Unparticle Physics

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Unparticle Physics Howard Georgi Center for the Fundamental Laws of Nature Jefferson Physical Laboratory Harvard University Sidney Coleman Unparticle stuff with scale dimension d looks like a non-integral number d of invisible massless particles. before I circulated the paper widely, I sent it to some of my smartest former students and grand-students, including Ann Nelson and Lisa Randall Unparticle stuff with scale dimension d looks like a non-integral number d of invisible massless particles. before I circulated the paper widely, I sent it to some of my smartest former students and grand-students, including Ann Nelson and Lisa Randall The attached hallucination came to me a few days ago and I have been in a trance since then trying to work out the details. I thought it was time to try it out on some of my friends. Since this is very possibly embarrassingly nuts, I would appreciate it if you could keep it to yourselves for a day or so. Several possibilities occur to me. 1 - It is trivially wrong for some reason. 2 - Everyone knows it already and is not interested. 3 - Some other type of bound kills these theories so that the unparticles can never be seen. I would be grateful for a little sanity check.

Could we see something really different at the LHC? We expect new particles! But could we see something else - not describable in the language of particles? Unparticles? A scale invariant shadow world? Maybe! Start with a review of scale invariance - then show how it might yield an example of unparticle physics. Scale invariance is common in mathematics — start there. Could we see something really different at the LHC? We expect new particles! But could we see something else - not describable in the language of particles? Unparticles? A scale invariant shadow world? Maybe! Start with a review of scale invariance - then show how it might yield an example of unparticle physics. Scale invariance is common in mathematics — start there. Could we see something really different at the LHC? We expect new particles! But could we see something else - not describable in the language of particles? Unparticles? A scale invariant shadow world? Maybe! Start with a review of scale invariance - then show how it might yield an example of unparticle physics. Scale invariance is common in mathematics — start there. trivial discrete scaling

X∞ 3 ³ 1´ ³3 ´ g(x) = 2−j h(frac(2jx)) h(x) = Θ x− Θ −x 4 2 4 j=−∞

1.0 0.5

0.8 0.4

0.6 0.3

0.4 0.2

0.2 0.1

0.2 0.4 0.6 0.8 1.0 0.1 0.2 0.3 0.4 0.5 trivial continuous y = x scaling

1.0 2.0

0.8 1.5

0.6 1.0 0.4

0.5 0.2

0.2 0.4 0.6 0.8 1.0 0.5 1.0 1.5 2.0 almost trivial continuous r = eθ/20 scaling

1.0 10

0.5 5

-1.0 -0.5 0.5 1.0 -10 -5 5 10

-0.5 -5

-1.0 -10 non-trivial discrete scaling with d = 1.2

X∞ 3 ³ 1´ ³3 ´ g(x) = 2−1.2j h(frac(2jx)) h(x) = Θ x− Θ −x 4 2 4 j=−∞

0.4 0.4 0.8

0.3 0.3 0.6

0.2 0.2 0.4

0.1 0.1 0.2

0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1.0 0.1 0.2 0.3 0.4 0.5 non-trivial continuous y = x1.5 scaling with d = 1.2

1.0 0.4 0.4 0.8

0.3 0.3 0.6

0.2 0.2 0.4

0.1 0.1 0.2

0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1.0 0.1 0.2 0.3 0.4 0.5 pleasing shapes z = (x2 + y2)α scale invariance is less common in physics scale invariance is less common in physics scale invariance is less common in physics scale invariance is less common in physics scale invariance is less common in physics scale invariance is less common in physics scale invariance is less common in physics scale invariance is less common in physics scale invariance is less common in physics scale invariance is less common in physics scale invariancescale invariance is less is less common common in physics in physics scale invariance is less common in physics

h¯ and c all dimensional quantities are tied together time and space and must scale together if time and space are scaled up, energy and momentum must be scaled down and there are massive particles h¯ and c all dimensional quantities are tied together time and space and must scale together if time and space are scaled up, energy and momentum must be scaled down and there are massive particles classically, particles = chunks of pµ

2 µ 2 0 p = pµp = m ~v = ~p/p (c = 1) in QM, p0, ~p → ω,~k - dispersion relation

µ 2 2 ~ 2 2 pµp = m → ω = k + m (¯h = 1) either way - ﬁxed non-zero m breaks scale invariance - you can’t scale energy and momentum or x and t without changing m classically, particles = chunks of pµ

2 µ 2 0 p = pµp = m ~v = ~p/p (c = 1) in QM, p0, ~p → ω,~k - dispersion relation

µ 2 2 ~ 2 2 pµp = m → ω = k + m either way - fixed non-zero m breaks scale invariance - you can’t scale energy and momentum or x and t without changing m but theories of free massless relativistic particles have scale invariance - here I understand the physics! - if a state of free massless particles exists with (Ej, ~pj) you can always make a “scaled” state with (λEj, λ~pj) theories of free massless relativistic particles have scale invariance - here I understand the physics! - if a state of free massless particles exists with (Ej, ~pj) you can always make a “scaled” state with (λEj, λ~pj) 2π P = |M |2 ρ if h¯ if f Fermi’s Golden Rule - ρf = density of final states - number of quantum states per unit volume - states in a cubical box with side ` with periodic boundary conditions - ~p = 2π~n/` # states d3p dρ(p) = = `3 (2π)3 “phase space” is a shorter phrase - conventional to make this relativistic 2π P = |M |2 ρ if h¯ if f Fermi’s Golden Rule - ρf = density of final states - number of quantum states per unit volume - states in a cubical box with side ` with periodic boundary conditions - ~p = 2π~n/` d3p d4p dρ(p) = = θ(p0) δ(p2) 2E(2π)3 (2π)3 “phase space” is a shorter phrase - conventional to make this relativistic massless particles phase space also scales d3p d4p dρ = = Θ(p0)δ(p2) 1 2E (2π)3 (2π)3

E = p0 = |~p | d3p d3p p → λp → λ2 2E (2π)3 2E (2π)3 this is trivial scaling - like y = x - dρ1 is the phase space for one massless particle massless particles phase space also scales d3p d4p dρ = = Θ(p0)δ(p2) 1 2E (2π)3 (2π)3

E = p0 = |~p | d3p d3p p → λp → λ2 2E (2π)3 2E (2π)3 this is trivial scaling - like y = x - dρ1 is the phase space for one massless particle now suppose you have two massless particles in your final state that you don’t see (neutrinos for example), so all you know is their total energy-momentum P you can combine the phase spaces for the massless particles to get the phase space for the combination P =total 2×massless m 4−momentum phase space z }| { z }| { z }|m {     Z X2 Y2 ³ ´ ³ ´ d4p  4   2 0 j  4 δ P − pj δ pj θ pj 3 d P j=1 j=1 (2π) 1 ³ ´ ³ ´ d4P ≡ dρ (P ) = θ P 0 θ P 2 2 8π (2π)4 assumes no other dependence on p1 and p2 - no δ-function - P 02 − P~ 2 can by anything greater than zero - the two-neutrino system can have any mass - the SYSTEM is not a particle - not too surprising P =total 2×massless m 4−momentum phase space z }| { z }| { z }|m {     Z X2 Y2 ³ ´ ³ ´ d4p  4   2 0 j  4 δ P − pj δ pj θ pj 3 d P j=1 j=1 (2π) 1 ³ ´ ³ ´ d4P ≡ dρ (P ) = θ P 0 θ P 2 2 8π (2π)4 assumes no other dependence on p1 and p2 - no δ-function - P 02 − P~ 2 can by anything greater than zero - the two-neutrino system can have any mass - the SYSTEM is not a particle - not too surprising P =total 2×massless m 4−momentum phase space z }| { z }| { z }|m {     Z X2 Y2 ³ ´ ³ ´ d4p  4   2 0 j  4 δ P − pj δ pj θ pj 3 d P j=1 j=1 (2π) 1 ³ ´ ³ ´ d4P ≡ dρ (P ) = θ P 0 θ P 2 2 8π (2π)4 assumes no other dependence on p1 and p2 - no δ-function - P 02 − P~ 2 can by anything greater than zero - the two-neutrino system can have any mass - the SYSTEM is not a particle - not too surprising we know this system is a 2-particle state - don’t we? can’t we see the particles individually? not necessarily, if the 2 particles always appear in exactly the same combination in all the physics! but we can see the 2-ness even if we can’t see the particles - missing E and ~p uniform dσ ∝ dρ (P ) = 2 in P 2 any single event just tells you that the missing stuff is not a single particle with zero mass because P 2 6= 0 - but the distribution of many events depends on the number of missing particles - phase space for more particles grows faster with P 2 dρn(P ) =     Z Xn Yn ³ ´ ³ ´ d4p  4   2 0 j  4 δ P − pj δ pj θ pj 3 d P j=1 j=1 (2π) 4 ³ ´ ³ ´ ³ ń−2 d P = A θ P 0 θ P 2 P 2 (2π)4 n (2π)4 16π5/2 Γ(n + 1/2) A = n (2π)2n Γ(n − 1) Γ(2n) 1 - you get information about how many massless particles you are making by measuring the differential cross-section dσ/d4P

2 - dρn(P ) scales

2n dρn(λP ) = λ dρn(P )

3 - but this can be confused by factors of P 2 in |M|2 1 - you get information about how many massless particles you are making by measuring the diﬀerential cross-section dσ/d4P

2 - dρn(P ) scales

2n dρn(λP ) = λ dρn(P )

3 - but this can be confused by factors of P 2 in |M|2 what happens if we see a scale invariant dρd, but d is not integral? a fractional number of unseen particles? not sure what that would mean! but what would it be? scale invariant unparticles with dimension d could this stuff exist? what happens if we see a scale invariant dρd, but d is not integral? a fractional number of unseen particles? not sure what that would mean! but what would it be? scale invariant unparticles with dimension d could this stuff exist? in fact non-trivial interacting scale invariant quantum field theories have been known for a long time - theorists know a lot about the correlation functions in Euclidean space I realized that I understood the field theory better than I understood the physics in fact non-trivial interacting scale invariant field theories have been known for a long time - theorists know a lot about the correlation functions in Euclidean space I realized that I understood the field theory better than I understood the physics QFT - “obvious” quantum extension of classical field theory like E&M the fields we are familiar with, like E~ (x) and B~ (x), are operators in the quantum mechanical Hilbert space of the world that create and destroy particles symmetries of quantum field theory are my favorite things in physics scale transformation in a field theory shrink coordinates by λ ⇒ x → x˜ = x/λ - fields get rescaled for the same reason that vector fields rotate when the coordinates rotate ...... E~ (x)...... 0 ...... x ...... x ...... o . → ...... o ...... ~ 0 0 ...... E (x ) ...... shrink coordinates by λ ⇒ x → x˜ = x/λ

O(x) → O˜(˜x) = λdO(λx˜) = λdO(x) d is scale “dimension” - for E~ (x) or B~ (x) describing free photons, this scale transformation is a symmetry for d = 2, the “engineering dimension” - not surprising since these ﬁelds create massless photons what does this have to do with unparticles? shrink coordinates by λ ⇒ x → x˜ = x/λ

O(x) → O˜(˜x) = λdO(λx˜) = λdO(x) h0| O(x) O(0) |0i = ∆(x) h0| O˜(˜x) O˜(0) |0i = ∆(˜x) λ2d h0| O(x) O(0) |0i = ∆(x/λ) ⇒ ∆(x) ∝ x−2d shrink coordinates by λ ⇒ x → x˜ = x/λ

O(x) → O˜(˜x) = λdO(λx˜) = λdO(x) h0| O(x) O(0) |0i ∝ x−2d insert intermediate unparticle states Z −ipx 2 = e |hU,P | O(0) |0i| dρU (P ) where |U,P i is an unparticle state and dρU (P ) is the density of unparticle states shrink coordinates by λ ⇒ x → x˜ = x/λ

O(x) → O˜(˜x) = λdO(λx˜) = λdO(x) h0| O(x) O(0) |0i ∝ x−2d Z −ipx 2 = e |hU,P | O(0) |0i| dρU (P ) hU,P | O(0) |0i = 1 (wave function)

2d ⇒ dρU (λP ) = λ dρU (P ) O(x) creates unparticle with dimension d shrink coordinates by λ ⇒ x → x˜ = x/λ

O(x) → O˜(˜x) = λdO(λx˜) = λdO(x) d is scale “dimension” - for fractional d, the unparticle stuff created by O(x) cannot be ordinary particles - what is it? More physical question might be easier - can it co-exist with the standard model? maybe if it couples to SM particles only at high energies - effective field theory Example - Banks-Zaks - ordinary Yang-Mills gauge theories like QCD with massless quarks but with the number of colors and flavors chosen to make the running slow - asymptotically free at large energies - at low energies, the gauge coupling gets stuck at a nonzero value - a nontrivial IR fixed point Just an example! Not the most interesting case - but familiar and understandable. Take the physics seriously and see what happens! Example - Banks-Zaks - ordinary Yang-Mills gauge theories like QCD with massless quarks but with the number of colors and flavors chosen to make the running slow - asymptotically free at large energies - at low energies, the gauge coupling gets stuck at a nonzero value - a nontrivial IR fixed point Just an example! Not the most interesting case - but familiar and understandable. Take the physics seriously and see what happens! Example - Banks-Zaks - ordinary Yang-Mills gauge theories like QCD with massless quarks but with the number of colors and flavors chosen to make the running slow - asymptotically free at large energies - at low energies, the gauge coupling gets stuck at a nonzero value - a nontrivial IR fixed point Just an example! Not the most interesting case - but familiar and understandable. Take the physics seriously and see what happens! at very high energies

standard fields of BZ fields model ↔↔ fields mass MU (Banks-Zaks) dimensional transmutation scale ΛU

dBZ −d OBZ → ΛU O O(x) → O˜(˜x) = λdO(x) below the scale MU

standard fields of BZ fields model ↔↔ fields mass MU (Banks-Zaks) dimensional transmutation scale ΛU

dBZ −d OBZ → ΛU O O(x) → O˜(˜x) = λdO(x) below the scale MU

standard 1 BZ ﬁelds model k OsmOBZ ﬁelds MU (Banks-Zaks) dimensional transmutation scale ΛU

dBZ −d OBZ → ΛU O O(x) → O˜(˜x) = λdO(x) below the scale MU

standard 1 BZ ﬁelds model k OsmOBZ ﬁelds MU (Banks-Zaks) dimensional transmutation scale ΛU

dBZ −d OBZ →ΛU O O(x) → O˜(˜x) = λdO(x) below the scale ΛU

standard 1 BZ ﬁelds model k OsmOBZ ﬁelds MU (Banks-Zaks) dimensional transmutation scale ΛU

dBZ −d OBZ →ΛU O O(x) → O˜(˜x) = λdO(x) below the scale ΛU

standard ΛdBZ −d unparticle model U O O k sm (physics) fields MU dimensional transmutation scale ΛU O(x) → O˜(˜x) = λdO(x) for MU is large enough, the unparticle stuff just doesn’t couple strongly enough to ordinary stuff to have been seen — but it could show up at larger energies What does unparticle stuff actually look like physically? This is too hard! By thinking about EFT, we have transformed the question into something that we can make some progress on. How does unparticle stuff begin to show up as the energy of our experiments is increased? What does unparticle stuff actually look like physically? This is too hard! By thinking about EFT, we have transformed the question into something that we can make some progress on. How does unparticle stuff begin to show up as the energy of our experiments is increased? What does unparticle stuff actually look like physically? This is too hard! By thinking about EFT, we have transformed the question into something that we can make some progress on. How does unparticle stuff begin to show up as the energy of our experiments is increased? what does this do?

dBZ −d ΛU ² Osm O where ² = k MU insertion in some standard model process

2 2 ² | hSMout| Osm |SMini hU| O |0i | results in production of unparticle stuff → missing energy and momentum in O(²2) — MISSING because (for one thing) seeing it again would require more ²s ...... 2 ...... U ...... ² ...... SMin ...... SMout ...... Feynman graph with one insertion — probability distribution is proportional to the phase space for scale invariant unparticle stuff which we already know goes like dρd(P ) which looks like the production of d massless particle. as promised the first amusing result

Unparticle stuff with scale dimension d looks like a non-integral number d of invisible massless particles. there is another effect that can appear in order ²2 as promised the first amusing result

Unparticle stuﬀ with scale dimension d looks like a non-integral number d of invisible massless particles. there is another eﬀect that can appear in order ²2 ...... 2 ...... SMin ...... SMout...... 2 . . . + ...... ² ² ...... SMin ...... SMout...... U . . . . . virtual unparticles! interference ∝ ²2× unparticle propagator Z eiP x h0| T (O(x) O(0)) |0i d4x

Z A ∞ ³ ´d−2 1 = i d M 2 dM 2 2π 0 P 2 − M 2 + i²

A ³ ´d−2 = i d −P 2 − i² 2 sin(dπ)

No pole - no ordinary propagation Crazy phase - oddest for d = (2j + 1)/2 References

[1] P. J. Fox, A. Rajaraman, and Y. Shirman, “Bounds on unparticles from the higgs sector,” Phys. Rev. D76 (2007) 075004.

[2] M. Bander, J. L. Feng, A. Rajaraman, and Y. Shirman, “Unparticles: Scales and high energy probes,” arXiv:0706.2677 [hep-ph].

[3] A. Delgado, J. R. Espinosa, and M. Quiros, “Unparticles-higgs interplay,” arXiv:0707.4309 [hep-ph].

[4] G. Cacciapaglia, G. Marandella, and J. Terning, “Colored unparticles,” arXiv:0708.0005 [hep-ph]. 1 eﬀective O O k sm BZ ﬁeld theory MU

dBZ −d dimensional OBZ → ΛU O transmution ˜ d scale O(x) → O(˜x) = λ O(x) invariance Does this make sense? Corrections? other interactions corrections to dimensional transmutation breaking of scale invariance Z S = L(x) d4x where L(x) = Z Ã ! ∞ 1 µ 1 2 2 2 ∂ φM ∂µφM − M φ dM 0 2 2 M M → M˜ = λM x → x˜ = x/λ ˜ φM (x) → φM˜ (˜x) = φM (λx˜) = φM (x) build unparticle field with d v u Z u ∞ tAd d−2 2 OU = M φM dM 0 2π Z S = L(x) d4x where L(x) = Z Ã ! ∞ 1 µ 1 2 2 2 ∂ φM ∂µφM − M φ dM 0 2 2 M M → M˜ = λM x → x˜ = x/λ ˜ φM (x) → φM˜ (˜x) = φM (λx˜) = φM (x) build unparticle field with d v u Z u ∞ tAd d−2 2 OU = M φM dM 0 2π build unparticle field with d v u Z u ∞ tAd d−2 2 OU = M φM dM 0 2π connection with extra dimensions - M like another momentum component - fractional power from warping - integral is a FT picking a position in the extra dimension look at the effects on unparticle physics of s OU for any small source s that might include a constant term build unparticle field with d v u Z u ∞ tAd d−2 2 OU = M φM dM 0 2π connection with extra dimensions - M like another momentum component - fractional power from warping - integral is a FT picking a position in the extra dimension look at the effects on unparticle physics of s OU for any small source s that might include a constant term Philosophical question - is this continuous mass formula “really” what unparticle physics “is”? No! - not independent degrees of freedom - but this is what it “looks like” not an explanation of anything but we can at least use the metaphor of a φM field for each M to talk about corrections to unparticle physics at low energies. Philosophical question - is this continuous mass formula “really” what unparticle physics “is”? No! - not independent degrees of freedom - but this is what it “looks like” not an explanation of anything but we can at least use the metaphor of a φM field for each M to talk about corrections to unparticle physics at low energies. Philosophical question - is this continuous mass formula “really” what unparticle physics “is”? No! - not independent degrees of freedom - but this is what it “looks like” not an explanation of anything but we can at least use the metaphor of a φM field for each M to talk about corrections to unparticle physics at low energies. Z S = L(x) d4x where L(x) = Z Ã ! ∞ 1 µ 1 2 2 2 ∂ φM ∂µφM − M φ dM 0 2 2 M M → M˜ = λM x → x˜ = x/λ ˜ φM (x) → φM˜ (˜x) = φM (λx˜) = φM (x) build unparticle field with d v u Z u ∞ tAd d−2 2 OU = M φM dM 0 2π Are this these limits reasonable? ...... possible d = 3/2 ...... unparticle mass ...... distribution ...... ? M → ΛU scale breaking transition to from high-energy BZ theory - also interactions - shows up in dependent on couplings to on detailed higher dimension dynamics unparticle fields Why don’t we see all these continuum states as different particles?