Light-front versus equal-time quantization in φ4
theory
Work supported in part by US DoE and the Minnesota Supercomputing Institute
and done in collaboration with M. Burkardt and J.R. Hiller.
Sophia S. Chabysheva
Department of Physics and Astronomy
University of Minnesota Duluth
6 Sep 2016
Overview
z systematic difference between light-front and equal-time
values for critical coupling
z
z
z
pointed out at Frascati
resolved by considering difference in mass renormalizations [M.
Burkardt, PRD 47, 4628 (1993)]
arXiv:1606.00026 to appear in PRD
z probability for higher Fock sectors should grow dramatically as
critical coupling is approached
z
z
z
not observed
sector-dependent mass did not help
seem to need basis with ∞ # of states
z details of methods provided by previous talk
Mass squared for unbroken phase
Critical coupling at g = 2.1 ± 0.05.
Critical couplings compared: ḡ ≡ π6 g
Method
LF sym. polys.
DLCQ
Quasi-sparse
eigenvector
Density matrix
Lattice
Uniform matrix
product
Renorm. H trunc.
ḡc
1.1±0.03
1.38
Reported by
this work
Harindranath & Vary
2.5
2.4954(4)
+0.025
2.70
−0.013
2.79±0.02
Lee & Salwen
Sugihara
2.766(5)
2.97(14)
Milsted et al.
Rychkov & Vitale
Schaich & Loinaz
Bosetti et al.
Mass renormalization
Bare mass renormalized by tadpole contributions in ET
quantization but not in LF quantization
[M. Burkardt, PRD 47, 4628 (1993)]
φ2
φ2
2
2
µLF = µET + λ h0| |0i − h0| |0ifree .
2
2
The vev’s of φ2 resum the tadpole contributions; the subscript free
indicates the vev with λ = 0.
→ need to calculate vev’s
Z ∞
X
φ2
1
h0| |0i → h0|φ(ǫ+ , ǫ− )
dP
|ψn (P)ihψn (P)|φ(0, 0)|0i.
2
2
0
n
φ(ǫ+ , ǫ− ) = e iP
− ǫ+ /2
φ(0, ǫ− )e −iP
− ǫ+ /2
.
Matrix elements
For the nth bound state
∗ a(P)
hψn (P)|φ(0, 0)|0i = h0|ψn1
h0|φ(ǫ+ , ǫ− )|ψn (P)iR
+
= h0|e i0ǫ
=
√ψn1
4πP
R
√dp a† (p)|0i
4πp
=
ψ∗
√ n1
4πP
−
2 +
√dp a(p)e −ipǫ /2 e −iMn ǫ /2P ψn1 a† (P)|0i
4πp
−
2 +
e −i(Pǫ +Mn ǫ /P)/2
For the one-particle free state
R
h0|a(P)φ(0, 0)|0i = h0|a(P)
√dp a† (p)|0i
4πp
=
√1
4πP
h0|φ(ǫ+ , ǫ− )a† (P)|0i
+ R
−
2 +
√dp a(p)e −ipǫ /2 e −iµ ǫ /2P a† (P)|0i
= h0|e i0ǫ
4πp
=
−
2 +
√ 1 e −i(Pǫ +µ ǫ /P)/2
4πP
vev’s
φ2
1X
h0| |0i =
2
2 n
Z
∞
0
dP
|ψn1 |2 −i(Pǫ− +Mn2 ǫ+ /P)/2
e
4πP
and
Z
1 −i(Pǫ− +µ2 ǫ+ /P)/2
1 ∞
φ2
dP
e
.
h0| |0ifree =
2
2 0
4πP
P
With use of 1 = n |ψn1 |2 ,
h0|
φ2
φ2
|0i − h0| |0ifree
2
2
2 +
X |ψn1 |2 Z ∞ dP
Mn2 ǫ+
−iPǫ− /2
−i 2P
−i µ2Pǫ
e
e
−e
=
8π
P
0
n
Mass shift
2
2
h0| φ2 |0i − h0| φ2 |0ifree
i
p
p
P n1 |2 h
K0 (Mn −ǫ2 + iη) − K0 (µ −ǫ2 + iη) .
= n |ψ4π
η is from a convergence factor; K0 (z) → − ln(z/2) − γ
h0|
X |ψn1 |2
φ2
φ2
Mn
|0i − h0| |0ifree = −
ln
≡ −∆/4π,
2
2
4π
µ
LF
n
µ2LF = µ2ET −
gET =
µ2
λ
∆ or ET
= 1 + gLF ∆.
4π
µ2LF
M2
gLF
1
gLF
M2
=
=
and
(1 + gLF ∆)
1 + gLF ∆ µ2LF
µ2ET /µ2LF
µ2ET
Relative probabilities
l
[
[
[
[
[
[
No indication of critical behavior at g = 2.1.
Computation of shift ∆ will diverge ∼ |ψ11 |2 ln(M1 /µLF ).
Plot of shift
Ʃ
J
Points obtained as extrapolations in the basis size.
Lines are linear and quadratic fits to shifts below g = 1,
extrapolated to the region of the critical coupling.
Plot of ET mass squared
(7
Different points at the same g value correspond to different
truncations of the basis.
Above g = 1, points begin to diverge.
Interpretation
z divergence from failure of |ψ11 | to go to zero with M1
z
the product |ψ11 |2 ln(M1 /µLF ) then diverges
z can use extrapolations from below g = 1 to estimate ∆ at the
critical coupling
z
z
∆(g = 2.1) = −0.47 ± 0.12
value is from the higher-order extrapolation, with the
lower-order extrapolation used to indicate the error.
z from the latest equal-time value for the critical coupling
[Rychkov & Vitale], gETc = π6 2.97 = 5.67, we extract a shift
of (gLFc /gETc − 1)/gLFc = −0.30
z consistent with the estimated value of the shift
Summary
z can understand difference between ET and LF by taking
different mass renormalizations into account
z calculation of the mass shift near critical coupling shows poor
behavior of computed eigenstates
z
z
z
relative probabilities of higher Fock states do not show critical
behavior
in fact, should → ∞ as denominator |ψ11 |2 → 0
the original hypothesis, that sector-dependent masses would
resolve the paradox, must be incorrect.
z correct representation near critical coupling apparently
requires method without Fock-space truncation
z
z
coherent-state basis
LFCC with nontrivial valence state: |1i + |3i
z ssc and jrh, PLB 711, 417 (2012)