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The many faces of (next-to) soft physics Chris White, Queen Mary University of London Solvay Workshop on Infrared Physics 1 / 44
The many faces of (next-to) softphysics
Chris White, Queen Mary University of London
Solvay Workshop on Infrared Physics
1 / 44
Overview
Brief introduction to (next-to-) soft divergences.
Applications in Collider Physics (mainly QCD).
Applications in high energy scattering (mainly gravity).
Outlook.
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Infrared divergences
In scattering amplitudes, get singularities due to soft orcollinear gauge bosons:
p
k
1
p · k=
1
|p||k |(1− cos θ).
Formal divergences cancelupon combining real andvirtual graphs (Block,Nordsieck).
Both soft and collinear radiation is universal.
Physics: it has an infinite wavelength, so cannot resolve theunderlying amplitude.
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Factorisation
Universality of soft / collinear radiation is expressed infactorisation formulae.
Example: consider a tree-level amplitude An+1({pi}, k) wheremomentum k becomes soft. We then get the soft theorems
limkµ→0
An+1({pi}, k) = S(0)({pi}, k)An({pi}),
where
S(0)QED =
n∑i=1
εµ(k)pµipi · k
, S(0)grav. =
n∑i=1
εµν(k)pµi pνi
pi · k
(Yennie, Frautschi, Suura; Weinberg).
All dependence on the soft momentum k is in the overallfactor S.
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Next-to-soft theorems
It is also possible to write such formulae at one order higher inthe k expansion (Cachazo, Strominger; Casali):
An+1({pi}, k) =[S(0) + S(1)
]An({pi}),
with
S(1)QED =
n∑i=1
εµkρJ(i)µρ
pi · k, S(1)
grav . =n∑
i=1
εµkρJ(i)µρ
pi · k,
where J(i)µν is the total angular momentum of (hard) particle i .
Next-to-next-to-soft also possible for gravity.
These and similar results have a surprisingly long history...
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History of next-to-soft physics
Next-to-soft effects were first studied in gauge theory (QED)by Low (1958).
He considered external scalars; generalised to fermions byBurnett and Kroll (1968).
Both groups only considered massive particles (no collineareffects).
Similar work in gravity by Gross, Jackiw (1968).
Del Duca (1990) generalised the Low-Burnett-Kroll result toinclude collinear effects.
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Path integral approach
Next-to-soft effects for massive particles considered usingworldline methods by Laenen, Stavenga, White (2008).
Can replace propagatorsfor external legs byquantum mechanics pathintegrals.
Leading term inperturbative expansion isclassical trajectory (softlimit).
First-order wobbles givenext-to-soft behaviour.
Also works for gravity (White, 2011).
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Applications
The tree-level (next-to)-soft theorems can be obtained usingWard identities associated with asymptotic symmetries.
This is the focus of much of this meeting!
However, the history of next-to-soft physics suggests thatthere are many other applications of next-to-soft physics.
Indeed, these have been reinvigorated by the recent work onnext-to-soft theorems.
The aim of this talk is to review some of these applications.
Key message: next-to-soft physics connects hep-th, hep-ph,hep-ex and gr-qc!
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Collider Physics
A major application of (next-to) soft physics is to colliderphysics.
We saw earlier that IR singularities cancel when real andvirtual diagrams are combined.
However, the cancellation can leave behind large contributionsto perturbative quantities.
Consider e.g. the production of a vector boson at a collider(“Drell-Yan production”):
Q
p
p_
Let z = Q2/s be the fraction of(squared) energy s carried bythe vector boson.
At LO, z = 1, and thus thecross-section is
dσ(0)
dz∝ δ(1− z).
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Drell-Yan production
At next-to-leading order (NLO), radiation can carry energy, sothat
0 ≤ z ≤ 1.
The NLO cross-section then turns out to be
dσ(1)qq
dz∼ αs
2π
[4(1 + z2)
(ln(1− z)
1− z
)+
− 21 + z2
1− zln(z)
+δ(1− z)
(2π2
3− 8
)].
It contains highly divergent terms as z → 1.
Looks like perturbation theory is in trouble!
Let’s go one order higher and see what happens...
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At NNLO the problem is even worse! One has
dσ(2)qq
dz∼ C 2
F
(αs
2π
)2[
128
(ln3(1− z)
1− z
)+
− 256
(ln(1− z)
1− z
)+
+ . . .
],
where . . . denotes terms suppressed by (1− z).
Logs get higher at higher orders in perturbation theory...
... which indeed breaks down as z → 1.
Precisely the regime where the vector boson is produced nearthreshold, so that extra radiation is soft / collinear!
The problem terms are echoes of IR singularities having beenpresent.
Thus, this problem affects many different scatteringprocesses...
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General structure of threshold logarithms
For heavy particles produced near threshold, we can define aξ, where ξ → 0 at threshold (e.g. ξ = (1− z)).
Then the general structure of any such cross-section is:
dσ
dξ=∑n,m
αn
[c
(0)nm
(lnm ξ
ξ
)+
+ c(1)nm lnm ξ + . . .
].
First set of terms correspond to (leading) threshold logs: puresoft and / or collinear.
Second set of terms is next-to-leading power (NLP) thresholdlogs: next-to-soft and / or collinear.
For ξ → 0, we need to rethink perturbation theory.
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Resummation
The solution to this problem is to somehow work out what thelarge logs are to all orders in αs .
Then we can sum them up to get a function of αs that isbetter behaved than any fixed order perturbation expansion.
Toy example: consider the function
e−αsx =∞∑n=0
αns (−x)n
n!.
Each term diverges as x →∞, but the all-order result iswell-behaved.
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Resummation approaches
Many approaches exist for resumming leading threshold logs.
There are many (hundreds?) of observables at e.g. the LHCfor which this is relevant.
Original diagrammatic approaches by e.g. Sterman; Catani,Trentadue),
Can also use Wilson lines (Korchemsky, Marchesini), or therenormalisation group (Forte, Ridolfi).
A widely used approach is to treat soft and collinear gluons asseparate fields in an effective theory: soft-collinear effectivetheory (SCET) (Becher, Neubert; Schwartz; Stewart).
All approaches have the factorisation of soft / collinearphysics at their heart.
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Soft-collinear factorisation
The general structure of an n-point amplitude is
An = Hn × S ×∏
i Ji∏i Ji
.
This is the virtual generalisation of the soft theorem.
Here Hn is the hard function, and is IR finite.
The soft and jet functions S and Ji collect soft / collinearsingularities respectively.
The eikonal jets Ji remove any double counting.
The soft and jet functions have universal definitions in termsof Wilson line operators.
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Resummation from factorisation
The soft-collinear factorisation formula leads directly toresummation of threshold effects.
Related ideas in other approaches (e.g. SCET).
Summing successive towers of threshold logs requirescalculating the soft and jet functions to a given order inperturbation theory.
State of the art is two loops (Sterman, Aybat, Dixon,Kidonakis, Mitov, Sung, Becher, Neubert, Beneke, Falgari,Schwinn, Ferroglia, Pecjak, Yang).
Progress towards three-loops and beyond (Gardi, Laenen,Stavenga, Smillie, White, Almelid, Duhr, Korchemsky, Henn,Huber, Grozin, Marquard, Correa, Maldacena, Sever).
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Next-to-leading power logs
To date, much less has been known about NLP effects.
Known for a while to be numerically significant e.g. in Higgsproduction (Kramer Laenen, Spira; Harlander, Kilgore; Catani,de Florian, Grazzini, Nason).
This has been confirmed by recent N3LO Higgs results(Anastasiou, Duhr, Dulat, Herzog, Mistlberger).
There are three good reasons to study NLP logs:1 Resummation of them will improve precision.2 Even without resummation, NLP logs may provide good
approximate NnLO cross-sections.3 Can improve the stability of numerical codes.
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Next-to-soft effects in particular scattering processes classifiedto all orders by (Almasy, Moch, Presti, Soar, Vermaseren,Vogt).
Can also be classified using the method of regions (Beneke,Smirnov, Pak, Jantzen) (see e.g. Bonocore, Laenen, Magnea,Vernazza, White).
None of the previous approaches is fully general - but stronghints of an underlying structure.
Can we predict NLP logs in an arbitrary process?
Can they be written in terms of universal functions (like LPeffects)?
Encouraging recent progress...
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SCET approach
It is well-known that LP effects can be described usingSoft-Collinear Effective Theory SCET (Stewart, Schwartz,Bauer, Fleming; Becher, Neubert).
The same language can be extended to NLP level.
Originally explored in B physics (Beneke, Campanario,Mannel, Pecjak).
Recent study for scattering amplitudes (Larkoski, Neill,Stewart).
Phenomenology explored by Feige, Kolodrubetz, Moult,Stewart, Rothen, Tackmann, Zhu; Boughezal, Liu, Petriello.
Recent resummation of leading NLP log for some observables(Stewart et. al.).
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Factorisation approach
The soft-collinear factorisation formula can be generalised tonext-to-leading power level (Bonocore, Laenen, Magnea,Melville, Vernazza, White).
This provides a loop-level generalisation of the next-to-softtheorem.
A new quantity appears at nex-to-soft level: the jet emissionfunction.
Has been calculated at one-loop level for quarks.
Non-trivial check: reproduces all NLP terms up to NNLO inDrell-Yan.
Observable loop-level corrections to the tree-level next-to-softtheorem!
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Universal NLO corrections
The tree-level next-to-soft theorem has an interestingapplication to production of N colour singlet particles (DelDuca, Laenen, Magnea, Vernazza, White).
...
p1
p2
p3
p4
pN + 2
j
i
Consider emission of anadditional gluon of momentumk, up to NLP level.
Next-to-soft theorems imply thegeneral NLP amplitude:
|ANLP|2 ∼p1 · p2
p1 · k p2 · k|ALO(p1 + δp1, p2 + δp2)|2,
where
δpα1,2 = −1
2
(p2,1 · kp1 · p2
− p1,2 · kp1 · p2
pα2 + kα).
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Universal NLO cross-sections
The formula works fully differentially.
Also leads to a universal form of the cross-section (same inquark or gluon channel):
1
σLO(zs)
d σNLP
dz=αs
π
(µ2
s
)ε [2−D0
ε+ 4D1(z)− 4 log(1− z)
],
where z → 1 at threshold.
Formula works if LO process is tree-level or loop induced.
Single, double and triple Higgs production are special cases.
Also checked for Drell-Yan, γγ and WW production.
New analytic information for double Higgs with full top mass!
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Collider Physics - Summary
Next-to-soft physics has a large number of applications incollider physics.
Typically this involves summing up large terms in perturbativecross-sections...
... or finding approximate forms for fixed-order cross-sections.
Such calculations improve the precision of theory predictionsat the LHC.
Current data demands this precision!
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Next-to-soft gravity
Much of this conference focuses on relating (next-to) softphysics with asymptotic symmetries in gravity.
However, (next)-to soft corrections have a different role toplay in understanding the conceptual structure of quantumgravity...
...and may even have phenomenological consequences!
More specifically, they are relevant to high energy scattering.
Many papers from the 1990s onwards (Amati, Ciafaloni,Veneziano, Colferai, Falcioni; ’t Hooft; Verlinde2; Jackiw,Kabat, Ortiz).
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Transplanckian scattering
More specifically, we will focus on 2→ 2 scattering in thehigh energy or Regge limit
s � |t|,
where s is the squared centre of mass energy, and |t| themomentum transfer.
Corresponds to scattering above the Planck scale in gravity.
Naıvely, we might think that non-renormalisability is aproblem.
However, in this limit infinite numbers of soft gravitons areexchanged, and the results are well-behaved!
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Can consider different regions in impact parameter b(conjugate to |t|), and energy E ∼
√s:
(see e.g. Giddings, Schmidt-Sommerfeld, Andersen).
Next-to-soft corrections probe unknown parts of this diagram.
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QCD meets gravity
Previous work on this topic focused on gravity only, includingpossible string theory corrections.
Recent studies have used QCD methods to analyse gravityscattering (Akhoury, Saotome, Sterman; Melville, Naculich,Schnitzer, White).
Idea is to develop a common language, that makes thestructure of both theories clear.
Let us look first at QCD...
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Wilson lines and the Regge limit
The Regge limit can be described by two Wilson linesseparated by a transverse distance (Korchemsky,Korchemskaya).
See also Balitsky; Caron-Huot.
1
2
3
4
b
Take particles of mass m,such that
s � −t � m2.
b is the (2-d) impactparameter (distance ofclosest approach).
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In the asymptotic high energy limit, the incoming / outgoingparticles follow classical straight line trajectories i.e. they donot recoil.
The only quantum behaviour they are allowed is to experiencea phase change.
However, gauge-covariance of the amplitude restricts thisphase to have the form (for each particle)
P exp
[igs
∫CdxµAµ(x)
],
where C is the spacetime contour of the particle.
This is a Wilson line!
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Korchemsky & Korchemskaya approach
The momentum space amplitude is then given by
A =
∫d2be−ib·q〈0|W(p1, 0)W(p2, z)|0〉,
where
W(p, z) = P exp
[igsp
µ
∫ ∞−∞
dsAµ(sp + z)
].
The momentum q is conjugate to the impact parameter, andsatisfies t ' −q2.
Can now calculate the position space amplitude at one-loop,using dimensional regularisation.
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The answer is (in d = 4− 2ε dimensions)
A(1) =g2s Γ(1− ε)
4π2−ε(µ2b2)ε
2ε
[iπT2
s + T2t log
(s
−t
)+
1
2
(log(− t
m2
)− iπ
) 4∑i=1
Ci
]+O(ε0),
whereT2
s = (T1 + T2)2, T2t = (T1 + T3)2
are quadratic colour operators for pure s- and t-channelexchanges; Ci the quadratic Casimir of particle i .
From the known properties of Wilson lines, we canimmediately exponentiate this!
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Position space amplitude
One then has
A = exp
{K
[iπT2
s + T2t log
(s
−t
)]+ . . .
}, K =
g2s Γ(1− ε)
4π2−ε(µ2b2)ε
2ε
There are two terms with non-trivial colour dependence:
(i) A t-channel term: ∝ T2t log( s
−t ).
(ii) A pure eikonal phase: ∝ iπT2s .
The former is responsible for Reggeisation of t-channelexchanges:
− iηµνq2− > − iηµν
q2
(s
−t
)αThe latter describes a spectrum of bound states (e.g.positronium).
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Eikonal phase and Regge trajectory
The eikonal phase comes from horizontal (crossed) ladderdiagrams, whereas the Regge trajectory comes from verticalladders.
(a) (b)
......
In QCD, the vertical ladders dominate.It is known that horizontal ladders dominate in gravity: theeikonal phase is enhanced by a factor s/(−t) w.r.t. theReggeisation term.The Wilson line approach gives an elegant view on this.
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Wilson lines for gravity
First, we need to find appropriate Wilson lines for gravity.
Here, we mean specifically the operator describing softgraviton emission.
The relevant quantity has appeared in various places(Brandhuber, Heslop, Spence, Travalgini; Naculich, Schnitzer;White):
exp
[iκ
2
∫Cds xµ xνhµν(x)
].
For straight line contours xµ = xµ0 + pµs, this becomes
exp
[iκ
2pµ pν
∫Cdshµν(x)
].
Closely related to its QCD counterpart!
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K& K approach for gravity
The Wilson line approach for the QCD Regge limit(Korchemsky, Korchemskaya) can be ported directly to gravity.
The momentum space gravity amplitude is given by
M =
∫d2be−ib·q〈0|Wg (p1, 0)Wg (p2, z)|0〉,
where
Wg (p, z) = exp
[iκ
2pµ pν
∫ ∞−∞
dshµν(sp + z)
].
Exponentiation of the one-loop calculation can be carried outas before.
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Position space gravity amplitude
One finds
M = exp
{−Kg (µ2b2)ε
[iπs + t log
(s
−t
)]+O(ε0)
},
Kg =(κ
2
)2 Γ(1− ε)8π2−ε .
The eikonal phase wins as s−t →∞, in contrast to QCD.
However, the structure of the result is basically the same, andcan be obtained by the procedure
gs →κ
2; T2
s,t → s, t; Ci → 0.
This is the BCJ double copy! (see also Akhoury, Saotome;Sabio Vera, Campillo, Vazquez-Mozo, Johansson).
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Next-to-soft corrections
Diagrammatic study of Regge limit by Akhoury, Saotome,Sterman.
Considered a light particle scattering on a black hole.
Next-to-soft corrections lead to a modifed eikonal phase:
χ→ χE + χNE,
where χNE ∝ Rs (Schwarzschild radius of black hole).
Correction corresponds to deflection angle of light particle(see also D’Appollonio, Di Vecchia, Russo, Veneziano;Bjerrum-Bohr, Donoghue, Holstein, Plante, Vanhove).
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Wilson line approach
Can also extend the Wilson line approach to next-to-soft level(Lanen, Stavenga, White).
Has been applied to the Regge limit in both QCD and gravity(Luna, Melville, Naculich, White).
General case of two massive particles.
In QCD, get a power-suppressed correction to the Reggetrajectory of the gluon.
In gravity, the correction to the NE phase corresponds to twosimultaneus deflection angles for the colliding particles (asconjectured by Andersen, Schmidt-Sommerfeld, Giddings).
Previous results of Akhoury, Saotome, Sterman emerge as aspecial case.
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High energy scattering - summary
Next-to-soft corrections are relevant to transplanckianscattering in gravity.
More generally, similar methods can be applied to understandradiation from scattering black holes.
Full solutions for colliding shockwaves / black holes are notalways known.
The next-to-soft calculation allows us to build them upperturbatively i.e. order-by-order in the deflection angle.
Methods exist for relating QCD and gravity results.
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Wilson lines and the double copy
For straight line contours, the Wilson lines for QCD andgravity are
P exp
[igsT
apµ∫
dsAaµ(x)
]↔ exp
[iκ
2pµ pν
∫Cdshµν(x)
].
To get gravity from QCD, we replace
gs →κ
2,
and replace a colour matrix Ta with a momentum pµ.
Vacuum expectation values of Wilson lines will then be relatedin QCD and gravity.
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This relationship is the known BCJ double copy foramplitudes (Bern, Carrasco, Johansson).
However, known resummation properties of Wilson lines meanthat this statement probes all orders in perturbation theory.
The Regge limit is one example where previous diagrammaticarguments for the double copy are made simpler by the Wilsonline langauge.
Another example is the all-order structure of infraredsingularities, first explored using a cumbersome diagrammaticargument by White, Oxburgh.
What else might gravitational Wilson lines be useful for?
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Conclusion
(Next-to)-soft physics has a large number of applications, indifferent areas of physics.
For hep-ph, hep-ex: increased precision for colliderobservables.
For hep-th, gr-qc: transplanckian scattering in gravity,radiation in black hole scattering.
Common languages for QCD and gravity (e.g. Wilson lines)make underlying structures / common behaviour clearer.
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Open questions
Can we resum next-to-leading power (NLP) threshold logs?
Other applications in precision physics?
Do next-to-soft methods help in calculating radiation fromscattering black holes?
What are gravitational Wilson lines useful for?
What does anything in this talk have to do with BMSsymmetry?
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Thanks for listening!
String Theory
QCD
(Super) Gravity(Super) Yang−Mills
Resummation
Regge limitAnalytic checks
of numerical
calculations
Approximate
cross−sections
higher−order
Transplanckian
scattering
Wilson lines
Double Copy
CHY equations
Ambitwistor strings
Increased stability
of NLO codes
Asymptotic
symmetries Black hole
entropy
Information
loss
Radiation and
memory effects
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