UV/IR Mixing and the Hierarchy Problem
David Rittenhouse Lab, UPenn Seth Koren EFI Oehme Fellow
University of Chicago
Broida Hall, UCSB
Based (mainly) on - IR Dynamics from UV Divergences: UV/IR Mixing, NCFT, and the Hierarchy Problem [1909.01365, JHEP] with N. Craig - The Hierarchy Problem: From the Fundamentals to the Frontiers [2009.11870, PhD thesis] Michelson Center for Physics, UChicago UMichigan HEP Seminar, 11/11/20 Fine-tuning problems come from asking big,
important questions Space
Time It just is
Why is there macroscopic structure?
The Hierarchy Problem There is no hierarchy problem in the Standard Model
Our toy model of the SM – a single scalar whose mass is an input parameter
1 1 푆 = න d4푥 − 휕 휙휕휇휙 − 푚2휙2 − 푉 휙 − 푔휙풪(휓 휓 ) 2 휇 2 0 푖 푖
푔2 1 푚2 = 푚2 + 푀2 phys 0 4휋 2 휖 푖 푖
The Higgs mass is an input so just choose the bare mass to give the right answer Hierarchy problem when Higgs mass is an output
Now imagine in the UV there is an SU(2) global symmetry 휓 Φa = Where we’ve measured 휙 to be very light, but 휓 must be heavy 휙 1 † 1 푆 = න d4푥 − 휕 Φ 휕휇Φ − 푀2Φ†Φ − 휆 Φ†ΣΣ†Φ − 푉 Φ 2 휇 2 0 0
2 2 푚2 = 푀2 + 휆 푣2 Tree-level 푚휓 = 푀0 휙 0 0 1 1 Loop-level 푚2 = 푀2 + 푀2 + ⋯ 푚2 = 푀2 + 푀2 + ⋯ + 휆 푣2 휓 0 휖 Σ 휙 0 휖 Σ 0 2 2 푚2 = 푀2 + 휆 푣2 Renormalized 푚휓 = 푀phys 휙 phys phys phys
2 푀푝ℎ푦푠 Fine-tuned unless m휙 ∼ scale of new physics 휆phys = −1.00000000000000000000000000001 × 2 푣푝ℎ푦푠 The Hierarchy Problem: From the Fundamentals to the Frontiers How to get a light scalar: Classic edition
Introduce UV structure to forbid large contributions, and IR dynamics to break that structure to the observed SM EFT
E.g. Supersymmetry
푄|fermionۧ = |bosonۧ 푄|bosonۧ = |fermionۧ Must be broken! Λ2 ∼ Λ2 × 0 + 푚2 − 푚 2 log 푚2
E.g. Extra dimensions
UV masses forbidden by gauge invariance
→ Either way, expect new strongly- interacting particles near the weak scale Data!
(With apologies to CMS) Data!
A maximalist interpretation of the results is that maybe there really is no new weak-scale physics. EFT expectations really are violated.
Iterate on familiar particle Innovate! physics ideas/strategies/searches
[Craig, SK, Trott ’17; Perhaps[Craig, the Garcia violation-Garcia, SK of‘18; EFT Alipour-fard, Craig, SK, Jiang ’18; expectationsCraig, Garcia results-Garcia, SK from ‘19; a Alipour-fard, Craig, Gori, SK, Redigolo ’18; physicalCraig, violation SK ’19; of EFT SK, McGehee ’19; Alipour-fard, SK ‘hopefully, …] Cohen, Craig, SK, McCullough, Tooby-Smith ‘20; Craig, SK, Öktem ‘soon, …]
(NB: I also do things/have interests unrelated to the hierarchy problem!) The EFT of Quantum Gravity
We have a great perturbative theory of quantum gravity! 18 푀pl ≈ 10 GeV 1 1 2 3 1 2 4 ℒ퐸퐻 = 2 −푔푅 ≈ 휕ℎ휕ℎ + 휕 h + 2 휕 h + ⋯ 푔휇휈 = 휂휇휈 + ℎ휇휈/푀pl 2푀pl 푀pl 푀pl Compare with Fermi theory of the weak interactions 1 ത 휇 푖 2 ≈ 300 GeV ℒ ≈ 휓푖훾휇휕 휓 + 퐺퐹휓푖휓푗휓푘휓푙 + 퐺퐹 휓푖휓푗휓푘휓푙휓푚휓푛 + ⋯ 퐺퐹
휓푖 휓푖
15 1 15 푠 ∼ 10 푠 ∼ 10 푀푝푙 퐺퐹 ? !
휓푗 휓푗 Noncommutativity
‘Quantize!’ [Snyder ‘47]!
UV/IR mixing is front and center! Separation of scales is violated! Noncommutative Field Theory
But how to do physics on such spaces?
Transfer the noncommutativity to the fields!
Introduce ‘star-product’ Noncommutative Field Theory
⇒
Vertices no longer permutation-invariant!
Phase factors of graphs reduce to a graph-topological statement [Filk ‘94]
Planar graphs: Solely an overall phase involving external momenta 푝 휇 휈 Nonplanar graphs: Additional phases for lines which cross ∼ 푒푖 푝 휃휇휈푘
푘 Thou Shalt Not Expand
The ‘theta-expanded’ NCFT removes all of the UV/IR mixing! Look at that exponential, Odysseus. You should expand it for low energies.
Much past work can be ignored from our perspective [Ulysses and the Sirens, Draper, 1909] Skeletons in the Closet
• Lorentz violation! • Not out the window; just like turning on a magnetic field in a lab • Folk theorems [Collins et al. ‘04] about empirical bounds not fully applicable See also [Calmet ‘04] • Unitarity of Lorentzian theory with timelike noncommutativity? • Failure well-understood from stringy perspective [Seiberg, Susskind, Toumbas ’00] • Field-theoretically, issue with formulation of nonlocal-in-time theories [Gomis et al. ‘00; Bahns et al. ’02; Bozkaya et al. ‘02; Liao & Sibold ’02; Rim & Yee ‘02; Denk & Schweda ‘03, Fischer & Putz ’03; Liao ‘04, …]
[Gomis, Mehen ’00] (But at the end I’ll mention some preliminary progress on these.) Euclidean φ4 [Minwalla, Seiberg, Van Raamsdonk ‘99]
Use Schwinger parameters ∞ 1 −훼 푘2+푚2 2 2 = න d훼 푒 푘 + 푚 0
1 − Regulate with 푒 Λ2훼 Completed the square 2휇휈 푝 ∘ 푘 ≡ −푝휇휃 푘휈 Λ2 Λ2 Planar ∼ 푔2 Λ2 − 푚2log + ⋯ Nonplanar ∼ 푔2 Λ2 − 푚2log eff + ⋯ 푚2 eff 푚2
Not regulator-dependent – can also see in dim reg [Craig, SK] [Minwalla, Seiberg, Van Raamsdonk ‘99] A new pole!
1 Loop 1PI quadratic effective action
with renormalized mass Nonperturbative in 휃!
A new light ‘particle’ with In Λ → ∞ limit nothing nearby to explain there are now its presence! two poles!
After Wick rotation, 휇휈 1 2 2 4 2 simple case 휃 ∼ ൗ 2 → 푝 ∝ 푔 Λ휃/푚 Λ휃 inaccessible in s-channel [Minwalla, Seiberg, Van Raamsdonk ‘99] Wilsonian Interpretation
Normally a renormalizable Wilsonian action must satisfy 1. Correlation functions are well defined in the Λ → ∞ limit 2. At finite Λ they differ from the limiting value by 푂(Λ−1) for all momenta
At small momenta (2) is badly violated here!
Can restore a Wilsonian interpretation by introducing a new field 휒 Yukawa theory
[c.f. Anisimov, Banks, Dine, Graesser ‘01]
However T is antiunitary!
So CPT ‘re-cycles’ the two interaction terms! Scalar Two-Point Function
2 2 2 Evaluation requires Planar ∼ −(푔1 + 푔2) Λ + ⋯ some cleverness and ‘lightcone Schwinger coordinates’
2 Nonplanar ∼ −푔1푔2 Λeff + ⋯ Fermion Two-Point Function
푝 ⋅ 훾 Planar ∼ − 푔2 + 푔2 푀 − log Λ2 + ⋯ 1 2 2
푝 ⋅ 훾 Nonplanar ∼ −푔 푔 푀 − log Λ2 + ⋯ 1 2 2 eff
Logarithmic UV sensitivity → only logarithmic IR feature Softly-broken Supersymmetry - Wess Zumino Look at the interplay between UV/IR mixing and UV finiteness
Compute one-loop two-point function again, with 푍 wavefunction renormalization and 훿푚2 mass correction
For Λ, Λeff large
The transmogrification accords with the intuition we’ve now built Softly-broken Wess Zumino
So a UV finite theory has no IR effects from UV/IR mixing A UV sensitive theory has this surprising IR feature
How do these connect? With soft breaking we can transition between the two.
Taking instead 푀 ≫ Λ, Λeff of the full result
EFT has been broken in a controlled way!
UV/IR mixing requires lack of UV finiteness! So not a module to tack on to theory with hierarchy problem. On the other hand, one says that in this theory there just never is a hierarchy problem.
Though perhaps we can have our cake and eat it too with noncommutative orbifold field theory? Some Early Words on Current Directions
Reformulate Lorentz invariant version of NCFT [Snyder ’47; Doplicher et al. ’95; Kase et al ’02; Carlson et al. ‘02; Heckman & Verlinde ’14; Much & Vergara ‘17, many large literatures in various directions,…] Easy example Conclusions
• Our EFT expectations have been violated! Perhaps by physical breakdown of EFT • In NCFT this can generate an IR scale ex nihilo • This behavior persists and with interesting properties as you go closer to the SM • UV/IR mixing requires UV sensitivity which puts this strategy in stark contrast to others • Lots of directions to investigate
Backup Slides Wilsonian Interpretation Redux
Then for any cutoff Λ, we have the behavior for Opposite sign from 흓ퟒ theory! small 푝 ∘ 푝
Introduce auxiliary field, or just talk about a modified Lorentzian behavior propagator after integrating with NC time involves it out some speculation New light pole at
[but see e.g. Bozkaya et al ‘02] Future Directions
In Wilsonian EFT, nonlocality for 푝 ≳ Λ ↔ 푥 ≲ 1/Λ
[Sheikh-Jabbari ‘99, Bigatti & Susskind Particles in NCFT are like rods of length 퐿 ∼ 푝휃 ‘00, Seiberg, Susskind, Toumbas ‘00, …] So ‘extra’ nonlocality for 1/Λ2 ≲ 푝휃2푝
Could this sort of nonlocality appear in other models? [originally Maggiore ‘93, ℏ Generalized Uncertainty Principle: Δ푥 ≳ + ℓ2Δ푝 review Tawfik & Diab, ‘15] Δ푝 푝
Similar claims in String Theory as well [Gross & Mende ‘88, Konishi, Paffuti, Provero ’90, Yoneya, ’00, …] Future Directions
푏 퐹 푧 = න d푥 푓(푥, 푧) 푎 Alternatively, try to understand general nonlocal theories
푏 푥푠(푧) [Tomboulis ’15, Tomboulis & Chin ‘18] look carefully at a disjoint class of nonlocal theories
Presumably the magic of NCFT is associated with ‘endpoint 푎 singularities’ Fine-tuning problems come from asking big, important questions
It just is
Why is there a macroscopic universe?
The Cosmological Constant Problem 퐸
Effective Field Theory 푈 1 푌 Landau pole
Focus on the important degrees of freedom! Quantum gravity Grand unification Right handed 휈?
Dark matter?
Higgs vev
Standard Model (+?) Correct your leading order description in a perturbative expansion depending on how much precision you want DFSZ axion? 푑풑 푚풗 1 푣2 푭 = = ≈ 푚풗 + 푚 2 풗 + ⋯ ℒ 휙 = ℒ 휙 푑푡 푣2 2 푐 푆푀 푆푀 푆푀 1 − 5 6 푐2 ℒ 휙 ℒ 휙 + 푆푀 + 푆푀 + ⋯ Λ Λ2 There is no hierarchy problem in the Standard Model
Our toy model of the SM – a single scalar whose mass is an input parameter
1 1 푆 = න d4푥 − 휕 휙휕휇휙 − 푚2휙2 − 푉 휙 − 푔휙풪(휓 휓 ) 2 휇 2 0 푖 푖
푔2 1 푚2 = 푚2 + 푀2 phys 0 4휋 2 휖 푖 푖
The Higgs mass is an input so just choose the bare mass to give the right answer Hierarchy problem when Higgs mass is an output
Now imagine in the UV there is an SU(2) global symmetry 휓 Φa = Where we’ve measured 휙 to be very light, but 휓 must be heavy 휙 1 † 1 푆 = න d4푥 − 휕 Φ 휕휇Φ − 푀2Φ†Φ − 휆 Φ†ΣΣ†Φ − 푉 Φ 2 휇 2 0 0
2 2 푚2 = 푀2 + 휆 푣2 Tree-level 푚휓 = 푀0 휙 0 0 1 1 Loop-level 푚2 = 푀2 + 푀2 + ⋯ 푚2 = 푀2 + 푀2 + ⋯ + 휆 푣2 휓 0 휖 Σ 휙 0 휖 Σ 0 2 2 푚2 = 푀2 + 휆 푣2 Renormalized 푚휓 = 푀phys 휙 phys phys phys
2 푀푝ℎ푦푠 휆 = −1.00000000000000000000000000001 × Fine-tuned unless m휙 ∼ scale of new physics phys 푣2 Waiter, there's Philosophy in my Physics
“Who cares? We can fit the data by tuning those parameters.”
A model being not literally impossible is a very low bar for a scientific theory To EFT or not to EFT? EFT Theory Space Gravity very generally motivates looking The Swampland at `UV/IR mixing’ effects! c.f. Cumrun
BH thermodynamics No QG 푇 ∝ 1/푀 c.f. Stephen 퐵퐻 퐵퐻 completion
Necessity of nonlocality in gravitational observables
BH info and firewalls?! c.f. Steve
풪(푥) c.f. Don NC Quantization Correspondence Principle Renormalizability Pole Accessibility
Dim Reg
UV Limit First ->
IR Limit First -> Strong UV/IR Duality 1 Commutative Λ2 → Noncommutative 푝 ∘ 푝
Is this always a necessary relationship? Not quite.
E.g. Self-interaction of complex scalar with global U(1) [Ruiz Ruiz, ‘02]
휆1 휆2
휆2
So are there other examples where it does occur? CPT Three Point Function
finite