text: 6 tation of the feedback model, in cosmological and galactic SFR-density relation (to the best of our knowledge, that has scale simulations the slope of the small-scale SFR - density not been demonstrated, though, so we only mention this as a relation is preserved in the large-scale Kennicutt-Schmidt re- possibility). If that was the case, however, it would be really lation. bad news for the observers, implying that all of the existing Is this conclusion robust in real life? Barring unexpected large-scale studies of the Kennicutt-Schmidt relation are, in problems with simulations, one should never forget that, some sense, misleading, as they obscure the true non-linear while the observations measure the instantaneous value of relation between the instantaneous gas density and the star the molecular gas density, any large-scale measure of star formation rate. formation rate always returns a time averaged value - over a Irrespectively of the proper interpretation of our results, it is time scale of about 20 Myr for UV observations and about clear that studying the linear small-scale relation between the − 4 5 Myr for H-α measurements. Hence, the observational SFR and molecular gas density is as important as exploring determinations of the large-scale Kennicutt-Schmidt relation the “constant eﬃciency per free-fall time” ansatz. always compare apples and oranges, as was convincingly shown by, for example, Schruba et al. (2010). The mismatch between the measurements of the instanta- neous gas density and of the time-averaged star formation rate We are grateful to Andrey Kravtsov, Diederik Kruijssen, may, potentially, cause the emergence of the large-scale lin- and the anonymous referee for the constructive criticism that ear Kennicutt-Schmidt relation from a non-linear small-scale signiﬁcantly improved the original manuscript. REFERENCES Agertz, O., Kravtsov, A. V., Leitner, S. N., & Gnedin, N. Y. 2013, ApJ, 770, Lada, C. J., Lombardi, M., & Alves, J. F. 2010, ApJ, 724, 687 25 Leroy, A. K., Bigiel, F., de Blok, W. J. G., Boissier, S., Bolatto, A., Brinks, Agertz, O., Teyssier, R., & Moore, B. 2011, MNRAS, 410, 1391 E., Madore, B., Munoz-Mateos, J.-C., Murphy, E., Sandstrom, K., Bigiel, F., Leroy, A., Walter, F., Brinks, E., de Blok, W. J. G., Madore, B., & Schruba, A., & Walter, F. 2012, AJ, 144, 3 Thornley, M. D. 2008, AJ, 136, 2846 Leroy, A. K., Walter, F., Brinks, E., Bigiel, F., de Blok, W. J. G., Madore, B., Bigiel, F., Leroy, A. K., Walter, F., Brinks, E., de Blok, W. J. G., Kramer, C., & Thornley, M. D. 2008, AJ, 136, 2782 Rix, H. W., Schruba, A., Schuster, K., Usero, A., & Wiesemeyer, H. W. Leroy, A. K., Walter, F., Sandstrom, K., Schruba, A., Munoz-Mateos, J.-C., 2011, ApJ, 730, L13+ Bigiel, F., Bolatto, A., Brinks, E., de Blok, W. J. G., Meidt, S., Rix, H.-W., Bolatto, A. D., Leroy, A. K., Jameson, K., Ostriker, E., Gordon, K., Lawton, Rosolowsky, E., Schinnerer, E., Schuster, K.-F., & Usero, A. 2013, ArXiv B., Stanimirovic, S., Israel, F. P., Madden, S. C., Hony, S., Sandstrom, e-prints K. M., Bot, C., Rubio, M., Winkler, P. F., Roman-Duval, J., van Loon, McKee, C. F. & Ostriker, E. C. 2007, ARA&A, 45, 565 J. T., Oliveira, J. M., & Indebetouw, R. 2011, ArXiv e-prints Padoan, P. & Nordlund, Å. 2011, ApJ, 730, 40 Daddi, E., Bournaud, F., Walter, F., Dannerbauer, H., Carilli, C. L., Schaye, J., Vecchia, C. D., Booth, C. M., Wiersma, R. P. C., Theuns, T., Dickinson, M., Elbaz, D., Morrison, G. E., Riechers, D., Onodera, M., Haas, M. R., Bertone, S., Duﬀy, A. R., McCarthy, I. G., & van de Voort, Salmi, F., Krips, M., & Stern, D. 2010, ApJ, 713, 686 F. 2009, MNRAS, 1888 Feldmann, R., Gnedin, N. Y., & Kravtsov, A. V. 2011, ApJ, 732, 115 Schruba, A., Leroy, A. K., Walter, F., Sandstrom, K., & Rosolowsky, E. —. 2012, ApJ, 758, 127 2010, ApJ, 722, 1699 Genzel, R., Tacconi, L. J., Gracia-Carpio, J., Sternberg, A., Cooper, M. C., Tacconi, L. J., Neri, R., Genzel, R., Combes, F., Bolatto, A., Cooper, M. C., Shapiro, K., Bolatto, A., Bouche´, N., Bournaud, F., Burkert, A., Combes, Wuyts, S., Bournaud, F., Burkert, A., Comerford, J., Cox, P., Davis, M., F., Comerford, J., Cox, P., Davis, M., Schreiber, N. M. F., Garcia-Burillo, Fo¨rster Schreiber, N. M., Garc´ıa-Burillo, S., Gracia-Carpio, J., Lutz, D., S., Lutz, D., Naab, T., Neri, R., Omont, A., Shapley, A., & Weiner, B. Naab, T., Newman, S., Omont, A., Saintonge, A., Shapiro Griﬃn, K., 2010, MNRAS, 407, 2091 Shapley, A., Sternberg, A., & Weiner, B. 2013, ApJ, 768, 74 Hopkins, P. F., Keres, D., Onorbe, J., Faucher-Giguere, C.-A., Quataert, E., Tasker, E. J. 2011, ApJ, 730, 11 Murray, N., & Bullock, J. S. 2013, ArXiv e-prints The Enzo Collaboration, Bryan, G. L., Norman, M. L., O’Shea, B. W., Abel, Hopkins, P. F., Quataert, E., & Murray, N. 2011, MNRAS, 417, 950 T., Wise, J. H., Turk, M. J., Reynolds, D. R., Collins, D. C., Wang, P., Kravtsov, A. V. 2003, ApJ, 590, L1 Skillman, S. W., Smith, B., Harkness, R. P., Bordner, J., Kim, J.-h., Kritsuk, A. G., Lee, C. T., & Norman, M. L. 2013, MNRAS, 436, 3247 Kuhlen, M., Xu, H., Goldbaum, N., Hummels, C., Kritsuk, A. G., Tasker, Kritsuk, A. G., Norman, M. L., & Wagner, R. 2011, ApJ, 727, L20 E., Skory, S., Simpson, C. M., Hahn, O., Oishi, J. S., So, G. C., Zhao, F., Krumholz, M. R. & Tan, J. C. 2007, ApJ, 654, 304 Cen, R., & Li, Y. 2013, ArXiv e-prints

text: tation

text: of

text: the

text: feedback

text: model,

text: in

text: cosmological

text: and

text: galactic

text: SFR-density

text: relation

text: (to

text: the

text: best

text: of

text: our

text: knowledge,

text: that

text: has

text: scale

text: simulations

text: the

text: slope

text: of

text: the

text: small-scale

text: SFR

text: density

text: not

text: been

text: demonstrated,

text: though,

text: so

text: we

text: only

text: mention

text: this

text: as

text: a

text: relation

text: is

text: preserved

text: in

text: the

text: large-scale

text: Kennicutt-Schmidt

text: re-

text: possibility).

text: If

text: that

text: was

text: the

text: case,

text: however,

text: it

text: would

text: be

text: really

text: lation.

text: bad

text: news

text: for

text: the

text: observers,

text: implying

text: that

text: all

text: of

text: the

text: existing

text: Is

text: this

text: conclusion

text: robust

text: in

text: real

text: life?

text: Barring

text: unexpected

text: large-scale

text: studies

text: of

text: the

text: Kennicutt-Schmidt

text: relation

text: are,

text: in

text: problems

text: with

text: simulations,

text: one

text: should

text: never

text: forget

text: that,

text: some

text: sense,

text: misleading,

text: as

text: they

text: obscure

text: the

text: true

text: non-linear

text: while

text: the

text: observations

text: measure

text: the

text: instantaneous

text: value

text: of

text: relation

text: between

text: the

text: instantaneous

text: gas

text: density

text: and

text: the

text: star

text: the

text: molecular

text: gas

text: density,

text: any

text: large-scale

text: measure

text: of

text: star

text: formation

text: rate.

text: formation

text: rate

text: always

text: returns

text: a

text: time

text: averaged

text: value

text: over

text: a

text: Irrespectively

text: of

text: the

text: proper

text: interpretation

text: of

text: our

text: results,

text: it

text: is

text: time

text: scale

text: of

text: about

text: Myr

text: for

text: UV

text: observations

text: and

text: about

text: clear

text: that

text: studying

text: the

text: linear

text: small-scale

text: relation

text: between

text: the

text: Myr

text: for

text: H-α

text: measurements.

text: Hence,

text: the

text: observational

text: SFR

text: and

text: molecular

text: gas

text: density

text: is

text: as

text: important

text: as

text: exploring

text: determinations

text: of

text: the

text: large-scale

text: Kennicutt-Schmidt

text: relation

text: the

text: “constant

text: eﬃciency

text: per

text: free-fall

text: time”

text: ansatz.

text: always

text: compare

text: apples

text: and

text: oranges,

text: as

text: was

text: convincingly

text: shown

text: by,

text: for

text: example,

text: Schruba

text: et

text: al.

text: The

text: mismatch

text: between

text: the

text: measurements

text: of

text: the

text: instanta-

text: neous

text: gas

text: density

text: and

text: of

text: the

text: time-averaged

text: star

text: formation

text: rate

text: We

text: are

text: grateful

text: to

text: Andrey

text: Kravtsov,

text: Diederik

text: Kruijssen,

text: may,

text: potentially,

text: cause

text: the

text: emergence

text: of

text: the

text: large-scale

text: lin-

text: and

text: the

text: anonymous

text: referee

text: for

text: the

text: constructive

text: criticism

text: that

text: ear

text: Kennicutt-Schmidt

text: relation

text: from

text: a

text: non-linear

text: small-scale

text: signiﬁcantly

text: improved

text: the

text: original

text: manuscript.

text: REFERENCES

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text: S˜ ∋ σ ℓ(σ) n g n ti = lif S ∋ s ℓ (s) n c Figure 3: The strategy of lifting a circular permutation s to an aﬃne permutation σ in order to calculate its length. Thick arrows denote the mappings that are constructed in this paper. tions: s2 = 1 for each i = 1, . . . , n, i s s = s s if i − j mod n (cid:54)= ±1, and i j j i s s s = s s s for each i = 1, . . . , n − 1. i i+1 i i+1 i i+1 While the length function for the symmetric group with respect to the Coxeter-type presentation is well-studied (see for example Humphreys (1990)), the additional generator provided by the circle makes the relevant theorems inapplicable. Fortunately, there is another way to view this group using the aﬃne symmetric group S˜ . n In the remainder of this section we show how to lift a circular permutation to the aﬃne sym- metric group in such a way that the length in the aﬃne symmetric group is the minimal number of 2-region inversions required to sort the circular permutation (see Figure 3). Because we know how to calculate length in the aﬃne symmetric group (see next section), we can then use this to ﬁnd the length of the circular permutation. We will in fact be working often in the extended aﬃne symmetric group S˜ext, which we introduce in Section . This group is deﬁned in the same way as n the aﬃne symmetric group, but without the restriction on the sum of the images of a permutation given by Equation (2) below. 3.2 The aﬃne symmetric group Recall that n := {1, 2, . . . , n}. Z Z Deﬁnition 3.1 (aﬃne permutation). A bijection σ : → is an aﬃne permutation if it satisﬁes the following two conditions: periodic: Z σ(i + n) = σ(i) + n for all i ∈ (1) (cid:88) (cid:88) balanced: n(n + 1) σ(i) = i = (2) ∈ ∈ 2 i n i n following Lusztig (1983). The aﬃne symmetric group S˜ can be realized as the group of aﬃne permutations. The n symmetric group S can be identiﬁed with a subgroup of S˜ by extending each permutation from Z n n n to obeying the periodicity requirement, and can also be obtained by projecting from S˜ by n reducing domain and codomain mod n. In particular the generators s˜ of S˜ are the periodic i Zn permutations s˜ (i + kn) = i + 1 + kn and s˜ (i + 1 + kn) = i + kn for all k ∈ , and s˜ (j) = j if i i i j (cid:54)≡ i or i + 1 mod n. As with the symmetric group in relation to the Coxeter presentation, there is an easy formula for the minimum number of (aﬃne) transpositions s˜ required to represent an aﬃne permutation, due i to Shi (1986) (see also Shi (1994)). In our context, using the length function in S would not take n into account the additional generator s ; the key idea here is that by lifting a circular permutation n up to the aﬃne symmetric group the length function in S˜ accounts for all n generators. n 7 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S˜

label: paragraph

font: default

text: σ

text: ℓ(σ)

text: n

text: g

text: n

text: ti

text: lif

text: S

text: s

text: ℓ

text: (s)

text: n

text: c

text: Figure

text: The

label: caption

text: strategy

text: of

text: lifting

text: a

text: circular

text: permutation

text: s

text: to

text: an

text: aﬃne

text: permutation

text: σ

text: in

text: order

text: to

text: calculate

text: its

text: length.

text: Thick

text: arrows

text: denote

text: the

text: mappings

text: that

text: are

text: constructed

text: in

text: this

text: paper.

text: tions:

text: for

text: each

text: i

text: n,

text: i

text: s

text: s

text: s

text: s

text: if

text: i

text: j

text: mod

text: n

text: and

text: i

text: j

text: j

text: i

text: s

text: s

text: s

text: s

text: s

text: s

text: for

text: each

text: i

text: n

text: i

text: i

text: i

text: While

text: the

text: length

text: function

text: for

text: the

text: symmetric

text: group

text: with

text: respect

text: to

text: the

text: Coxeter-type

text: presentation

text: is

text: well-studied

text: (see

text: for

text: example

text: Humphreys

text: the

text: additional

text: generator

text: provided

text: by

text: the

text: circle

text: makes

text: the

text: relevant

text: theorems

text: inapplicable.

text: Fortunately,

text: there

text: is

text: another

text: way

text: to

text: view

text: this

text: group

text: using

text: the

text: aﬃne

text: symmetric

text: group

text: S˜

text: n

text: In

text: the

text: remainder

text: of

text: this

text: section

text: we

text: show

text: how

text: to

text: lift

text: a

text: circular

text: permutation

text: to

text: the

text: aﬃne

text: sym-

text: metric

text: group

text: in

text: such

text: a

text: way

text: that

text: the

text: length

text: in

text: the

text: aﬃne

text: symmetric

text: group

text: is

text: the

text: minimal

text: number

text: of

text: inversions

text: required

text: to

text: sort

text: the

text: circular

text: permutation

text: (see

text: Figure

text: Because

text: we

text: know

text: how

text: to

text: calculate

text: length

text: in

text: the

text: aﬃne

text: symmetric

text: group

text: (see

text: next

text: section),

text: we

text: can

text: then

text: use

text: this

text: to

text: ﬁnd

text: the

text: length

text: of

text: the

text: circular

text: permutation.

text: We

text: will

text: in

text: fact

text: be

text: working

text: often

text: in

text: the

text: extended

text: aﬃne

text: symmetric

text: group

text: S˜ext,

text: which

text: we

text: introduce

text: in

text: Section

text: This

text: group

text: is

text: deﬁned

text: in

text: the

text: same

text: way

text: as

text: n

text: the

text: aﬃne

text: symmetric

text: group,

text: but

text: without

text: the

text: restriction

text: on

text: the

text: sum

text: of

text: the

text: images

text: of

text: a

text: permutation

text: given

text: by

text: Equation

text: below.

text: The

text: aﬃne

text: symmetric

text: group

text: Recall

text: that

text: n

text: n}.

text: Z

text: Z

text: Deﬁnition

text: (aﬃne

text: permutation).

text: A

text: bijection

text: σ

text: is

text: an

text: aﬃne

text: permutation

text: if

text: it

text: satisﬁes

text: the

text: following

text: two

text: conditions:

text: periodic:

text: Z

text: σ(i

text: n)

text: σ(i)

text: n

text: for

text: all

text: i

text: balanced:

text: n(n

text: σ(i)

text: i

text: i

text: n

text: i

text: n

text: following

text: Lusztig

text: The

text: aﬃne

text: symmetric

text: group

text: S˜

text: can

text: be

text: realized

text: as

text: the

text: group

text: of

text: aﬃne

text: permutations.

text: The

text: n

text: symmetric

text: group

text: S

text: can

text: be

text: identiﬁed

text: with

text: a

text: subgroup

text: of

text: S˜

text: by

text: extending

text: each

text: permutation

text: from

text: Z

text: n

text: n

text: n

text: to

text: obeying

text: the

text: periodicity

text: requirement,

text: and

text: can

text: also

text: be

text: obtained

text: by

text: projecting

text: from

text: S˜

text: by

text: n

text: reducing

text: domain

text: and

text: codomain

text: mod

text: n.

text: In

text: particular

text: the

text: generators

text: s˜

text: of

text: S˜

text: are

text: the

text: periodic

text: i

text: Zn

text: permutations

text: s˜

text: (i

text: kn)

text: i

text: kn

text: and

text: s˜

text: (i

text: kn)

text: i

text: kn

text: for

text: all

text: k

text: and

text: s˜

text: (j)

text: j

text: if

text: i

text: i

text: i

text: j

text: i

text: or

text: i

text: mod

text: n.

text: As

text: with

text: the

text: symmetric

text: group

text: in

text: relation

text: to

text: the

text: Coxeter

text: presentation,

text: there

text: is

text: an

text: easy

text: formula

text: for

text: the

text: minimum

text: number

text: of

text: (aﬃne)

text: transpositions

text: s˜

text: required

text: to

text: represent

text: an

text: aﬃne

text: permutation,

text: due

text: i

text: to

text: Shi

text: (see

text: also

text: Shi

text: In

text: our

text: context,

text: using

text: the

text: length

text: function

text: in

text: S

text: would

text: not

text: take

text: n

text: into

text: account

text: the

text: additional

text: generator

text: s

text: the

text: key

text: idea

text: here

text: is

text: that

text: by

text: lifting

text: a

text: circular

text: permutation

text: n

text: up

text: to

text: the

text: aﬃne

text: symmetric

text: group

text: the

text: length

text: function

text: in

text: S˜

text: accounts

text: for

text: all

text: n

text: generators.

text: n

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3. As (i, k) ∈ I (σ) is equivalent to rn ≤ σ(k)−σ(i) < (r+1)n and (j, k) ∈ I (σ) is equivalent to r s −(s+1)n < σ(j)−σ(k) ≤ −sn, one has (r−s−1)n < σ(j)−σ(i) < (r−s+1)n, respectively (s−r−1)n < σ(i)−σ(j) < (s−r+1)n. If i < j, the former implies (i, j) ∈ I − − (σ)∪I − (σ), r s 1 r s while in the case j < i, the latter implies (j, i) ∈ I − − (σ) ∪ I − (σ). s r 1 s r The idea now is the following: Assume that α > 1 is the maximal integer such that I− (σ) ∪ α I (σ) is non-empty. This means that |α| is the maximal size of a crossing number in σ. We α will deﬁne two transformations that reduce this, by removing the pairs with crossing numbers −α (respectively α) without increasing the absolute value of any crossing number, except from 0 to 1. The proof of this claim uses the transitivity constraints in Lemma 3.6 and the maximality of α. (cid:40) Given an extended aﬃne permutation σ and a subset S ⊆ n, we deﬁne the extended aﬃne permutation σS by setting σ(i) + n if i ∈ S, σS(i) = σ(i) otherwise. That is, σS has the image of each element of S shifted by n. Note that this does not change the circular permutation that they correspond to.  Lemma 3.7. For an extended aﬃne permutation σ, if κi,j(σ) = r, then one has r + 1 if i ∈/ S and j ∈ S, κ (σS) = r − 1 if i ∈ S and j ∈/ S, i,j r otherwise. Proof. The claims follow immediately from Deﬁnition 3.4. The following two propositions show how to choose a subset S ⊆ n in such a way that the largest crossing number for σS is strictly lower than that of σ. (cid:8) (cid:9) Proposition 3.8. Let σ be an extended aﬃne permutation, and assume that α > 1 is the maximal integer such that I (σ) ∪ I− (σ) is non empty. α α ∅ For S = i ∈ n | ∃ j ∈ n , (i, j) ∈ I (σ) one has I (σS) = . Moreover, for any (i, j) ∈ P α α with κ (σ) (cid:54)= 0, one has |κ (σS)| ≤ |κ (σ)|. i,j i,j i,j Proof. Assume that (i, j) ∈ I (σ), so that κ (σ) = r, and suppose r i,j |κ (σS)| > |κ (σ)| = |r| > 0. i,j i,j Then, by Lemma 3.7, if r > 0 we have i ∈/ S and j ∈ S, and if r < 0 we have i ∈ S and j ∈/ S. If r > 0 then there exists k ∈ {j + 1, . . . , n} with (j, k) ∈ S = I (σ), so (i, k) ∈ I (σ) ∪ α α+r I (σ) by Lemma 3.6 contradicting the maximality of α. α+r+1 If r < 0 then there exists k ∈ {i + 1, . . . , n} such that (i, k) ∈ S = I (σ). By Lemma 3.6, either α j < k and (j, k) ∈ I − − (σ) ∪ I − (σ), or k < j and (k, j) ∈ I − − (σ) ∪ I − (σ). Since r < 0, α r 1 α r r α 1 r α the only situation compatible with the maximality of α is r = −1 and (j, k) ∈ I (σ), but in this α case j ∈ S, which is a contradiction. By Lemma 3.7, it only remains to show that κ (σS) (cid:54)= κ (σ) for any (i, j) ∈ I (σ). As i,j i,j α (i, j) ∈ I (σ) implies i ∈ S, one can only have κ (σS) = κ (σ) if j ∈ S, that is, if there exists α i,j i,j k ∈ {j + 1, . . . , n} with (j, k) ∈ S = I (σ). But then, by Lemma 3.6, (i, k) ∈ I (σ) ∪ I (σ), α 2α 2α+1 again contradicting the maximality of α. A similar argument deals with the case in which I (σ) is empty: α (cid:8) (cid:9) Proposition 3.9. Let σ be an extended aﬃne permutation, and assume that α > 1 is the maximal ∅ integer such that I (σ) ∪ I− (σ) is non empty. Moreover assume that I (σ) = . α α ∅ α For S = j ∈ n | ∃ i ∈ n , (i, j) ∈ I− (σ) one has I− (σS) = . Moreover, for any (i, j) ∈ P α α with κ (σ) (cid:54)= 0, one has |κ (σS)| ≤ |κ (σ)|. i,j i,j i,j Proof. The proof is similar to that of Proposition 3.8. 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: list

text: As

text: (i,

text: k)

text: I

text: (σ)

text: is

text: equivalent

text: to

text: rn

text: σ(k)−σ(i)

text: and

text: (j,

text: k)

text: I

text: (σ)

text: is

text: equivalent

text: to

text: r

text: s

text: σ(j)−σ(k)

text: −sn,

text: one

text: has

text: σ(j)−σ(i)

text: respectively

text: σ(i)−σ(j)

text: If

text: i

text: j,

text: the

text: former

text: implies

text: (i,

text: j)

text: I

text: (σ)∪I

text: (σ),

text: r

text: s

text: r

text: s

text: while

text: in

text: the

text: case

text: j

text: i,

text: the

text: latter

text: implies

text: (j,

text: i)

text: I

text: (σ)

text: I

text: (σ).

text: s

text: r

text: s

text: r

text: The

text: idea

text: now

text: is

text: the

text: following:

text: Assume

text: that

text: α

text: is

text: the

text: maximal

text: integer

text: such

text: that

text: I−

text: (σ)

text: α

text: I

text: (σ)

text: is

text: non-empty.

text: This

text: means

text: that

text: |α|

text: is

text: the

text: maximal

text: size

text: of

text: a

text: crossing

text: number

text: in

text: σ.

text: We

text: α

text: will

text: deﬁne

text: two

text: transformations

text: that

text: reduce

text: this,

text: by

text: removing

text: the

text: pairs

text: with

text: crossing

text: numbers

text: −α

text: (respectively

text: α)

text: without

text: increasing

text: the

text: absolute

text: value

text: of

text: any

text: crossing

text: number,

text: except

text: from

text: to

text: The

text: proof

text: of

text: this

text: claim

text: uses

text: the

text: transitivity

text: constraints

text: in

text: Lemma

text: and

text: the

text: maximality

text: of

text: α.

text: Given

text: an

text: extended

text: aﬃne

text: permutation

text: σ

text: and

text: a

text: subset

text: S

text: n,

text: we

text: deﬁne

text: the

text: extended

text: aﬃne

text: permutation

text: σS

text: by

text: setting

text: σ(i)

text: n

text: if

text: i

text: S,

text: σS(i)

text: σ(i)

text: otherwise.

text: That

text: is,

text: σS

text: has

text: the

text: image

text: of

text: each

text: element

text: of

text: S

text: shifted

text: by

text: n.

text: Note

text: that

text: this

text: does

text: not

text: change

text: the

text: circular

text: permutation

text: that

text: they

text: correspond

text: to.

text: Lemma

text: For

text: an

text: extended

text: aﬃne

text: permutation

text: σ,

text: if

text: κi,j(σ)

text: r,

text: then

text: one

text: has

text: r

text: if

text: i

text: S

text: and

text: j

text: S,

text: κ

text: (σS)

text: r

text: if

text: i

text: S

text: and

text: j

text: S,

text: i,j

text: r

text: otherwise.

text: Proof.

text: The

text: claims

text: follow

text: immediately

text: from

text: Deﬁnition

text: The

text: following

text: two

text: propositions

text: show

text: how

text: to

text: choose

text: a

text: subset

text: S

text: n

text: in

text: such

text: a

text: way

text: that

text: the

text: largest

text: crossing

text: number

text: for

text: σS

text: is

text: strictly

text: lower

text: than

text: that

text: of

text: σ.

text: Proposition

text: Let

text: σ

text: be

text: an

text: extended

text: aﬃne

text: permutation,

text: and

text: assume

text: that

text: α

text: is

text: the

text: maximal

text: integer

text: such

text: that

text: I

text: (σ)

text: I−

text: (σ)

text: is

text: non

text: empty.

text: α

text: α

text: For

text: S

text: i

text: n

text: j

text: n

text: (i,

text: j)

text: I

text: (σ)

text: one

text: has

text: I

text: (σS)

text: Moreover,

text: for

text: any

text: (i,

text: j)

text: P

text: α

text: α

text: with

text: κ

text: (σ)

text: one

text: has

text: |κ

text: (σS)|

text: |κ

text: (σ)|.

text: i,j

text: i,j

text: i,j

text: Proof.

text: Assume

text: that

text: (i,

text: j)

text: I

text: (σ),

text: so

text: that

text: κ

text: (σ)

text: r,

text: and

text: suppose

text: r

text: i,j

text: |κ

text: (σS)|

text: |κ

text: (σ)|

text: |r|

text: i,j

text: i,j

text: Then,

text: by

text: Lemma

text: if

text: r

text: we

text: have

text: i

text: S

text: and

text: j

text: S,

text: and

text: if

text: r

text: we

text: have

text: i

text: S

text: and

text: j

text: S.

text: If

text: r

text: then

text: there

text: exists

text: k

text: {j

text: n}

text: with

text: (j,

text: k)

text: S

text: I

text: (σ),

text: so

text: (i,

text: k)

text: I

text: (σ)

text: α

text: α+r

text: I

text: (σ)

text: by

text: Lemma

text: contradicting

text: the

text: maximality

text: of

text: α.

text: If

text: r

text: then

text: there

text: exists

text: k

text: {i

text: n}

text: such

text: that

text: (i,

text: k)

text: S

text: I

text: (σ).

text: By

text: Lemma

text: either

text: α

text: j

text: k

text: and

text: (j,

text: k)

text: I

text: (σ)

text: I

text: (σ),

text: or

text: k

text: j

text: and

text: (k,

text: j)

text: I

text: (σ)

text: I

text: (σ).

text: Since

text: r

text: α

text: r

text: α

text: r

text: r

text: α

text: r

text: α

text: the

text: only

text: situation

text: compatible

text: with

text: the

text: maximality

text: of

text: α

text: is

text: r

text: −1

text: and

text: (j,

text: k)

text: I

text: (σ),

text: but

text: in

text: this

text: α

text: case

text: j

text: S,

text: which

text: is

text: a

text: contradiction.

text: By

text: Lemma

text: it

text: only

text: remains

text: to

text: show

text: that

text: κ

text: (σS)

text: κ

text: (σ)

text: for

text: any

text: (i,

text: j)

text: I

text: (σ).

text: As

text: i,j

text: i,j

text: α

text: (i,

text: j)

text: I

text: (σ)

text: implies

text: i

text: S,

text: one

text: can

text: only

text: have

text: κ

text: (σS)

text: κ

text: (σ)

text: if

text: j

text: S,

text: that

text: is,

text: if

text: there

text: exists

text: α

text: i,j

text: i,j

text: k

text: {j

text: n}

text: with

text: (j,

text: k)

text: S

text: I

text: (σ).

text: But

text: then,

text: by

text: Lemma

text: (i,

text: k)

text: I

text: (σ)

text: I

text: (σ),

text: α

text: 2α

text: again

text: contradicting

text: the

text: maximality

text: of

text: α.

text: A

text: similar

text: argument

text: deals

text: with

text: the

text: case

text: in

text: which

text: I

text: (σ)

text: is

text: empty:

text: α

text: Proposition

text: Let

text: σ

text: be

text: an

text: extended

text: aﬃne

text: permutation,

text: and

text: assume

text: that

text: α

text: is

text: the

text: maximal

text: integer

text: such

text: that

text: I

text: (σ)

text: I−

text: (σ)

text: is

text: non

text: empty.

text: Moreover

text: assume

text: that

text: I

text: (σ)

text: α

text: α

text: α

text: For

text: S

text: j

text: n

text: i

text: n

text: (i,

text: j)

text: I−

text: (σ)

text: one

text: has

text: I−

text: (σS)

text: Moreover,

text: for

text: any

text: (i,

text: j)

text: P

text: α

text: α

text: with

text: κ

text: (σ)

text: one

text: has

text: |κ

text: (σS)|

text: |κ

text: (σ)|.

text: i,j

text: i,j

text: i,j

text: Proof.

text: The

text: proof

text: is

text: similar

text: to

text: that

text: of

text: Proposition

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Contents 1 Introduction 1 2 Regge–Wheeler equation 2 2.1 Spheroidal harmonics 2 2.2 Eﬀective potential 3 2.3 Positivity properties 4 2.4 Super-radiance 6 3 Non-super-radiant modes (m < m∗) 7 3.1 Zero-angular-momentum modes (m = 0) 7 (cid:54) 3.2 Non-zero-angular-momentum modes (m = 0) 10 3.2.1 Negative-angular-momentum modes (m < 0) 11 ∈ 3.2.2 Low-lying positive-angular-momentum modes (m (0, m∗) ) 12 3.3 Summary (non-super-radiant modes) 13 ≥ 4 Super-radiant modes (m m∗) 14 ∈ 4.1 Low-lying super-radiant modes (m [m∗, 2m∗) ) 16 ≥ 4.2 Highly super-radiant modes (m 2m∗) 18 4.3 Summary (super-radiant modes) 19 5 Discussion 20 1 Introduction The spacetime geometry of a black hole, in the region that interpolates between the horizon and spatial inﬁnity, (the domain of outer communication), generically acts as a potential barrier that partially reﬂects both ingoing and outgoing excitations. (See for instance [1–4].) In the case of outgoing excitations (Hawking quanta) the resulting transmission probabilities are called “greybody factors”. Calculation of these greybody factors, when practical, is based on analyzing the excitations in terms of a Regge– Wheeler equation, (or closely related variant thereof), which in the non-super-radiant case reduces the problem to a one-dimensional barrier-penetration problem. Even then, ﬁnding exact solutions is mostly impractical, and one typically resorts either to making semi-analytic or numerical estimates, or to deriving rigorous analytic – 1 – ##LTLine## ##LTLine##

text: Contents

text: Introduction

text: Regge–Wheeler

text: equation

text: Spheroidal

text: harmonics

text: Eﬀective

text: potential

text: Positivity

text: properties

text: Super-radiance

text: Non-super-radiant

text: modes

text: (m

text: m∗)

text: Zero-angular-momentum

text: modes

text: (m

text: Non-zero-angular-momentum

text: modes

text: (m

text: Negative-angular-momentum

text: modes

text: (m

text: Low-lying

text: positive-angular-momentum

text: modes

text: (m

text: m∗)

text: Summary

text: (non-super-radiant

text: modes)

text: Super-radiant

text: modes

text: (m

text: m∗)

text: Low-lying

text: super-radiant

text: modes

text: (m

text: [m∗,

text: Highly

text: super-radiant

text: modes

text: (m

text: Summary

text: (super-radiant

text: modes)

text: Discussion

text: Introduction

text: The

text: spacetime

text: geometry

text: of

text: a

text: black

text: hole,

text: in

text: the

text: region

text: that

text: interpolates

text: between

text: the

text: horizon

text: and

text: spatial

text: inﬁnity,

text: (the

text: domain

text: of

text: outer

text: communication),

text: generically

text: acts

text: as

text: a

text: potential

text: barrier

text: that

text: partially

text: reﬂects

text: both

text: ingoing

text: and

text: outgoing

text: excitations.

text: (See

text: for

text: instance

text: In

text: the

text: case

text: of

text: outgoing

text: excitations

text: (Hawking

text: quanta)

text: the

text: resulting

text: transmission

text: probabilities

text: are

text: called

text: “greybody

text: factors”.

text: Calculation

text: of

text: these

text: greybody

text: factors,

text: when

text: practical,

text: is

text: based

text: on

text: analyzing

text: the

text: excitations

text: in

text: terms

text: of

text: a

text: Regge–

text: Wheeler

text: equation,

text: (or

text: closely

text: related

text: variant

text: thereof),

text: which

text: in

text: the

text: non-super-radiant

text: case

text: reduces

text: the

text: problem

text: to

text: a

text: one-dimensional

text: barrier-penetration

text: problem.

text: Even

text: then,

text: ﬁnding

text: exact

text: solutions

text: is

text: mostly

text: impractical,

text: and

text: one

text: typically

text: resorts

text: either

text: to

text: making

text: semi-analytic

text: or

text: numerical

text: estimates,

text: or

text: to

text: deriving

text: rigorous

text: analytic

text: ##LTLine##

text: ##LTLine##


text: Apply this general result to our speciﬁc situation − − U (r) = V (ω m(cid:36))2 , (3.19) (cid:96)m (cid:96)m by choosing − h(r) = ω m(cid:36). (3.20) (This construction is now as close as one can get to the Case I bound of refer- ence [10].) Note this choice for h(r) is, since (cid:36) = a/(a2 + r2), always monotonic as a function of r. In contrast, (remember that ω > 0 and a > 0), we see that this h(r) is positive throughout the domain of outer communication if and only if ω > mΩ , which + is completely equivalent to m < ω/Ω , or m < m∗. This is easily recognized as the + quite standard condition that the mode does not suﬀer from super-radiant instability. Let us now see where we can go with this. 3.2.1 Negative-angular-momentum modes (m < 0) First note that in this situation, for the speciﬁc function h(r) chosen above, we have ∞ h( ) ω 1 −∞ = − = − < 1. (3.21) h( ) ω mΩ 1 mΩ /ω + + (cid:12) (cid:20) (cid:21)(cid:12) (cid:12) (cid:12) Then (cid:12) (cid:12) (cid:12) (cid:12) ∞ 1 h( ) 1 − ln −∞ = ln(1 mΩ /ω). (3.22) + 2 h( ) 2 (cid:90) − (cid:90) (cid:90) Also in this case we have ω mΩ > h(r) > ω, so + ∞ | | ∞ | | ∞ + U (r) + h(r)2 + V + V (cid:96)m dr∗ = (cid:96)m dr∗ < (cid:96)m dr∗. (3.23) −∞ 2h(r) −∞ 2h(r) −∞ 2ω (cid:26) (cid:90) (cid:27) Then ∞ ≥ 1 − + V T sech2 ln(1 mΩ /ω) + (cid:96),m<0 dr∗. (3.24) (cid:96),m<0 + 2 −∞ 2ω – 11 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Apply

text: this

text: general

text: result

text: to

text: our

text: speciﬁc

text: situation

label: equation

text: U

text: (r)

text: V

text: (ω

text: by

text: choosing

text: h(r)

text: ω

text: (This

text: construction

text: is

text: now

text: as

text: close

text: as

text: one

text: can

text: get

text: to

text: the

text: Case

text: I

text: bound

text: of

text: refer-

text: ence

text: Note

text: this

text: choice

text: for

text: h(r)

text: is,

text: since

text: always

text: monotonic

text: as

text: a

text: function

text: of

text: r.

text: In

text: contrast,

text: (remember

text: that

text: ω

text: and

text: a

text: we

text: see

text: that

text: this

text: h(r)

text: is

text: positive

text: throughout

text: the

text: domain

text: of

text: outer

text: communication

text: if

text: and

text: only

text: if

text: ω

text: mΩ

text: which

text: is

text: completely

text: equivalent

text: to

text: m

text: ω/Ω

text: or

text: m

text: m∗.

text: This

text: is

text: easily

text: recognized

text: as

text: the

text: quite

text: standard

text: condition

text: that

text: the

text: mode

text: does

text: not

text: suﬀer

text: from

text: super-radiant

text: instability.

text: Let

text: us

text: now

text: see

text: where

text: we

text: can

text: go

text: with

text: this.

text: Negative-angular-momentum

text: modes

text: (m

text: First

text: note

text: that

text: in

text: this

text: situation,

text: for

text: the

text: speciﬁc

text: function

text: h(r)

text: chosen

text: above,

text: we

text: have

text: h(

text: ω

text: h(

text: ω

text: mΩ

text: mΩ

text: /ω

text: Then

text: h(

text: ln

text: mΩ

text: /ω).

text: h(

text: Also

text: in

text: this

text: case

text: we

text: have

text: ω

text: mΩ

text: h(r)

text: ω,

text: so

text: U

text: (r)

text: V

text: V

text: dr∗

text: dr∗

text: dr∗.

text: 2ω

text: Then

text: V

text: T

text: mΩ

text: /ω)

text: dr∗.

text: 2ω

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: “Bounding the Greybody Factors for Non-rotating Black Holes”, Int. J. Mod. Phys. D22 (2013) 1350058, DOI: 10.1142/S0218271813500582 arXiv:1211.4070 [math-ph]. [7] T. Ngampitipan and P. Boonserm, “Bounding the greybody factors for the Reissner–Nordstr¨om black holes”, J. Phys. Conf. Ser. 435 (2013) 012027 [arXiv:1301.7527 [math-ph]], ICAST2012. [8] P. Boonserm, T. Ngampitipan and M. Visser, “Regge-Wheeler equation, linear stability, and greybody factors for dirty black holes”, Phys. Rev. D 88 (2013) 041502 [arXiv:1305.1416 [gr-qc]]. [9] T. Ngampitipan , P. Boonserm, and M. Visser, “Rigorous bound on transmission probability for m = 0 scalar excitations of Kerr–Newman black holes”, ICAPM2014 (International Conference on Applied Physics and Mathematics), Singapore. [10] M. Visser, “Some general bounds for 1-D scattering”, Phys. Rev. A 59 (1999) 427 [quant-ph/9901030]. [11] P. Boonserm and M. Visser, “Transmission probabilities and the Miller–Good transformation”, J. Phys. A 42, 045301 (2009) [arXiv:0808.2516 [math-ph]]. [12] P. Boonserm and M. Visser, “Bounding the Bogoliubov coeﬃcients”, Annals Phys. 323 (2008) 2779 [arXiv:0801.0610 [quant-ph]]. [13] P. Boonserm and M. Visser, “Analytic bounds on transmission probabilities”, Annals Phys. 325 (2010) 1328 [arXiv:0901.0944 [math-ph]]. [14] P. Boonserm and M. Visser, “Reformulating the Schr¨odinger equation as a Shabat–Zakharov system”, J. Math. Phys. 51 (2010) 022105 [arXiv:0910.2600 [math-ph]]. [15] K. D. Kokkotas, R. A. Konoplya and A. Zhidenko, “Quasinormal modes, scattering and Hawking radiation of Kerr-Newman black holes in a magnetic ﬁeld”, Phys. Rev. D 83, 024031 (2011) [arXiv:1011.1843 [gr-qc]]. [16] R. A. Konoplya and A. Zhidenko, “Quasinormal modes of black holes: From astrophysics to string theory”, Rev. Mod. Phys. 83 (2011) 793 [arXiv:1102.4014 [gr-qc]]. [17] R. A. Konoplya and A. Zhidenko, “A massive charged scalar ﬁeld in the Kerr-Newman background I: quasinormal modes, late-time tails and stability”, – 22 –

text: “Bounding

label: reference

text: the

text: Greybody

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text: Holes”,

text: Int.

text: J.

text: Mod.

text: Phys.

text: DOI:

text: [math-ph].

text: T.

text: Ngampitipan

text: and

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text: Reissner–Nordstr¨om

text: black

text: holes”,

text: J.

text: Phys.

text: Conf.

text: Ser.

text: [math-ph]],

text: P.

text: Boonserm,

text: T.

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text: M.

text: Visser,

text: “Regge-Wheeler

text: equation,

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text: Phys.

text: Rev.

text: D

text: [gr-qc]].

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text: black

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text: Conference

text: on

text: Applied

text: Physics

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text: general

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text: Schr¨odinger

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text: system”,

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text: modes,

text: scattering

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text: Kerr-Newman

text: black

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text: magnetic

text: ﬁeld”,

text: Phys.

text: Rev.

text: D

text: [gr-qc]].

text: R.

text: A.

text: Konoplya

text: and

text: A.

text: Zhidenko,

text: “Quasinormal

text: modes

text: of

text: black

text: holes:

text: From

text: astrophysics

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text: string

text: theory”,

text: Rev.

text: Mod.

text: Phys.

text: [gr-qc]].

text: R.

text: A.

text: Konoplya

text: and

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text: “A

text: massive

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text: in

text: the

text: Kerr-Newman

text: background

text: I:

text: quasinormal

text: modes,

text: late-time

text: tails

text: and

text: stability”,


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text: Phys.

text: Rev.

text: D

text: [gr-qc]].

text: K.

text: D.

text: Kokkotas

text: and

text: B.

text: G.

text: Schmidt,

text: “Quasinormal

text: modes

text: of

text: stars

text: and

text: black

text: holes”,

text: Living

text: Rev.

text: Rel.

text: V.

text: Ferrari,

text: “Stellar

text: Perturbations”,

text: Lecture

text: Notes

text: in

text: Physics

text: L.

text: Samuelsson

text: and

text: N.

text: Andersson,

text: “Neutron

text: Star

text: Asteroseismology.

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text: Oscillations

text: in

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text: Cowling

text: Approximation”,

text: Mon.

text: Not.

text: Roy.

text: Astron.

text: Soc.

text: V.

text: Ferrari,

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text: waves

text: from

text: perturbed

text: stars”,

text: Bulletin

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text: Astronomical

text: Society

text: of

text: India

text: [gr-qc].

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text: Newman

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text: Spinning-Particle

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text: GROUPS OF PL HOMEOMORPHISMS OF CUBES 9 The (real vector) space of homogeneous quasimorphisms on G is denoted Q(G). A function satisfying the second condition but not the ﬁrst is said to be (simply) a quasimorphism. If φ : G → R is any quasimorphism, the function φ¯ : G → R deﬁned by φ¯(g) = lim →∞ φ(gn)/n is a homogeneous quasimorphism, and satisﬁes − ≤ n φ¯ φ D(φ). This operation is called homogenization of quasimorphisms. A homogeneous quasimorphism has defect 0 if and only if it is a homomorphism R to G. Thus D descends to a norm on the quotient space Q(G)/H1(G; ). It is a fact that Q/H1 with the defect norm is a Banach space. It is known that Q/H1 vanishes on any subgroup G of PL(I, ∂I). By contrast, we show that the subgroup PL (I2, ∂I2) admits an inﬁnite dimensional Q/H1. ω The proof is by explicit construction, and based on a general method due to Gambaudo-Ghys [12]. The construction is as follows. First, ﬁx some n and let B · · · n denote the braid group on n strands, and ﬁx n distinct points x0, , x0 in the → R 1 n interior of I2. Let µ : B be any function. n ∈ Now, since Homeo(I2, ∂I2) is path-connected and simply-connected, for any g PL (I2, ∂I2) there is a unique homotopy class of isotopy g from g to Id. ω · · · t For any (generic) n-tuple of distinct points x , , x in the interior of I2, and ∈ · · · ∈ 1 n any g PL (I2, ∂I2), let γ(g; x , , x ) B be the braid obtained by ﬁrst ω 1 n n moving the x0 in a straight line to the x , then composing with the isotopy g from i i t x to g(x ), then ﬁnally moving the g(x ) in a straight line back to the x0 (note that i i (cid:90) i i γ does not depend on the choice of g , since Homeo(I2, ∂I2) is simply-connected). t Now we deﬁne · · · · · · Φ (g) = µ(γ(g; x , , x ))dω(x )dω(x ) dω(x ) µ ×···× 1 n 1 2 n I2 I2 where the integral is taken over the subset of the product of n copies of (the interior of) I2 consisting of distinct n-tuples of points where γ is well-deﬁned. Lemma 3.5.5. Suppose µ is a quasimorphism on B . Then Φ is a quasimorphism n µ on PL (I2, ∂I2) ω Proof. Changing the vector x0 to a new vector y0 changes γ by multiplication by one of ﬁnitely many elements of B ; since µ is a quasimorphism, this changes Φ n µ by a bounded amount. ∈ · · · For any two g, h PL (I2, ∂I2) and generic x , , x we have ω 1 n · · · · · · · · · γ(gh; x , , x ) = γ(h; x , , x )γ(g; h(x ), , h(x )) 1 n 1 n 1 n × · · · × Now integrate over I2 I2 and use the fact that µ is a quasimorphism to see (cid:3) that Φ is a quasimorphism. µ The homogenization Φ is a homogeneous quasimorphism, and one may check µ that it is nontrivial whenever µ is nontrivial on B . In fact, it is easiest to check n this for the special case of homogeneous quasimorphisms on B that vanish on n reducible elements. Such quasimorphisms are plentiful; for example, any “counting quasimorphism” arising from a weakly properly discontinuous action of B on a § n hyperbolic simplicial complex. See [7], 3.6 for more details. ##LTLine## ##LTLine##

text: GROUPS

text: OF

text: PL

text: HOMEOMORPHISMS

text: OF

text: CUBES

text: The

text: (real

text: vector)

text: space

text: of

text: homogeneous

text: quasimorphisms

text: on

text: G

text: is

text: denoted

text: Q(G).

text: A

text: function

text: satisfying

text: the

text: second

text: condition

text: but

text: not

text: the

text: ﬁrst

text: is

text: said

text: to

text: be

text: (simply)

text: a

text: quasimorphism.

text: If

text: φ

text: G

text: R

text: is

text: any

text: quasimorphism,

text: the

text: function

text: φ¯

text: G

text: R

text: deﬁned

text: by

text: φ¯(g)

text: lim

text: φ(gn)/n

text: is

text: a

text: homogeneous

text: quasimorphism,

text: and

text: satisﬁes

text: n

text: φ¯

text: φ

text: D(φ).

text: This

text: operation

text: is

text: called

text: homogenization

text: of

text: quasimorphisms.

text: A

text: homogeneous

text: quasimorphism

text: has

text: defect

text: if

text: and

text: only

text: if

text: it

text: is

text: a

text: homomorphism

text: R

text: to

text: G.

text: Thus

text: D

text: descends

text: to

text: a

text: norm

text: on

text: the

text: quotient

text: space

text: It

text: is

text: a

text: fact

text: that

text: with

text: the

text: defect

text: norm

text: is

text: a

text: Banach

text: space.

text: It

text: is

text: known

text: that

text: vanishes

text: on

text: any

text: subgroup

text: G

text: of

text: PL(I,

text: ∂I).

text: By

text: contrast,

text: we

text: show

text: that

text: the

text: subgroup

text: PL

text: admits

text: an

text: inﬁnite

text: dimensional

text: ω

text: The

text: proof

text: is

text: by

text: explicit

text: construction,

text: and

text: based

text: on

text: a

text: general

text: method

text: due

text: to

text: Gambaudo-Ghys

text: The

text: construction

text: is

text: as

text: follows.

text: First,

text: ﬁx

text: some

text: n

text: and

text: let

text: B

text: n

text: denote

text: the

text: braid

text: group

text: on

text: n

text: strands,

text: and

text: ﬁx

text: n

text: distinct

text: points

text: in

text: the

text: R

text: n

text: interior

text: of

text: Let

text: µ

text: B

text: be

text: any

text: function.

text: n

text: Now,

text: since

text: is

text: path-connected

text: and

text: simply-connected,

text: for

text: any

text: g

text: PL

text: there

text: is

text: a

text: unique

text: homotopy

text: class

text: of

text: isotopy

text: g

text: from

text: g

text: to

text: Id.

text: ω

text: t

text: For

text: any

text: (generic)

text: n-tuple

text: of

text: distinct

text: points

text: x

text: x

text: in

text: the

text: interior

text: of

text: and

text: n

text: any

text: g

text: PL

text: let

text: γ(g;

text: x

text: x

text: B

text: be

text: the

text: braid

text: obtained

text: by

text: ﬁrst

text: ω

text: n

text: n

text: moving

text: the

text: in

text: a

text: straight

text: line

text: to

text: the

text: x

text: then

text: composing

text: with

text: the

text: isotopy

text: g

text: from

text: i

text: i

text: t

text: x

text: to

text: g(x

text: then

text: ﬁnally

text: moving

text: the

text: g(x

text: in

text: a

text: straight

text: line

text: back

text: to

text: the

text: (note

text: that

text: i

text: i

text: i

text: i

text: γ

text: does

text: not

text: depend

text: on

text: the

text: choice

text: of

text: g

text: since

text: is

text: simply-connected).

text: t

text: Now

text: we

text: deﬁne

text: Φ

text: (g)

text: µ(γ(g;

text: x

text: x

text: ))dω(x

text: )dω(x

text: dω(x

text: µ

text: n

text: n

text: where

text: the

text: integral

text: is

text: taken

text: over

text: the

text: subset

text: of

text: the

text: product

text: of

text: n

text: copies

text: of

text: (the

text: interior

text: of)

text: consisting

text: of

text: distinct

text: n-tuples

text: of

text: points

text: where

text: γ

text: is

text: well-deﬁned.

text: Lemma

text: Suppose

text: µ

text: is

text: a

text: quasimorphism

text: on

text: B

text: Then

text: Φ

text: is

text: a

text: quasimorphism

text: n

text: µ

text: on

text: PL

text: ω

text: Proof.

text: Changing

text: the

text: vector

text: to

text: a

text: new

text: vector

text: changes

text: γ

text: by

text: multiplication

text: by

text: one

text: of

text: ﬁnitely

text: many

text: elements

text: of

text: B

text: since

text: µ

text: is

text: a

text: quasimorphism,

text: this

text: changes

text: Φ

text: n

text: µ

text: by

text: a

text: bounded

text: amount.

text: For

text: any

text: two

text: g,

text: h

text: PL

text: and

text: generic

text: x

text: x

text: we

text: have

text: ω

text: n

text: γ(gh;

text: x

text: x

text: γ(h;

text: x

text: x

text: )γ(g;

text: h(x

text: h(x

text: n

text: n

text: n

text: Now

text: integrate

text: over

text: and

text: use

text: the

text: fact

text: that

text: µ

text: is

text: a

text: quasimorphism

text: to

text: see

text: that

text: Φ

text: is

text: a

text: quasimorphism.

text: µ

text: The

text: homogenization

text: Φ

text: is

text: a

text: homogeneous

text: quasimorphism,

text: and

text: one

text: may

text: check

text: µ

text: that

text: it

text: is

text: nontrivial

text: whenever

text: µ

text: is

text: nontrivial

text: on

text: B

text: In

text: fact,

text: it

text: is

text: easiest

text: to

text: check

text: n

text: this

text: for

text: the

text: special

text: case

text: of

text: homogeneous

text: quasimorphisms

text: on

text: B

text: that

text: vanish

text: on

text: n

text: reducible

text: elements.

text: Such

text: quasimorphisms

text: are

text: plentiful;

text: for

text: example,

text: any

text: “counting

text: quasimorphism”

text: arising

text: from

text: a

text: weakly

text: properly

text: discontinuous

text: action

text: of

text: B

text: on

text: a

text: n

text: hyperbolic

text: simplicial

text: complex.

text: See

text: for

text: more

text: details.

text: ##LTLine##

text: ##LTLine##


text: ∗ Abstract In this short review, we present the current status about the theoretical and exper- imental studies for some important semileptonic decays of B/B mesons. We ﬁrstly gave s→ − − a brief introduction for the experimental measurements for B/B P (l+l , l ν¯ , νν¯) de- ∗ (cid:48) s → ∗ l− cays, the BaBar’s R(D) and R(D ) anomaly, the P deviation for B0 K 0µ+µ decay. 5 We then made a careful discussion about the evaluations for the relevant form factors in the light-cone QCD sum rule (LCSRs), the heavy quark effective theory, and the perturba- tive QCD factorization approach. By using the form factors calculated in the perturbative (pQCD) approach, we then calculate and show the pQCD predictions for the decay rates of many semileptonic decays of B/B mesons. We also made careful phenomenological s analysis for these pQCD predictions and found, in general, the following points: (a) For all the considered B/B semileptonic decays, the next-to-leading order (NLO) pQCD predic- s tions for their decay rates agree well with the data and those from other different theoretical ∗ methods; (b) For R(D) and R(D ), the pQCD predictions agree very well with the data, the ∗ BaBar’s anomaly of R(D( )) are therefore explained successfully in the standard model by employing the pQCD approach; and (c) We deﬁned several new ratios Rl,τ and Rl,τ , they → D ∗ D∗ s may be more sensitive to the QCD dynamics which controls the B/B (D( ), D( )) tran- s s sitions than the old ratios, we therefore strongly suggest LHCb and the forthcoming Super-B experiments to measure these new ratios. Key Words B/B meson semileptonic decays; The pQCD factorization approach; Form fac- s tors; Branching ratios; LHCb experiments 1. INTRODUCTION As is well-known, the semileptonic (SL) decays of B and B meson are very important pro- s cesses in testing the standard model (SM) and in searching for the signal and/or evidence of the new physics (NP) beyond the standard model: such as the extractions of the Cabbibo-Kobayashi- | | | | Maskawa matrix elements V and V , and the determinations of the form factors F (q2) for ub cb 0,+,T the B/B transitions to the pion, kaon or other light mesons [1–4]. Since the Spring of 2012, the s R ∗ → ∗ − BaBar’s anomaly about the ratio (D( )) [5, 6] invoked intensive studies for B D( )l ν¯ decays l in the framework of the SM and various new physics (NP) models, for example, in Refs. [7–15]. → ∗ − The B K µ+µ anomaly observed by LHCb experiments [16–18] also stimulate many inter- esting studies [19–23]. We here will present a short review about the experimental measurements and the theoretical studies for the following B/B semileptonic decay modes: s → − ∗ ∗ − B/B P (l+l , lν, νν¯), (D( ), D( ))l ν¯ , (1) s → ∗ − s l B K µ+µ , (2) (cid:48) where l = (e, µ, τ) are leptons and P = (K, π, η, η ) are light pseudoscalar mesons. For those → considered B/B P lν¯ decays, as illustrated in Fig.1, the ”Tree” Feynman diagrams provide the s → − dominant leading order (LO) contribution. For those B/B P l+l and P νν¯ decays, however, s the dominant LO standard model contributions come from those electroweak penguin diagrams ∗ ##LTLine##

text: Abstract

label: abstract

text: In

text: this

text: short

text: review,

text: we

text: present

text: the

text: current

text: status

text: about

text: the

text: theoretical

text: and

text: exper-

text: imental

text: studies

text: for

text: some

text: important

text: semileptonic

text: decays

text: of

text: B/B

text: mesons.

text: We

text: ﬁrstly

text: gave

text: s→

text: a

text: brief

text: introduction

text: for

text: the

text: experimental

text: measurements

text: for

text: B/B

text: P

text: (l+l

text: l

text: ν¯

text: νν¯)

text: de-

text: s

text: l−

text: cays,

text: the

text: BaBar’s

text: R(D)

text: and

text: R(D

text: anomaly,

text: the

text: P

text: deviation

text: for

text: K

text: 0µ+µ

text: decay.

text: We

text: then

text: made

text: a

text: careful

text: discussion

text: about

text: the

text: evaluations

text: for

text: the

text: relevant

text: form

text: factors

text: in

text: the

text: light-cone

text: QCD

text: sum

text: rule

text: (LCSRs),

text: the

text: heavy

text: quark

text: effective

text: theory,

text: and

text: the

text: perturba-

text: tive

text: QCD

text: factorization

text: approach.

text: By

text: using

text: the

text: form

text: factors

text: calculated

text: in

text: the

text: perturbative

text: (pQCD)

text: approach,

text: we

text: then

text: calculate

text: and

text: show

text: the

text: pQCD

text: predictions

text: for

text: the

text: decay

text: rates

text: of

text: many

text: semileptonic

text: decays

text: of

text: B/B

text: mesons.

text: We

text: also

text: made

text: careful

text: phenomenological

text: s

text: analysis

text: for

text: these

text: pQCD

text: predictions

text: and

text: found,

text: in

text: general,

text: the

text: following

text: points:

text: (a)

text: For

text: all

text: the

text: considered

text: B/B

text: semileptonic

text: decays,

text: the

text: next-to-leading

text: order

text: (NLO)

text: pQCD

text: predic-

text: s

text: tions

text: for

text: their

text: decay

text: rates

text: agree

text: well

text: with

text: the

text: data

text: and

text: those

text: from

text: other

text: different

text: theoretical

text: methods;

text: (b)

text: For

text: R(D)

text: and

text: R(D

text: the

text: pQCD

text: predictions

text: agree

text: very

text: well

text: with

text: the

text: data,

text: the

text: BaBar’s

text: anomaly

text: of

text: R(D(

text: are

text: therefore

text: explained

text: successfully

text: in

text: the

text: standard

text: model

text: by

text: employing

text: the

text: pQCD

text: approach;

text: and

text: (c)

text: We

text: deﬁned

text: several

text: new

text: ratios

text: Rl,τ

text: and

text: Rl,τ

text: they

text: D

text: D∗

text: s

text: may

text: be

text: more

text: sensitive

text: to

text: the

text: QCD

text: dynamics

text: which

text: controls

text: the

text: B/B

text: (D(

text: D(

text: tran-

text: s

text: s

text: sitions

text: than

text: the

text: old

text: ratios,

text: we

text: therefore

text: strongly

text: suggest

text: LHCb

text: and

text: the

text: forthcoming

text: Super-B

text: experiments

text: to

text: measure

text: these

text: new

text: ratios.

text: Key

text: Words

text: B/B

text: meson

text: semileptonic

text: decays;

text: The

text: pQCD

text: factorization

text: approach;

text: Form

text: fac-

text: s

text: tors;

text: Branching

text: ratios;

text: LHCb

text: experiments

text: INTRODUCTION

text: As

text: is

text: well-known,

text: the

text: semileptonic

text: (SL)

text: decays

text: of

text: B

text: and

text: B

text: meson

text: are

text: very

text: important

text: pro-

text: s

text: cesses

text: in

text: testing

text: the

text: standard

text: model

text: (SM)

text: and

text: in

text: searching

text: for

text: the

text: signal

text: and/or

text: evidence

text: of

text: the

text: new

text: physics

text: (NP)

text: beyond

text: the

text: standard

text: model:

text: such

text: as

text: the

text: extractions

text: of

text: the

text: Cabbibo-Kobayashi-

text: Maskawa

text: matrix

text: elements

text: V

text: and

text: V

text: and

text: the

text: determinations

text: of

text: the

text: form

text: factors

text: F

text: for

text: ub

text: cb

text: the

text: B/B

text: transitions

text: to

text: the

text: pion,

text: kaon

text: or

text: other

text: light

text: mesons

text: Since

text: the

text: Spring

text: of

text: the

text: s

text: R

text: BaBar’s

text: anomaly

text: about

text: the

text: ratio

text: (D(

text: invoked

text: intensive

text: studies

text: for

text: B

text: D(

text: )l

text: ν¯

text: decays

text: l

text: in

text: the

text: framework

text: of

text: the

text: SM

text: and

text: various

text: new

text: physics

text: (NP)

text: models,

text: for

text: example,

text: in

text: Refs.

text: The

text: B

text: K

text: µ+µ

text: anomaly

text: observed

text: by

text: LHCb

text: experiments

text: also

text: stimulate

text: many

text: inter-

text: esting

text: studies

text: We

text: here

text: will

text: present

text: a

text: short

text: review

text: about

text: the

text: experimental

text: measurements

text: and

text: the

text: theoretical

text: studies

text: for

text: the

text: following

text: B/B

text: semileptonic

text: decay

text: modes:

text: s

text: B/B

text: P

text: (l+l

text: lν,

text: νν¯),

text: (D(

text: D(

text: ))l

text: ν¯

text: s

text: s

text: l

text: B

text: K

text: µ+µ

text: where

text: l

text: (e,

text: µ,

text: τ)

text: are

text: leptons

text: and

text: P

text: (K,

text: π,

text: η,

text: η

text: are

text: light

text: pseudoscalar

text: mesons.

text: For

text: those

text: considered

text: B/B

text: P

text: lν¯

text: decays,

text: as

text: illustrated

text: in

text: the

text: ”Tree”

text: Feynman

text: diagrams

text: provide

text: the

text: s

text: dominant

text: leading

text: order

text: (LO)

text: contribution.

text: For

text: those

text: B/B

text: P

text: l+l

text: and

text: P

text: νν¯

text: decays,

text: however,

text: s

text: the

text: dominant

text: LO

text: standard

text: model

text: contributions

text: come

text: from

text: those

text: electroweak

text: penguin

text: diagrams

text: ##LTLine##


text: 15 [4] Straub D M (2013) Heavy Flavour Theory. Lectures given at CERN-FermiLab HCPSS, 2013, CERN, Geneva → ∗ [5] Lees J P et al BABAR Collaboration (2012) Evidence for an Excess of B¯ D( )τν¯ Decays. Phys τ Rev Lett 109: 101802 → ∗ [6] Lees J P et al BABAR Collaboration (2013) Measurement of an excess of B¯ D( )τν¯ decays and τ implications for charged Higgs bosons. Phys Rev D 88:072012 → [7] Bailey J A et al (2012) Reﬁning new-physics searches in B Dτν decay with lattice QCD. Phys Rev Lett 109(07): 071802 → → [8] Bailey J A et al Fermilab Lattice and MILC Collaborations (2012) B D /B D semileptonic → − s s form-factor ratios and their application to BR(B0 µ+µ . Phys Rev D 85(11):114502; Errotem s 86(03): 039904 [9] Fajfer S, Kamenik J F, Nisandzic I et al (2012) Implications of Lepton Flavor Universality Violations in B Decays. Phys Rev Lett 109(16):161801 → → ∗ → [10] Crivellin A, Greub C, Kokulu A (2012) Explaining B Dτν, B D τν and B τν in a 2HDM of type III. Phys Rev D 86(05):054014 → → [11] Celis A, Jung M, Li X Q et al (2013) Sensitivity to charged scalars in B Dτν and B τν decays. J High Energy Phys 2013(01): 054 → [12] Datta A, Duraisamy M, Ghosh D (2012) Diagnosing new physics in b cτν decays in the light of the recent BABAR result. Phys Rev D 86(03): 034027 [13] Choudhury D, Ghosh D K, Kundu A (2012) B decay anomalies in an effective theory. Phys Rev D 86(11):114037 [14] Faustov R N, Galkin V O (2012) Exclusive weak B decays involving τ lepton in the relativistic quark model. Mod Phys Lett A 27(31): 1250183. → → ∗ [15] Fajfer S and Nisandzic I (2013) Theory of B τν and B D τν. Conference: C12-09-28; arXiv:1301.1167 [hep-ph] [16] Aaij A et al. LHCb collaboration, (2013) Differential branching fraction and angular analysis of the → ∗ − decay B0 K 0µ+µ . J High Energy Phys 08: 131 [17] Aaij A et al. LHCb collaboration, (2013) Measurement of form-factor independent observables in → ∗ − the decay B0 K 0µ+µ . Phys Rev Lett 111:191801. arXiv:1308.1707[hep-ex]. → [18] Serra N, (2013) Studies of electroweak penguin transitions of b sµµ. LHCb-TALK-2013-208. Talk given at EPSHEP 2013, 18-24 July, 2013, Stockholm, Sweden [19] Albrecht J (2013) Heavy Flavour Experiment. Lectures presented at CERN-FermiLab HCPSS-2013, Aug.28 - Sept.6, 2013, CERN, Geneva → [20] Ja¨ger S, Camalich J M (2013) On B V ll at small dilepton invariant mass, power corrections, and new physics. J High Energy Phys 05:043. arXiv:1212.2263[hep-ph]. → ∗ − [21] Descotes-Genona S, Matiasb J, Virtob J (2013) Understanding the B K µ+µ Anomaly. Phys Rev D 88:074002. arXiv:1307.5683[hep-ph]. → ∗ − [22] Altmannshofera W, Straub D M (2013) New Physics in B K µ+µ . Eur Phys J C 73: 2646 (cid:48) → ∗ − [23] Gaulda R, Goertzb F, Haischc U (2014) On minimal Z explanations of the B K µ+µ anomaly. Phys Rev D 89: 015005 → → [24] del Amo Sanchez P et al. BABAR Collaboration (2011) Study of B πlν¯ and B ρlν¯ decays | | and determination of V . Phys Rev D 83: 032007 ub → − [25] del Amo Sanchez P et al. BABAR Collaboration (2011) Measurement of the B0 π l+ν and → (cid:48) → − → B+ η( )l+ν branching fractions, the B0 π l+ν and B+ ηl+ν form-factor shapes, and | | determination of V . Phys Rev D 83:052011 ub [26] Adam N E et al. CLEO Collaboration (2007) Study of Exclusive Charmless Semileptonic B Decays

text: Straub

text: D

text: M

text: Heavy

text: Flavour

text: Theory.

text: Lectures

text: given

text: at

text: CERN-FermiLab

text: HCPSS,

text: CERN,

text: Geneva

text: Lees

text: J

text: P

text: et

text: al

text: BABAR

text: Collaboration

text: Evidence

text: for

text: an

text: Excess

text: of

text: B¯

text: D(

text: )τν¯

text: Decays.

text: Phys

text: τ

text: Rev

text: Lett

text: Lees

text: J

text: P

text: et

text: al

text: BABAR

text: Collaboration

text: Measurement

text: of

text: an

text: excess

text: of

text: B¯

text: D(

text: )τν¯

text: decays

text: and

text: τ

text: implications

text: for

text: charged

text: Higgs

text: bosons.

text: Phys

text: Rev

text: D

text: Bailey

text: J

text: A

text: et

text: al

text: Reﬁning

text: new-physics

text: searches

text: in

text: B

text: Dτν

text: decay

text: with

text: lattice

text: QCD.

text: Phys

text: Rev

text: Lett

text: Bailey

text: J

text: A

text: et

text: al

text: Fermilab

text: Lattice

text: and

text: MILC

text: Collaborations

text: B

text: D

text: /B

text: D

text: semileptonic

text: s

text: s

text: form-factor

text: ratios

text: and

text: their

text: application

text: to

text: µ+µ

text: Phys

text: Rev

text: D

text: Errotem

text: s

text: Fajfer

text: S,

text: Kamenik

text: J

text: F,

text: Nisandzic

text: I

text: et

text: al

text: Implications

text: of

text: Lepton

text: Flavor

text: Universality

text: Violations

text: in

text: B

text: Decays.

text: Phys

text: Rev

text: Lett

text: Crivellin

text: A,

text: Greub

text: C,

text: Kokulu

text: A

text: Explaining

text: B

text: Dτν,

text: B

text: D

text: τν

text: and

text: B

text: τν

text: in

text: a

text: of

text: type

text: III.

text: Phys

text: Rev

text: D

text: Celis

text: A,

text: Jung

text: M,

text: Li

text: X

text: Q

text: et

text: al

text: Sensitivity

text: to

text: charged

text: scalars

text: in

text: B

text: Dτν

text: and

text: B

text: τν

text: decays.

text: J

text: High

text: Energy

text: Phys

text: Datta

text: A,

text: Duraisamy

text: M,

text: Ghosh

text: D

text: Diagnosing

text: new

text: physics

text: in

text: b

text: cτν

text: decays

text: in

text: the

text: light

text: of

text: the

text: recent

text: BABAR

text: result.

text: Phys

text: Rev

text: D

text: Choudhury

text: D,

text: Ghosh

text: D

text: K,

text: Kundu

text: A

text: B

text: decay

text: anomalies

text: in

text: an

text: effective

text: theory.

text: Phys

text: Rev

text: D

text: Faustov

text: R

text: N,

text: Galkin

text: V

text: O

text: Exclusive

text: weak

text: B

text: decays

text: involving

text: τ

text: lepton

text: in

text: the

text: relativistic

text: quark

text: model.

text: Mod

text: Phys

text: Lett

text: A

text: Fajfer

text: S

text: and

text: Nisandzic

text: I

text: Theory

text: of

text: B

text: τν

text: and

text: B

text: D

text: τν.

text: Conference:

text: [hep-ph]

text: Aaij

text: A

text: et

text: al.

text: LHCb

text: collaboration,

text: Differential

text: branching

text: fraction

text: and

text: angular

text: analysis

text: of

text: the

text: decay

text: K

text: 0µ+µ

text: J

text: High

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text: observables

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text: decay

text: K

text: 0µ+µ

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text: EPSHEP

text: July,

text: Stockholm,

text: Sweden

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text: J

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text: Flavour

text: Experiment.

text: Lectures

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text: CERN,

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text: S,

text: Camalich

text: J

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text: new

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text: Understanding

text: the

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text: Anomaly.

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text: Sanchez

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text: al.

text: BABAR

text: Collaboration

text: Study

text: of

text: B

text: πlν¯

text: and

text: B

text: ρlν¯

text: decays

text: and

text: determination

text: of

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text: l+ν

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text: η(

text: )l+ν

text: branching

text: fractions,

text: the

text: π

text: l+ν

text: and

text: B+

text: ηl+ν

text: form-factor

text: shapes,

text: and

text: determination

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text: 8 TABLE II. The pQCD predictions for the branching ratios of those considered decay modes with l = (e, µ). The theoretical predictions based on other methods [115–117] and currently available data or the world averages [110, 114, 118]. The upper limits are at the 90%C.L. Decay modes pQCD Others Data → − − Br(B¯0 π+l ν¯ )(10 4) 1.42+−0.52 1.44+−0.05 → − l − 0.44 0.05 Br(B¯0 π+τ ν¯ )(10 4) 0.90+−0.33 − → − τ − 0.27 Br(B π0l ν¯ )(10 4) 0.76+−0.28 0.77+−0.05 − → − l − 0.24 0.05 Br(B π0τ ν¯ )(10 4) 0.49+−0.18 τ 0.15 → − − Br(B¯0 π0l+l )(10 8) 0.91+−0.34 < 12 → − − 0.29 Br(B¯0 π0τ+τ )(10 8) 0.28+−0.09 → − 0.09 Br(B¯0 π0νν¯)(10 8) 7.30+−2.72 < 22000 − → − − − 2.24 ± Br(B π l+l )(10 8) 1.95+−0.73 2.0+−0.2 2.3 0.6 − → − − − 0.60 0.2 Br(B π τ+τ )(10 8) 0.60+−0.21 − → − − 0.17 Br(B π νν¯)(10 8) 15.7+−5.8 < 10000 4.3 → − − Br(B¯0 K¯ 0l+l )(10 7) 5.12+−1.91 4.7+−0.6 → − − 1.57 0.2 Br(B¯0 K¯ 0τ+τ )(10 7) 1.20+−0.42 → − 0.35 Br(B¯0 K¯ 0νν¯)(10 6) 4.11+−1.54 < 56 − → − − − 1.26 ± Br(B K l+l )(10 7) 5.50+−2.06 5.8+−2.4 5.1 0.5 − → − − − 1.69 2.0 Br(B K τ+τ )(10 7) 1.29+−0.46 − → − − 0.37 Br(B K νν¯)(10 6) 4.42+−1.66 4.4+−1.5 < 13 1.35 1.5 → − − Br(B¯0 K+l ν¯ )(10 4) 1.27+−0.60 s → − l − 0.43 Br(B¯0 K+τ ν¯ )(10 4) 0.78+−0.34 s → −τ − 0.27 Br(B¯0 K0l+l )(10 8) 1.63+−0.73 1.4 s → − − 0.58 Br(B¯0 K0τ+τ )(10 8) 0.43+−0.18 0.3 s → − 0.15 Br(B¯0 K0νν¯)(10 7) 1.31+−0.58 1.0 s 0.47 theoretical methods [39–41, 44, 45, 50]. The pQCD predictions for the NLO twist-2 contri- ∼ bution to the form factors is 20% of the total value. → − − → − → − − → (2) For the considered decay modes B¯0 π+l ν¯ , B π0l ν¯ , B¯0 K¯ 0l+l , and B − − l l K l+l , the NLO pQCD predictions for their decay rates agree very well with currently available experimental measurements. (cid:0) (cid:1) → (3) From the direct comparison between the pQCD predictions for the branching ratio Br(B¯0 − | | π+l ν¯ ) and the measured value, the value of V can be extracted directly: V = l ± × − ub ub 3.80+−0.66(th.) 0.13 10 3. 0.60 − → (cid:48) − (4) For the branching ratios Br(B η( )l ν¯ ), the NLO twist-2 contribution to the transition ∼ l form factors can provide 25% enhancement to the LO pQCD results, which leads to a better agreement of the pQCD predictions with the measured values. (5) Analgous to the ratio R(D), we here also deﬁned several ratios of the branching ratios R , R and R , calculated and listed the pQCD predictions for their values and the ν C N1,N2,N3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: II.

text: The

text: pQCD

text: predictions

text: for

text: the

text: branching

text: ratios

text: of

text: those

text: considered

text: decay

text: modes

text: with

text: l

text: (e,

text: µ).

text: The

text: theoretical

text: predictions

text: based

text: on

text: other

text: methods

text: and

text: currently

text: available

text: data

text: or

text: the

text: world

text: averages

text: The

text: upper

text: limits

text: are

text: at

text: the

text: Decay

label: table

text: modes

text: pQCD

text: Others

text: Data

text: π+l

text: ν¯

text: l

text: π+τ

text: ν¯

text: τ

text: Br(B

text: ν¯

text: l

text: Br(B

text: π0τ

text: ν¯

text: τ

text: π0τ+τ

text: Br(B

text: π

text: l+l

text: Br(B

text: π

text: τ+τ

text: Br(B

text: π

text: K¯

text: K¯

text: 0τ+τ

text: K¯

text: Br(B

text: K

text: l+l

text: Br(B

text: K

text: τ+τ

text: Br(B

text: K

text: K+l

text: ν¯

text: s

text: l

text: K+τ

text: ν¯

text: s

text: −τ

text: s

text: s

text: s

text: theoretical

text: methods

text: The

text: pQCD

text: predictions

text: for

text: the

text: NLO

text: contri-

text: bution

text: to

text: the

text: form

text: factors

text: is

text: of

text: the

text: total

text: value.

text: For

text: the

text: considered

text: decay

text: modes

text: π+l

text: ν¯

text: B

text: ν¯

text: K¯

text: and

text: B

text: l

text: l

text: K

text: l+l

text: the

text: NLO

text: pQCD

text: predictions

text: for

text: their

text: decay

text: rates

text: agree

text: very

text: well

text: with

text: currently

text: available

text: experimental

text: measurements.

text: From

text: the

text: direct

text: comparison

text: between

text: the

text: pQCD

text: predictions

text: for

text: the

text: branching

text: ratio

text: π+l

text: ν¯

text: and

text: the

text: measured

text: value,

text: the

text: value

text: of

text: V

text: can

text: be

text: extracted

text: directly:

text: V

text: l

text: ub

text: ub

text: For

text: the

text: branching

text: ratios

text: Br(B

text: η(

text: )l

text: ν¯

text: the

text: NLO

text: contribution

text: to

text: the

text: transition

text: l

text: form

text: factors

text: can

text: provide

text: enhancement

text: to

text: the

text: LO

text: pQCD

text: results,

text: which

text: leads

text: to

text: a

text: better

text: agreement

text: of

text: the

text: pQCD

text: predictions

text: with

text: the

text: measured

text: values.

text: Analgous

text: to

text: the

text: ratio

text: R(D),

text: we

text: here

text: also

text: deﬁned

text: several

text: ratios

text: of

text: the

text: branching

text: ratios

text: R

text: R

text: and

text: R

text: calculated

text: and

text: listed

text: the

text: pQCD

text: predictions

text: for

text: their

text: values

text: and

text: the

text: ν

text: C

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