Download Some possible consequences of the HUGE magnetic fields

Some possible consequences of the HUGE
magnetic fields generated at the LHC
Talk by Poul Olesen
CP3-Origins, Odense, November 2012
Work done in part with Jan Ambjørn, NBI
JA+PO, PL B 257(1991) 201; PO, arXiv: 1207.7045
November 9, 2012
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• HUGE magnetic fields at the LHC: ⇒ Blackboard
• Higgs → γγ: ⇒ Blackboard.
• Propagators and magnetic fields: Perturbative calculations should anyhow
be done in the presence of an external field. Propagators are not just the
perturbative ones!
• W-condensation:
How it works for a homogeneous magnetic field in the electroweak sector
of the standard model.
• Does it work in the LHC circumstances? The field is now space-time dependent!
• Also HUGE magnetic fields in heavy ion collisions. Many consequences,
many papers!
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PROPAGATORS AND MAGNETIC FIELDS
• In the di-gamma decay of the Higgs one uses
(iγ µ pµ + mq )
p2
and (unitary gauge)
−i(gµν −
−i
,
− m2q + iǫ
pµ pν
)
m2W
p2 − m2W + iǫ
(1)
.
(2)
as well as simple vertices. Summary: Huang, Tang, Wu, arXiv:1109.4846.
• These propagators change in a magnetic field. For example, in a homogeneous magnetic
field along the z-axis the quark propagator is changed into
!
∂
SF (x, x ) = γ ( µ − ieAµ ) − mq DF (x, x′ ),
∂x
µ
′
(3)
where DF should satisfy
∂
∂2
− e2 H 2 x2 − m2q − ieγ1 γ2 H DF (x, x′ ) = 0.
∇ − 2 − 2ieHx
∂t
∂y
!
2
(4)
D was computed many years ago, J. Geheniau, Physica 16 (1950)822,
Z ∞+iǫ
i
2
2
′
DF (x, x ) = 2 Φ(x, x )
dα e(i/2) (s α+mq /α) ×
4π
0
eH
eH
eH
′ 2
′ 2
cot
− γ1 γ2
) exp (i/4) (x − x ) + (y − y ) eH (2α/eH − cot eH/2α) (5)
× (
2α
2α
2α
′
with s2 = (t − t′ )2 − (x − x′ )2 − (y − y ′ )2 − (z − z ′ )2 ,
Φ is an overall gauge dependent phase factor.
H → 0 ⇒ the square bracket → 1 and
1
8π 2
Z
d4 xipx
Z
0
∞+iǫ
dα e(i/2)
(αs2 +m2 /α)
=
p2
−i
.
− m2 + iǫ
(6)
• For a√large field H the propagator becomes centered around small transverse distances
≤ 1/ eH.
LIFE WITH A MAGNETIC FIELD IS MUCH MORE COMPLICATED
THAN WITHOUT. BUT MORE INTERESTING!
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W-CONDENSATION IN A STRONG HOMOGENEOUS MAGNETIC FIELD
• W-condensation occurs in the electroweak theory. First, take for simplicity Lagrangian
for a massive field
1 2
1 ex 2 1
ex
) − |Dµ Wν −Dν Wµ |2 −m2W Wµ† Wµ −iefµν
Wµ† Wν + gW
[Wµ†2 Wν2 −(Wµ† Wµ )2 ]
L = − (fµν
4
2
2
(7)
Higgs fixed at its vacuum value φ0 .
Dµ = ∂µ − ieAex
µ ,
(8)
e = gW sin θ,
(9)
1 2 2
m2W = gW
φ0
2
(10)
ex
−iefµν
Wµ†
(11)
ex
m2W Wµ† Wµ + iefµν
Wµ† Wν
(12)
• “Anomalous” magnetic moment
unbounded from below. Ignoring W 4 -terms this means instability, Niels Kjær
Nielsen + PO, NP B144 (1978)376. Quantum fluctuations around Wµ = 0 ⇒ energy can be gained by going to Wµ 6= 0.
There is an effective mass term
With f12 = H the eigenvalues and eigenvectors are
m2W ± eH
(13)
(W1 , W2 ) = (W, iW )
(14)
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Electroweak Theory
• Electroweak Lagrangian:
LEW = −
1
1 2
1 2
|Dµ Wν − Dν Wµ |2 − fµν
− Zµν
− (∂φ)2
2
4
4
g2
1
Z 2 + 2λφ20 φ2
− g 2 φ2 Wµ† Wµ −
2
4 cos2 θ µ
−ig(fµν sin θ + Zµν cos θ)Wµ† Wν
1 2
− gW
[Wµ†2 Wν2 − (Wµ† Wµ )2 ] + λ(φ + φ40 )
2
Dµ = ∂µ − ig(Aµ sin θ + Zµ cos θ)
(15)
(16)
• STATIC SOLUTION: Energy can be written a la Bogomolny’i as
g2
(φ2 − φ20 )2 + T D, T D = total derivatives.
E = () + λ −
8 cos2 θ
2
!
(17)
Look for periodic static solutions. The integral over T D vanishes. For
λ≥
energy larger than zero (no tachyon).
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g2
8 cos2 θ
(18)
A SIMPLE BOGOMOLNY’I TYPE OF SOLUTION:
• If λ =
g2
8 cos2 θ
(unrealistic) we get nice eqs. (JA+PO, NPB 315 (1989) 606):
−∂ 2 log |W (x1 , x2 )| =
∂ 2 log φ =
g2 2
φ + 2g 2|W |2,
2
g2
(φ − φ20 ) + g 2 |W |2
4 cos2 θ
(19)
(20)
W = W1 , W2 = iW1 , W3 = W0 = 0.
• These coupled eqs can be solved and the solution is periodic. Having the solution, the
fields are found from
f12 =
gφ20
gφ20
m2
+ 2g sin θ|W |2,
= W
2 sin θ
2 sin θ
e
(21)
g
(φ2 − φ20 ) + 2g cos θ|W |2 ,
2 cos θ
(22)
Z12 =
Solution has been generalized, Chemodub, Doorsselaere, Verschelde, arXiv:1203.3071
• For a periodic solution the phase in W = |W |eiχ is topologically non-trivial. Use
1
1
Ai = ǫij ∂j (log |W | + 2 cos2 θ log φ) + ∂i χ.
e
e
⇒
Z
cell
f12 =
I
Ai dxi =
6
I
∂i χdxi /e = 2π/e
(23)
(24)
COMPARISON WITH SUPERCONDUCTIVITY:
• W is an order parameter. In superconductor, let ψ be order parameter, then
∂ 2 log |ψ|2 = e2 (|ψ|2 − ψ02 ).
(25)
Somewhat similar to coupled eqs for |W | and φ. Important differences: In superc.
we have solution |ψ| = ψ0 . |W | cannot be constant, since
−∂ 2 log |W (x1 , x2 )| =
g2 2
φ + 2g 2|W |2,
2
(26)
implies that |W |2 = −φ2 /4, and we are back at unstable magnetic field.
• Anti-Lenz law/anti screening:
∂i fij = 2g sin θǫij ∂i |W |2 (non − Abelian)
(27)
f12 = m2W /e + 2e|W |2 (non − Abelian)
(28)
∂i fij = −2eǫij ∂i |ψ|2 (Abelian)
(29)
In superconductor
f12 = Hc − 2e|ψ|2 (Abelian)
W-condensate is an upside-down superconductor!
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(30)
DISAPPEARENCE OF THE W CONDENSATE AT EVEN LARGER MAGNETIC FIELDS
Threshold for W-condensation occurs at magnetic field m2W /e ≈ 1024 Gauss.
W-condensation does not continue for arbitrarily large fields: For simplicity ignore the Z
field. The potential involving the Higgs and the W field is
−2ef12 |W |2 + g 2 φ2 |W |2 − 2λφ20 φ2 + g 2 |W |4 + λ(φ4 + φ40 )
(31)
Kinetic terms ignored. Can be taken into account...but let us proceed heuristically:
• If ef12 less than m2w = g 2φ20 /2 then minimum at
φ = φ0 , W = 0, ⇒ no W − condensate
(32)
• If f12 ≥ m2W /e we get a W-condensate. Minimizing wrt W gives
g 2 |W |2max = ef12 − g 2φ2 /2.
(33)
As |W |2 increases with the magnetic field, the value of φ2 will decrease from φ20 because
of the terms g 2φ2 |W |2 − 2λφ20φ2 in the potential ⇒
φ2min =
m2H − ef12 2
φ0 , m2H = 4λφ20 , m2W = g 2 φ20 /2
2
2
mH − mW
(34)
Thus W-condensation exists for
m2W /e ≤ f¯12 ≤ m2H /e.
(35)
At upper lt the SU(2)×UY (1) symmetry is restored. Supported by rigorous argument.
Above upper lt the W field becomes a pure gauge. The magnetic field turns into a Ymagnetic UY (1) field
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DOES W-CONDENSATION WORK IN THE LHC CIRCUMSTANCES?
• The magnetic field is now space-time dependent. S. Schramm, B. Mueller and A.
Schramm, PL B 277 (1992) 512: IT DOES NOT WORK!
Beam direction z-axis, fields are pancake like, so take (b=impact parameter)
Hx = (Ze/b) δ(z), Ay = (Ze/b)θ(−z), Ax = Az = A0 = 0.
(36)
Factor in front fixed by demanding that the integral over Hx is the same for this field
as it is for the right field. They compute the Greens function of the W’s. Poles at
E 2 = m2W + kx2 + ky2 −
1
(2ky + 3Ze2 /b)
16
(37)
Minimum for kx = 0 and ky 6= 0, giving
3
E 2 ≥ m2W − (Ze2 /b)2
4
(38)
W condense if the right hand side can be negative ⇒
b≤
2πα
∼ 0.045m−1
W
mW
(39)
This localization requires an additional kinetic energy contribution of order ∼ 20mW
which by far exceeds the interaction of the magnetic field.
• HM!
• IN ANY CASE THE MAGNETIC FIELD INFLUENCES THE RELEVANT W AND QUARK PROPAGATORS! So standard calculations need modifications!!
• The excess rate for HIGGS→ γγ may be an indication that standard perturbation theory needs to be modified: IF W CONDENSE THERE ARE
MORE W’S THAN PERTURBATIVELY EXPECTED ⇒ MORE PHOTONS Note that the parameter τ = m2H /4m2W is replaced by
m2H m2H − eH
4m2W m2H − m2W
(40)
This parameter becomes smaller than the original τ , and the rate goes up for smaller
τ.
• Would be interesting to invent experimental tests. A very expensive way: build a p − p̄
collider! The magnetic field would change considerably.
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HEAVY ION-MANY MORE QUARKS AND GLUONS-LARGE MAGNETISM
D. Kharzeev, L. McLerran, H. Warringa, arXiv:0711.0950:
• In classical QCD vacuum the field configurations must be a pure gauge, U ∈SU(3).
Different vacua are characterized by the topological invariant winding number
nw =
1
24π 2
Z
d3 xǫijk Tr[(U † ∂i U)(U † ∂j U)(U † ∂k U)].
(41)
The winding number
Qw = nw (t = ∞) − nw (t = −∞)
(42)
counts transitions from one classical vacuum to another.
• Configurations with non-zero Qw break the charge-parity symmetry of QCD.
• Large magnetic fields provides a mechanism whereby configurations with Qw 6= 0 can
separate charge−”The Chiral Magnetic Effect”.
• Large H means all particles in the lowest Landau level. Their spin is aligned along
H and they can only move along H. Quarks with opposite charges have spin aligned
in different directions. With H in the z direction positively charged right handed and
negatively charged left handed fermions will move up. Similarly, positively charged
left handed and negatively charged right handed fermions move down.
• Fermions interact with the field and some of them will change helicity,
NLf − NRf = 2Qw
(43)
Fermions can only change helicity by reversing their momenta, since spin flip is energetically suppressed in a large magnetic field. Taking again the magnetic field in the z direction, then positively charged right handed and negatively charged left handed fermions
will move upwards. Also positively charged left handed and negatively charged right
handed quarks move downwards, as before. But there is now a difference between the
numbers of right and left handed quarks, so a current is produced along the direction
of H.
• The chiral magnetic effect is summarized by the formula
j ∝ qf2 µH
(44)
where µ is the “chiral chemical potential” which induces the difference between right
and left handed particles.
• Sufficiently large magnetic fields cause preferential emission of charged particles along
the direction of angular momentum.
• Many other works on the effects of large magnetic fields in heavy ion physics. See for
example Bzdak and Skokov, “Anisotropy of photon production: Initial eccentricity or
magnetic field”, arXiv:1208.5502.
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