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Feynman periods of the primitive log-divergent $\phi^4$ graphs
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Numbers
$\ell$
$k$ 
$P_{\ell,k}$
1
1:
1
comment: The one-loop period is exactly $1$, since $\int_0^\infty (1+x)^{-2}\,dx=1$.
3
1:
7.212341418957565712398428969068699944589917754042993290753629332051029234717878541118735241656011549
comment: The wheel with three spokes has period $6\zeta(3)$; its completion is $K_5$.
4
1:
20.73855510286739852662730972914068336114161839003825623948385355807607179572562969120086213114266673
5
1:
55.58525391567849582954383993547004637294247877615878369125273290350953149224423491095359541137605423
6
1:
168.3374099947818120221992652310452261615817830343413111001642314728438889695634110236660356133644905
6
2:
132.2435331106599532911895377706974446396400339725113552093427868101366762097601933904237556132148444
6
3:
107.7110248410231180583722503561263587603598987578266720565794474901873518987992160820808774505448753
6
4:
71.50608179656193002641539411089698478143109329211826553247240062436675412167508399642771996076380609
7
1:
527.7450517669760781880817243095484153944325728868963235207046209578394531988766050710124691973809891
7
2:
380.8878295341900847984585830831078412702630570893434483879072982479418073341519268734440951804095206
7
3:
336.0670721108467360236940229108821051345490437057062089387806496422942362998827550174096090983607144
7
4:
294.0353141852964238424980465745848549988891621270650576783364730744721065151666263135955366474079997
7
5:
254.7630095958172824868903598894810828121238346719011924984897905955131567840471463905533204974813036
7
6:
273.4825742584976819206301428976800769562808388412317070187975289769604112338032474363209813232727242
7
7:
294.0353141852964238424980465745848549988891621270650576783364730744721065151666263135955366474079997
equals: #7,4
7
8:
183.0324200304989017170119123092288951484063911120159691631312951542424372471548436590605773016962163
7
9:
216.9193755875438330514675513675736680028050131928873156158513091731920664797373687400492926075776002
7
10:
254.7630095958172824868903598894810828121238346719011924984897905955131567840471463905533204974813036
equals: #7,5
7
11:
200.3575664292754469676345909901000737950363376631638406063975188978537920781316404289320172248664268
8
1:
1716.210576104444687375826151479229291045696081716694894254581784501458783240012664038923897143808121
8
2:
1145.592929599713130090219511911411448267325484966987091830717829045881549251298166500046596199840045
8
3:
1105.107697390380571481750718398180520990357606630164969906863691080393088885859715568466805242627290
8
4:
966.8308019861797060577711510960472003418160280528182909657418457194083989634621674211698586767987366
8
5:
844.5125186033249391551602012080025663235677568424293073378251400712683200219181261845747194468507093
8
6:
904.2808243578157996899656671139994051607494437202343474230631730723491404179423009925618140377085746
8
7:
847.6461156397111251207670837670689902453735503393825637271084556248374782788666005729678024107691507
8
8:
847.6461156397111251207670837670689902453735503393825637271084556248374782788666005729678024107691507
equals: #8,7
8
9:
904.2808243578157996899656671139994051607494437202343474230631730723491404179423009925618140377085746
equals: #8,6
8
10:
735.7641034682980549642713858006820306576874821086949067028760493025807628282160405507049179739600302
8
11:
805.3473885075099739727415538185746087726949567579216701644803181268229282378132785018514020962671983
8
12:
688.8983612960199708142608425087004241645878076060838230346406528073870470840754183872818805470518271
8
13:
742.9770903664300002188614578326673385751282088052628089324441266088040245398648843715976386693655588
8
14:
749.8186229956812217736182060128660712173130131258009834868211701392831566066139841555241716217257988
8
15:
805.3473885075099739727415538185746087726949567579216701644803181268229282378132785018514020962671983
equals: #8,11
8
16:
633.4389145496823178678219700957963039453785996197679838312899327078674105086378127067211423128229185
8
17:
589.3545104346694327066720075186179443939767234974940214844170099364716567105597506655601766605375850
8
18:
641.7233582974217226531476315556340108583909248665495260338280841884983438913861462810457600701220101
8
19:
598.6176907501957880831491838922227405778251308263097466704485970130748131212354747562848479177079046
8
20:
641.3466996208366990680369537053461304570227885162204770580325857947593268016544605111353658231972142
8
21:
742.9770903664300002188614578326673385751282088052628089324441266088040245398648843715976386693655588
equals: #8,13
8
22:
735.7641034682980549642713858006820306576874821086949067028760493025807628282160405507049179739600302
equals: #8,10
8
23:
589.3545104346694327066720075186179443939767234974940214844170099364716567105597506655601766605375850
equals: #8,17
8
24:
414.8739757227826753249599638601435919304156469062053418619242229318754721286995725398631195888808506
8
25:
641.7233582974217226531476315556340108583909248665495260338280841884983438913861462810457600701220101
equals: #8,18
8
26:
500.4451522168274199439817404597915425327646511359569892712665968568167584894666339398386619564313894
8
27:
598.6176907501957880831491838922227405778251308263097466704485970130748131212354747562848479177079046
equals: #8,19
8
28:
500.4451522168274199439817404597915425327646511359569892712665968568167584894666339398386619564313894
equals: #8,26
8
29:
553.2737946123595615783838778346382293655804335760081532437544517696097756320517417490654862183827582
8
31:
460.0885382469178784340499342388281161611020551401644375902796027026761853387372443853020919470483634
8
32:
470.7201255349266341730116527622521722939816239536112680414903904332287281945892074669657903302813201
8
33:
468.0384989926273064971558626167955267868934217060796900029816742642386560470725882837049578333326579
8
34:
470.7201255349266341730116527622521722939816239536112680414903904332287281945892074669657903302813201
equals: #8,32
9
1:
5719.825832938927641832660374636609918538463236728558168112522790807077983871086049739079833168525281
9
2:
3619.751093039732474155284588430971826239200369433264540170322191849439623480964325618574034681393348
9
3:
3311.906323872714309018462110152284723196929797198046628008829182756919720382582543930981133597292717
9
4:
2933.689352782508151399073473343808647121395993936881821895215404905939848048014985005598130352200934
9
5:
2588.209695695248061858229420862193942671214335478973569691523944693927828319822613834705647169142649
9
6:
2631.471115417906959910432192129564649923112485630470265079038075692137797079757480337718985585929869
9
7:
2763.030811157260163366100259245807652422215015868087455911107993612807410487489193642023483140826602
9
8:
2456.011422957318994363549593176856546641396345587414905215473223245136793952241183732457639255149113
9
9:
2456.011422957318994363549593176856546641396345587414905215473223245136793952241183732457639255149113
equals: #9,8
9
10:
2631.471115417906959910432192129564649923112485630470265079038075692137797079757480337718985585929869
equals: #9,6
9
11:
2149.782567292816128388176645327866503019275932910802220563235743482788481992323932398140070314911580
9
12:
2302.616629860738494199796173109908732014060933304729359995981905394999869830798651248513019763896987
9
13:
2009.751778696508782168449772285384414428876522409159146724711678053737261639065974971882402946112107
9
14:
2132.916142116134107214527460332361236435824399847667091563196344184023700917085541548489636489385448
9
15:
2158.868393778857763694739850235251416290229704133154435151554503625071288095500297579519652038618031
9
16:
2302.616629860738494199796173109908732014060933304729359995981905394999869830798651248513019763896987
equals: #9,12
9
17:
2572.249136201250713799685822270242503289615630740653397612896180440963505991325031560632035040429101
9
18:
2274.359382501803965186991992898682050761123384401318962931654860293368248509740002954151266161369664
9
19:
1966.114700194103239001629140125818727726411431190869402099670774485064508771381526299466567581246911
9
20:
2124.408288179873053949273477793887911825275852462174218324357023044643623717271367734161525138023875
9
21:
2124.408288179873053949273477793887911825275852462174218324357023044643623717271367734161525138023875
equals: #9,20
9
22:
2274.359382501803965186991992898682050761123384401318962931654860293368248509740002954151266161369664
equals: #9,18
9
23:
1830.009850396601992693984241067554743819180925735956063832771174793030773173206355727161607235406178
9
24:
2124.408288179873053949273477793887911825275852462174218324357023044643623717271367734161525138023875
equals: #9,20
9
25:
1830.009850396601992693984241067554743819180925735956063832771174793030773173206355727161607235406178
equals: #9,23
9
26:
2274.359382501803965186991992898682050761123384401318962931654860293368248509740002954151266161369664
equals: #9,18
9
27:
1966.114700194103239001629140125818727726411431190869402099670774485064508771381526299466567581246911
equals: #9,19
9
28:
2572.249136201250713799685822270242503289615630740653397612896180440963505991325031560632035040429101
equals: #9,17
9
29:
2024.164223243966833326723140372495800745543937418585121332591299146658748704151552369458079535265582
9
30:
1698.212514235212099264720668180748927335012163120846686658933080768845552306694939039502478548720118
9
31:
1861.606680497207873453609345679883168832779530485214088702735489754200698274141503486590197977092797
9
32:
1870.627073354638213384928345655719098844456788159897511067740937361070754987309260848740384079350733
9
33:
1870.627073354638213384928345655719098844456788159897511067740937361070754987309260848740384079350733
equals: #9,32
9
34:
1861.606680497207873453609345679883168832779530485214088702735489754200698274141503486590197977092797
equals: #9,31
9
35:
2024.164223243966833326723140372495800745543937418585121332591299146658748704151552369458079535265582
equals: #9,29
9
36:
1752.408898835993987322392519622811596143740214459745149909203360599141113308858501230698339634442745
9
37:
1894.233379467969840990777264522310714993868449811536867244264658872161175069946302883606553618973509
9
38:
1834.641238173770781972132650701901864029414156337798059149456764909000311995107711364452850370506002
9
39:
2051.571895525863715234522483757949607107770277420463506008047508807555455431040975612037675741376369
9
40:
2019.607198793633248448469066081354000732064634235082141352672607031471635904505230892909243381948767
9
41:
2052.088252957104395435412606352775072176049904606777353869135425159504120511771898884733648935517663
9
42:
1625.857572492288917220677767368048882441270326420868624416187769997594613308432547636392267526188229
9
43:
1582.832645117142702004621781065431715755905441845569390604663885604490658020123217916024724162051779
9
44:
1745.193409005156881017148187487585914890956779098839077635137782885301586767789790368638867179903183
9
45:
1619.593946330972017046235072281619067524975550741879193889049048275449754472581251847219109583318061
9
46:
1750.030816059881531204737138760334025401620308515240635435340040033073419972541129257374683203020571
9
47:
1721.126714006066136049899099453072978124508480337588844573010688575436668948175401199270437479786711
9
48:
2247.313279898006574929840090545897196023334135201741129997557845724702686074595164824810282249098128
9
49:
2177.933247483338816934354125332663459858981085549225322570934890541381001890299504473709384071336700
9
50:
1745.193409005156881017148187487585914890956779098839077635137782885301586767789790368638867179903183
equals: #9,44
9
52:
2051.571895525863715234522483757949607107770277420463506008047508807555455431040975612037675741376369
equals: #9,39
9
54:
2019.607198793633248448469066081354000732064634235082141352672607031471635904505230892909243381948767
equals: #9,40
9
55:
2024.164223243966833326723140372495800745543937418585121332591299146658748704151552369458079535265582
equals: #9,29
9
56:
1625.857572492288917220677767368048882441270326420868624416187769997594613308432547636392267526188229
equals: #9,42
9
58:
2149.782567292816128388176645327866503019275932910802220563235743482788481992323932398140070314911580
equals: #9,11
9
59:
1582.832645117142702004621781065431715755905441845569390604663885604490658020123217916024724162051779
equals: #9,43
9
60:
1745.193409005156881017148187487585914890956779098839077635137782885301586767789790368638867179903183
equals: #9,44
9
62:
1619.593946330972017046235072281619067524975550741879193889049048275449754472581251847219109583318061
equals: #9,45
9
65:
1830.009850396601992693984241067554743819180925735956063832771174793030773173206355727161607235406178
equals: #9,23
9
66:
2132.916142116134107214527460332361236435824399847667091563196344184023700917085541548489636489385448
equals: #9,14
9
67:
1488.218371697741217579387192859478436626636039325181515457466859240505634433869018228134890864441561
9
69:
1238.958365010244911339059330605850206998024018031490287575384387436570165448196468141905358798327605
9
71:
1861.606680497207873453609345679883168832779530485214088702735489754200698274141503486590197977092797
equals: #9,31
9
72:
1562.127250868547443682474688537139789739464494980112400767165437562281454518824999187771840048504287
9
73:
1435.860779495584724448852779117936139242442906820256686837821366235007796384314504810663879827213267
9
74:
1605.304547791520435956701416295971216297359321278577053295134570973972284384601738995092152161108930
9
75:
1752.408898835993987322392519622811596143740214459745149909203360599141113308858501230698339634442745
equals: #9,36
9
87:
1509.757448482941763270251961608596338704167421210889379117458177374339624936357536594421648819862982
9
88:
1750.030816059881531204737138760334025401620308515240635435340040033073419972541129257374683203020571
equals: #9,46
9
90:
1625.857572492288917220677767368048882441270326420868624416187769997594613308432547636392267526188229
equals: #9,42
9
92:
1605.304547791520435956701416295971216297359321278577053295134570973972284384601738995092152161108930
equals: #9,74
9
108:
1074.773096415984967101015517191463571576305415523055254153651487384953286276539097165172475524934926
9
109:
1057.275577334488460856971136729338366637373632427669329489099409593484430505289305445036598719021359
9
110:
1057.275577334488460856971136729338366637373632427669329489099409593484430505289305445036598719021359
equals: #9,109
9
111:
1074.773096415984967101015517191463571576305415523055254153651487384953286276539097165172475524934926
equals: #9,108
9
112:
1509.757448482941763270251961608596338704167421210889379117458177374339624936357536594421648819862982
equals: #9,87
9
113:
1509.757448482941763270251961608596338704167421210889379117458177374339624936357536594421648819862982
equals: #9,87
9
134:
1189.402377090919078041134534708894251522757382233739995194384089452176390143651234349791106161633597
9
136:
1183.350821661750256742169863719678392071684474900415408402091071141920529190818965074898821151933164
9
148:
1170.673742709633638131271730389598223883117948517522078480851543770347473497927466141187367752479764
9
149:
1183.350821661750256742169863719678392071684474900415408402091071141920529190818965074898821151933164
equals: #9,136
9
150:
1170.673742709633638131271730389598223883117948517522078480851543770347473497927466141187367752479764
equals: #9,148
9
156:
1216.295990822750490669541041441938524341810029539144660541500076133694980587733978365085662131636387
9
159:
1189.402377090919078041134534708894251522757382233739995194384089452176390143651234349791106161633597
equals: #9,134
10
1:
19448.14852821966187457669697850921379213614026830742858304694720558565239070941326928480565301075542
10
2:
11795.40023080513061109609347582831406197471678238993780714216026648419991761365085215851037894011116
10
3:
10464.76922351336642731527444825402107148172753701458095157062912333917118216255508490151159978069228
10
4:
9300.764935848848285752382246721441110968293351769369157883769277805710146552522994143571936618454078
10
5:
8236.687196758556059105546993632223907947152916118356570553993201068496545624170958461077554252879870
10
6:
9861.593063523201885952337919078791495895866492629489337265162288340321767544076544730473157556553375
10
7:
7944.083170121483578636537726001624458165891355027317767242987778393215588606678472823753643995169209
10
8:
8362.489314896220526960184552718265425011509845663948142905096113971049941114044465874178572432318550
10
9:
7463.545356142233133100248029004990437453453159951436443996237649577001113551762040401892400717957200
10
10:
7463.545356142233133100248029004990437453453159951436443996237649577001113551762040401892400717957200
equals: #10,9
10
11:
7944.083170121483578636537726001624458165891355027317767242987778393215588606678472823753643995169209
equals: #10,7
10
12:
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11
1112:
13836.47652529348656308842227821322014344038419919127889865790008920522595654568506114648869002597083
equals: #11,516
11
1162:
13836.47652529348656308842227821322014344038419919127889865790008920522595654568506114648869002597083
equals: #11,516
11
1173:
12694.00184020405650714038622801399155108260628569066133580871754345447601082515765247031353918022360
equals: #11,594
11
1185:
12694.00184020405650714038622801399155108260628569066133580871754345447601082515765247031353918022360
equals: #11,594
11
1668:
16233.70831234543096834210998319168732135659151109961343044731633493270267450318741298833019497594288
equals: #11,200
11
1669:
16521.67434177890063750215508653187593507655956909059886663514184748937674286455705874189316456689656
equals: #11,199
11
2243:
13696.03769519541520344114283385565141647096450300695630231082867264387104055610211394483875287697391
equals: #11,662
Definition
For a primitive log-divergent $\phi^4$ graph $G$, the Feynman period is $P(G)=\int_{(0,\infty)^{N_G-1}}\left.\Psi_G^{-2}\right|_{x_{N_G}=1}\,dx_1\cdots dx_{N_G-1}$ [1].
Parameters
$\ell$
—   Loop order ($\ell\geq1$ is the loop order of the uncompleted graph.)
$k$
—   Census index ($k\geq1$ is an index present at loop order $\ell$ in Schnetz's census.)
Formulas
(1)
$\Psi_G(x)=\sum_{T\text{ spanning tree of }G}\prod_{e\notin T}x_e.$
(2)
For the $\ell$-loop zig-zag graph $Z_\ell$, $P(Z_\ell)=\frac{4(2\ell-2)!}{\ell!(\ell-1)!}\left(1-\frac{1-(-1)^\ell}{2^{2\ell-3}}\right)\zeta(2\ell-3)$ for $\ell\geq3$, where $\zeta$ is the Riemann zeta function [2]. The census entry $P_{\ell,1}$ is a zig-zag period for $3\leq\ell\leq11$.
(3)
$P_{6,4}=12P_{4,1}P_{3,1}-16P_{6,3}$. The completed graph of $P_{6,4}$ is $K_{4,4}$ [1], equation (5.3).
Comments
(4)
The graph $G$ is connected, and each vertex has degree at most four. Writing $N_G$ for its number of edges and $h_G$ for its first Betti number, primitive logarithmic divergence means $N_G=2h_G$ and $N_\gamma>2h_\gamma$ for every nonempty strict edge subgraph $\gamma$. There is one variable $x_e$ per edge; the graph polynomial $\Psi_G$ is defined in (1).
(5)
The label $P_{\ell,k}$ is the label in the Panzer–Schnetz ancillary census [3]: $\ell$ is the loop order before completion and $k$ distinguishes its census entries. Distinct labels remain distinct rows even when their exact source expressions coincide.
(6)
The edge lists in the attached census-graphs.json and [3] describe the completed graphs. Each completion is 4-regular and has $\ell+2$ vertices and $\ell+3$ loops. Removing any vertex gives an $\ell$-loop graph with the same period, by the completion theorem [1], Theorem 5.2. In particular, the completion of $P_{3,1}$ is $K_5$.
(7)
The multiple zeta values of length 2 and length 3 use the convention $\zeta(s_1,\ldots,s_r)=\sum_{n_1>\cdots>n_r>0}\prod_{j=1}^r n_j^{-s_j}$ with $s_1\geq2$. The ancillary expressions use this convention, together with multiple polylogarithms at $-1$ and $e^{\pm i\pi/3}$. The single-index case is the Riemann zeta function.
(8)
Entries marked FAIL in the source have no evaluable expression and are omitted. No transcendence assertion is made for individual periods.
Programs
(P1)
Sage
# P_{6,3} from the ancillary multiple-zeta expression.
import numberdb.sage as numberdb
from sage.libs.pari import pari
from sage.rings.rational_field import QQ
from sage.rings.real_mpfr import RealField
R = RealField(numberdb.bits(150))
zeta = lambda *s: R(pari.zetamult(list(s), precision=R.precision()))
P_6_3 = QQ(432)/5*zeta(5,3) - QQ(58)/2625*R.pi()**8 + 252*zeta(3)*zeta(5)
print(P_6_3)
References
[1]
Erik Panzer and Oliver Schnetz, The Galois coaction on $\phi^4$ periods, Communications in Number Theory and Physics 11 (2017), 657–705. (arXiv) (doi)
[2]
Francis Brown and Oliver Schnetz, Single-valued multiple polylogarithms and a proof of the zig-zag conjecture, Journal of Number Theory 148 (2015), 478–506. (arXiv) (doi)
Links
Similar tables
Multiple zeta values of length 2 —   Depth-two values entering the period expressions.
Multiple zeta values of length 3 —   Depth-three values entering the period expressions.
Values of the Riemann zeta function at rational numbers —   Single zeta values occurring in zig-zag periods.
Data properties
Entries are of type: real number
Table is complete: no (it holds all 852 entries with an evaluable expression in the version-2 ancillary census through eleven loops, including the exact one-loop period; these are the source’s explicitly evaluated graphs, and all census labels are retained)
How well the digits are known: heuristic (agreement-checked)
digits: 100 (Unknown key)
How they were obtained:

The attached generator parses the version-2 ancillary file [3], pinned by SHA-256, and evaluates each of its 299 distinct expressions using PARI zetamult and polylogmult [4] at 150 and 200 working decimal digits. All 852 census labels are retained. The one-loop integral is exactly 1; every other entry is written to 100 significant digits. The smallest measured agreement between working precisions is 147.72 significant digits. Every recomputed value agrees with the source decimal within 0.499 units of its last printed place.

more

Rational coefficients are exact. To avoid PARI rejecting a root of unity rounded just outside the unit disk, each nonreal root is approached from inside by a relative displacement of 16 binary units at the working precision. This displacement shrinks in the second computation; the resulting error claim is heuristic, not a certified enclosure of the graph integral.

The attached checks use nonzero controls for the multiple-zeta convention, alternating sums and sixth-root polylogarithms. They check every completion’s degree, loop count and nontrivial edge cuts, identify the nine zig-zag completions as circulant graphs, and compare their periods with the zig-zag formula using independent Arb zeta evaluations. They also check the relation for the completed graph $K_{4,4}$ and all declared equalities. The same numerical checks are applied to values read back from the draft. The computation evaluates the source’s formulas; it does not independently integrate all 852 graphs.



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