New Skins for an Old Ceremony The Conformal Bootstrap and the Ising - - PowerPoint PPT Presentation
New Skins for an Old Ceremony The Conformal Bootstrap and the Ising Model Sheer El-Showk CEA Saclay Based on: arXiv:1203.6064 with M. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, A. Vichi arXiv:1211.XXXX with M. Paulos November 8, 2012
The Conformal Bootstrap and the Ising Model Sheer El-Showk
CEA Saclay
Based on: arXiv:1203.6064 with M. Paulos, D. Poland, S. Rychkov,
arXiv:1211.XXXX with M. Paulos
November 8, 2012 DAMTP, Cambridge
◮ Conformal symmetry very powerful tool that goes largely unused in D > 2. ◮ Completely non-perturbative tool to study field theories
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Does not require SUSY, large N, or weak coupling.
◮ In D = 2 conformal symmetry enhanced to Virasoro symmetry
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Allows us to completely solve some CFTs (c < 1).
◮ How far could we get in D = 2 with only “global” conformal group?
◮ Use only “global” conformal group, valid in all D. ◮ Constrain “landscape of CFTs” in D = 2, 3 using conformal bootstrap. ◮ Result:
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CFTs (e.g. Ising model) sit at boundary of solution space.
◮ Use to solve (partially) spectrum & OPE of 2D Ising without Virasoro. ◮ The Hope: Apply this to D = 3 Ising model?
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◮ Conformal symmetry very powerful tool that goes largely unused in D > 2. ◮ Completely non-perturbative tool to study field theories
1
Does not require SUSY, large N, or weak coupling.
◮ In D = 2 conformal symmetry enhanced to Virasoro symmetry
1
Allows us to completely solve some CFTs (c < 1).
◮ How far could we get in D = 2 with only “global” conformal group?
◮ Use only “global” conformal group, valid in all D. ◮ Constrain “landscape of CFTs” in D = 2, 3 using conformal bootstrap. ◮ Result:
1
CFTs (e.g. Ising model) sit at boundary of solution space.
◮ Use to solve (partially) spectrum & OPE of 2D Ising without Virasoro. ◮ The Hope: Apply this to D = 3 Ising model?
2
◮ Conformal symmetry very powerful tool that goes largely unused in D > 2. ◮ Completely non-perturbative tool to study field theories
1
Does not require SUSY, large N, or weak coupling.
◮ In D = 2 conformal symmetry enhanced to Virasoro symmetry
1
Allows us to completely solve some CFTs (c < 1).
◮ How far could we get in D = 2 with only “global” conformal group?
◮ Use only “global” conformal group, valid in all D. ◮ Constrain “landscape of CFTs” in D = 2, 3 using conformal bootstrap. ◮ Result:
1
CFTs (e.g. Ising model) sit at boundary of solution space.
◮ Use to solve (partially) spectrum & OPE of 2D Ising without Virasoro. ◮ The Hope: Apply this to D = 3 Ising model?
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1 Motivation 2 CFT Refresher ◮ What is crossing symmetry (and why is it useful)? 3 Results ◮ Using our techniques to “solve” 2d Ising. 4 Methodology ◮ The bootstrap in (gory) detail. 5 The (Near) Future ◮ Solving (??) the 3D Ising Model. 6 Conclusions/Comments 3
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Conformal symmetry: SO(1, D − 1) × R1,D−1
+ D (Dilatations) + Kµ (Special conformal) Higher weight representation built on primary operators O: Primary operators: Kµ O(0) = 0 Descendents: Pµ1 . . . Pµn O(0) All dynamics of descendants fixed by those of primaries.
◮ Primaries O called quasi-primaries in D = 2. ◮ Descendents are with respect to “small” conformal group: L0, L±1. ◮ Viraso descendents L−2O are primaries in our language. ◮ In this talk we always mean small conformal group (i.e. for descendants,
conformal blocks, primaries, . . . ).
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CFT Background
CFT defined by specifying:
◮ Spectrum S = {Oi} of primary operators dimensions, spins: (∆i, li) ◮ Operator Product Expansion (OPE)
Oi(x) · Oj(0) ∼
Ck
ij D(x, ∂x)Ok(0)
Oi are primaries. Diff operator D(x, ∂x) encodes descendent contributions. This data fixed all correlatiors in the CFT:
◮ 2-pt & 3-pt fixed:
OiOj = δij x2∆i , OiOjOk ∼ Cijk
◮ Higher pt functions contain no new dynamical info:
12 D(x12,∂x2)Ok(x2)
O3(x3)O4(x4)
34 D(x34,∂x4)(x3)Ol(x4)
12Cl 34D(x12,x34,∂x2,∂x4)Ok(x2)Ol(x4)
CFT Background
This procedure is not unique: φ1φ2φ3φ4 Consistency requires equivalence of two different contractions
Ck
12Ck 34 G12;34 ∆k,lk(x1, x2, x3, x4) =
Ck
14Ck 23 G14;23 ∆k,lk(x1, x2, x3, x4)
Functions Gab;cd
∆k,lk are conformal blocks (of “small” conformal group): ◮ Encode contribution of operator Ok to double OPE contraction. ◮ Entirely kinematical: all dynamical information is in Ck ij. ◮ Crossing symmetry give non-perturbative constraints on (∆k, Ck ij).
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Constraints from Crossing Symmetry
Constraining the spectrum
Figure: A Putative Spectrum in D = 3
Unitarity Bound Gap
2 4 L 1 2 3 4 5 6
∆ ≥ D − 2 2 (l = 0), ∆ ≥ l + D − 2 (l ≥ 0)
◮ “Carve” landscape of CFTs
by imposing gap in scalar sector.
◮ Fix lightest scalar: σ. ◮ Vary next scalar: ǫ. ◮ Spectrum otherwise
unconstrained: allow any
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σ four-point function: σ1σ2σ3σ4 Crossing symmetric values of σ-ǫ
0.50 0.55 0.60 0.65 0.70 0.75 0.80 Σ 1.0 1.2 1.4 1.6 1.8
Blue = solution exists. White = No solution exists.
◮ Only certain values of σ, ǫ consistent
with crossing symmetry.
◮ Solutions to crossing:
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white region ⇒ 0 solutions.
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blue region ⇒ ∞ solutions.
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boundary ⇒ 1 solution (unique)!
◮ Ising model special in two ways:
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On boundary of allowed region.
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At kink in boundary curve.
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To check our technique lets apply to 2d Ising model.
◮ Get exacty same plot as above in 2d with “kink” at Ising point. ◮ Completely solvable theory. ◮ Using Virasoro symmetry can compute full spectrum & OPE.
Can we reproduce using crossing symmetry & only “global” conformal group? Approach:
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Use same methods (even code) that works in 3d.
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Apply to 2d Ising.
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Look at unique solution on boundary.
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Fix σ using value at “kink” or by hand.
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Compute spectrum & OPE from unique solution on boundary.
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Compare to known exact results.
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Exact (Virasoro) results compared to unique solution at “kink” on boundary: Spin 0
L Bootstrap Virasoro ∆ Error Bootstrap Virasoro OPE Error ∆ ∆ (in %) OPE OPE (in %) 1. 1 0.0000106812 0.500001 0.5 0.000140121 4.00145 4 0.03625 0.0156159 0.015625 0.0582036 8.035 8 0.4375 0.00019183 0.000219727 12.6962 12.175 12 1.45833 3.99524 × 10−6 6.81196 × 10−6 41.3496 But not just low spin: Spin 8 L Bootstrap Virasoro ∆ Error Bootstrap Virasoro OPE Error ∆ ∆ (in %) OPE OPE (in %) 8 8. 8 1.25 × 10−9 0.000731575 0.000734387 0.382872 8 9.01 9 0.111111 0.000283447 0.000273566 3.61199 8 12.02 12 0.166667 0.0000205853 0.0000193865 6.18355
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Mileage from Crossing Symmetry?
◮ 12 OPE coefficients to < 1% error. ◮ Spectrum better:
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In 2d Ising expect operators at L, L + 1, L + 4.
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We find this structure up to L = 20 ∼ 38 operator dimensions < 1% error!
0.001 0.01 0.1 1. 10. 100. 1 2 3 4
OPE Error
0.0001 0.01 1. 5 10 15
Spec Error 14
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Bootstrap
So how do we enforce crossing symmetry in practice? Consider four identical scalars: φ(x1)φ(x2)φ(x3)φ(x4) dim(φ) = ∆φ Recall crossing symmetry constraint:
(Ck
φφ)2 G12;34 ∆k,lk(xi) =
(Ck
φφ)2 G14;23 ∆k,lk(xi)
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Bootstrap
So how do we enforce crossing symmetry in practice? Consider four identical scalars: φ(x1)φ(x2)φ(x3)φ(x4) dim(φ) = ∆φ Move everything to LHS:
(Ck
φφ)2 G12;34 ∆k,lk(xi) −
(Ck
φφ)2 G14;23 ∆k,lk(xi) = 0
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Bootstrap
So how do we enforce crossing symmetry in practice? Consider four identical scalars: φ(x1)φ(x2)φ(x3)φ(x4) dim(φ) = ∆φ Express as sum of functions with positive coefficients:
(Ck
φφ)2 pk
[G12;34
∆k,lk(xi) − G14;23 ∆k,ll(xi)]
= 0
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Bootstrap
So how do we enforce crossing symmetry in practice? Consider four identical scalars: φ(x1)φ(x2)φ(x3)φ(x4) dim(φ) = ∆φ
(Ck
φφ)2 pk
[G12;34
∆k,lk(xi) − G14;23 ∆k,ll(xi)]
= 0 How can we solve a complicated functional equation? Answer: Taylor Expand! p1F1(x0) + p2F2(x0) + p3F3(x0) · · · + x (p1F′
1(x0) + p2F′ 2(x0) + p3F′ 3(x0) . . . )+
x2 (p1F′′
1 (x0) + p2F′′ 2 (x0) + p3F′′ 3 (x0) . . . ) + · · · = 0
Every term must vanish independently so rewrite as a vector equation: p1 (F1, F′
1, F′′ 1 , . . . )
+p2 (F2, F′
2, F′′ 2 , . . . )
+p3 (F3, F′
3, F′′ 3 , . . . )
+ · · · = 0
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Bootstrap
So how do we enforce crossing symmetry in practice? Consider four identical scalars: φ(x1)φ(x2)φ(x3)φ(x4) dim(φ) = ∆φ
(Ck
φφ)2 pk
[G12;34
∆k,lk(xi) − G14;23 ∆k,ll(xi)]
= 0 How can we solve a complicated functional equation? Every term must vanish independently so rewrite as a vector equation: p1 (F1, F′
1, F′′ 1 , . . . )
+p2 (F2, F′
2, F′′ 2 , . . . )
+p3 (F3, F′
3, F′′ 3 , . . . )
+ · · · = 0
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Since pi ≥ 0 if the vectors {v1, v2, . . . } span a cone then no solutions!
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Thus can prove crossing symmetry has no solutions even without knowing Ck
ij.
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Problem reduces to a geometric question about cones in derivative space.
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Number of constraints equals number of Taylor coefficients (components of v).
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L0 L 2 L 4 L6 L8 L10 0.6 0.4 0.2 0.2 0.4 0.6 0.8 F 3 , 0 1.0 0.5 0.5 F 1 , 1
Why does this work?
◮ Consider φφφφ with ∆(φ) = 0.515. ◮ We plot e.g. (F(1,1), F(3,0)) ◮ Fix a choice of spectrum {∆k, lk}
∆ = ∆unitarity l = 0 to 10
◮ Vectors for “free” Fock spectrum:
⇒ Took ∆(φ) > 0.5 so not free ⇒ Inconsistent! ⇒ vectors lie inside “cone” ⇒ no crossing symmetry.
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L0 L 2 L 4 L6 L8 L10 0.6 0.4 0.2 0.2 0.4 0.6 0.8 F 3 , 0 1.0 0.5 0.5 F 1 , 1
Why does this work?
◮ Consider φφφφ with ∆(φ) = 0.515. ◮ We plot e.g. (F(1,1), F(3,0)) ◮ Add more ∆ up to ∆unitarity + ǫ.
∆ = ∆unitarity to ∆unitarity + ǫ l = 0 to 10
◮ Colored lines represent range of ∆ we
allow in spectrum.
◮ For ǫ small
⇒ vectors still inside “cone” ⇒ no crossing symmetry.
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L0 L 2 L 4 L6 L8 L10 0.6 0.4 0.2 0.2 0.4 0.6 0.8 F 3 , 0 1.0 0.5 0.5 F 1 , 1
Why does this work?
◮ Consider φφφφ with ∆(φ) = 0.515. ◮ We plot e.g. (F(1,1), F(3,0)) ◮ Colored lines represent range of ∆ we
allow in spectrum.
◮ For ǫ large enough
⇒ vectors span plane. ⇒ In particular can find pk ≥ 0
⇒ crossing sym. can be satisfied.
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L0 L2 L4 L6 L8 L10 L12 L14 L16 L18 L20 L22 L24 L26 L28 L30 0 0.76 0 2.091 1.0 0.5 0.5 F 3 , 0 1.0 0.5 0.5 F 1 , 1
Why does this work?
◮ Consider φφφφ with ∆(φ) = 0.515. ◮ We plot e.g. (F(1,1), F(3,0)) ◮ Colored lines represent range of ∆ we
allow in spectrum.
◮ When ǫ big enough
⇒ vectors no longer in “cone” ⇒ crossing sym. can be satisfied. ⇒ Requires 0.76 ≤ ∆0 ≤ 2.099.
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L0 L 2 L 4 L6 L8 L10 0.6 0.4 0.2 0.2 0.4 0.6 0.8 F 3 , 0 1.0 0.5 0.5 F 1 , 1
Uniqueness of “Boundary Functional”
– Consider ∆0 < 0.76
◮ No combination of vecs give a zero.
pi vi = 0 for pi > 0 – Consider ∆0 > 0.76
◮ Many ways to for vects to give zero. ◮ Families of possible {pi}. ◮ Neither spectrum nor OPE fixed.
– Consider ∆0 = 0.76
◮ Only one way to form zero. ◮ Non-zero pi fixed ⇒ unique spectrum. ◮ Value of pi := (Ck
ii)2 fixed ⇒ unique OPE.
◮ Non-zero pi:
∆ ∼ 0.76, L = 0 ∆ = 2, L = 2 – NOTE: Num operators = num components of vi
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L0 L 2 L 4 L6 L8 L10 0.6 0.4 0.2 0.2 0.4 0.6 0.8 F 3 , 0 1.0 0.5 0.5 F 1 , 1
Uniqueness of “Boundary Functional”
– Consider ∆0 < 0.76
◮ No combination of vecs give a zero.
pi vi = 0 for pi > 0 – Consider ∆0 > 0.76
◮ Many ways to for vects to give zero. ◮ Families of possible {pi}. ◮ Neither spectrum nor OPE fixed.
– Consider ∆0 = 0.76
◮ Only one way to form zero. ◮ Non-zero pi fixed ⇒ unique spectrum. ◮ Value of pi := (Ck
ii)2 fixed ⇒ unique OPE.
◮ Non-zero pi:
∆ ∼ 0.76, L = 0 ∆ = 2, L = 2 – NOTE: Num operators = num components of vi
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L0 L2 L4 L6 L8 L10 0.6 0.4 0.2 0.2 0.4 0.6 0.8 F 3 , 0 1.0 0.5 0.5 F 1 , 1
Uniqueness of “Boundary Functional”
– Consider ∆0 < 0.76
◮ No combination of vecs give a zero.
pi vi = 0 for pi > 0 – Consider ∆0 > 0.76
◮ Many ways to for vects to give zero. ◮ Families of possible {pi}. ◮ Neither spectrum nor OPE fixed.
– Consider ∆0 = 0.76
◮ Only one way to form zero. ◮ Non-zero pi fixed ⇒ unique spectrum. ◮ Value of pi := (Ck
ii)2 fixed ⇒ unique OPE.
◮ Non-zero pi:
∆ ∼ 0.76, L = 0 ∆ = 2, L = 2 – NOTE: Num operators = num components of vi
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L0 L2 L4 L6 L8 L10 0.6 0.4 0.2 0.2 0.4 0.6 0.8 F 3 , 0 1.0 0.5 0.5 F 1 , 1
Uniqueness of “Boundary Functional”
– Consider ∆0 < 0.76
◮ No combination of vecs give a zero.
pi vi = 0 for pi > 0 – Consider ∆0 > 0.76
◮ Many ways to for vects to give zero. ◮ Families of possible {pi}. ◮ Neither spectrum nor OPE fixed.
– Consider ∆0 = 0.76
◮ Only one way to form zero. ◮ Non-zero pi fixed ⇒ unique spectrum. ◮ Value of pi := (Ck
ii)2 fixed ⇒ unique OPE.
◮ Non-zero pi:
∆ ∼ 0.76, L = 0 ∆ = 2, L = 2 – NOTE: Num operators = num components of vi
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Using E-expansion, Monte Carlo and other techniques find partial spectrum: Field: σ ǫ ǫ′ Tµν Cµνρλ Dim (∆): 0.5182(3) 1.413(1) 3.84(4) 3 5.0208(12) Spin (l): 2 4 Only 5 operators and no OPE coefficients known for 3d Ising! Lots of room for improvement!
Compute these anomolous dimensions (and many more) and OPE coefficients using the bootstrap applied along the boundary curve.
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Computing 3d Spectrum from Boundary Functional?
A first problem: what point on the boundary? what is correct value of σ?
Ising
0.50 0.55 0.60 0.65 0.70 0.75 0.80 Σ 1.0 1.2 1.4 1.6 1.8
Ε Ising
0.510 0.515 0.520 0.525 0.530Σ 1.38 1.39 1.40 1.41 1.42 1.43 1.44
Ε
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In D = 2 we know σ by other means.
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“Kink” is not so sharp when we zoom in.
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Gets sharper as we increase number of constraints ⇒ should taylor expand to higher order!
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What’s left to do?
◮ Fix dimension of σ in 3d Ising using “kink” or other features. ◮ Use boundary functional to compute spectrum, OPE for 3d Ising. ◮ Compare with lattice or experiment! ◮ Additional constraints: add another correlator
σσǫǫ.
◮ Study spectrum, OPE as a function of spacetime dimension. ◮ CT and Cσσǫ appear in both correlators ⇒ should give strong constraints.
◮ How specific is this structure to Ising model? ◮ Can we impose more constraints and find new “kinks” for other CFTs? ◮ Can any CFT be “solved” by imposing a few constraints (gaps) and then solving
crossing symmetry?
◮ What about SCFTs? Need to know structure of supersymmetric conformal blocks.
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What’s left to do?
◮ Technology still begin refined ⇒ lots to do! ◮ Why is Ising model on boundary? Why at a “kink”? ◮ Do these features have physical meanings or artifacts of method? ◮ Only just begun to take advantage of conformal symmetry in D > 2.
◮ Generalized Free Field CFTs are dual to free (N ∼ ∞) fields in AdS
[Heemskerk et al, SE and Papadodimas]
◮ Higher spin GFFs are “multi-particle states” in bulk:
O ∼ φ∂{µ1 . . . ∂µn}φ with ∆O = n + 2∆φ and ∆φ > D−2
2 .
◮ Tentative result: Bound on gap for any spins is saturated by GFFs. ◮ If true then: leading 1/N2 always negative!
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What’s left to do?
◮ Technology still begin refined ⇒ lots to do! ◮ Why is Ising model on boundary? Why at a “kink”? ◮ Do these features have physical meanings or artifacts of method? ◮ Only just begun to take advantage of conformal symmetry in D > 2.
◮ Generalized Free Field CFTs are dual to free (N ∼ ∞) fields in AdS
[Heemskerk et al, SE and Papadodimas]
◮ Higher spin GFFs are “multi-particle states” in bulk:
O ∼ φ∂{µ1 . . . ∂µn}φ with ∆O = n + 2∆φ and ∆φ > D−2
2 .
◮ Tentative result: Bound on gap for any spins is saturated by GFFs. ◮ If true then: leading 1/N2 always negative!
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