text: 13 → → TABLE VI. Error budgets for the B π(cid:96)ν and B K(cid:96)ν form factors at three representative q2 values in the range of s simulated lattice momenta. For convenience, we also show the corresponding pion or kaon energy, E . Errors are given in %. P The total error is obtained by adding the individual errors in quadrature. fBπ fBπ fBsK fBsK + 0 + 0 E [GeV] 0.85 0.50 0.27 0.85 0.50 0.27 1.07 0.77 0.53 1.07 0.77 0.53 P q2 [GeV2] 19.0 22.6 25.1 19.0 22.6 25.1 17.6 20.8 23.4 17.6 20.8 23.4 f(q2) 1.21 2.27 4.11 0.46 0.68 0.92 0.99 1.64 2.77 0.48 0.63 0.81 Statistics 7.9 5.9 12.4 7.3 4.6 3.3 4.1 3.4 3.2 3.4 2.7 2.6 Chiral-continuum extrapolation 6.3 5.0 6.2 10.9 7.6 5.8 3.2 2.8 2.5 5.0 4.9 5.1 Light-quark mass m 0.3 0.2 0.2 0.4 0.3 0.2 0.1 0.1 0.1 0.1 0.1 0.1 ud Strange-quark mass m 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.1 0.0 0.0 0.0 s Lattice-scale uncertainty 2.0 2.0 2.0 2.2 2.2 2.2 2.0 2.0 2.0 2.2 2.2 2.2 RHQ parameter tuning 0.9 0.9 0.8 1.0 1.0 1.0 0.9 0.9 0.9 1.0 1.0 1.0 Renormalization factor 0.8 0.8 0.7 1.6 1.6 1.7 0.9 0.8 0.8 1.6 1.6 1.7 Finite volume 0.5 0.4 0.3 0.7 0.5 0.4 0.2 0.2 0.2 0.2 0.1 0.1 Heavy-quark discretization errors 1.8 1.8 1.8 1.7 1.7 1.7 1.8 1.8 1.8 1.7 1.7 1.7 Light-quark & gluon discretization errors 1.1 1.1 1.1 1.1 1.1 1.1 1.3 1.3 1.3 1.3 1.3 1.3 Isospin breaking 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 Total (%) 10.6 8.4 14.3 13.6 9.6 7.6 6.2 5.5 5.3 7.1 6.7 6.8 → vidual contributions from the three RHQ parameters in B K both fBsK and fBsK change by 0.1%. s + 0 quadrature to obtain the total systematic error due the 2. Strange sea-quark mistuning lattice spacing. We examined the slopes with respect to the RHQ pa- → → Our preferred chiral-continuum ﬁt employs SU(2) chi- rameters for both B π and B K, and found s ral perturbation theory, in which the strange quark mass them to be consistent. We therefore base our estimates is integrated out, so our ﬁt function has no explicit depen- for the systematic uncertainty due to the lattice spac- → dence on m . Further, at each lattice spacing, results for s ing on the slopes obtained for the B K form fac- s → the form factors are only available at a single value of the tors because the smaller statistical errors in B K s strange sea-quark mass, so we cannot directly compute enable the slopes to be resolve→d more precisely. Figure 9 the strange sea-quark mass dependence of f(cid:107) and f⊥. We shows the slopes of the B K(cid:96)ν form factors with { }s ≈ therefore study the light sea-quark mass dependence and respect to the m a, c , ζ on the a 0.11 fm ensem- 0 P use it to bound the strange sea-quark mass dependence. ble with am = 0.005. For this slope estimate, we use l We cannot resolve any light sea-quark mass dependence the unimproved heavy-light vector current from Eq. (10). within statistical uncertainties, and expect the strange We ﬁnd the largest slopes at p(cid:126) = 2π(2, 0, 0) for f⊥ and sea-quark mass dependence to be even smaller. Thus we p(cid:126) = 2π(1, 1, 0) for f(cid:107). Following the procedure outlined take the error due to mistuning the strange sea-quark above, we estimate lattice-spacing errors in f⊥ and f(cid:107) mass to be negligible. of 1.9 % and 2.2 %, respectively. In the continuum this 3. Valence strange-quark mass uncertainty corresponds to errors on f (f ) of 2.0% (2.2%) which → + 0 → we take for both Bs K and B π. → C. Light- and strange-quark mass uncertainties The Bs K form factors have explicit strange valence-quark mass dependence. The strange-quark masses employed in our simulations diﬀer slightly from Here we estimate the error in the form factors due to the physical, tuned values a m˜ phys = 0.0379(11) and the uncertainty in the light-quark mass and the mistun- 24 s a m˜ phys = 0.0280(7) [16]. To study the valence strange- ing of the strange sea quark. For clarity we discuss sepa- 32 s → quark mass dependence, we calculated the B K form rately each place where the light- or strange-quark mass ≈ s factors on the a 0.11 fm, am = 0.005, ensemble with enters the analysis. l two additional spectator-quark masses of a m˜ = 0.033 1. u/d-quark mass uncertainty 24 s and 0.043. Figure 10 shows the valence-quark mass → dependence of the B K form factors; we observe s We obtain the physical form factors f⊥ and f(cid:107) af- the largest slopes for f(cid:107) at p = (0, 0, 0) and for f⊥ at ter the chiral-continuum ﬁt by evaluating Eqs. (31) and p = (1, 0, 0). Multiplication of these measured slopes (32) at the physical average u/d-quark mass a32m˜ puhdys = by the discrepancy between t≡he sim−ulated and tuned 0.00102(5). We estimate the error in the form factors due strange-quark masses, ∆(m ) (m˜ m˜ phys) = 0.004, s s s to the light-quark mass uncertainty by varying m˜ phys by leads to estimates for the error due to mistuning the → ud plus/minus one sigma. For B π the central value shifts valence strange-quark mass of about 0.1% for f and − − + by 0.2 0.3% for fBπ and 0.2 0.4% for fBπ, while for below this for f (which we consider as negligible). + 0 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TABLE

text: VI.

text: Error

text: budgets

text: for

text: the

text: B

text: and

text: B

text: form

text: factors

text: at

text: three

text: representative

text: values

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text: s

text: simulated

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text: momenta.

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text: convenience,

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text: fBπ

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text: ]%[ πB+f Δ ]%[ KsB+f Δ ]%[ πB0f Δ ]%[ KsB0f Δ 15 15 15 varying g varying f π omitting zero momentum 10 10 omitting a2 term omitting M2 term π omitting a2 and M2 terms π analytic analytic omitting a2 term 5 5 0 0 18 20 22 24 26 18 20 22 24 26 q2 [GeV2] q2 [GeV2] varying g 6 6 varying f π varying M (0+) omitting zeBr*o momentum omitting largest momentum 4 4 omitting a2 term omitting M2 term π omitting a2 and M2 terms π 2 2 analytic analytic omitting a2 term 0 0 17 18 19 20 21 22 23 17 18 19 20 21 22 23 q2 [GeV2] q2 [GeV2] → → FIG. 8. Relative change of the form-factor central value under the considered ﬁt variations for B π(cid:96)ν (upper) and B K(cid:96)ν s (lower). In each plot, the shaded band shows the statistical uncertainty of the preferred ﬁt. The three vertical lines show the location of the synthetic data points used in the subsequent extrapolation to q2 = 0. TABLE VII. Percentage errors from mismatches in the action and current for the bottom quark on the 243 and 323 ensembles. For this estimate, we calculate the mismatch functions for the nonperturbatively tuned parameters of the RHQ action from Table II. We estimate the size of operators using HQET power counting with Λ = 500 MeV and the coupling constant QCD αMS(1/a) = 1/3 on the 243 ensemble and 0.22 on the 323 ensembles. To obtain the total, we add the individual errors in s quadrature, including each contribution the number of times that operator occurs. Contribution E is counted twice, and 3 is counted twice for f(cid:107) and four times for f⊥. The deﬁnitions of operators “E,” “X ,” “X ,” “Y,” and “3” and expressions for 1 2 the mismatch functions are given in Appendix B of Ref. [14]. O O O (a2) error (a2) errors (α2a) error s from action from current from current Total (%) E X X Y 3 f(cid:107) f⊥ ≈ 1 2 a 0.11 fm 0.55 0.67 1.27 1.34 1.48 2.97 3.64 ≈ a 0.086 fm 0.42 0.46 0.85 0.91 0.55 1.65 1.82 E. Heavy-quark discretization errors we use the same power-counting approach as in our com- panion papers on bottomonium masses and splittings [18] and B -meson decay constants [14]. The RHQ action gives rise to nontrivial lattice-spacing (s) ∼ dependence in the form factors in the region m a 1. 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text: 19 parametrizations. Two such parametrizations commonly → TABLE IX. Normalized statistical (upper) and systematic → used to extrapolate the B π(cid:96)ν form factor are by (lower) correlation matrices for the B K(cid:96)ν form factors s Boyd, Grinstein, and Lebed (BGL) [24] and Bourrely, at three representative q2 values. Caprini, and Lellouch (BCL) [25]. fBsK fBsK The change of var(cid:112)iables from q2 (cid:112)to z is given by + 0 q2 [GeV2] 17.6 20.8 23.4 17.6 20.8 23.4 (cid:112) (cid:112) 17.6 1.000 0.868 0.828 0.799 0.754 0.702 − − − f+BsK 20.8 0.868 1.000 0.783 0.677 0.799 0.764 z(q2, t0) = 1 − q2/t+ 1 − t0/t+ , (39) 23.4 0.828 0.783 1.000 0.615 0.703 0.708 1 q2/t + 1 t /t + 0 + 17.6 0.799 0.677 0.615 1.000 0.828 0.755 ≡ ≡ − fBsK 20.8 0.754 0.799 0.703 0.828 1.000 0.974 where t (M +M )2 and t− (M M )2. This 0 23.4 0.702 0.764 0.708 0.755 0.974 1.000 transform+ation mB(asp) s theπsemileptonic reBg(iso) n 0 <π q2 < t− → onto a unit circle in the complex z plane. The B π(cid:96)ν → f+BsK f0BsK and Bs K(cid:96)ν form factors can then be expanded as a q2 [GeV2] 17.6 20.8 23.4 17.6 20.8 23.4 simple power series in z: (cid:88) 17.6 1.000 0.939 0.921 0.865 0.843 0.808 fBsK 20.8 0.939 1.000 0.913 0.794 0.860 0.835 ∞ + 23.4 0.921 0.914 1.000 0.760 0.806 0.801 P (q2)φ (q2, t )f (q2) = a(k)(t )z(q2, t )k, (40) 17.6 0.865 0.794 0.760 1.000 0.889 0.840 i i 0 i i 0 0 fBsK 20.8 0.843 0.860 0.806 0.889 1.000 0.983 k=0 0 { } 23.4 0.808 0.835 0.801 0.840 0.983 1.000 where i = 0, + for the scalar and vector form fac- tors, respectively. The free parameter t in Eq. (39) de- | | 0 termines the range of z in the semileptonic region, and and Lebed (BGL) as a cross-check. Next, in Sec. we ex- hence can be chosen to accelerate the series convergence. trapolate our synthetic lattice data to q2 = 0; we present The “Blaschke factor” Pi(q2) must be chosen to vanish our preferred results for f and f in Tables XI and XII at any subthreshold poles to preserve the correct ana- + 0 as coeﬃcients of the z-expansion and the matrix of cor- lytic stru−cture of fi(q2). For f+, the relevant state is the relations between them. JP = 1 meson, while for f0, the relevant state is the Use of the z-parametrization to describe semileptonic JP∗ = 0+ meson. As discussed earlier in Sec. , the scalar form factors has several advantages over other functional B meson has not been observed experimentally, but is forms used in| t|he literature [54, 55]. Because the abso- predicted to have a mass well above the Bπ prod→uction lute value of z is small in the semileptonic region, and threshold [44, 45]. Thus the functions Pi for B π(cid:96)ν the z-coeﬃcients are constrained to be small by unitarity are typically taken in the literature to be and heavy-quark symmetry, one needs only the ﬁrst few terms in the expansion to accurately describe the form PBπ(q2) = z(q2, M ∗) , PBπ(q2) = 1 . (41) factor shape with a negligible truncation error. More- + B 0 over, as the precisions of both the lattice calculations and experimental measurements improve, one may easily Finally, the outer function φ (q2, t ) can be any ana- i 0 include higher-order terms in z as needed. Finally, com- lytic function of q2; diﬀerent choices for φ correspond to i parisons of the z-expansion parameters resulting from diﬀerent z-parametrizations. → → ﬁts to diﬀerent theoretical or experimental data sets en- The form factors that describe B π(cid:96)ν (B K(cid:96)ν) s able a meaningful quantitative comparison of the shapes, in the range 0 < q2 < t−, when analytically continued, while a combined ﬁt to lattice an|d ex|perimental data en- also describe Bπ (BK) production for q2 > t+. The ables a clean determination of Vub . The z-expansion coeﬃcients of the z-expansion are therefore bounded by has therefore been a|dopt|ed as the preferred method for the fact that the rate of production of Bπ (BK) states obtaining exclusive Vub by experimentalists on Babar is less →than the production rate of all states coupling to and Belle, the Heavy Flavor Averaging Group, and the the b u current. In Ref. [24], Boyd, Grinstein, and Particle Data Group [2–5, 10, 56]. Lebed choose the outer function φ so that the unitarity i A. z-expansions of semileptonic form factors constraint on the series coeﬃcients takes a particularly simple form: → → (cid:88) (cid:16) (cid:17) The B π(cid:96)ν and B K(cid:96)ν form factors are ana- s lytic functions of q2 except at physical poles and branch N cuts above the production threshold. Therefore, given a a(k) 2 ∼< 1, (42) i suitable change of variables, they can be expressed as a k=0 convergent power series (see, e.g., [24, 25, 57–60]). Uni- tarity and heavy-quark power counting bound the size where this holds for any value of N. The explicit of the series coeﬃcients. In the literature, the new vari- functions for φBGL and φBGL and their numerical val- + 0 able is called z, and the class of functions are called z- ues can be found in Ref. [60]. When using the BGL ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: parametrizations.

text: Two

text: such

text: parametrizations

text: commonly

text: TABLE

text: IX.

text: Normalized

label: caption

text: statistical

text: (upper)

text: and

text: systematic

text: used

text: to

text: extrapolate

text: the

text: B

text: form

text: factor

text: are

text: by

text: (lower)

text: correlation

text: matrices

text: for

text: the

text: B

text: form

text: factors

text: s

text: Boyd,

text: Grinstein,

text: and

text: Lebed

text: (BGL)

text: and

text: Bourrely,

text: at

text: three

text: representative

text: values.

text: Caprini,

text: and

text: Lellouch

text: (BCL)

text: fBsK

label: table

text: fBsK

text: The

text: change

text: of

label: equation

text: from

text: z

text: is

text: given

text: by

text: f+BsK

text: t

text: /t

text: fBsK

text: where

text: t

text: (M

text: +M

text: and

text: t−

text: (M

text: M

text: This

text: transform+ation

text: mB(asp)

text: s

text: theπsemileptonic

text: reBg(iso)

text: n

text: <π

text: t−

text: onto

text: a

text: unit

text: circle

text: in

text: the

text: complex

text: z

text: plane.

text: The

text: B

text: f+BsK

text: and

text: Bs

text: form

text: factors

text: can

text: then

text: be

text: expanded

text: as

text: a

text: simple

text: power

text: series

text: in

text: z:

text: fBsK

text: P

text: t

text: )f

text: a(k)(t

text: t

text: )k,

text: i

text: i

text: i

text: i

text: fBsK

text: where

text: i

text: for

text: the

text: scalar

text: and

text: vector

text: form

text: fac-

text: tors,

text: respectively.

text: The

text: free

text: parameter

text: t

text: in

text: Eq.

text: de-

text: termines

text: the

text: range

text: of

text: z

text: in

text: the

text: semileptonic

text: region,

text: and

text: and

text: Lebed

text: (BGL)

text: as

text: a

text: cross-check.

text: Next,

text: in

text: Sec.

text: we

text: ex-

text: hence

text: can

text: be

text: chosen

text: to

text: accelerate

text: the

text: series

text: convergence.

text: trapolate

text: our

text: synthetic

text: lattice

text: data

text: to

text: we

text: present

text: The

text: “Blaschke

text: factor”

text: must

text: be

text: chosen

text: to

text: vanish

text: our

text: preferred

text: results

text: for

text: f

text: and

text: f

text: in

text: Tables

text: XI

text: and

text: XII

text: at

text: any

text: subthreshold

text: poles

text: to

text: preserve

text: the

text: correct

text: ana-

text: as

text: coeﬃcients

text: of

text: the

text: z-expansion

text: and

text: the

text: matrix

text: of

text: cor-

text: lytic

text: stru−cture

text: of

text: For

text: f+,

text: the

text: relevant

text: state

text: is

text: the

text: relations

text: between

text: them.

text: JP

text: meson,

text: while

text: for

text: the

text: relevant

text: state

text: is

text: the

text: Use

text: of

text: the

text: z-parametrization

text: to

text: describe

text: semileptonic

text: JP∗

text: meson.

text: As

text: discussed

text: earlier

text: in

text: Sec.

text: the

text: scalar

text: form

text: factors

text: has

text: several

text: advantages

text: over

text: other

text: functional

text: B

text: meson

text: has

text: not

text: been

text: observed

text: experimentally,

text: but

text: is

text: forms

text: used

text: in|

text: t|he

text: literature

text: Because

text: the

text: abso-

text: predicted

text: to

text: have

text: a

text: mass

text: well

text: above

text: the

text: Bπ

text: prod→uction

text: lute

text: value

text: of

text: z

text: is

text: small

text: in

text: the

text: semileptonic

text: region,

text: and

text: threshold

text: Thus

text: the

text: functions

text: Pi

text: for

text: B

text: the

text: z-coeﬃcients

text: are

text: constrained

text: to

text: be

text: small

text: by

text: unitarity

text: are

text: typically

text: taken

text: in

text: the

text: literature

text: to

text: be

text: and

text: heavy-quark

text: symmetry,

text: one

text: needs

text: only

text: the

text: ﬁrst

text: few

text: terms

text: in

text: the

text: expansion

text: to

text: accurately

text: describe

text: the

text: form

text: M

text: factor

text: shape

text: with

text: a

text: negligible

text: truncation

text: error.

text: More-

text: B

text: over,

text: as

text: the

text: precisions

text: of

text: both

text: the

text: lattice

text: calculations

text: and

text: experimental

text: measurements

text: improve,

text: one

text: may

text: easily

text: Finally,

text: the

text: outer

text: function

text: φ

text: t

text: can

text: be

text: any

text: ana-

text: i

text: include

text: higher-order

text: terms

text: in

text: z

text: as

text: needed.

text: Finally,

text: com-

text: lytic

text: function

text: of

text: diﬀerent

text: choices

text: for

text: φ

text: correspond

text: to

text: i

text: parisons

text: of

text: the

text: z-expansion

text: parameters

text: resulting

text: from

text: diﬀerent

text: z-parametrizations.

text: ﬁts

text: to

text: diﬀerent

text: theoretical

text: or

text: experimental

text: data

text: sets

text: en-

text: The

text: form

text: factors

text: that

text: describe

text: B

text: (B

text: s

text: able

text: a

text: meaningful

text: quantitative

text: comparison

text: of

text: the

text: shapes,

text: in

text: the

text: range

text: t−,

text: when

text: analytically

text: continued,

text: while

text: a

text: combined

text: ﬁt

text: to

text: lattice

text: an|d

text: ex|perimental

text: data

text: en-

text: also

text: describe

text: Bπ

text: (BK)

text: production

text: for

text: t+.

text: The

text: ables

text: a

text: clean

text: determination

text: of

text: Vub

text: The

text: z-expansion

text: coeﬃcients

text: of

text: the

text: z-expansion

text: are

text: therefore

text: bounded

text: by

text: has

text: therefore

text: been

text: a|dopt|ed

text: as

text: the

text: preferred

text: method

text: for

text: the

text: fact

text: that

text: the

text: rate

text: of

text: production

text: of

text: Bπ

text: (BK)

text: states

text: obtaining

text: exclusive

text: Vub

text: by

text: experimentalists

text: on

text: Babar

text: is

text: less

text: →than

text: the

text: production

text: rate

text: of

text: all

text: states

text: coupling

text: to

text: and

text: Belle,

text: the

text: Heavy

text: Flavor

text: Averaging

text: Group,

text: and

text: the

text: the

text: b

text: u

text: current.

text: In

text: Ref.

text: Boyd,

text: Grinstein,

text: and

text: Particle

text: Data

text: Group

text: Lebed

text: choose

text: the

text: outer

text: function

text: φ

text: so

text: that

text: the

text: unitarity

text: i

text: A.

text: z-expansions

text: of

text: semileptonic

text: form

text: factors

text: constraint

text: on

text: the

text: series

text: coeﬃcients

text: takes

text: a

text: particularly

text: simple

text: form:

text: The

text: B

text: and

text: B

text: form

text: factors

text: are

text: ana-

text: s

text: lytic

text: functions

text: of

text: except

text: at

text: physical

text: poles

text: and

text: branch

text: N

text: cuts

text: above

text: the

text: production

text: threshold.

text: Therefore,

text: given

text: a

text: a(k)

text: i

text: suitable

text: change

text: of

text: variables,

text: they

text: can

text: be

text: expressed

text: as

text: a

text: convergent

text: power

text: series

text: (see,

text: e.g.,

text: Uni-

text: tarity

text: and

text: heavy-quark

text: power

text: counting

text: bound

text: the

text: size

text: where

text: this

text: holds

text: for

text: any

text: value

text: of

text: N.

text: The

text: explicit

text: of

text: the

text: series

text: coeﬃcients.

text: In

text: the

text: literature,

text: the

text: new

text: vari-

text: functions

text: for

text: φBGL

text: and

text: φBGL

text: and

text: their

text: numerical

text: val-

text: able

text: is

text: called

text: z,

text: and

text: the

text: class

text: of

text: functions

text: are

text: called

text: z-

text: ues

text: can

text: be

text: found

text: in

text: Ref.

text: When

text: using

text: the

text: BGL

text: ##LTLine##

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text: 3 values that span the range accessible in our numerical simulations; for clarity, we discuss each source of system- The vector and scalar form factors can be easily ob- atic uncertainty in a separate subsection. In Section we tained from f(cid:107) and f⊥ via extrapolate our form-factor data to q2 = 0 using a model- (cid:104) (cid:0) (cid:1) (cid:105) (cid:112) independent z parametrization. We present our results for f (q2) and f (q2) as the coeﬃcients of the series in 1 − + 0 f (q2) = f(cid:107)(E ) + M E f⊥(E ) , z and the matrix of correlations between them; this pro- + (cid:112)2M P B(s) P P B(s) (cid:104)(cid:0) (cid:1) vides a model-independent parametrization of the form (6) factors valid over the entire allowed kinematic range. We illustrate the phenomenological utility of our form-factor f (q2) = (cid:0) 2M−B(s)(cid:1) M (cid:105) − E f(cid:107)(E ) results in Sec. . First, in Sec. , we perform a combined 0 M2 M2 B(s) P P z-ﬁt of our numerical B → π(cid:96)ν form-factor data with B(s−) P the experimental measurements of the b|ranc|hing frac- + EP2 MP2 f⊥(EP ) . (7) tion from BaBar and Belle to determine V . Next, in ub Sec. , we make predictions for Standard-Model observ- → → ables for the decay processes B π(cid:96)ν and B K(cid:96)ν B. Actions and parameters s with (cid:96) = µ, τ in anticipation of future experimental mea- surements. Section concludes with a comparison of our We use the (2 + 1)-ﬂavor domain-wall fermion and results with other lattice determinations, and with an Iwasaki gauge-ﬁeld ensembles generated by the RBC and outlook for the future. UKQCD collaborations [15, 16]. We perform measure- II. LATTICE CALCULATION ments at ﬁve diﬀerent light sea-quark masses m and at ≈ − ≈ l two lattice spacings of a 0.11 fm (a 1 1.729 GeV) ≈ − ≈ Here we present the setup of our numerical lattice cal- and a 0.086 fm (a 1 2.281 GeV). The light sea-quark (cid:46) culation. masses m correspond to pion masses of 289 MeV (cid:46) l A. Form factors M 422 MeV. The up and down sea-quark masses are π degenerate and the strange sea-quark mass m is tuned → → h The B π(cid:96)ν and B K(cid:96)ν semileptonic form factors within 10% of its physical value. The spatial volumes s → ≥ parametrize the hadronic matrix element of the b u are approximately (2.6 fm)3, such that M L 4. We V ≡ π vector current µ uγ(cid:32)µb: (cid:33) summarize the simulation parameters in Table I. In the valence sector we use for the light quarks the − domain-wall action [35, 36] and generate propagators (cid:104) |V | (cid:105) − M2 M2 P µ B = f (q2) pµ + pµ B(s) P qµ with periodic boundary conditions in space and time and (s) + B(s) P q2 with the same domain-wall height (M = 1.8) and extent 5 − M2 M2 of the ﬁfth dimension (Ls = 16) as in the sea sector. We + f (q2) B(s) P qµ , (2) generate both unitary light valence-quark propagators 0 q2 with the same mass as the light sea quarks and propa- gators with a mass close to the physical strange quark. where f+(q2) and f0(q2) are the vector and scalar form On the coarser ensembles we choose ams = 0.0343 and factors, respectively. It is convenient in lattice simula- on the ﬁner ensembles am = 0.0272. s tions to instead calculate the form factors f(cid:107)(E ) and P For the bottom quarks, we use the Columbia version f⊥(E ), which a(cid:113)re deﬁned(cid:2) by (cid:3) P of the relativistic heavy quark (RHQ) action [17] to con- trol heavy-quark discretization errors introduced by the O (cid:104)P|Vµ|B (cid:105) = 2M vµf(cid:107)(E ) + pµ⊥f⊥(E ) , (3) large lattice b-quark mass. We use the anisotropic (a) (s) B(s) P P improved Wilson-clover action with the following three parameters: the bare-quark mass m a, clover coeﬃcient 0 where E≡P is the outgoing light pseudoscalar meson en- cP , and anisotropy parameter ζ. In this work we use the ergy, vµ pµ /M is the B -meson velocity, and RHQ parameters tuned nonperturbatively in Ref. [18] to pµ⊥ ≡ pµ − (Bp(s) · v)Bv(µs). In the B(s) -meson rest frame, reproduce the experimentally measured Bs-meson mass which wPe will uPse for our simulation(ss), f(cid:107) and f⊥ are pro- and hyperﬁne splitting; we list their values in Table II. portional to the hadronic matrix elements of the tempo- We reduce autocorrelations between our lattices by ral and spatial vector currents: shifting the gauge ﬁelds by a random 4-vector before cre- ating the sources for the valence-quark propagators used (cid:112) (cid:104) |V | (cid:105) in the 2-point and 3-point correlation functions. This P 0 B f(cid:107)(E ) = (s) , (4) random 4-vector shift is equivalent to placing the sources P (cid:112)2M at random positions in spacetime but simpliﬁes the sub- (cid:104) |V | B(s)(cid:105) sequent analysis. On the ﬁner ensembles, we double the P i B 1 f⊥(E ) = (s) . (5) statistics by using two sources per conﬁguration sepa- P 2M pi B(s) P rated by half the lattice temporal extent. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: values

text: that

text: span

text: the

text: range

text: accessible

text: in

text: our

text: numerical

text: simulations;

text: for

text: clarity,

text: we

text: discuss

text: each

text: source

text: of

text: system-

text: The

text: vector

text: and

text: scalar

text: form

text: factors

text: can

text: be

text: easily

text: ob-

text: atic

text: uncertainty

text: in

text: a

text: separate

text: subsection.

text: In

text: Section

text: we

text: tained

text: from

text: and

text: f⊥

text: via

text: extrapolate

text: our

text: form-factor

text: data

text: to

text: using

text: a

text: model-

text: independent

text: z

text: parametrization.

text: We

text: present

text: our

text: results

text: for

text: f

text: and

text: f

text: as

text: the

text: coeﬃcients

text: of

text: the

text: series

text: in

text: f

text: M

text: E

text: f⊥(E

text: z

text: and

text: the

text: matrix

text: of

text: correlations

text: between

text: them;

text: this

text: pro-

text: P

text: B(s)

text: P

text: P

text: B(s)

text: vides

text: a

text: model-independent

text: parametrization

text: of

text: the

text: form

text: factors

text: valid

text: over

text: the

text: entire

text: allowed

text: kinematic

text: range.

text: We

text: illustrate

text: the

text: phenomenological

text: utility

text: of

text: our

text: form-factor

text: f

text: M

text: E

text: results

text: in

text: Sec.

text: First,

text: in

text: Sec.

text: we

text: perform

text: a

text: combined

text: B(s)

text: P

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text: Modern Studies of the Deuteron: from the Lab Frame to the Light Front Werner Boeglin and Misak Sargsian Department of Physics, Florida International University, Miami, FL 33199 (Dated: May 27, 2020) Abstract We review the recent progress made in studies of deuteron structure at small internucleon dis- tances. This progress is largely facilitated by the new generation of experiments in deuteron elec- trodisintegration carried out at unprecedentedly high momentum transfer. The theoretical analysis of these data conﬁrms the onset of the high energy eikonal regime in the scattering process which allows one to separate long range nuclear eﬀects from the eﬀects genuinely related to the short distance structure of the deuteron. Our conclusion is that for the ﬁrst time the deuteron is probed at relative momenta beyond 300 MeV/c without dominating long range eﬀects. As a result, at these large nucleon momenta the cross section is sensitive to the nuclear dynamics at sub-fermi distances. Due to large internal momenta involved we are dealing with the relativistic bound state that is best described by the light-cone momentum distribution of nucleons in the deuteron. We present the ﬁrst attempt of extracting the deuteron light-cone momentum distribution function from data and discuss the importance of this quantity for studies of QCD structure of the bound nucleon in deep inelastic scattering oﬀ the deuteron. We conclude the review giving an outlook of the next generation of high energy experiments which will extend our reach to much smaller distances in the deuteron. PACS numbers: 21.45.Bc,25.30.-c,27.10.+h,11.80.Fv 1

text: Modern

label: title

text: Studies

text: of

text: the

text: Deuteron:

text: from

text: the

text: Lab

text: Frame

text: to

text: the

text: Light

text: Front

text: Werner

label: author

text: Boeglin

text: and

text: Misak

text: Sargsian

text: Department

text: of

text: Physics,

text: Florida

text: International

text: University,

text: Miami,

text: FL

text: (Dated:

text: May

label: date

text: Abstract

text: We

label: abstract

text: review

text: the

text: recent

text: progress

text: made

text: in

text: studies

text: of

text: deuteron

text: structure

text: at

text: small

text: internucleon

text: dis-

text: tances.

text: This

text: progress

text: is

text: largely

text: facilitated

text: by

text: the

text: new

text: generation

text: of

text: experiments

text: in

text: deuteron

text: elec-

text: trodisintegration

text: carried

text: out

text: at

text: unprecedentedly

text: high

text: momentum

text: transfer.

text: The

text: theoretical

text: analysis

text: of

text: these

text: data

text: conﬁrms

text: the

text: onset

text: of

text: the

text: high

text: energy

text: eikonal

text: regime

text: in

text: the

text: scattering

text: process

text: which

text: allows

text: one

text: to

text: separate

text: long

text: range

text: nuclear

text: eﬀects

text: from

text: the

text: eﬀects

text: genuinely

text: related

text: to

text: the

text: short

text: distance

text: structure

text: of

text: the

text: deuteron.

text: Our

text: conclusion

text: is

text: that

text: for

text: the

text: ﬁrst

text: time

text: the

text: deuteron

text: is

text: probed

text: at

text: relative

text: momenta

text: beyond

text: MeV/c

text: without

text: dominating

text: long

text: range

text: eﬀects.

text: As

text: a

text: result,

text: at

text: these

text: large

text: nucleon

text: momenta

text: the

text: cross

text: section

text: is

text: sensitive

text: to

text: the

text: nuclear

text: dynamics

text: at

text: sub-fermi

text: distances.

text: Due

text: to

text: large

text: internal

text: momenta

text: involved

text: we

text: are

text: dealing

text: with

text: the

text: relativistic

text: bound

text: state

text: that

text: is

text: best

text: described

text: by

text: the

text: light-cone

text: momentum

text: distribution

text: of

text: nucleons

text: in

text: the

text: deuteron.

text: We

text: present

text: the

text: ﬁrst

text: attempt

text: of

text: extracting

text: the

text: deuteron

text: light-cone

text: momentum

text: distribution

text: function

text: from

text: data

text: and

text: discuss

text: the

text: importance

text: of

text: this

text: quantity

text: for

text: studies

text: of

text: QCD

text: structure

text: of

text: the

text: bound

text: nucleon

text: in

text: deep

text: inelastic

text: scattering

text: oﬀ

text: the

text: deuteron.

text: We

text: conclude

text: the

text: review

text: giving

text: an

text: outlook

text: of

text: the

text: next

text: generation

text: of

text: high

text: energy

text: experiments

text: which

text: will

text: extend

text: our

text: reach

text: to

text: much

text: smaller

text: distances

text: in

text: the

text: deuteron.

text: PACS

text: numbers:


text: )3mf(  der distributions clearly demonstrate the onset of the eikonal regime for FSI, Fig.3. Thus one expects that the momentum dependence of the cross section of the reaction (1) measured at θ ≤ 400 directions and p > 300 MeV/c momenta will be related to the genuine high nq r momentum component of the deuteron wave function. Such a dependence was extracted from the experimental data of Ref.[32] in which the re- duced cross section (6) was measured for ﬁxed Q2 = 3.5 GeV2 for recoil nucleon momenta up to 550 MeV/c. The ability of this experiment to separate narrow θ bins in the momentum nq distribution was important for separating regions with diﬀerent degrees of FSI contribution. ✓ = 35 ± 5 ✓ = 45 ± 5 ✓ = 75 ± 5 nq nq nq p (GeV /c) m FIG. 4: The experimental reduced cross sections (momentum distributions) for three values of the recoil angle θ . The solid lines indicate calculations[24] including FSI and the dashed ones are nq PWIA calculations. As an illustration in Fig.4 the missing momentum dependence for ﬁxed recoil (neutron) angles θ = 35, 45 and 75 degrees are shown for the reduced cross section deﬁned in Eq.(6). nq For comparison, theoretical calculations of Ref.[24] with and without FSI are also shown. For the large recoil angle, (θ = 750) corresponding to an electron kinematics of x ≈ 1 nq B FSI dominates for missing momenta above 0.3 GeV/c and the sensitivity to the choice of the deuteron wave function is greatly reduced. In fact the momentum dependence in this case strongly resembles that of the low and intermediate energies (Fig.2). The momentum dependence of the reduced cross section at large p qualitatively changes m at small angles. As Fig.4 shows for θ = 350 and 450 FSI contributes signiﬁcantly less and nq the reduced cross section is much more sensitive to the details of the high momentum distribution, allowing us for the ﬁrst time to discriminate between diﬀerent models of the deuteron wave function. 15 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: der

text: distributions

text: clearly

text: demonstrate

text: the

text: onset

text: of

text: the

text: eikonal

text: regime

text: for

text: FSI,

text: Thus

text: one

text: expects

text: that

text: the

text: momentum

text: dependence

text: of

text: the

text: cross

text: section

text: of

text: the

text: reaction

text: measured

text: at

text: θ

text: directions

text: and

text: p

text: MeV/c

text: momenta

text: will

text: be

text: related

text: to

text: the

text: genuine

text: high

text: nq

text: r

text: momentum

text: component

text: of

text: the

text: deuteron

text: wave

text: function.

text: Such

text: a

text: dependence

text: was

text: extracted

text: from

text: the

text: experimental

text: data

text: of

text: in

text: which

text: the

text: re-

text: duced

text: cross

text: section

text: was

text: measured

text: for

text: ﬁxed

text: for

text: recoil

text: nucleon

text: momenta

text: up

text: to

text: MeV/c.

text: The

text: ability

text: of

text: this

text: experiment

text: to

text: separate

text: narrow

text: θ

text: bins

text: in

text: the

text: momentum

text: nq

text: distribution

text: was

text: important

text: for

text: separating

text: regions

text: with

text: diﬀerent

text: degrees

text: of

text: FSI

text: contribution.

text: nq

text: nq

text: nq

text: p

text: (GeV

text: /c)

text: m

text: FIG.

text: The

text: experimental

text: reduced

text: cross

text: sections

text: (momentum

text: distributions)

text: for

text: three

text: values

text: of

text: the

text: recoil

text: angle

text: θ

text: The

text: solid

text: lines

text: indicate

text: including

text: FSI

text: and

text: the

text: dashed

text: ones

text: are

text: nq

text: PWIA

text: calculations.

text: As

text: an

text: illustration

text: in

text: the

text: missing

text: momentum

text: dependence

text: for

text: ﬁxed

text: recoil

text: (neutron)

text: angles

text: θ

text: and

text: degrees

text: are

text: shown

text: for

text: the

text: reduced

text: cross

text: section

text: deﬁned

text: in

text: nq

text: For

text: comparison,

text: theoretical

text: calculations

text: of

text: with

text: and

text: without

text: FSI

text: are

text: also

text: shown.

text: For

text: the

text: large

text: recoil

text: angle,

text: (θ

text: corresponding

text: to

text: an

text: electron

text: kinematics

text: of

text: x

text: nq

text: B

text: FSI

text: dominates

text: for

text: missing

text: momenta

text: above

text: GeV/c

text: and

text: the

text: sensitivity

text: to

text: the

text: choice

text: of

text: the

text: deuteron

text: wave

text: function

text: is

text: greatly

text: reduced.

text: In

text: fact

text: the

text: momentum

text: dependence

text: in

text: this

text: case

text: strongly

text: resembles

text: that

text: of

text: the

text: low

text: and

text: intermediate

text: energies

text: The

text: momentum

text: dependence

text: of

text: the

text: reduced

text: cross

text: section

text: at

text: large

text: p

text: qualitatively

text: changes

text: m

text: at

text: small

text: angles.

text: As

text: shows

text: for

text: θ

text: and

text: FSI

text: contributes

text: signiﬁcantly

text: less

text: and

text: nq

text: the

text: reduced

text: cross

text: section

text: is

text: much

text: more

text: sensitive

text: to

text: the

text: details

text: of

text: the

text: high

text: momentum

text: distribution,

text: allowing

text: us

text: for

text: the

text: ﬁrst

text: time

text: to

text: discriminate

text: between

text: diﬀerent

text: models

text: of

text: the

text: deuteron

text: wave

text: function.

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text: to the speciﬁcs of these reactions only integrated properties of the deuteron momentum distribution are accessed. As a result the ability of using these reactions to probe the deuteron at extremely large internal momenta is rather restricted. The most direct way of probing the internal structure of the deuteron is to study the exclusive (cid:48) e + d → e + p + n (1) reaction in which one of the nucleons is struck by the incoming electron and the other is a spectator to the reaction. First such exclusive experiments were attempted in the early 60’s using electron probes of a few 100 MeV. Due to the very small duty factor of these early accelerators and the extremely small cross section of the coincidence reaction only measurements at small nucleon momenta were possible[14, 15]. Starting in the 70s, when the duty factor of electron accelerators increased, attempts to probe larger nucleon momenta were carried out providing ﬁrst cross section data for nucleon momenta up to 300 MeV/c [16, 17] and in the 80s cross section measurements at internal nucleon momenta beyond 300 MeV/c were carried out at SACLAY[18, 19]. The theoretical analysis of these data established that due to the lack of resolving power of the probe (Q2 (cid:28) M2 GeV2) the cross section of the reaction (1) at internal relative pn N momenta ≥ 300 MeV/c was dominated by long range processes such as ﬁnal state interaction, intermediate isobar contributions as well as meson exchange currents. The situation changed only after the appearance of the high energy and high duty cycle CEBAF accelerator, were it was, for the ﬁrst time, possible to perform dedicated large momentum transfer (Q2 > M2 GeV2) experiments of the reaction (1). N In this paper we review the chronology of the experimental studies of exclusive electro- disintegration of the deuteron (1) and elaborate how the increase of the transferred momen- tum made it possible to access and probe the smaller internal distances in the deuteron. We demonstrate the qualitative changes that high momentum transfer processes bring into the observables and the emergence of the new theoretical approximation in describing the reaction (1) in the high energy limit. One of the main changes in the high energy limit is the emergence of the light-front dynamics in probing the relativistic internal structure of the deuteron and the possibility of extracting the light-cone momentum distribution of the bound nucleon in the deuteron. We present the ﬁrst such extraction using the recent high momentum transfer data from Jeﬀerson Lab. We conclude the review demonstrating how the eikonal regime allows an extension of the deuteron studies to internal momenta in and above the GeV region. Such momenta can be experimentally studied at Jeﬀerson Lab after the 12 GeV energy upgrade. II. THE MODERN VIEW OF THE DEUTERON We start with decomposing the deuteron state vector into the Fock states restricted by the total spin, S=1 and isospin, T=0 quantum numbers of the deuteron: Ψ = Ψ + Ψ + Ψ ∗ + Ψ + Ψ · · · (2) d pn ∆∆ NN hc NNπ 3

text: to

text: the

text: speciﬁcs

text: of

text: these

text: reactions

text: only

text: integrated

text: properties

text: of

text: the

text: deuteron

text: momentum

text: distribution

text: are

text: accessed.

text: As

text: a

text: result

text: the

text: ability

text: of

text: using

text: these

text: reactions

text: to

text: probe

text: the

text: deuteron

text: at

text: extremely

text: large

text: internal

text: momenta

text: is

text: rather

text: restricted.

text: The

text: most

text: direct

text: way

text: of

text: probing

text: the

text: internal

text: structure

text: of

text: the

text: deuteron

text: is

text: to

text: study

text: the

text: exclusive

text: e

text: d

text: e

text: p

text: n

text: reaction

text: in

text: which

text: one

text: of

text: the

text: nucleons

text: is

text: struck

text: by

text: the

text: incoming

text: electron

text: and

text: the

text: other

text: is

text: a

text: spectator

text: to

text: the

text: reaction.

text: First

text: such

text: exclusive

text: experiments

text: were

text: attempted

text: in

text: the

text: early

text: using

text: electron

text: probes

text: of

text: a

text: few

text: MeV.

text: Due

text: to

text: the

text: very

text: small

text: duty

text: factor

text: of

text: these

text: early

text: accelerators

text: and

text: the

text: extremely

text: small

text: cross

text: section

text: of

text: the

text: coincidence

text: reaction

text: only

text: measurements

text: at

text: small

text: nucleon

text: momenta

text: were

text: Starting

text: in

text: the

text: when

text: the

text: duty

text: factor

text: of

text: electron

text: accelerators

text: increased,

text: attempts

text: to

text: probe

text: larger

text: nucleon

text: momenta

text: were

text: carried

text: out

text: providing

text: ﬁrst

text: cross

text: section

text: data

text: for

text: nucleon

text: momenta

text: up

text: to

text: MeV/c

text: and

text: in

text: the

text: cross

text: section

text: measurements

text: at

text: internal

text: nucleon

text: momenta

text: beyond

text: MeV/c

text: were

text: carried

text: out

text: at

text: The

text: theoretical

text: analysis

text: of

text: these

text: data

text: established

text: that

text: due

text: to

text: the

text: lack

text: of

text: resolving

text: power

text: of

text: the

text: probe

text: the

text: cross

text: section

text: of

text: the

text: reaction

text: at

text: internal

text: relative

text: pn

text: N

text: momenta

text: MeV/c

text: was

text: dominated

text: by

text: long

text: range

text: processes

text: such

text: as

text: ﬁnal

text: state

text: interaction,

text: intermediate

text: isobar

text: contributions

text: as

text: well

text: as

text: meson

text: exchange

text: currents.

text: The

text: situation

text: changed

text: only

text: after

text: the

text: appearance

text: of

text: the

text: high

text: energy

text: and

text: high

text: duty

text: cycle

text: CEBAF

text: accelerator,

text: were

text: it

text: was,

text: for

text: the

text: ﬁrst

text: time,

text: possible

text: to

text: perform

text: dedicated

text: large

text: momentum

text: transfer

text: experiments

text: of

text: the

text: reaction

text: N

text: In

text: this

text: paper

text: we

text: review

text: the

text: chronology

text: of

text: the

text: experimental

text: studies

text: of

text: exclusive

text: electro-

text: disintegration

text: of

text: the

text: deuteron

text: and

text: elaborate

text: how

text: the

text: increase

text: of

text: the

text: transferred

text: momen-

text: tum

text: made

text: it

text: possible

text: to

text: access

text: and

text: probe

text: the

text: smaller

text: internal

text: distances

text: in

text: the

text: deuteron.

text: We

text: demonstrate

text: the

text: qualitative

text: changes

text: that

text: high

text: momentum

text: transfer

text: processes

text: bring

text: into

text: the

text: observables

text: and

text: the

text: emergence

text: of

text: the

text: new

text: theoretical

text: approximation

text: in

text: describing

text: the

text: reaction

text: in

text: the

text: high

text: energy

text: limit.

text: One

text: of

text: the

text: main

text: changes

text: in

text: the

text: high

text: energy

text: limit

text: is

text: the

text: emergence

text: of

text: the

text: light-front

text: dynamics

text: in

text: probing

text: the

text: relativistic

text: internal

text: structure

text: of

text: the

text: deuteron

text: and

text: the

text: possibility

text: of

text: extracting

text: the

text: light-cone

text: momentum

text: distribution

text: of

text: the

text: bound

text: nucleon

text: in

text: the

text: deuteron.

text: We

text: present

text: the

text: ﬁrst

text: such

text: extraction

text: using

text: the

text: recent

text: high

text: momentum

text: transfer

text: data

text: from

text: Jeﬀerson

text: Lab.

text: We

text: conclude

text: the

text: review

text: demonstrating

text: how

text: the

text: eikonal

text: regime

text: allows

text: an

text: extension

text: of

text: the

text: deuteron

text: studies

text: to

text: internal

text: momenta

text: in

text: and

text: above

text: the

text: GeV

text: region.

text: Such

text: momenta

text: can

text: be

text: experimentally

text: studied

text: at

text: Jeﬀerson

text: Lab

text: after

text: the

text: GeV

text: energy

text: upgrade.

text: II.

text: THE

text: MODERN

text: VIEW

text: OF

text: THE

text: DEUTERON

text: We

text: start

text: with

text: decomposing

text: the

text: deuteron

text: state

text: vector

text: into

text: the

text: Fock

text: states

text: restricted

text: by

text: the

text: total

text: spin,

text: and

text: isospin,

text: quantum

text: numbers

text: of

text: the

text: deuteron:

text: Ψ

text: Ψ

text: Ψ

text: Ψ

text: Ψ

text: Ψ

text: d

text: pn

text: NN

text: hc

text: NNπ