein and Kobach [27] found about a 10%difference when switching between CLN [28] and BGL parametrizations of the form factors. Atthe time of FPCP 2018, I thought the so-called V cb puzzle was largely resolved [29] as the exclusivevalue of | V cb | using the BGL parametrization was quite compatible with the inclusive value, andthe difference between CLN parametrized value and the inclusive value was only 1 . σ . However,the situation subsequently became more murky. At first, Belle [30] put out a preprint in September,2018, that supported the notion that there is a 10% difference when switching between CLN andBGL. Then, in April, 2019, version 3 of that preprint found that CLN and BGL parametrizationsare quite compatible. A month earlier, an analysis from BaBar [31], using an unbinned fit, an angu-13 QCD Impact on Determination of CKM Matrix
Steven Gottlieb lar analysis, the BGL parametrization found | V cb | = ( . ± . ) × − . This result is in goodagreement with previous values of | V cb | based on exclusive decays.More details can be found in Andrew Lytle’s review in these proceedings, and in a recent paperby Gambino, Jung, and Schacht [33]. After performing a number of fits to Belle data, the latter findabout a 2 σ difference between exclusive and inclusive values for | V cb | . Thus, the puzzle remainsuntil there are improvements in the lattice QCD calculations and the experiments. The form factorcalculations will likely be improved by four competing groups: FNAL/MILC, HPQCD, JLQCD,and LANL/SWME. The interested reader should check the proceedings for their contributions.Figure 15 displays current results using either BGL or CLN parametrizations. Figure 15:
FLAG has summarized recent results for | V ub | and | V cb | [2]. On the left, the BGL parametrizationfor the B → D ∗ form factor is used. On the right, CLN is used. Further details may be found in Ref. [2].
6. Third Row
Neutral B meson mixing is a loop level process that depends upon V td or V ts depending onwhether we consider B or B s meson mixing. Figure16 from a recent paper by the Fermilab Latticeand MILC Collaborations [34] shows the 1-loop diagrams responsible for neutral B meson mixingin the standard model. The line labeled q can either be a d or s quark. Both the mass difference andlifetime difference of the two resulting eigenstates are measured in experiments. A CP violatingphase is also determined. A short distance expansion of the loops results in an effective weakHamiltonian involving 4-quark operators. Because of the GIM and loop suppression of the mixing,this is a good place to look for BSM physics, which results in either operators that do not appear in14 QCD Impact on Determination of CKM Matrix
Steven Gottlieb
Figure 16:
Box diagrams that contribute to neutral B meson mixing from Ref. [34]. the standard model effective Hamiltonian, or modification of the coefficient of the standard modeloperator.Briefly, it is conventional to define the effective Hamiltonian in terms of eight operators, whereonly the first one appears in the standard model. H eff = ∑ i = C i O qi + ∑ i = ˜ C i ˜ O qi , (6.1)The first operator is given by O q = ¯ b α γ µ Lq α ¯ b β γ µ Lq β (6.2)and its matrix element is conventionally defined as [35] (cid:104) O q (cid:105) ( µ ) = c f B q M B q B ( ) B q ( µ ) , (6.3)where the last factor is called the bag parameter and would be 1 in the vacuum saturation approxi-mation. It is also convenient to define a renormalization-group-invariant bag parameter ˆ B ( ) B q :ˆ B ( ) B q = α s ( µ ) − γ / ( β ) (cid:20) + α s ( µ ) π (cid:18) β γ − β γ β (cid:19)(cid:21) B ( ) B q ( µ ) (6.4)In the standard model, we can express the difference in the masses of the mixing eigenstates as ∆ M q = G F m W M B q π S ( x t ) η B | V ∗ tq V tb | f B q ˆ B ( ) B q . (6.5)Details can be found in Ref. [34]. The key point is that the measurable mass difference is propor-tional to | V tq | and the bag parameter (or that bag parameter multiplied by the square of the decayconstant). Figure 17 contains the FLAG [2] summary of f B d (cid:113) ˆ B B d and f B s (cid:113) ˆ B B s . There are threeresults that are averaged for N f = + N f =
2. FLAG does not calculate theimpact of these results on the determination of the CKM matrix elements V td and V ts ; however,Ref. [34] does. That paper also contains results for BSM operators.Using experimental results for B ( s ) , mixing, it is found in Ref. [34] that | V td | = . ( )( )( )( ) × − , (6.6) | V ts | = . ( . )( . )( . )( . ) × − , (6.7) | V td / V ts | = . ( )( )( )( ) , (6.8)15 QCD Impact on Determination of CKM Matrix
Steven Gottlieb
180 220 260 =+=
ETM 13BFLAG average = HPQCD 06AHPQCD 09FNAL/MILC 11 ARBC/UKQCD 14AFNAL/MILC 16FLAG average for = +
220 260 300 MeV
ETM 13B for = HPQCD 06AHPQCD 09FNAL/MILC 11 ARBC/UKQCD 14AFNAL/MILC 16our average for = +
Figure 17:
FLAG [2] results for f B d (cid:113) ˆ B B d and f B s (cid:113) ˆ B B s . where in each case the first error is from the error in the lattice mixing matrix element, the seconderror comes from the error in the experimental mass difference, the third error comes from errorsin parameters used in Eq. (6.5), and the final error is from the lack of charm quarks in the sea. Onecan see that the lattice errors are dominant, even in the ratio of CKM matrix elements in which thehadronic uncertainties are suppressed. The errors on each quantity are 4.3%, 3.2%, and 1.6%, fromEq. 6.6 to Eq. 6.8, respectively.
7. Summary
Table 1 shows the value of the magnitude of each CKM matrix element except V tb . The secondcolumn has the value, and the third, the percentage error. We see that we have sub percent accuracyin the first row, but the other elements have errors ranging from 1.4% to 4.3%. The final columnindicates the source of each result. This is particularly pertinent for | V ud | and | V us | as we havealready remarked on the tensions with unitarity for the first row, and what happens if one considersone leptonic and semileptonic decays without input from nuclear β -decay on | V ud | .
8. Prospects
Yogi Berra is purported to have said, “It is tough to make predictions, especially about thefuture.” I trust you will keep that in mind as you read this section.We have seen that in a number of quantities the theory error is the limiting factor in determin-ing a CKM matrix element. Even in cases for which the experimental and theoretical errors arecurrently comparable, we can expect that BESIII, Belle II, and LHCb will reduce the experimentalerrors. The Belle II Physics Book [36] details what is expected to be accomplished at Belle II inmany topics. In order to improve the future precision of CKM matrix elements, it is essential for16
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Quantity value percentage error Comment | V ud | | V ud | l2 & K l3 ) | V us | | V us | l2 & K l3 ) | V cd | | V cs | | V ub | × | V cb | × | V td | × | V ts | × Table 1:
Summary of CKM matrix elements determined with input from lattice QCD. increased theoretical precision. It is assumed in this work that there will be a factor of five im-provement in errors from lattice QCD in ten years. Figure 85 of Ref. [36] predicts the error on V ub from the study B → π (cid:96) ν , taking into account both increased integrated luminosity and presumedimprovement in lattice QCD precision in five and ten years. The figure makes clear how impor-tant it is to improve our calculations to make the best use of data coming from Belle II. Failure toimprove the theoretical input could increase the error on V ub by a factor of two or more.There has also been a recent report on future prospects of the LHC [37] where we can lookforward to both the high-luminosity and higher energy enhancements of the machine. This reportassumes a factor of three improvement in theoretical precision. A white paper from USQCD onflavor physics [38] will also interest the reader. We see that there is both a need and an expectationthat the lattice QCD community will continue to improve our calculations.
9. Conclusions
Over the past few years, there has been very significant progress in using lattice QCD tocalculate standard model parameters such as quark masses, the strong coupling α s , and matrixelements need to determine the CKM matrix. A number of quantities are now available at the sub-percent level, and we expect the precision to increase by a factor of 3–5 over the next 5–10 yearsfor form factors. Thus, the interplay between theory and experiment will provide more and morestringent tests of the Standard Model (and, perhaps, evidence of new physics). We are getting tothe level of precision at which electromagnetic corrections are important. The Rome-Southamptongroup has shown leadership in the area and their work was presented in the parallel sessions.Finally, BESIII, Belle II, and LHCb all have a large role to play in the future of flavor physics.I, for one, can hardly wait for their new results! Acknowledgments
I am grateful to the local organizers and the International Advisory Committee for offeringme the opportunity to give this talk. I owe a great debt to my friends and collaborators in the17
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Fermilab Lattice and MILC Collaborations. I am thankful to all the members of the Flavour LatticeAveraging Group, whose work I have relied upon. In particular, working with the other membersof the two working groups that deal with decay of hadrons containing charm or bottom quarks, and B meson mixing has been a pleasure. Thanks to Zech Gelzer for preparing Fig. 12. Finally, thiswork was supported by the U.S. Department of Energy through grant DE-SC0010120. References [1] S. Aoki et al. , Eur. Phys. J. C , no. 2, 112 (2017) doi:10.1140/epjc/s10052-016-4509-7[arXiv:1607.00299 [hep-lat]].[2] S. Aoki et al. [Flavour Lattice Averaging Group], arXiv:1902.08191 [hep-lat].[3] W. J. Marciano, Phys. Rev. Lett. , 231803 (2004) doi:10.1103/PhysRevLett.93.231803[hep-ph/0402299].[4] M. Tanabashi et al. [Particle Data Group], Phys. Rev. D , no. 3, 030001 (2018).doi:10.1103/PhysRevD.98.030001[5] M. Moulson, PoS CKM , 033 (2017) doi:10.22323/1.291.0033 [arXiv:1704.04104[hep-ex]].[6] A. Bazavov et al. , Phys. Rev. D , no. 7, 074512 (2018) doi:10.1103/PhysRevD.98.074512[arXiv:1712.09262 [hep-lat]].[7] A. Bazavov et al. [Fermilab Lattice and MILC Collaborations], Phys. Rev. D , no. 11, 114509(2019) doi:10.1103/PhysRevD.99.114509 [arXiv:1809.02827 [hep-lat]].[8] M. Artuso et al. [CLEO Collaboration], Phys. Rev. Lett. , 251801 (2005)doi:10.1103/PhysRevLett.95.251801 [hep-ex/0508057].[9] C. Aubin et al. , Phys. Rev. Lett. , 122002 (2005) doi:10.1103/PhysRevLett.95.122002[hep-lat/0506030].[10] M. Ablikim et al. [BESIII Collaboration], Phys. Rev. Lett. , no. 7, 071802 (2019)doi:10.1103/PhysRevLett.122.071802 [arXiv:1811.10890 [hep-ex]].[11] Y. S. Amhis et al. [HFLAV Collaboration], arXiv:1909.12524 [hep-ex].[12] V. Lubicz et al. [ETM Collaboration], Phys. Rev. D , no. 5, 054514 (2017) Erratum: [Phys. Rev. D , no. 9, 099902 (2019)] doi:10.1103/PhysRevD.96.054514, 10.1103/PhysRevD.99.099902[arXiv:1706.03017 [hep-lat]].[13] H. Na, C. T. H. Davies, E. Follana, G. P. Lepage and J. Shigemitsu, Phys. Rev. D , 114506 (2010)doi:10.1103/PhysRevD.82.114506 [arXiv:1008.4562 [hep-lat]].[14] H. Na, C. T. H. Davies, E. Follana, J. Koponen, G. P. Lepage and J. Shigemitsu, Phys. Rev. D ,114505 (2011) doi:10.1103/PhysRevD.84.114505 [arXiv:1109.1501 [hep-lat]].[15] L. Riggio, G. Salerno and S. Simula, Eur. Phys. J. C , no. 6, 501 (2018)doi:10.1140/epjc/s10052-018-5943-5 [arXiv:1706.03657 [hep-lat]].[16] W. Detmold, C. Lehner and S. Meinel, Phys. Rev. D , no. 3, 034503 (2015)doi:10.1103/PhysRevD.92.034503 [arXiv:1503.01421 [hep-lat]]. QCD Impact on Determination of CKM Matrix
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