Gradient Flow Exact
Renormalization GroupandScalar Field Theories
Junichi Haruna(Kyoto University)
2023/1/21 @ FRG at RIKEN 2023

Abstract• We discuss some aspects of Gradient Flow Exact
Renormalization Group, a new frame work to define theWilsonian effective action via the gradient flow
• We study the fixed point structure of GFERG flow
associated with a general polynomial gradient flow
equation for scalar field theories• We also investigate the RG flow of 𝑂 3 non-linear
sigma model with the Wess-Zumino term by GFERG, as aloophole of the above discussion

2/35Contents• Introduction (3)• Review of GFERG (8)• Fixed point structure of scalar field theories (10)• O(3) non-linear sigma model in two dimensions (9)• Conclusion and Future direction (5)3/35Contents• Introduction (3)• Review of GFERG (8)• Fixed point structure of scalar field theories (10)• O(3) non-linear sigma model in two dimensions (9)• Conclusion and Future direction (5)4/35

Review ofGradient Flow ExactRenormalization GroupExact Renormalization Group (ERG)• A framework to study physics under varying the energyscale• The Wilson action 𝑆𝜏 is intuitively defined byintegrating out higher momentum modes of the fields:

𝑒 −𝑆𝜏 ≔ න𝐷𝜙𝑝>Λ 𝑒 −𝑆0(Λ ≔ Λ0 𝑒 −𝜏 , Λ0 : cutoff)• 𝜏-dependence of the Wilson action 𝑆𝜏 is described bya differential equation →“ERG equation”
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Wilson-Polchinski equation• A typical example of the ERG equation:𝜕𝜏 𝑆𝜏 = ඲ ൝

2𝑝2 +𝑝𝜕𝜏𝑆𝜏

=𝐷 + 2 − 𝜂𝜏𝜕𝛿𝑆𝜏+ 𝑝𝜇𝜙 𝑝2𝜕𝑝𝜇 𝑖
𝛿𝜙𝑖 𝑝2𝑆𝜂𝛿𝛿𝑆𝜏𝛿𝑆𝜏𝜏𝜏
2+ 2𝑝 + 1 −−ቋ2 𝛿𝜙𝑖 𝑝 𝛿𝜙𝑖 −𝑝
𝛿𝜙𝑖 𝑝 𝛿𝜙𝑖 −𝑝𝑆𝜏

+𝑆𝜏

−𝑆𝜏𝑆𝜏

• This equation defines a renormalization procedure
non-perturbatively(𝜂𝜏 : anomalous dimension)[J.Polchinski Nucl.Phys.B 231 (1984) 269-295](We work on the dimensionless framework and 𝐷-dimensional Euclidean space)
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Gauge invariance in ERG• The Wilson action with the naive UV cutoff isinconsistent with gauge invariance• The gauge transformation mixes higher and lower
momentum modes:

𝐴𝜇𝑎 𝑝 → 𝐴𝜇𝑎 𝑝 − 𝑝𝜇 𝜔𝑎 𝑝 − 𝑖𝑓 𝑎𝑏𝑐 න 𝜔𝑏 𝑝 − 𝑞 𝐴𝑐 𝑞𝑞• Can we define a Wilson action in a manifestly
gauge invariant manner?

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Renormalization and Diffusion• The solution to the WP equation can be written
in the following form:′𝑒 𝑆𝜏 𝜙 = 𝑠Ƹ𝜙−1 න𝐷𝜙 ′ ෑ 𝛿(𝜙𝑖 𝑥 − 𝑒 𝜏 𝐷−2 Τ2 𝑍 1Τ2 𝜑𝑖′ (𝑡, 𝑥𝑒 𝜏 )) 𝑠Ƹ 𝜙′ 𝑒 𝑆𝜏=0[𝜙 ]𝑥,𝑖(𝑠Ƹ𝜙 ≔ exp1
𝛿2− 2 ‫ 𝑥 𝜙𝛿 𝑥׬‬2𝑖:“scrambler")• 𝜑′ is a solution to the diffusion equation:
𝜕𝑡 𝜑′ 𝑡, 𝑥 = 𝜕𝑥2 𝜑′ 𝑡, 𝑥 , 𝜑′ 0, 𝑥 = 𝜙 ′ 𝑥where 𝑡 ≔ 𝑒 2𝜏 − 1• This representation implies that the coarse-graining
by the diffusion can be used to define an RG flow
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Gradient Flow (GF)
• This one-parameter deformation of fields via
the diffusion equation has been studied
in the context of “gradient flow"• The gradient flow is a method to construct composite
operators without the equal-point singularity• Correlation functions are UV finite with wave
function renormalization:−𝑛/2𝑍𝑡

𝜑 𝑡, 𝑥1 𝜑 𝑡, 𝑥2 ⋯ 𝜑 𝑡, 𝑥𝑛

𝜙 <∞even for the equal point case (e.g. 𝑥1 = 𝑥2 )[F.Capponi,L.Debbio,S.Ehret,R.Pellegrini,A.Rago 1512.02851]
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Gradient Flow ERG (GFERG)• GF equation for gauge fields[M.Lüscher,P.Weisz 1101.0963]′ + 𝛼 𝐷′ 𝜕 𝐵′
𝜕𝑡 𝐵𝜇′ = 𝐷𝜈′ 𝐺𝜈𝜇0 𝜇 𝜈 𝜈with 𝐵𝜇′ 0, 𝑥 = 𝐴𝜇′ 𝑥• 𝑆𝜏 for gauge fields can be defined via the GF equation[H.Sonoda and H.Suzuki 2012.03568]

𝑎𝑎𝑒 −𝑆𝜏 𝐴𝜇 ≔ 𝑠𝐴Ƹ −1 න 𝐷𝐴′𝜇 ෑ 𝛿 𝐴𝜇𝑎 𝑥 − 𝑒 𝜏 𝐷−2 /2 𝐵𝜇′𝑎 𝑡, 𝑥 ′ 𝑒 𝜏′𝑎𝑠Ƹ𝐴′ 𝑒 −𝑆𝜏=0 𝐴𝜇𝑥 ′ ,𝑎,𝜇𝛿21(𝑠𝐴Ƹ ≔ exp − 2 ‫ 𝑥 𝑎𝐴𝛿 𝑥 𝑎𝐴𝛿 𝑥׬‬:“scrambler" for gauge fields)
𝜇𝜇• GFERG defines an RG flow in a gauge-invariance way!
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Contents• Introduction (3)• Review of GFERG (8)• Fixed point structure of scalar field theories (10)• O(3) non-linear sigma model in two dimensions (9)• Conclusion and Future direction (5)12/35

Fixed Point Structure ofGFERG forScalar Field Theories
Based on arXiv:2201.04111, PTEP 2022, No.3, 033B03(2022) with Y.Abe (Wisconsin), Y.Hamada (KEK)Our motivation• The appropriate gradient flow eq. highly depends on
details of the theory, such as its symmetry andinteractions• The GFERG flow depends on the form of the GF eq.,
and then becomes different for each theory• This means that the GFERG eq. for general scalar
field theories is no longer given by the WP eq.
• Is GFERG consistent with the conventional ERG?

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Our motivation• The gradient flow equation for the 𝑂 𝑁 non-linear
sigma model𝜕𝑡 𝜑𝑖 = 𝜕𝑥2 𝜑𝑖 − 𝜑𝑗 𝜕𝑥2 𝜑𝑗 𝜑𝑖[H.Makino, H.Suzuki 1410.7538]• But this model and the linear sigma model in three
dimension belong to the same universality class• “Does GFERG give the same prediction as theconventional ERG for the IR behavior of a theory?"• These facts strongly motivate us to study
the fixed points and the critical exponents in GFERG

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Our work• We study the fixed points of the general GFERG eq.
for scalar fields→ The fixed points of the WP eq. appear
in the 𝜏 → ∞ limit along the GFERG flow

• We show the GFERG eq. has a similar RG flow structurearound a fixed point→ Scaling dimensions of relevant or marginal
operators are the same, while those of irrelevant
operators can be different• GFERG gives the same prediction as the conventional
ERG for the IR behavior of a theory

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General gradient flow equation• General form of gradient flow equations∞

𝜕𝜏 𝜑𝑖 = 𝜕𝑥2 𝜑𝑖 + ෍ න
𝑛=𝑛min 𝑥1 ,…,𝑥𝑛𝑖 ,…,𝑖𝑛𝑓𝑖 1𝑥; 𝑥1 , … , 𝑥𝑛 ; 𝜕𝑥1 , … , 𝜕𝑥𝑛 𝜑𝑖1 𝜏, 𝑥1 ⋯ 𝜑𝑖𝑛 𝜏, 𝑥𝑛

WP partextra terms• Counterpart of WP eq. in GFERG (GFERG eq.)𝜕𝜏 𝑒 −𝑆𝜏 Φ = (WP part)∞𝑛−1+ ෍ 𝜆𝑛=𝑛min𝛿𝛿𝑖1 ,…,𝑖𝑛𝜏 නቊ𝑓𝑖𝜙𝑖1 𝑥1 +𝛿𝜙𝑥
𝛿𝜙𝑖1 𝑥1
𝑖𝑥,𝑥1 ,…,𝑥𝑛
× 𝜙𝑖𝑛 𝑥𝑛 +×⋯𝛿𝛿𝜙𝑖𝑛 𝑥𝑛

ቋ 𝑒 −𝑆𝜏 Φwhere 𝜆 𝜏 ≔ 𝑒 −𝜏 𝐷−2 Τ2 𝑍𝜏−1Τ2
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Fixed point• The fixed-point action 𝑆 ∗ is defined as𝜕𝜏 𝑆 ∗ = 0 and 𝑆 ∗ ≔ lim 𝑆𝜏𝜏→∞• 𝑆 ∗ satisfies∞𝛿𝛿
𝑖 ,…,𝑖ቊ𝑓𝑖 1 𝑛 𝜙𝑖1 𝑥1 +𝛿𝜙𝑖1 𝑥1𝑥,𝑥1 ,…,𝑥𝑛 𝛿𝜙𝑖 𝑥

0 = (WP part) + ෍ 𝜆𝑛−1 (∞) න𝑛=𝑛min

× 𝜙𝑖𝑛 𝑥𝑛 +• Asymptotic behavior of 𝜆 𝜏𝜆 𝜏 ∼ exp −𝜏 𝐷 − 2 + 𝜂 /2𝛿
𝛿𝜙𝑖𝑛 𝑥𝑛×⋯∗

ቋ 𝑒 −𝑆 Φ𝜂𝑑≔log 𝑍𝜏 ቚ2 𝑑𝜏𝜏=∞• 𝜆 ∞ should vanish from the cluster decomposition
principle at 𝑆 ∗⇒𝑆 ∗ satisfies the fixed-point condition of the WP eq.
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RG flow around fixed point• Let us study the 𝜏-dependence of 𝑆𝜏 after a long time𝜏 = 𝜏0 ≫ 1 so that 𝜆 𝜏0 ∼ 𝑒 −𝜏0 𝐷−2+𝜂 /2 ≪ 1• Consider perturbating 𝑆𝜏 from a fixed point 𝑆 ∗at 𝜏 = 𝜏0 as𝑆𝜏0 = 𝑆 ∗ + ෍ 𝛿𝑐 𝐴 𝒪𝐴𝐴

𝛿𝑐 𝐴 ≪ 1
𝒪𝐴 : a complete set of operators(defined in the next slide)19/35

Solution in the 𝜏 → ∞ limit• We have two small quantities: 𝛿𝑐 𝐴 , 𝜆0 ≔ 𝑒 −𝜏0 𝐷−2+𝜂 /2• Solution to the GFERG eq. up to the leading order𝑆𝜏= 𝑆∗ + ෍′𝑛−1𝐴𝑥𝜏𝐴𝛿𝑐 𝑒− 𝜆0 min′ Τ2′−𝑛−1𝐷−2+𝜂𝜏𝑥
𝜏
𝐴min𝑒−𝑒ℎ 𝐴 𝒪𝐴

𝐴where 𝜏 = 𝜏0 + 𝜏 ′𝒪𝐴 : eigenoperators of the linearized WP eq.
around 𝑆 ∗𝑥𝐴 : eigenvalue of 𝒪𝐴𝑛min : minimum order of the non-linear terms
in the GF eq.
ℎ 𝐴 : some constant
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Scaling dimension• Scaling Dimension 𝑑𝐴
𝑑𝐴 = max 𝑥𝐴 , − 𝑛𝑚 − 1 (𝐷 − 2 + 𝜂)/2𝑥𝐴 ≥ 0 (relevant or marignal)⟹ 𝑑𝐴 = 𝑥𝐴

𝑥𝐴 < 0 (irrelevant)⟹ 𝑑𝐴 = − min( 𝑥𝐴 , 𝑛𝑚 − 1 𝐷 − 2 + 𝜂 /2)• 𝑑𝐴 of relevant or marginal operators are 𝑥𝐴⇒ critical exponents are the same as those ofthe WP eq.
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Contents• Introduction (3)• Review of GFERG (8)• Fixed point structure of scalar field theories (10)• O(3) non-linear sigma model in two dimensions (9)• Conclusion and Future direction (5)22/35

GFERG analysis ofO(3) non-linear sigma modelin two dimensions
Based on a preliminary work with Hiroshi Suzuki(Kyushu)Our Motivation• 𝑂 3 non-linear sigma model is a loophole of theprevious study• This model is gapless with the Wess-Zumino term with
𝜃 = 𝜋 → non-trivial fixed point• Its gradient flow equation
𝜕𝑡 𝜑𝑖 = 𝜕𝑥2 𝜑𝑖 − 𝜑𝑗 𝜕𝑥2 𝜑𝑗 𝜑𝑖[H.Makino, H.Suzuki 1410.7538]• It is interesting to study its RG flow by GFERG

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Wilson action• We want to treat bare fields rather than renormalized
ones1−2• We should rescale 𝜙𝑖 (𝑥) → 𝜆 𝜏 𝜙𝑖 (𝑥) in the definitionof the Wilsonian effective action of GFERG:′𝑒 𝑆𝜏 𝜙 = 𝑠𝜆Ƹ −1𝜏 −1Τ2𝜙 න𝐷𝜙 ′ ෑ 𝛿(𝜙𝑖 𝑥 − 𝜑𝑖′ (𝑡, 𝑥𝑒 𝜏 )) 𝑠Ƹ 𝜙′ 𝑒 𝑆𝜏=0[𝜙 ]𝑥where 𝜑𝑖′ is the solution to the GF equation:
𝜕𝑡 𝜑𝑖′ = 𝜕𝑥2 𝜑𝑖′ − 𝜑𝑗′ 𝜕𝑥2 𝜑𝑗′ 𝜑𝑖′

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Constraint along RG flow
• Constraint of the fields of 𝑂 𝑁 NL sigma model
𝜙𝑖2 = 1• 𝑆𝜏 preserves it in the sense of the (modified)correlation functions:≪ (𝜙𝑖2 − 1)𝜙𝑖1 ⋯ 𝜙𝑖𝑛 ≫𝑆𝜏 = 0where ≪ 𝒪 𝜙 ≫𝑆𝜏 ≔ න𝐷𝜙𝑖 𝑠Ƹ𝜆 𝜏 −1Τ2𝜙 𝒪 𝜙 𝑒 𝑆𝜏 𝜙(Recall:𝑒 𝑆𝜏 𝜙= 𝑠𝜆Ƹ −1𝜏 −1Τ2 𝜙 න𝐷𝜙 ′ ෑ 𝛿(𝜙𝑖 𝑥𝑥′]′𝜏𝑆[𝜙𝜏=0− 𝜑𝑖 (𝑡, 𝑥𝑒 )) 𝑠Ƹ 𝜙′ 𝑒)• GFERG can define an RG flow which preserves theinformation of the target space
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GFREG equation• GFERG equation𝜕𝜏 𝑒 𝑆𝜏 = WP part +𝛿𝑥 𝛿𝜙𝑖 𝑥

2න

𝜙𝑗 𝑥 + 𝜆 𝜏 2

𝛿
𝛿𝜙𝑗 𝑥

𝜕 2 𝜙𝑗 𝑥 + 𝜆 𝜏 2

𝛿
𝛿𝜙𝑗 𝑥

𝜙𝑖 𝑥 + 𝜆 𝜏 2

𝛿
𝛿𝜙𝑖 𝑥𝑒 𝑆𝜏

• We assume 𝜆 ∞ takes a finite non-zero value 𝑔• Fixed point condition

0 = WP part +𝛿𝑥 𝛿𝜙𝑖 𝑥+2 න

𝜙𝑗 𝑥 + 𝑔2𝛿
𝛿𝜙𝑗 𝑥

𝜕 2 𝜙𝑗 𝑥 + 𝑔2

𝛿
𝛿𝜙𝑗 𝑥

𝜙𝑖 𝑥 + 𝑔2

𝛿
𝛿𝜙𝑖 𝑥𝑒𝑆∗

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Ansatz
2 + 𝑚2
1𝑝
𝑆 ∗ = 2 න −2𝑝2
𝜙𝑖 −𝑝 𝜙𝑖 𝑝222𝑔 𝑝 𝑒+𝑝 +𝑚𝑖𝜃
෩ 𝑖 𝑥 𝜕𝜇 Φ
෩𝑗 𝑥 𝜕𝜈 Φ
෩ 𝑘 (𝑥)+ න 𝜖 𝑖𝑗𝑘 𝜖𝜇𝜈 Φ3 𝑥ℎ
෩ 𝑖 𝑥 𝜕𝜈 Φ
෩𝑗 𝑥 𝜕𝜌 Φ෩ 𝑖 𝑥 𝜕𝜎 Φ෩𝑗 𝑥+ න 𝜖𝜇𝜈 𝜖𝜌𝜎 𝜕𝜇 Φ4 𝑥
෩𝑖 𝑝 ≔where Φ2𝑒 −𝑝

2𝑒 −2𝑝 +𝑝2 +𝑚2

𝜙𝑖 𝑝• We consider only mass and kinetic terms, and these
cubic and quartic interactions(“truncation method”)
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Fixed Point Condition
0 = −𝑔−2 𝑚2 + 2 3𝑚2 − 1 𝑐1
0 = 2𝑔2 ℎ𝑐2 + 2 𝑚2 − 4 𝑐0 + 12𝑐1 − 10𝑔4 ℎ𝑐3
0 = 𝜃(𝑚2 + 𝑔2 6 − 4𝑚2 𝑐1 )ℎ0 = 1 + 𝑚2 1 + 5𝑚2 + 8𝑔2 2 𝑚2 − 1 𝑐0 + 𝑐12+ 2𝜃 2 𝑔2 − 10𝑔4 𝜃 2 𝑐1

where 𝑐𝑖 𝑖 = 1,2,3 is some constant including 𝑚2• These constants arise from the loop integrals such as𝑐0 ≔ න

𝑝𝑒𝑒 −2𝑝2−2𝑝2 + 𝑝2 + 𝑚 2

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Fixed Point• Unfortunately, the solution to the all fixed pointcondition is just the Gaussian one:𝑔=𝜃=ℎ=0• If we neglect the condition from the mass term, there
is a non-trivial fixed point:𝑚2 ≃ 2.82, 𝑔−2 ≃ 9.29 × 10−3 , ℎ ≃ −2.39 × 103

with 𝜃 = 3/4• We can discuss 𝜃 = 𝜋 can be a fixed pointfrom the periodicity (𝜃 → 𝜃 + 2𝜋) and CP symmetry→ Our ansatz may be bad..
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Contents• Introduction (3)• Review of GFERG (8)• Fixed point structure of scalar field theories (10)• O(3) non-linear sigma model in two dimensions (9)• Conclusion and Future direction (5)31/35Conclusions andFuture Directions

Conclusion 1• Studied the fixed point structure of the GFERG
equation associated with a general gradient flow
equation for scalar field theories• Showed that the fixed points are the same as those ofthe Wilson-Polchinski equation in general• Discussed that the GFERG equation has a similar RG
flow structure around a fixed point to the WPequation• GFERG gives the same prediction as the conventional
ERG for the IR behavior of a theory

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Conclusion 2• Studied an RG flow of 𝑂(3) non-linear sigma model in
two-dimensions• Wrote down its GFERG equation, which preserves theconstraint 𝜙𝑖2 = 1, i.e., the information of the target
space• Discussed that the fixed point and found there is anon-trivial fixed point if we neglect the condition
for the mass term, as well as the Gaussian one

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Future Direction• (Non-abelian) gauge theory• Gravity and asymptotic safety“Wilson action with manifest diffeomorphism
invariance"• Scalar field theories with a non-trivial target spacein two dimensions
e.g.) 𝑂 𝑁 NL sigma model, 𝑪𝑷𝑁−1 model• Effects of topological terms in QFT

Thank you!
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